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Icing Forecasting for Power Transmission Lines Based on a Wavelet Support Vector Machine Optimized by a Quantum Fireworks Algorithm

Icing Forecasting for Power Transmission Lines Based on a Wavelet Support Vector Machine... applied sciences Article Icing Forecasting for Power Transmission Lines Based on a Wavelet Support Vector Machine Optimized by a Quantum Fireworks Algorithm Tiannan Ma *, Dongxiao Niu and Ming Fu Departemnt of Economics and Management, North China Electric Power University, Beijing 102206, China; ndx@ncepu.edu.cn (D.N.); fm@ncepu.edu.cn (M.F.) * Correspondence: matiannan_1234@ncepu.edu.cn; Tel.: +86-185-1562-1058 Academic Editor: Fan-Gang Tseng Received: 22 December 2015; Accepted: 4 February 2016; Published: 17 February 2016 Abstract: Icing on power transmission lines is a serious threat to the security and stability of the power grid, and it is necessary to establish a forecasting model to make accurate predictions of icing thickness. In order to improve the forecasting accuracy with regard to icing thickness, this paper proposes a combination model based on a wavelet support vector machine (w-SVM) and a quantum fireworks algorithm (QFA) for prediction. First, this paper uses the wavelet kernel function to replace the Gaussian wavelet kernel function and improve the nonlinear mapping ability of the SVM. Second, the regular fireworks algorithm is improved by combining it with a quantum optimization algorithm to strengthen optimization performance. Lastly, the parameters of w-SVM are optimized using the QFA model, and the QFA-w-SVM icing thickness forecasting model is established. Through verification using real-world examples, the results show that the proposed method has a higher forecasting accuracy and the model is effective and feasible. Keywords: icing forecasting; support vector machine; fireworks algorithm 1. Introduction In recent years, a variety of extreme weather phenomena have occurred on a global scale, causing overhead power transmission line icing disasters to happen frequently, accompanied by power outages and huge economic losses due to the destruction of a large number of fixed assets. Since the first power transmission line icing accident recorded in 1932, several serious icing disasters have occurred one after another throughout the world [1], such as Canada’s freezing rain disaster in 1998, which led to direct economic losses of 1 billion dollars and indirect losses of 30 billion dollars for the power system. The power system failure caused by freezing rain was also a tremendous shock to the production and life of the region. In China, the earliest recorded icing disaster occurred in 1954. After that year, several large areas of freezing rain and snow disasters have successively occurred in China, especially Southern China’s freezing weather in January 2008 [2]. That disaster caused tremendous damage to China’s power system, directly resulting in 8709 tower collapses, more than 27,000 line breakages, and 1497 substation outages of 110 kV and above lines. The direct property loss for the State Grid Corporation amounted to 10.45 billion RMB, and the investment in post-disaster electricity reconstruction and transformation was 39 billion RMB. According to the lessons learned from previous grid freezing accidents, and given the current development situation of China's grid and the conditions of global climate change, the grid in China will again be tested by the future large-scale icing disasters. It is always preferable to take preventative action instead of reacting to events after an accident has Appl. Sci. 2016, 6, 54; doi:10.3390/app6020054 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 54 2 of 23 already happened. Therefore, icing forecasting research with regard to overhead power transmission lines have important practical application value. At present, there are many studies on icing forecasting for power transmission lines. According to the hydrodynamic movement law and heat transfer mechanisms, domestic and foreign scholars have established a variety of transmission line icing forecasting models that consider meteorological factors, environmental factors, and various line parameters. Generally these models can be divided into three categories: mathematical and physical models, statistical models and intelligent forecasting models. Mathematical and physical models involve simulation modeling and forecasting of the icing growth process by observing the physical process and mathematical equations of icing formation, notably including the Goodwin model [3], the Makkonen model [4] and so on. These models are established based on the experimental data, but there are differences between the experimental data and the practical data, so the forecasting results of these models are not ideal. Statistical models process historical data based on statistical theory and traditional statistical methods and do not consider the physical process of icing formation, e.g., the multiple linear regression icing model [5]. However, transmission line icing is affected by a variety of factors, and a multiple linear regression model cannot take all factors into account, meaning the icing forecasting accuracy is greatly reduced. Intelligent forecasting models combine modern computer technology with mathematical sciences, and are able to handle high-dimensional nonlinear problems through their powerful learning and processing capabilities, which can improve the prediction accuracy. The common intelligent forecasting models are support vector machines (SVM) and Back Propagation(BP) neural network. BP neural network has a strong nonlinear fitting ability to create the non-linear relationship between the output and a variety of impact factors. With its strong learning ability and forecasting capability, the non-linear output can arbitrarily approximate the actual value. For example, paper [6] presented a short-term icing prediction model based on a three layer BP neural network, and the results showed that the BP forecasting model is accurate for transmission lines of different areas. Paper [7] presented an ice thickness prediction model based on fuzzy neural network, the tested result demonstrated its forecasting abilities of better learning and mapping. However, the single BP model easily falls into the local optimum, and cannot always reach the expected accuracy. For this problem, some scholars adopted optimization algorithms to optimize the parameters of BP neural network, thereby improving the prediction accuracy. For instance, Du and Zheng et al. used Genetic Algorithm (GA) to optimize BP network and built the GA-BP ice thickness prediction model [8], and this model proved that the GA-BP model was more effective than the BP model in ice thickness prediction for transmission lines. Although this combination model could improve the prediction accuracy of BP neural networks, some scholars started to use SVM to build the icing forecasting model for transmission lines due to its slow calculating speed and overall poor performance. SVM can establish the non-linear relationship between various factors and ice thickness, and has better nonlinear mapping ability and generalization ability. In addition, SVM has a strong learning ability and can quickly approach the target value through continuous repetitive learning. Therefore, theSVM model is more widely used in transmission line research. For example, paper [9–11] introduced the icing forecasting model based on the Support Vector Regression learning algorithm, and obtained the ideal results. As is known, the standard SVM uses a Gaussian kernel function to solve the support vector by quadratic programming, and the quadratic programming will involve the calculation of an m order matrix (m is sample number). When m is bigger, the processing ability of the Gaussian function is more unsatisfactory. It will take much computing time, thereby seriously affecting the learning accuracy and predictive accuracy of the algorithm. In transmission line icing forecasting, there are many influencing factors, so a large amount of input data will make the SVM algorithm become unfeasible using the traditional Gaussian kernel for such a large-scale training sample. In view of this problem, this paper will replace the Gaussian kernel function with the wavelet kernel function, and establishes the wavelet support vector machine (w-SVM) for icing forecasting. Using wavelet kernel function in place of the Gaussian kernel is mainly based on the following considerations [12]: (1) the wavelet kernel function Appl. Sci. 2016, 6, 54 3 of 23 has the fine characteristic of progressively describing the data, and using the SVM of the wavelet kernel function can approximate any function with high accuracy while the traditional Gaussian kernel function cannot; (2) the wavelet kernel functions are orthogonal or nearly orthogonal, and the traditional Gaussian kernel functions are relevant, even redundant; (3) the wavelet kernel function has multi-resolution analyzing ability for wavelet signals, so the nonlinear processing capacity of the wavelet kernel function is better than that of Gaussian kernel function, which can improve the generalization ability of the support vector machine regression model. The forecasting performance of the w-SVM model largely depends on the values of its parameters; however, most researchers choose the parameters of SVM only by subjective judgment or experience. Therefore, the parameters values of SVM need to be optimized by meta-heuristic algorithms. Currently, several algorithms have been successfully applied to determine the control parameters of SVM, such as genetic algorithms [13], particle swarm optimization [14], differential evolution [15] and so on. However, those algorithms have the defects of being hard to control, achieving the global optima slowly, etc. In this paper, we use the fireworks optimization algorithm (FA) proposed by Tan and Zhu in 2010 [16] to determine the parameter values of w-SVM. The advantage of using FA over other techniques is that it can be easily realized and be able to reach global optimal with greater convergence speed. Besides, in order to strengthen the optimization ability and obtain better results, this paper also attempts to improve the FA model by using a quantum evolutionary algorithm. The rest of this paper is organized as follows: In Section 2, the quantum fireworks algorithm (QFA) and w-SVM are presented in detail. Also, in this section, a hybrid icing forecasting model (QFA-w-SVM) that combines the QFA and w-SVM models is established; In Section 3, several real-world cases are selected to verify the robustness and feasibility of QFA-w-SVM, and the computation, comparison and discussion of the numerical cases are discussed in detail; Section 4 concludes this paper. 2. Experimental Section 2.1. Quantum Fireworks Algorithm 2.1.1. Fireworks Algorithm A fireworks algorithm (FA) [16] is used to simulate the whole process of the explosion of fireworks. When the fireworks explosion generates a lot of sparks, the sparks can continue to explode to generate new sparks, resulting in beautiful and colorful patterns. In an FA, each firework can be regarded as a feasible solution of the solution space for an optimization problem, and the fireworks explosion process can be seen as a searching process for the optimal solution. In a particular optimization problem, the algorithm needs to take into account the number of sparks of each fireworks explosion, how wide the explosion radius is, and how to select an optimal set of fireworks and sparks for the next explosion (searching process). The most important three components of FA are the explosion operator, mutation operator and selection strategy. (1) Explosion operator. The number of sparks each fireworks explosion generates and explosion radius are calculated based on fitness value of fireworks. For the fireworks x pi  1, 2, , Nq, the calculation formula for the number of sparks S and explosion radius R are: i i y  fpx q # max i S  M  (1) py  fpx qq # max i1 fpx q y # i min R  R  (2) p f px q y q # i min i1 Appl. Sci. 2016, 6, 54 4 of 23 In the above formula, y , y represent the maximum and minimum fitness value of the current max min population, respectively; f px q is the fitness value of fireworks x ; and M is a constant to adjust the i i number of explosive sparks. R is a constant to adjust the size of the fireworks explosion radius. # is the minimum machine value to be used to avoid zero operation. (2) Mutation operator. The purpose of mutation operator is to increase the diversity of the sparks population; the mutation sparks of fireworks algorithm is obtained by Gaussian mutation, namely Gaussian mutation sparks. Select fireworks x to make the Gaussian mutation, and the k-dimensional Gaussian mutation is: x  x  e (3) ik ik Where x ˆ representsk-dimensional mutation fireworks, e represents the Gaussian distribution. ik In the fireworks algorithm, the new generated explosion fireworks and mutation sparks may fall out of the search space, which makes it necessary to map it to a new location, using the following formula: x ˆ  x |x ˆ | % x  x (4) ik LB,k ik UB,k LB,k where x , x are the upper and lower search spaces, respectively; and % denotes the modulo UB,k LB,k operator for floating-point number. (3) Selection strategy. In order to transmit the information to the next generation, it is necessary to select a number of individuals as the next generation. Assume that K individuals are selected; the number of population is N and the best individual is always determined to become the fireworks of next generation. Other N  1 fireworks are randomly chosen using a probabilistic approach. For fireworks x , its probability ppx q of being chosen is i i calculated as follows: ¸ ¸ Rpx q ppx q  ° , Rpx q  d x  x  ||x  x || (5) i i i j i j x PK x PK j j x PK where Rpxq is the sum of the distances between all individuals in the current candidate set. In the candidate set, if the individual has a higher density, that is, the individual is surrounded by other candidates, the probability of the individual selected will be reduced. If Fpxq is the objective function of fireworks algorithm, the steps of the algorithm are shown as follows: (1) Parameters initialized; Randomly select N fireworks and initialize their coordinates. (2) Calculate the fitness value fpx q of each firework, and calculate the blast radius R and i i generated sparks number of each firework. Randomly select dimension z-dimensional coordinates to update coordinates, coordinate updating formula is as follows: x ˆ  x R  U p1, 1q, U(–1,1) ik ik i stands for the uniform distribution on [–1,1]. (3) Generate M Gaussian mutation sparks; randomly select sparks x , use the Gaussian mutation formula to obtain M Gaussian mutation sparks x ˆ , and save those sparks into the Gaussian mutation ik sparks population. (4) Choose N individuals as the fireworks of next generation by using probabilistic formula from fireworks, explosion sparks and Gaussian mutation sparks population. (5) Stop condition. If the stop condition is satisfied, then output the optimal results; if not, return step (2) and continue to cycle. 2.1.2. Quantum Evolutionary Algorithm The development of quantum mechanics impels quantum computing to be increasingly applied in various fields. In quantum computing, the expression of a quantum state is a quantum bit, and usually quantum information is expressed by using the 0 and 1 binary method. The basic quantum states are “0” state and “1” state. In addition, the state can be an arbitrary linear superposition state Appl. Sci. 2016, 6, 54 5 of 23 between “0” and “1”. That is to say, the two states can exist at the same time, which challenges the classic bit expression method in classical mechanics to a large extent [17]. The superposition state of quantum state can be presented as shown in Equation (6). 2 2 |y ¡ a|0 ¡ b|1 ¡ , |a| |b|  1 (6) where |0 ¡ and |1 ¡ are the two quantum states a and b are the probability amplitudes. |a| represents the probability at a quantum state of |0 ¡ and |b| represents the probability at a quantum state of |1 ¡. In QFA, the updating proceeds by quantum rotating gate, and the adjustment is: a cospqq sinpqq a (7) b sinpqq cospqq b cospqq sinpqq in which set U  , U is quantum rotating gate, q is quantum rotating angle, sinpqq cospqq and q  arctanpa{bq. A quantum evolutionary algorithm is proposed based on a probabilistic search, as the conception of qubits and quantum superposition means a quantum evolutionary algorithm has many advantages, such as better population diversity, strong global search capability, especially great robustness, and the possibility of combining with other algorithms. 2.1.3. Quantum Fireworks Algorithm Parameters Initialized In the solution space, randomly generate N fireworks, and initialize their coordinates. Here, we use the probability amplitude of quantum bits to encode the current position of fireworks, and the encoding method is used by the following formula: cospq qcospq q cospq q i1 i2 in P  (8) sinpq qsinpq q sinpq q i1 i2 in where q  2prandpq, randpq is a random number between 0 and 1; i  1, 2, , m; j  1, 2, , n; m is ij the number of fireworks, n is the number of solution space. Therefore, the corresponding probability amplitude of individual fireworks for the quantum states |0 ¡ and |1 ¡ are as follows: P  pcospq q, cospq q cospq qq (9) ic i1 i2 in P  psinpq q, sinpq q sinpq qq (10) is i1 i2 in Solution Space Conversion The searching process of a fireworks optimization algorithm is carried out on the actual parameter space ra, bs. Due to the probability amplitude of fireworks location being in the range of r0, 1s, it needs to k 1 be decoded into the actual parameter space ra, bs to search in the fireworks algorithm. Let the Dq th jd j j k1 quantum bit for the individual a is a , b , and its corresponding conversion equations be: jd i i j j j X  rb p1 a q a p1  a qs i f randpq   P (11) i i id ic i i j j j X  rb p1 b q a p1  b qs i f randpq ¥ P (12) i i id is i i 2 Appl. Sci. 2016, 6, 54 6 of 23 where randpq is a random number between r0, 1s; X is the actual parameter value in j th dimension ic position when the quantum state of i th fireworks individual is |0 ¡ , X is the actual parameter value is in j th dimension position when the quantum state of the i th fireworks individual is |1 ¡ . b and a are i i the lower and upper limits. Assuming the FA is searching in two-dimensional space, that means j  1, 2. Initialize the position of population: InitX_ axis; InitY_axis; and the position of individuals can be determined as follows: i f randpq   P : id 1 1 Xpiq  X_axis rb p1 a q a p1  a qs (13) i i i i 2 2 Ypiq  Y_axis rb p1 a q a p1  a qs (14) i i i i i f randpq ¥ P : id 2 2 Xpiq  X_axis rb p1 b q a p1  b qs (15) i i i i 2 2 Ypiq  Y_axis rb p1 b q a p1  b qs (16) i i i i Calculate the Fitness Value fpx q of Each Individual, and Obtain the Explosion Radius and Generated Sparks Number S y  fpx q # max i S  M  (17) py  fpx qq # max i i1 fpx q y # i min R  R  (18) p f px q y q # min i1 Individual Position Updating The individual position update is operated by using a quantum rotating gate using the following equation: k1 k1 k1 a cosq sinq a jd jd jd jd (19) k1 k1 k1 k b sinq cosq b jd jd jd jd k1 k1 where a and b are the probability amplitude of j th fireworks individual in k 1 th iteration for jd jd k1 d dimension space; q is the rotating angle, which can be get from equation: jd k1 k1 k k q  spa , b qDq (20) jd jd jd jd k1 k k where spa , b q determines the rotating angle direction and is the Dq rotating angle increment. jd jd jd k 1 In order to adapt to operation mechanism of fireworks algorithm, we convert the updated a jd k1 and b to a solution space. jd d k1 k1 X  rb p1 a q a p1  a qs i f randpq   P j j id jc jd jd (21) k1 k1 X  rb p1 b q a p1  b qs i f randpq ¥ P j j id js jd jd Then calculate the positional offset amount: d d d d h  R  X , h  R  X (22) i i jc jc js js Appl. Sci. 2016, 6, 54 7 of 23 i f randpq   P , then d  1 id k1 k1 Xpjq  Xpiq h  Xpiq R  rb p1 a q a p1  a qs i j j jc jd jd (23) d k1 k1 Ypjq  Ypiq h  Ypiq R  rb p1 a q a p1  a qs i j i jc jd jd i f randpq ¥ P : then d  2 id d k1 k1 Xpjq  Xpiq h  Xpiq R  rb p1 b q a p1  b qs i j j js jd jd (24) k1 k1 Ypjq  Ypiq h  Ypiq R  rb p1 b q a p1  b qs i j j js jd jd Detection of cross-border. If the generated explosion sparks exceed the possible domain boundary, the position of sparks can be updated with the following equations: Xpjq  X |Xpjq| %pX  X q LB,k UB,k LB,k (25) Ypjq  Y |Ypjq| %pY  Y q LB,k UB,k LB,k Individual Mutation Operation The main reason for the premature convergence and local optimum of the fireworks group is that the diversity of the population is lost in the process of population search. In the quantum fireworks algorithm, in order to increase the diversity of the population, the Gauss mutation in the original algorithm is replaced by a quantum mutation. Randomly select fireworks x , and generate M quantum mutation sparks, and its operation formula is shown as follows: cosp  q q 01 cospq q sinpq q ij ij ij (26) 10 sinpq q cospq q ij ij sinp  q q ij Let the probability of individual be P , and randpq be a random number between r0, 1s; if randpq   P , the mutation can be operated with the above formula and the probability amplitude in quantum bit is changed; finally, the mutated individual can be converted into the solution space and save it to the mutation sparks population. 1 p p X  rb p1 cosp  q qq a p1  cosp  q qqs i f randpq   P j ij j ij id jc 2 2 2 (27) 1 p p X  rb p1 sinp  q qq a p1  sinp  q qqs i f randpq ¥ P j ij j ij id js 2 2 2 i f randpq   P , then d  1 id d d ˆ ˆ ˆ ˆ ˆ XpMq  XpMq X YpMq  YpMq X jc jc i f randpq ¥ P : then d  2 id d d ˆ ˆ ˆ ˆ ˆ XpMq  XpMq X YpMq  YpMq X js js Detection of cross-border. If the generated explosion sparks exceed the possible domain boundary, the position of sparks can be updated by the following equations: ˆ ˆ  ˆ ˆ XpMq  X XpMq %pX  X q LB,k UB,k LB,k (28) ˆ  ˆ ˆ ˆ YpMq  Y YpMq %pY  Y q LB,k UB,k LB,k Appl. Sci. 2016, 6, 54 8 of 23 (6) Choose N individuals as the fireworks of the next generation by using probabilistic formula ppx q from fireworks, explosion sparks and Gaussian mutation sparks population. (7) Stop condition. If the stop condition is satisfied, then output the optimal results; if not, return to step (2) and continue to cycle. 2.2. Wavelet Support Vector Machine 2.2.1. Basic Theory of Support Vector Machine (SVM) A support vector machine, proposed by Vapnik, is a kind of feed forward network [18]; its main purpose is to establish a hyper-plane to make the input vector project into another high-dimensional space. Given a set of data tpx , d qu , where x is the input vector; d is the expected output; it is i i i i1 further assumed that the estimate value of d is y, which is obtained by the projection of a set of nonlinear functions: y  w f pxq  w fpxq (29) j j j0 T T where fpxq  rf pxq, f pxq, , f pxqs , w  rw , w , , w s ; Let f pxq  1; w represent the bias b, 0 1 m 0 1 m 0 0 and the minimization risk function can be described as follows: 1 T Fpw, x, x q  w w C px x q (30) i1 The minimization risk function must satisfy the conditions: d  w fpx q ¤ # x ' i i i & 1 w fpx q d ¤ # x i i (31) x ¥ 0 ' i % 1 x ¥ 0 where i  1, 2, , N, x and x are slack variables; loss function is a #-insensitive loss function, C is a constant. Establish the Lagrange function and obtain: N N ° ° 1 1 1 T T Jpw, x, x , a, a , g, g q  C px x q w w  a rw fpx q d # x s i i i i i i1 i1 (32) N N ° ° 1 1 1 1 a rd  w fpx q # x s pg x g x q i i i i i i i i i1 i1 1 1 1 1 where a and a are Lagrange multiplier; take the partial derivative of variables w, x, x , a, a , g, g and obtain: w  pa  a qfpx q i i (33) i1 1 1 g  C  a ; g  C  a i i i i The above problem can be converted into a dual problem: N N N N ¸ ¸ ¸ ¸ 1 1 1 1 1 maxQ a , a  d pa  a q # pa a q pa  a qpa  a qf px qfpx q i i i i i j i j i i i i j i1 i1 i1 j1 ' pa  a q  0 i1 0 ¤ a ¤ C (34) 0 ¤ a ¤ C i  1, 2, , N Appl. Sci. 2016, 6, 54 9 of 23 Solve the equation and obtain: w  pa  a qfpxq i1 T T T T Then Fpx, wq  w x  pa  a qf pxqfpxq, let Kpx , xq  f pxqfpxq be the kernel function. i1 In this paper, we choose the wavelet kernel function to replace the Gaussian kernel function, and the construction of the wavelet kernel function will be introduced in detail in Section 2.2.2. 2.2.2. Construction of Wavelet Kernel Function The kernel function kpx, x q to SVM is the inner product of the image of two input space data points in the spatial characteristic. It has two important features: first, the symmetric function to inner 1 1 product kernel variables is kpx, x q  kpx , xq; second, the sum of the kernel function on the same plane is a constant. In general, only if the kernel function satisfies the following two theorems, can it become the kernel of support vector machine [19]. Mercer Lemma kpx, x q represents a continuous symmetric kernel, which can be expanded into a series as: 1 1 kpx, x q  l g pxqg px q (35) i i i i1 where l is positive, in order to ensure the above expansion is absolutely uniform convergence, the sufficient and necessary condition is: 1 1 1 n kpx, x qgpxqgpxqdxdx ¥ 0, x, x P R (36) gpq  0 For all gpq needs to satisfy the condition: , gpx q stands for the expansion of the g pxqdx   8 characteristic function, l stands for eigenvalue and all are positive, thus the kpx, x q is positive definite. Smola and Scholkopf Lemma If the support vector machine’s kernel function has meet the Mercer Lemma, then it only needs to prove kpx, x q to satisfy the follow formula: n{2 Fpxqpwq  p2pq exppJpw  xqqkpxqdx ¥ 0, x P R (37) Construction of Wavelet Kernel 2 1 ˆ ˆ If the wavelet kernel satisfies the condition: ypxq P L pRq X L pRq and ypxq  0,ypxq is the Fourier transform of ypxq, the ypxq can be defined as [20]: x  m 1{2 y pxq  psq yp q, x P R (38) s,m where s means contraction-expansion factor, and m is horizontal floating coefficient, s ¡ 0, m P R. When the function fpxq, fpxq P L pRq, the wavelet transform to fpxq can be defined as: x  m 1{2 Wps, mq  s fpxqy p qdx (39) 8 Appl. Sci. 2016, 6, 54 10 of 23 where y pxq is the complex conjugation of ypxq. The wavelet transform Wps, mq is reversible and also can be used to reconstruct the original signal, so: » » 8 8 ds fpxq  C Wps, mqy pxq dm (40) s,m 8 8 ³ ypwq ³ j 8 Among them C dw   8, ypwq  ypxqexppJwxqdx. |w| For the above equation, is a constant. The theory of wavelet decomposition is to approximate the function group by the linear combination of the wavelet function. Assuming ypxq is a one-dimensional function, based on tensor theory, the multi-wavelet function can be defined as: lxd d y pxq  P ypx q, x P R , x P R (41) l i i i1 The horizontal floating kernel function can be built as: x  x 1 i kpx, x q  P yp q, s ¡ 0 (42) i1 s In support vector machines, the kernel function should satisfy the Fourier transform; therefore, only when the wavelet kernel function satisfies the Fourier transform can it be used for support vector machines. Thus, the following formula needs to be proved. l{2 Frkspwq  p2pq exppJpwxqqkpxqdx ¥ 0 (43) In order to keep the generality of wavelet kernel function, choose the Morlet mother wavelet as follows: ypxq  cosp1.75xqexppx {2q (44) N N s can be figured out. Where x P R , s, x P R , the multi-dimensional wavelet function is an allowable multidimensional support vector machine functions. #    + 2 2 1 1 ||x  x || ||x  x || i i 1 i i kpx, x q  P cos 1.75p q exp 2 2 i1 2s 2s (45) 1.75X i 2 P cosp qexpp||x || {2s q i1 Proof: exppJwxqkpxqdx # + 1 1 x  x ||x  x || i i i i exppJwxq p P cosr1.75p qsexpp q qdx R 2 i1 s 2s exppj1.75x {sq expp1.75x {sq i i P exppJw x q p q expp||x || {2s qdx i i i i i1 2 #    + 2 2 1 ||x || 1.75j ||x || 1.75j i i P exp  p  jw sq x exp  p  jw sq x dx i i i i i 2 2 i1 2 2s s 2s s # + 2 2 |s| 2p p1.75  w sq p1.75 w sq i i P expp q expp q 2 2 2 i1 2 2 |s| p1.75  w sq p1.75 w sq i i Then Fpxqpwq  P p qpexpp q expp qq i1 2 2 2 |s|  0 ñ Fpxqpwq ¥ 0 ñ wavelet kernel function is an allowable support vector kernel function. Appl. Sci. 2016, 6, 54 11 of 23 2.3. Quantum Fireworks Algorithm for Parameters Selection of Wavelet Support Vector Machine (w-SVM) Model It is extremely important to select the parameters of w-SVM which can affect the fitting and learning ability. In this paper, the constructed quantum fireworks algorithm (QFA) is used for selecting the appropriate parameters of the w-SVM model in order to improve the icing forecasting accuracy. The flowchart of QFA for parameter selection of the fireworks the w-SVM model is shown in Figure 1 and the details of the QFA-w-SVM model are shown as follows: (1) Initialize the parameters. Initialize the number of fireworks N, the explosion radius A1, the number of explosive sparks M, the mutation probability P , the maximum number of iterations id Maxgen, the upper and lower bound of the solution space V and V respectively, and so on. up down (2) Separate the sample date as training samples and test samples, then normalize the sample data. (3) Initialize the solution space. a In the solution space, randomly initialize N positions, that is, N fireworks. Each of the fireworks has two dimensions, that is, C and s. b Use the probability amplitude of quantum bits to encode current position of fireworks according to Equations (9)–(10). c Converse the Solution space according to Equations (11)–(12). d Input the training samples, use w-SVM to carry out a training simulation for each fireworks, and calculate the value of the fitness function corresponding to each of the fireworks. (4) Initialize the global optimal solution by using the above initialized solution space, including the global optimal phase, the global optimal position quantization of fireworks, the global best fireworks, and the global best fitness value. (5) Start iteration and stop the cycle when the maximum number of iterations Maxgen is achieved. a According to the fitness value of each firework, calculate the corresponding explosion radius Rpiq and the number of sparks Spiq generated by each explosion. The purpose of calculating Rpiq and Spiq is to obtain the optimal fitness values, which means that if the fitness value is smaller, the explosion radius is larger, and the number of sparks generated by each explosion is bigger. Therefore, more excellent fireworks can be retained as much as possible. b Generate the explosive sparks. When each fireworks explodes, carry out the solution space conversion for each explosion spark and control their spatial positions through cross border detection. c Generate M Gaussian mutation sparks. Carry out a fireworks mutation operation according to Equations (26)–(27), resulting in M mutation sparks. d Use the initial fireworks, explosive sparks and Gauss mutation sparks to establish a group, and the fitness value of each firework in the new group is calculated by using w-SVM. e Update the global optimal phase, the global optimal position quantization of fireworks, the global best fireworks, and the global best fitness value. f Use roulette algorithm to choose the next generation of fireworks, and obtain the next generation of N fireworks. g Use w-SVM to carry out a training simulation for each fireworks, and update the value of the fitness function corresponding to each of the fireworks. (6) After all iterations, the best fireworks can be obtained, which correspond to the best parameters of C and s. Appl. Sci. 2016, 6, 54 12 of 23 Appl. Sci. 2016, 6, 54  13 of 24  Start Initialize the global Use w-SVM to calculate the optimal solution fitness of each Fireworks in the new group Parameter setting Iteration number<maxGen Construct training Update global optimum samples and test samples YES Calculate the explosion radius Sample data Use Roulette algorithm to select R(i), the number of sparks normalization the next generation N fireworks generated by each explosion S(i) Use w-SVM to update the fitness Initialize N fireworks Generate explosion sparks function value of each fireworks No Generate Gauss mutation Obtain the fireworks with the Fireworks qua ntization sparks optimal fitness encoding Space so lutio n conversion Use optimal parameters to carry out w-SVM s imulation prediction Calculate the fitness function value of each fireworks using w-SVM End Figure 1. The flowchart of the quantum fireworks algorithm (QFA)-w-SVM (wavelet Support Vector Figure 1. The flowchart of the quantum fireworks algorithm (QFA)‐w‐SVM (wavelet Support Vector  Machine) model. Machine) model.  3. Case Study and Results Analysis 3. Case Study and Results Analysis  3.1. Data Selection 3.1. Data Selection  Transmission line icing is affected by many factors, which mainly include wind direction, light Transmission line icing is affected by many factors, which mainly include wind direction, light  intensity, air pressure, altitude, condensation level, terrain, alignments, wires hanging height, wire intensity, air pressure, altitude, condensation level, terrain, alignments, wires hanging height, wire  stiffness, wire diameter, load current and so on. However, the necessary meteorological conditions stiffness, wire diameter, load current and so on. However, the necessary meteorological conditions  are: (1) the relative air humidity must be above 85%; (2) the wind speed should be greater than 1 m/s; are: (1) the relative air humidity must be above 85%; (2) the wind speed should be greater than 1 m/s;  (3) the temperature needs to reach 0 C and below. In addition, the impact factors, which have the (3) the temperature needs to reach 0 °C and below. In addition, the impact factors, which have the  greater correlations with the line icing, mainly include: wind direction, light intensity and air pressure, greater  correlations  with  the  line  icing,  mainly  include:  wind  direction,  light  intensity  and  air  etc. In general, when the wind direction is parallel to the wire or the angle between the wire and the pressure, etc. In general, when the wind direction is parallel to the wire or the angle between the wire  wire is less than 45, the extent of line icing is lighter; when the wind direction is vertical to the wire and the wire is less than 45, the extent of line icing is lighter; when the wind direction is vertical to  or the angle between the wire and the wire is more than 45, the extent of line icing is more severe. the wire or the angle between the wire and the wire is more than 45, the extent of line icing is more  Similarly, the lower the light intensity is, the more severe the line icing is. severe. Similarly, the lower the light intensity is, the more severe the line icing is.  In this paper, three power transmission lines, named “Qianpingxian-95”, “Fusha-I-xian” and In this paper, three power transmission lines, named “Qianpingxian‐95”, ”Fusha‐Ӏ‐xian” and  “Yangtongxian” in PingXi, ChangSha and ZhaoYang of Hunan province, respectively, are selected as the “Yangtongxian” in PingXi, ChangSha and ZhaoYang of Hunan province, respectively, are selected  case studies to demonstrate the effectiveness, feasibility and robustness of the proposed method. The as  the  case  studies  to  demonstrate  the  effectiveness,  feasibility  and  robustness  of  the  proposed  data from the above mentioned three transmission lines are provided by Key Laboratory of Disaster method.  The  data  from  the  above  mentioned  three  transmission  lines  are  provided  by  Key  Prevention and Mitigation of Power Transmission and Transformation Equipment (Changsha, China). Laboratory  of  Disaster  Prevention  and  Mitigation  of  Power  Transmission  and  Transformation  As is known, a large freezing disaster occurred in the south of China, which caused huge damage Equipment (Changsha, China).  to the power grid system. Hunan province, located in the southern part of China, was one of the As  is  known,  a  large  freezing  disaster  occurred  in  the  south  of  China,  which  caused  huge  most serious areas affected by this disaster. During the disaster period, one third of 500 and 200 kV damage to the power grid system. Hunan province, located in the southern part of China, was one of  substations were out of action in the Hunan power grid. According to the statistics, there were 481 line the most serious areas affected by this disaster. During the disaster period, one third of 500 and 200  breakages of 500 kV transmission lines, 673 line breakages of 200 kV transmission lines, 142 tower kV substations were out of action in the Hunan power grid. According to the statistics, there were  Appl. Sci. 2016, 6, 54  14 of 24  Appl. Sci. 2016, 6, 54  14 of 24  Appl. Sci. 2016, 6, 54 13 of 23 481 line breakages of 500 kV transmission lines, 673 line breakages of 200 kV transmission lines, 142  481 line breakages of 500 kV transmission lines, 673 line breakages of 200 kV transmission lines, 142  tower collapses of 500 kV AC and DC transmission lines, 633 tower collapses of 220 kV transmission  collapses of 500 kV AC and DC transmission lines, 633 tower collapses of 220 kV transmission lines, tower collapses of 500 kV AC and DC transmission lines, 633 tower collapses of 220 kV transmission  lines, and 1203 tower collapses of 110 kV transmission lines. Moreover, in the Hunan area, which  and 1203 tower collapses of 110 kV transmission lines. Moreover, in the Hunan area, which suffered lines, and 1203 tower collapses of 110 kV transmission lines. Moreover, in the Hunan area, which  suffered  from  the  influence  of  topography,  it  is  easy  to  form  the  stationary  front  due  to  the  from the influence of topography, it is easy to form the stationary front due to the mountains block, suffered  from  the  influence  of  topography,  it  is  easy  to  form  the  stationary  front  due  to  the  mountains block, when cold air enters the area. Therefore, it is due to their certain typicality that we  when cold air enters the area. Therefore, it is due to their certain typicality that we select those mountains block, when cold air enters the area. Therefore, it is due to their certain typicality that we  select those three transmission lines in Hunan province as cases to verify the validity and robustness  three transmission lines in Hunan province as cases to verify the validity and robustness of the select those three transmission lines in Hunan province as cases to verify the validity and robustness  of the proposed method.  proposed method. of the proposed method.  Case 1: the data on “Qianpingxian‐95” are from 10 January 2008 to 15 February 2008, which has  Case 1: the data on “Qianpingxian-95” are from 10 January 2008 to 15 February 2008, which has Case 1: the data on “Qianpingxian‐95” are from 10 January 2008 to 15 February 2008, which has  221 data groups for the training and testing in total. The former 180 data groups are regarded as a  221 data groups for the training and testing in total. The former 180 data groups are regarded as a 221 data groups for the training and testing in total. The former 180 data groups are regarded as a  training set and the last 41 groups of data are a testing set. The input vectors of SVM are average  training set and the last 41 groups of data are a testing set. The input vectors of SVM are average training set and the last 41 groups of data are a testing set. The input vectors of SVM are average  temperature, relative humidity, wind speed, wind direction, and sunlight intensity, and the output  temperature, relative humidity, wind speed, wind direction, and sunlight intensity, and the output temperature, relative humidity, wind speed, wind direction, and sunlight intensity, and the output  vector is ice thickness. The sample data are shown in Figure 2.  vector is ice thickness. The sample data are shown in Figure 2. vector is ice thickness. The sample data are shown in Figure 2.  Case 2: the data of ”Fusha‐Ӏ‐xian” are from 12 January 2008 to 25 February 2008, which has 287  Case 2: the data of “Fusha-I-xian” are from 12 January 2008 to 25 February 2008, which has Case 2: the data of ”Fusha‐Ӏ‐xian” are from 12 January 2008 to 25 February 2008, which has 287  data groups. The former 240 groups are taken as a training set and the remaining 47 groups as a  287 data groups. The former 240 groups are taken as a training set and the remaining 47 groups as a data groups. The former 240 groups are taken as a training set and the remaining 47 groups as a  testing set. The input vectors are same as that of Case 1. The sample data are shown in Figure 3.  testing set. The input vectors are same as that of Case 1. The sample data are shown in Figure 3. testing set. The input vectors are same as that of Case 1. The sample data are shown in Figure 3.  Case 3: the data of “Yangtongxian” are from 8 January 2008 to 24 February 2008, whose data  Case 3: the data of “Yangtongxian” are from 8 January 2008 to 24 February 2008, whose data Case 3: the data of “Yangtongxian” are from 8 January 2008 to 24 February 2008, whose data  groups  total  up  to  329.  The  former  289  groups  are  taken  as  a  training  set  and  the  remaining   groups total up to 329. The former 289 groups are taken as a training set and the remaining 40 groups groups  total  up  to  329.  The  former  289  groups  are  taken  as  a  training  set  and  the  remaining   40 groups as a testing set. The input vectors are still same as that of Case 1. The sample data are  as a testing set. The input vectors are still same as that of Case 1. The sample data are shown in 40 groups as a testing set. The input vectors are still same as that of Case 1. The sample data are  shown in Figure 4.  Figure 4. shown in Figure 4.  Figure 2. The original sample data charts.  Figure 2. The original sample data charts. Figure 2. The original sample data charts.  Figure Figure 3. 3. Origi Original nal data data of of “Fusha “Fusha-I-xian”. ‐Ӏ‐xian”.  Figure 3. Original data of “Fusha‐Ӏ‐xian”.  Appl. Sci. 2016, 6, 54 14 of 23 Appl. Sci. 2016, 6, 54  15 of 24  Figure 4. Original data charts of “Yangtongxian”.  Figure 4. Original data charts of “Yangtongxian”. 3.2. Data Pre‐Treatment  3.2. Data Pre-Treatment Before the calculation, the data must be screened and normalized to put them in the range of 0  Before the calculation, the data must be screened and normalized to put them in the range of 0 to to 1 using the following formula:  1 using the following formula: y  y i min y  y Z  z  i  1, 2 , 3 ,..., n i min (46)  Z  tz u  i  1, 2, 3, ..., n (46) i y  y max min y  y   max min where    and    are the maximum and minimum value of sample data, respectively. The  y y max min where y and y are the maximum and minimum value of sample data, respectively. The values max min values of each data are in the range [0,1] for eliminating the dimension influence.  of each data are in the range [0,1] for eliminating the dimension influence. Furthermore,  this  paper  will  use  a  quantum  fireworks  algorithm  (QFA)  to  optimize  the  Furthermore, this paper will use a quantum fireworks algorithm (QFA) to optimize the parameters parameters  C,   of the wavelet support vector machine to find optimal parameters to improve the  C, s of the wavelet support vector machine to find optimal parameters to improve the prediction prediction accuracy. In the parameters optimization, we adopt the mean square error (MSE) as the  accuracy. In the parameters optimization, we adopt the mean square error (MSE) as the fitness function fitness function to realize the process of QFA, and the formula of MSE is as follows:  to realize the process of QFA, and the formula of MSE is as follows: ° y  y  i i (47)  y  y i 1 i function (i )  i1 n f unctionpiq  (47) where    is the actual value;  y   is the prediction value; and  n  is the sample number.  where y is the actual value; y is the prediction value; and n is the sample number. 3.3. Case Study 1  3.3. Case Study 1 In this case, “Qianpingxian‐95”, which is a 220kV high voltage line, is selected to performed the  In this case, “Qianpingxian-95”, which is a 220kV high voltage line, is selected to performed simulation. After the above preparation, apply the constructed model to verify its feasibility and  the simulation. After the above preparation, apply the constructed model to verify its feasibility robustness.  and robustness. Firstly, initialize the parameters of QFA. Let the maximum iteration number  Maxgen  200 ,  Firstly, initialize the parameters of QFA. Let the maximum iteration number Maxgen  200, PopNum 40 M  100 population  number  ,  Sparks  number  determination  constant  ,  explosion  population number PopNum  40, Sparks number determination constant M  100, explosion radius 510 ˆ 5 10 determination radius determconstant ination con R  stant 150  ,R the bor 150 der , the of parameter border of pa C ra bemr2eter 2 s,  the be  [2 border 2 of ] , parameter the border sof  5 5 55 be r2 2 s, the upper and lower limits of searching space of fireworks individual be V  512 and up parameter    be  [2 2 ] , the upper and lower limits of searching space of fireworks individual  V  512, respectively, and mutation rate P  0.05. Then use the steps of QFA to optimize the down id be  V  512  and  , respectively, and mutation rate  P  0.05 . Then use the steps  V  512 up down id parameters of w-SVM and obtain C  18.3516, s  0.031402. Finally, we predict the icing thickness of of  QFA  to  optimize  the  parameters  of  w‐SVM  and  obtain  C 18.3516,    0.031402 .  Finally,  we  the testing sample after putting the optimal parameters into the w-SVM regression model. predict the icing thickness of the testing sample after putting the optimal parameters into the w‐SVM  Figure 5 shows the optimization process of the quantum fireworks algorithm (QFA). As we can regression model.  see, the proposed model obtains the optimal value when the iteration number is 35, and the optimal Figure 5 shows the optimization process of the quantum fireworks algorithm (QFA). As we can  value is 0.11; this illustrates that the proposed algorithm can obtain the global optimum with a fast see, the proposed model obtains the optimal value when the iteration number is 35, and the optimal  convergence speed. Figure 6 shows the forecasting results of the proposed method. This paper also value is 0.11; this illustrates that the proposed algorithm can obtain the global optimum with a fast  convergence speed. Figure 6 shows the forecasting results of the proposed method. This paper also  Appl. Sci. 2016, 6, 54  16 of 24  Appl. Sci. 2016, 6, 54 15 of 23 Appl. Appl. Sci.  Sci. 2016  2016, 6, ,6 54 , 54    1616 of of 24 24    selects w‐SVM optimized by a regular fireworks algorithm (FA‐w‐SVM), SVM optimized by particle  selects selects w w‐SV ‐SVMM optimized  optimized by by a a reg  reguular lar firework  fireworkss al algo gorit rithhmm (FA  (FA‐w‐w‐SV ‐SVM), M), SV  SVMM opti  optimi mizzeedd by by part  particle icle   selects w-SVM optimized by a regular fireworks algorithm (FA-w-SVM), SVM optimized by particle swarm  optimization  algorithm  (PSO‐SVM),  SVM  model,  and  multiple  linear  regression  model  swarm  optimization  algorithm  (PSO‐SVM),  SVM  model,  and  multiple  linear  regression  model  swarm  optimization  algorithm  (PSO‐SVM),  SVM  model,  and  multiple  linear  regression  model  swarm (MLR) optimization to make a coalgorithm mparison,  (PSO-SVM), and the convergence SVM model,  curves and multiple of FA‐wlinear ‐SVM ran egrdession  PSO‐SV model M ar(MLR) e also  (ML (MLRR) )to to ma  makkee a a co compa mparis risoonn, ,and  and the  the convergence  convergence curves  curves of of FA  FA‐w‐w‐SV ‐SVMM an  andd PSO  PSO‐SV ‐SVMM ar aree al also so   to make a comparison, and the convergence curves of FA-w-SVM and PSO-SVM are also shown in shown in Figure 5. The forecasting results of four models are shown in Figure 7.  shown in Figure 5. The forecasting results of four models are shown in Figure 7.  shown in Figure 5. The forecasting results of four models are shown in Figure 7.  Figure 5. The forecasting results of four models are shown in Figure 7.    Figure 5. The iteration process of QFA‐w‐SVM. PSO‐SVM: particle swarm optimization algorithm.  Figure Figure 5. 5. The  The it iteration eration process  process of of QFA  QFA‐w‐w‐SV ‐SVM. M. PSO  PSO‐SVM ‐SVM: :particle  particle swarm  swarm optimization  optimization al algori gorithm. thm.   Figure 5. The iteration process of QFA-w-SVM. PSO-SVM: particle swarm optimization algorithm. Actual value A QF A ctua cA tua l- v w- l v aS lu aV lu eM e QFA-w-SVM QFA-w-SVM 1 6 11 16 21 26 31 36 41 1 6 11 16 21 26 31 36 41 1 6 11 16 21 26 31 36 41 Figure 6. Forecasting value of QFA‐w‐SVM.  Figure 6. Forecasting value of QFA-w-SVM. Figure 6. Forecasting value of QFA‐w‐SVM.  Figure 6. Forecasting value of QFA‐w‐SVM.  8 Actual value 88 Ac Ac tuta ula l 7.5 FA-w-SVM value value 7.5 7.5 7 FA-w-SVM FA-w-SVM PSO-SVM 6.5 PS PS O-SVM O-SVM 6.5 6.5 5.5 5.5 5.5 4.5 4.5 4.5 3.5 3.5 3.5 1 6 11 16 21 26 31 36 41 1 6 11 16 21 26 31 36 41   1 6 11 16 21 26 31 36 41 Figure 7. Forecasting value of each method. MLR: multiple linear regression model. Figure 7. Forecasting value of each method. MLR: multiple linear regression model.  Figure 7. Forecasting value of each method. MLR: multiple linear regression model.  Figure 7. Forecasting value of each method. MLR: multiple linear regression model.  Forecasting results/mm Forecasting results/mm Forecasting results/mm Forecasting results/mm Forecasting results/mm Forecasting results/mm Appl. Sci. 2016, 6, x Appl. Sci. 2016, 6, 54 16 of 23 For further analysis, this paper will use the relative error (RE), the mean absolute percentage error (MAPE), mean square error (MSE) and average absolute error (AAE) to evaluate the results of prediction. For further analysis, this paper will use the relative error (RE), the mean absolute percentage error (MAPE), mean square error (MSE) and average absolute error (AAE) to evaluate the results Y − Y ′ 1 Y − Y ′ i i i i RE = ×100% MAPE = × 100% of prediction. Y n Y i i 1 1 Y  Y 1 Y  Y i i i i (48) RE   100% MAPE   100% n n n 1 1 Y n Y i i 1 ′ ′ MSE = () Y −Y AAE = ( Y −Y ) ( Y )   i i i i i (48) n n n n n i=1  i== 11   i ° ° ° 1 1 1 1 1 MSE  Y  Y AAE  p Y  Y q{p Y q i i i i i n n n Y Y ′ i1 i1 i1 where is the actual value of power load; is the forecasting value of power load; i i i = 1,2, , n where Y is the actual value of power load; Y is the forecasting value of power load; i  1, 2, , n. The relative errors (RE) value of the QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR The relative errors (RE) value of the QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR models models are shown in Figure 8. It can be clearly seen that the RE curve of QFA-w-SVM is the lowest are shown in Figure 8. It can be clearly seen that the RE curve of QFA-w-SVM is the lowest among among the other four models, which demonstrates that the accuracy of the proposed algorithm is the other four models, which demonstrates that the accuracy of the proposed algorithm is much much higher than other mentioned algorithms. The RE curve of the MLR model is the highest, and higher than other mentioned algorithms. The RE curve of the MLR model is the highest, and this this demonstrates the forecasting results based on the MLR model are not satisfied here. demonstrates the forecasting results based on the MLR model are not satisfied here. 18% QFA-w-SVM FA-w-SVM PSO-SVM SVM 15% MLR 12% 9% 6% 3% 0% 1 4 7 101316 1922 252831 34 3740 Figure 8. Forecasting errors of each model. Figure 8. Forecasting errors of each model. The relative error range [−3%,+3%] is always regarded as a standard to evaluate the The relative error range [3%,+3%] is always regarded as a standard to evaluate the performance performance of a forecasting model. As we can see from Figure 7, 35 points of QFA-w-SVM means of a forecasting model. As we can see from Figure 7, 35 points of QFA-w-SVM means 85% forcasting 85% forcasting points are in the range [−3%,+3%], and only 6 forecasting points are out of this scope. points are in the range [3%,+3%], and only 6 forecasting points are out of this scope. In the model of In the model of FA-w-SVM, only 8 out of 41 points are in the range [−3%,+3%], which means that FA-w-SVM, only 8 out of 41 points are in the range [3%,+3%], which means that 78% of forecasting 78% of forecasting points are out of the scope. In addition, only 5 forecasting points of PSO-SVM and points are out of the scope. In addition, only 5 forecasting points of PSO-SVM and one forecasting one forecasting point of SVM are in the scope of [−3%,+3%]; none of the forecasting points of the point of SVM are in the scope of [3%,+3%]; none of the forecasting points of the MLR model are in MLR model are in the scope. These results demonstrate that the QFA-w-SVM model has a better the scope. These results demonstrate that the QFA-w-SVM model has a better performance in icing performance in icing forecasting when compared with other models. In addition, the maximum and forecasting when compared with other models. In addition, the maximum and minimum relative minimum relative errors (MaxRE and MinRE) can also reflect the forecasting accuracy of icing errors (MaxRE and MinRE) can also reflect the forecasting accuracy of icing forecasting models. Firstly, forecasting models. Firstly, both the MaxRE and MinRE values of combined models are smaller than both the MaxRE and MinRE values of combined models are smaller than those of single models, those of single models, which proves the optimization algorithm can facilitate improving accuracy of which proves the optimization algorithm can facilitate improving accuracy of w-SVM through finding w-SVM through finding optimal parameters. Secondly, the MaxRE and MinRE values of QFA-w-SVM optimal parameters. Secondly, the MaxRE and MinRE values of QFA-w-SVM model are 3.61% and model are 3.61% and 0.89%, respectively, both of which are the smallest among the five icing 0.89%, respectively, both of which are the smallest among the five icing forecasting models, which forecasting models, which illustrates that QFA-w-SVM has a better nonlinear fitting ability in illustrates that QFA-w-SVM has a better nonlinear fitting ability in sample training. Thirdly, in the sample training. Thirdly, in the FA-w-SVM model, the MaxRE and MinRE values are 5.91% and FA-w-SVM model, the MaxRE and MinRE values are 5.91% and 1.02%, respectively, both of which are 1.02%, respectively, both of which are smaller than that of PSO-SVM model(6.75% and 1.82%), which smaller than that of PSO-SVM model(6.75% and 1.82%), which means FA-w-SVM has better forecasting means FA-w-SVM has better forecasting accuracy. Finally, both the MaxRE and MinRE values of accuracy. Finally, both the MaxRE and MinRE values of SVM are 8.73% and 2.73%, respectively, both of SVM are 8.73% and 2.73%, respectively, both of which are smaller than that of MLR, that mean the 17 Appl. Sci. 2016, 6, 54  18 of 24  nonlinear processing ability is better than that of MLR; in other words, the robustness of SVM is  much Appl.  stronge Sci. 2016 r ,tha 6, 54n that of the MLR model.  17 of 23 The MAPE, MSE and AAE values of the above mentioned five models are shown in Figure 9.  The experimental results show that the forecasting effect of the SVM model obviously precedes the  which are smaller than that of MLR, that mean the nonlinear processing ability is better than that of multiple  linear  regression  model;  this illustrates  that  the  intelligent  forecasting  model  has strong  MLR; in other words, the robustness of SVM is much stronger than that of the MLR model. learning ability and nonlinear mapping ability which the multiple linear regression model cannot  The MAPE, MSE and AAE values of the above mentioned five models are shown in Figure 9. match.  In  addition,  the  MAPE  value  of  single  SVM  is  4.988%,  which  is  much  higher  than  that  The experimental results show that the forecasting effect of the SVM model obviously precedes the multiple linear regression model; this illustrates that the intelligent forecasting model has strong obtained by hybrid QFA‐w‐SVM, FA‐w‐SVM and PSO‐SVM models (which are 1.56%, 2.83% and  learning ability and nonlinear mapping ability which the multiple linear regression model cannot 2.776%, respectively); this result proves that the optimization algorithm can help single SVM to find  match. In addition, the MAPE value of single SVM is 4.988%, which is much higher than that obtained the optimal parameters, and heighten the learning capacity and forecasting accuracy. Furthermore,  by hybrid QFA-w-SVM, FA-w-SVM and PSO-SVM models (which are 1.56%, 2.83% and 2.776%, the MAPEs of QFA‐w‐SVM and FA‐w‐SVM are 1.56% and 2.83%, respectively; this shows that the  respectively); this result proves that the optimization algorithm can help single SVM to find the QFA model can greatly improve the optimization performance of regular FA model through the  optimal parameters, and heighten the learning capacity and forecasting accuracy. Furthermore, the combination  of  quantum  optimization  algorithm,  which  makes  the  algorithm  can  easily  get  the  MAPEs of QFA-w-SVM and FA-w-SVM are 1.56% and 2.83%, respectively; this shows that the QFA optimal results. Meanwhile, the MAPE value of PSO‐SVM model is 2.776%, which is higher than that  model can greatly improve the optimization performance of regular FA model through the combination of  QFA‐w‐SVM;  this  illustrates  that  QFA‐w‐SVM  model  has  a  better  global  convergence  of quantum optimization algorithm, which makes the algorithm can easily get the optimal results. performance compared with PSO‐SVM, and also proves that w‐SVM has better nonlinear mapping  Meanwhile, the MAPE value of PSO-SVM model is 2.776%, which is higher than that of QFA-w-SVM; capability compared with SVM.  this illustrates that QFA-w-SVM model has a better global convergence performance compared with PSO-SVM, and also proves that w-SVM has better nonlinear mapping capability compared with SVM. 40% MAPE MSE AAE 30% 20% 10% 0% QFA-w-SVM FA-w-SVM SVM MLR PSO-SVM Figure 9. Values of mean absolute percentage error (MAPE), mean square error (MSE) and average Figure 9. Values of mean absolute percentage error (MAPE), mean square error (MSE) and average  absolute error (AAE). absolute error (AAE).  Furthermore, the MSE values of QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR models are Furthermore, the MSE values of QFA‐w‐SVM, FA‐w‐SVM, PSO‐SVM, SVM and MLR models  0.0174, 0.0466, 0.0590, 0.1094 and 0.3581, respectively, and the AAE values of those five models are 0.024, 0.0389, 0.0439, 0.0602 and 0.1092, respectively. As we know, when the value of MSE and AAE is are 0.0174, 0.0466, 0.0590, 0.1094 and 0.3581, respectively, and the AAE values of those five models  smaller, the prediction effect of model is more ideal. It can be clearly seen that the MSE value and AAE are 0.024, 0.0389, 0.0439, 0.0602 and 0.1092, respectively. As we know, when the value of MSE and  value of QFA-w-SVM is the smallest among other models, which directly demonstrates the feasibility AAE is smaller, the prediction effect of model is more ideal. It can be clearly seen that the MSE value  and effectiveness of QFA-w-SVM model, and QFA can improve the optimization performance of the and AAE value of QFA‐w‐SVM is the smallest among other models, which directly demonstrates the  regular FA to help SVM find the optimal parameters to improve the forecasting accuracy. feasibility  and  effectiveness  of  QFA‐w‐SVM  model,  and  QFA  can  improve  the  optimization  performance of the regular FA to help SVM find the optimal parameters to improve the forecasting  3.4. Case Study 2 accuracy.  “Fusha-I-xian”, which is a representative 500 kV transmission line between Fuxing and Shapin of Hunan province, is chosen to prove the robustness and stability of the proposed QFA-w-SVM icing 3.4. Case Study 2  forecasting model. The sample data of “Fusha-I-xian” are also predicted by the five models and the results of five models are also used to make a comparison. ”Fusha‐Ӏ‐xian”, which is a representative 500kV transmission line between Fuxing and Shapin  Figure 10 shows the iteration trend of the QFA-w-SVM searching of optimization parameters. of Hunan province, is chosen to prove the robustness and stability of the proposed QFA‐w‐SVM  As can be seen from Figure 3, the convergence can be seen in generation 39 with the optimal MSE icing forecasting model. The sample data of ”Fusha‐Ӏ‐xian” are also predicted by the five models and  value of 0.1124, and the parameters of w-SVM are obtained with C  23.68, s  0.1526, respectively. the results of five models are also used to make a comparison.  Figure 10 shows the iteration trend of the QFA‐w‐SVM searching of optimization parameters.   As can be seen from Figure 3, the convergence can be seen in generation 39 with the optimal MSE  Appl. Sci. 2016, 6, 54  19 of 24  Appl. Sci. 2016, 6, 54  19 of 24    0.1526 value  of  0.1124,  and  the  parameters  of  w‐SVM  are  obtained  with  C  23.68 ,  ,  respectively. In FA‐w‐SVM model, the convergence can be seen in generation 53 with the optimal    0.1526 value  of  0.1124,  and  the  parameters  of  w‐SVM  are  obtained  with  C  23.68 ,  ,  C  31.58   0.0496 MSE value of 0.1307, and the parameters of w‐SVM are obtained with  ,  . In the  respectively. Appl. Sci. 2016 In  ,FA 6, 54‐w‐SVM model, the convergence can be seen in generation 53 with18 the of 23 optimal  PSO‐SVM  model,  the  convergence  can  be  seen  in  generation  58  with  the  optimal  MSE  value  of  C  31.58   0.0496 MSE value of 0.1307, and the parameters of w‐SVM are obtained with  ,  . In the  C  19.48   0.5962 0.1324, and the parameters of SVM are obtained with  ,  . This proves that the  PSO‐SVM  model,  the  convergence  can  be  seen  in  generation  58  with  the  optimal  MSE  value  of  In FA-w-SVM model, the convergence can be seen in generation 53 with the optimal MSE value of proposed  QFA‐w‐SVM  model  can  find  the  global  optimal  value  with  the  faster  convergence  0.1324, 0.1307,  and the and  the para parameters meters of of SV w-SVM M are are obtained obtained with  witCh C31.58  19. , s 48, 0.0496   0. . 59 In 62 the. PSO-SVM This proves model,  that the  compared with FA‐w‐SVM and PSO‐SVM models. This also validates the stability of the proposed  the convergence can be seen in generation 58 with the optimal MSE value of 0.1324, and the parameters proposed  QFA‐w‐SVM  model  can  find  the  global  optimal  value  with  the  faster  convergence  icing forecasting model.  of SVM are obtained with C  19.48, s  0.5962. This proves that the proposed QFA-w-SVM model can compared with FA‐w‐SVM and PSO‐SVM models. This also validates the stability of the proposed  find the global optimal value with the faster convergence compared with FA-w-SVM and PSO-SVM icing forecasting model.  models. This also validates the stability of the proposed icing forecasting model. Figure 10. The convergence curves of algorithms.  Figure 11 shows the forecasting results of the five models. The forecasting errors of the five  Figure 10. The convergence curves of algorithms. Figure 10. The convergence curves of algorithms.  models are shown in Figure 12. It can be seen from the comparison of the forecasting curve and the  actual value that the forecasting results of ice thickness of those five models have approximation to  Figure 11 shows the forecasting results of the five models. The forecasting errors of the five models Figure 11 shows the forecasting results of the five models. The forecasting errors of the five  actual  curve.  Among  them,  the  proposed  QFA‐w‐SVM  has  the  best  fitting  learning  and  fitting  are shown in Figure 12. It can be seen from the comparison of the forecasting curve and the actual models are shown in Figure 12. It can be seen from the comparison of the forecasting curve and the  ability, and 44 forecasting points mean that almost 94% of points are in the scope of [−3,+3]; only  value that the forecasting results of ice thickness of those five models have approximation to actual actual value that the forecasting results of ice thickness of those five models have approximation to  threecurve.  forecaAmong sting points them, are the out proposed  of this QF sc A-w-SVM ope. In FA has‐wthe ‐SVbest M model, fitting learning 27 forecand astin fitting g points ability  mea , and ns that  actual  curve.  Among  them,  the  proposed  QFA‐w‐SVM  has  the  best  fitting  learning  and  fitting  44 forecasting points mean that almost 94% of points are in the scope of [3,+3]; only three forecasting nearly 57% points are in the range [−3,+3], and 20 points fall out this scope. In PSO‐SVM model, 20  ability, points  and ar 44 e out forec ofa this stin scope. g points In F A-w-SVM mean tha model, t almo 27stfor  94 ecasting % of points points are means  in that the nearly scope57%  of [points −3,+3]; only  forecasting points means that nearly 43% of points are in the scope, and 27 points are out of the  are in the range [3,+3], and 20 points fall out this scope. In PSO-SVM model, 20 forecasting points three forecasting points are out of this scope. In FA‐w‐SVM model, 27 forecasting points means that  range. In SVM model, only 2 points are in this range, and the rest of the forecasting points are not in  means that nearly 43% of points are in the scope, and 27 points are out of the range. In SVM model, nearly 57% points are in the range [−3,+3], and 20 points fall out this scope. In PSO‐SVM model, 20  this scope. In MLR model, none of points are in this scope. This reveals that QFA‐w‐SVM model has  only 2 points are in this range, and the rest of the forecasting points are not in this scope. In MLR forec a higher asting forec  poiansttsing  mea  accnus rathcyat tha  nearly n other  43%  mentioned  of points mo  are de lin s,  an thed  it sco alpsoe,  has and stron  27 po gerin robustness ts are out  and of the     model, none of points are in this scope. This reveals that QFA-w-SVM model has a higher forecasting range nonlinear . In SVM  fitting  model,  abilit only y.   2 points are in this range, and the rest of the forecasting points are not in  accuracy than other mentioned models, and it also has stronger robustness and nonlinear fitting ability. this scope. In MLR model, none of points are in this scope. This reveals that QFA‐w‐SVM model has  a higher forecasting accuracy than other mentioned models, and it also has stronger robustness and  Original data QFA-w-SVM FA-w-SVM nonlinear fitting ability.  PSO-SVM SVM MLR Original data QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR 1 6 11 16 21 26 31 36 41 46 Figure 11. Forecasting results of the five models. Figure 11. Forecasting results of the five models.  1 6 11 16 21 26 31 36 41 46 Figure 11. Forecasting results of the five models.  Foreasting results/mm Foreasting results/mm Appl. Sci. 2016, 6, 54  20 of 24  Appl. Sci. 2016, 6, 54  20 of 24  Appl. Sci. 2016, 6, 54 19 of 23 10% QFA-w-SVM FA-w-SVM PSO-SVM 10% SVM MLR QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR 8% 8% 6% 6% 4% 4% 2% 2% 0% 0% 1 6 11 16 21 26 31 36 41 46 1 6 11 16 21 26 31 36 41 46 Figure 12. Forecasting errors of each model.  Figure 12. Forecasting errors of each model. Figure 12. Forecasting errors of each model.  The values of MAPE, MSE, AAE of the five models in icing forecasting of ”Fusha‐Ӏ‐xian” are  The values of MAPE, MSE, AAE of the five models in icing forecasting of “Fusha-I-xian” are The values of MAPE, MSE, AAE of the five models in icing forecasting of ”Fusha‐Ӏ‐xian” are  shown in Figure 13. It can be seen that the proposed QFA‐w‐SVM model still has the smallest MAPE,  shown in Figure 13. It can be seen that the proposed QFA-w-SVM model still has the smallest MAPE, shown in Figure 13. It can be seen that the proposed QFA‐w‐SVM model still has the smallest MAPE,  MSE  and  AAE  values,  which  are  1.9%,  0.026  and  0.018,  respectively.  This  again  reveals  that  the  MSE and AAE values, which are 1.9%, 0.026 and 0.018, respectively. This again reveals that the MSE  and  AAE  values,  which  are  1.9%,  0.026  and  0.018,  respectively.  This  again  reveals  that  the  proposed QFA‐w‐SVM model has the best performance in the icing forecasting results. In addition,  proposed QFA-w-SVM model has the best performance in the icing forecasting results. In addition, the proposed QFA‐w‐SVM model has the best performance in the icing forecasting results. In addition,  the MAPE, MSE and AAE values of FA‐w‐SVM are 3.01%, 0.063 and 0.028, respectively, which are  MAPE, MSE and AAE values of FA-w-SVM are 3.01%, 0.063 and 0.028, respectively, which are both the MAPE, MSE and AAE values of FA‐w‐SVM are 3.01%, 0.063 and 0.028, respectively, which are  both lar large gerr than  than that  tha oft QF of A-w-SVM, QFA‐w‐SV bM, ut smaller  but smaller than the  tha rest n the of thr  rest ee  models. of threeIn models. PSO-SVM,  In PSO the MAPE, ‐SVM, the  both larger than that of QFA‐w‐SVM, but smaller than the rest of three models. In PSO‐SVM, the  MSE and AAE values are 3.45%, 0.099and 0.034, respectively, which are smaller than those of SVM and MAPE, MSE and AAE values are 3.45%, 0.099and 0.034, respectively, which are smaller than those of  MAPE, MSE and AAE values are 3.45%, 0.099and 0.034, respectively, which are smaller than those of  MLR models. That proves that the combined algorithms have better forecasting performance and the SVM  and  MLR  models.  That  proves  that  the  combined  algorithms  have  better  forecasting  optimization algorithms can help single regression model to achieve a better accuracy through finding SVM  and  MLR  models.  That  proves  that  the  combined  algorithms  have  better  forecasting  performance and the optimization algorithms can help single regression model to achieve a better  better parameters. This result agrees with the one presented in Section 3.3. performance and the optimization algorithms can help single regression model to achieve a better  accuracy through finding better parameters. This result agrees with the one presented in Section 3.3.  accuracy through finding better parameters. This result agrees with the one presented in Section 3.3.  30% 30% MAPE MSE AAE MAPE MSE AAE 20% 20% 10% 10% 0% QFA-w-SVM FA-SVM PSO-SVM SVM MLR 0% QFA-w-SVM FA-SVM PSO-SVM SVM MLR   Figure 13. Values of MAPE, MSE, AAE. Figure 13. Values of MAPE, MSE, AAE.  Figure 13. Values of MAPE, MSE, AAE.  3.5. Case Study 3 3.5. Case Study 3  In this case, the icing data from a 110 kV transmission line called “Yangtongxian” is selected to 3.5. Case Study 3  In this case, the icing data from a 110 kV transmission line called “Yangtongxian” is selected to  validate the robustness and stability of the proposed QFA-w-SVM icing forecasting model. Similarly, In this case, the icing data from a 110 kV transmission line called “Yangtongxian” is selected to  validate  the  robustness  and  stability  of  the  proposed  QFA‐w‐SVM  icing  forecasting  model.  the five models are still used to make a comparison for prediction results. validate  the  robustness  and  stability  of  the  proposed  QFA‐w‐SVM  icing  forecasting  model.  Similarly, the five models are still used to make a comparison for prediction results.  Similarly, the five models are still used to make a comparison for prediction results.  Figure 14 shows the convergence plots of QFA‐w‐SVM, FA‐w‐SVM and PSO‐SVM. As shown  Figure 14 shows the convergence plots of QFA‐w‐SVM, FA‐w‐SVM and PSO‐SVM. As shown  in  the  figure,  QFA‐w‐SVM  obtains  the  optimal  value  after  40  iterations  while  FA‐w‐SVM  and  in  the  figure,  QFA‐w‐SVM  obtains  the  optimal  value  after  40  iterations  while  FA‐w‐SVM  and  PSO‐SVM  models  converge  to  global  solutions  after  53  and  57  iterations,  respectively.  In  other  PSO‐SVM  models  converge  to  global  solutions  after  53  and  57  iterations,  respectively.  In  other  Appl. Sci. 2016, 6, 54 20 of 23 Appl. Sci. 2016, 6, 54  21 of 24  Appl. Sci. 2016, 6, 54  21 of 24  Figure 14 shows the convergence plots of QFA-w-SVM, FA-w-SVM and PSO-SVM. As shown in words,  the  QFA‐w‐SVM still  has  the fastest  convergence  speed and  stronger global  optimization  the figure, QFA-w-SVM obtains the optimal value after 40 iterations while FA-w-SVM and PSO-SVM ability compared with other models.  words,  the  QFA‐w‐SVM still  has  the fastest  convergence  speed and  stronger global  optimization  models converge to global solutions after 53 and 57 iterations, respectively. In other words, the ability compared with other models.  QFA-w-SVM still has the fastest convergence speed and stronger global optimization ability compared with other models. Figure 14. The convergence curves of algorithms.  Figure 14. The convergence curves of algorithms. Figure 14. The convergence curves of algorithms.  The forecasting results and relative errors are shown in Figures 15 and 16, respectively. From  The forecasting results and relative errors are shown in Figures 15 and 16 respectively. From the The forecasting results and relative errors are shown in Figures 15 and 16, respectively. From  the  figures,  it  is  obvious  that  the  proposed  QFA‐w‐SVM  model  still  has  better  accuracy  in  icing  figures, it is obvious that the proposed QFA-w-SVM model still has better accuracy in icing forecasting, the  figures,  it  is  obvious  that  the  proposed  QFA‐w‐SVM  model  still  has  better  accuracy  in  icing  forecasting,  which  demonstrates  that  the  proposed  model  is  superior  for  solving  the  icing  which demonstrates that the proposed model is superior for solving the icing forecasting problem forecasting,  which  demonstrates  that  the  proposed  model  is  superior  for  solving  the  icing  forecasting problem compared to other algorithms, because it can achieve global optima in fewer  compared to other algorithms, because it can achieve global optima in fewer iterations and is a critical forecasting problem compared to other algorithms, because it can achieve global optima in fewer  iterations and is a critical factor in convergence process of algorithms.  factor in convergence process of algorithms. iterations and is a critical factor in convergence process of algorithms.  5 5 Original data QFA-w-SVM FA-w-SVM Original data QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR PSO-SVM SVM MLR 1 6 11 16 21 26 31 36 1 6 11 16 21 26 31 36 Figure 15. Forecasting results of each model.  Figure 15. Forecasting results of each model. Figure 15. Forecasting results of each model.  Forecasting results/mm Forecasting results/mm Appl. Sci. 2016, 6, 54  22 of 24  Appl. Sci. 2016, 6, 54 21 of 23 Appl. Sci. 2016, 6, 54  22 of 24  QFA-w-SVM FA-w-SVM PSO-SVM 12% QFA-w-SVM FA-w-SVM PSO-SVM 12% SVM MLR SVM MLR 10% 10% 8% 8% 6% 6% 4% 4% 2% 2% 0% 0% 1 6 11 16 21 26 31 36 1 6 11 16 21 26 31 36 Figure 16. Forecasting errors of each model.  Figure 16. Forecasting errors of each model. Figure 16. Forecasting errors of each model.  Figure 17 shows the MAPE, MSE, and AAE values of the five models; it is clear that the values  Figure 17 shows the MAPE, MSE, and AAE values of the five models; it is clear that the values of Figure 17 shows the MAPE, MSE, and AAE values of the five models; it is clear that the values  of MAPE, MSE, AAE of the proposed QFA‐w‐SVM model are the smallest among the five models,  MAPE, MSE, AAE of the proposed QFA-w-SVM model are the smallest among the five models, which of MAPE, MSE, AAE of the proposed QFA‐w‐SVM model are the smallest among the five models,  which again demonstrates that the proposed model performs with great robustness and stability.  again demonstrates that the proposed model performs with great robustness and stability. which again demonstrates that the proposed model performs with great robustness and stability.  20% 20% MAPE MSE AAE MAPE MSE AAE 16% 16% 12% 12% 8% 8% 4% 4% 0% QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR 0% QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR Figure 17. MAPE, MSE, AAE values of each model. Figure 17. MAPE, MSE, AAE values of each model.  Figure 17. MAPE, MSE, AAE values of each model.  In summary, the proposed QFA-w-SVM model can greatly close the gap between forecasting In summary, the proposed QFA‐w‐SVM model can greatly close the gap between forecasting  values and original data, which means it outperforms the FA-w-SVM, PSO-SVM, SVM, and MLR values and original data, which means it outperforms the FA‐w‐SVM, PSO‐SVM, SVM, and MLR  In summary, the proposed QFA‐w‐SVM model can greatly close the gap between forecasting  models in icing forecasting with different voltage transmission lines. Moreover, the QFA-w-SVM models in icing forecasting with different voltage transmission lines. Moreover, the QFA‐w‐SVM  values and original data, which means it outperforms the FA‐w‐SVM, PSO‐SVM, SVM, and MLR  which uses QFA to select the parameters of the SVM model, can effectively make a great improvement which  uses  QFA  to  select  the  parameters  of  the  SVM  model,  can  effectively  make  a  great  models in icing forecasting with different voltage transmission lines. Moreover, the QFA‐w‐SVM  for icing forecasting accuracy. It might be a promising alternative for icing thickness forecasting. improvement for icing forecasting accuracy. It might be a promising alternative for icing thickness  which  uses  QFA  to  select  the  parameters  of  the  SVM  model,  can  effectively  make  a  great  forecasting.  4. Conclusions improvement for icing forecasting accuracy. It might be a promising alternative for icing thickness  4. Conclusions  forecasting.  For better forecasting accuracy for icing thickness, this paper proposes an intelligent method for a wavelet support vector machine (w-SVM) based on a quantum fireworks optimization algorithm For better forecasting accuracy for icing thickness, this paper proposes an intelligent method for  4. Conclusions  a wavelet support vector machine (w‐SVM) based on a quantum fireworks optimization algorithm  For better forecasting accuracy for icing thickness, this paper proposes an intelligent method for  a wavelet support vector machine (w‐SVM) based on a quantum fireworks optimization algorithm  Appl. Sci. 2016, 6, 54 22 of 23 (QFA). Firstly, the regular fireworks optimization algorithm is improved through combination with a quantum optimization algorithm and proposes the quantum fireworks optimization algorithm (QFA); the steps of QFA are listed in detail. Secondly, in order to make use of the advantages of kernel function of SVM, the wavelet kernel function is applied to SVM instead of a Gaussian kernel function. Finally, this paper uses the proposed QFA model to optimize the parameters of w-SVM, and builds the icing thickness forecasting model of QFA-w-SVM. During the application of icing thickness forecasting, this paper also gives full consideration to the impact factors and selects the temperature, humidity, wind speed, wind direction and sunlight as the main impact factors. Through the numerical calculations of a self-written program, the empirical results show that the proposed forecasting model of QFA-w-SVM has great robustness and stability in icing thickness forecasting and is effective and feasible. Author Contributions: Drafting of the manuscript: Tiannan Ma and Dongxiao Niu; Implementation of numerical simulations and preparations of figures: Tiannan Ma and Ming Fu; Finalizing the manuscript: Tiannan Ma. Planning and supervision of the study: Dongxiao Niu and Tiannan Ma. Acknowledgments: This work is supported by the Natural Science Foundation of China (Project Nos. 71471059). Conflicts of Interest: The authors declare no conflict of interest. Reference 1. Yin, S.; Lu, Y.; Wang, Z.; Li, P.; Xu, K.J. Icing Thickness forecasting of overhead transmission line under rough weather based on CACA-WNN. Electr. Power Sci. Eng. 2012, 28, 28–31. 2. Jia-Zheng, L.U.; Peng, J.W.; Zhang, H.X.; Li, B.; Fang, Z. Icing meteorological genetic analysis of hunan power grid in 2008. Electr. Power Constr. 2009, 30, 29–32. 3. Goodwin, E.J.I.; Mozer, J.D.; Digioia, A.M.J.; Power, B.A. Predicting ice and snow loads for transmission line design. 1983, 83, 267–276. 4. Makkonen, L. 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Zarnani, A.; Musilek, P.; Shi, X.; Ke, X.D.; He, H.; Greiner, R. Learning to predict ice accretion on electric power lines. Eng. Appl. Artif. Intell. 2012, 25, 609–617. [CrossRef] 10. Huang, X.-T.; Xu, J.-H.; Yang, C.-S.; Wang, J.; Xie, J.-J. Transmission line icing prediction based on data driven algorithm and LS-SVM. Autom. Electr. Power Syst. 2014, 38, 81–86. 11. Li, Q.; Li, P.; Zhang, Q.; Ren, W.P.; Cao, M.; Gao, S.F. Icing load prediction for overhead power lines based on SVM. In Proceedings of the 2011 International Conference on IEEE Modelling, Identification and Control (ICMIC), Shanghai, China, 26–29 June 2011; pp. 104–108. 12. Li, J.; Dong, H. Modeling of chaotic systems using wavelet kernel partial least squares regression method. Acta Phys. Sin. 2008, 57, 4756–4765. 13. Wu, Q. Hybrid model based on wavelet support vector machine and modified genetic algorithm penalizing Gaussian noises for power load forecasts. Expert Syst. Appl. 2011, 38, 379–385. [CrossRef] 14. Liao, R.J.; Zheng, H.B.; Grzybowski, S.; Yang, L.J. Particle swarm optimization-least squares support vector regression based forecasting model on dissolved gases in oil-filled power transformers. Electr. Power Syst. Res. 2011, 81, 2074–2080. [CrossRef] 15. Dos Santosa, G.S.; Justi Luvizottob, L.G.; Marianib, V.C.; Dos Santos, L.C. Least squares support vector machines with tuning based on chaotic differential evolution approach applied to the identification of a thermal process. Expert Syst. Appl. 2012, 39, 4805–4812. [CrossRef] Appl. Sci. 2016, 6, 54 23 of 23 16. Tan, Y.; YU, C.; Zheng, S.; Ding, K. Introduction to fireworks algorithm. Int. J. Swarm Intell. Res. 2013, 4, 39–70. [CrossRef] 17. Gao, J.; Wang, J. A hybrid quantum-inspired immune algorithm for multiobjective optimization. Appl. Math. Comput. 2011, 217, 4754–4770. [CrossRef] 18. Wang, J.; Li, L.; Niu, D.; Tan, Z. An annual load forecasting model based on support vector regression with differential evolution algorithm. Appl. Energy 2012, 94, 65–70. [CrossRef] 19. Dai, H.; Zhang, B.; Wang, W. A multiwavelet support vector regression method for efficient reliability assessment. Reliab. Eng. Syst. Saf. 2015, 136, 132–139. [CrossRef] 20. Kang, J.; Tang, L.W.; Zuo, X.Z.; Li, H.; Zhang, X.H. Data prediction and fusion in a sensor network based on grey wavelet kernel partial least squares. J. Vib. Shock 2011, 30, 144–149. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Icing Forecasting for Power Transmission Lines Based on a Wavelet Support Vector Machine Optimized by a Quantum Fireworks Algorithm

Applied Sciences , Volume 6 (2) – Feb 17, 2016

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applied sciences Article Icing Forecasting for Power Transmission Lines Based on a Wavelet Support Vector Machine Optimized by a Quantum Fireworks Algorithm Tiannan Ma *, Dongxiao Niu and Ming Fu Departemnt of Economics and Management, North China Electric Power University, Beijing 102206, China; ndx@ncepu.edu.cn (D.N.); fm@ncepu.edu.cn (M.F.) * Correspondence: matiannan_1234@ncepu.edu.cn; Tel.: +86-185-1562-1058 Academic Editor: Fan-Gang Tseng Received: 22 December 2015; Accepted: 4 February 2016; Published: 17 February 2016 Abstract: Icing on power transmission lines is a serious threat to the security and stability of the power grid, and it is necessary to establish a forecasting model to make accurate predictions of icing thickness. In order to improve the forecasting accuracy with regard to icing thickness, this paper proposes a combination model based on a wavelet support vector machine (w-SVM) and a quantum fireworks algorithm (QFA) for prediction. First, this paper uses the wavelet kernel function to replace the Gaussian wavelet kernel function and improve the nonlinear mapping ability of the SVM. Second, the regular fireworks algorithm is improved by combining it with a quantum optimization algorithm to strengthen optimization performance. Lastly, the parameters of w-SVM are optimized using the QFA model, and the QFA-w-SVM icing thickness forecasting model is established. Through verification using real-world examples, the results show that the proposed method has a higher forecasting accuracy and the model is effective and feasible. Keywords: icing forecasting; support vector machine; fireworks algorithm 1. Introduction In recent years, a variety of extreme weather phenomena have occurred on a global scale, causing overhead power transmission line icing disasters to happen frequently, accompanied by power outages and huge economic losses due to the destruction of a large number of fixed assets. Since the first power transmission line icing accident recorded in 1932, several serious icing disasters have occurred one after another throughout the world [1], such as Canada’s freezing rain disaster in 1998, which led to direct economic losses of 1 billion dollars and indirect losses of 30 billion dollars for the power system. The power system failure caused by freezing rain was also a tremendous shock to the production and life of the region. In China, the earliest recorded icing disaster occurred in 1954. After that year, several large areas of freezing rain and snow disasters have successively occurred in China, especially Southern China’s freezing weather in January 2008 [2]. That disaster caused tremendous damage to China’s power system, directly resulting in 8709 tower collapses, more than 27,000 line breakages, and 1497 substation outages of 110 kV and above lines. The direct property loss for the State Grid Corporation amounted to 10.45 billion RMB, and the investment in post-disaster electricity reconstruction and transformation was 39 billion RMB. According to the lessons learned from previous grid freezing accidents, and given the current development situation of China's grid and the conditions of global climate change, the grid in China will again be tested by the future large-scale icing disasters. It is always preferable to take preventative action instead of reacting to events after an accident has Appl. Sci. 2016, 6, 54; doi:10.3390/app6020054 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 54 2 of 23 already happened. Therefore, icing forecasting research with regard to overhead power transmission lines have important practical application value. At present, there are many studies on icing forecasting for power transmission lines. According to the hydrodynamic movement law and heat transfer mechanisms, domestic and foreign scholars have established a variety of transmission line icing forecasting models that consider meteorological factors, environmental factors, and various line parameters. Generally these models can be divided into three categories: mathematical and physical models, statistical models and intelligent forecasting models. Mathematical and physical models involve simulation modeling and forecasting of the icing growth process by observing the physical process and mathematical equations of icing formation, notably including the Goodwin model [3], the Makkonen model [4] and so on. These models are established based on the experimental data, but there are differences between the experimental data and the practical data, so the forecasting results of these models are not ideal. Statistical models process historical data based on statistical theory and traditional statistical methods and do not consider the physical process of icing formation, e.g., the multiple linear regression icing model [5]. However, transmission line icing is affected by a variety of factors, and a multiple linear regression model cannot take all factors into account, meaning the icing forecasting accuracy is greatly reduced. Intelligent forecasting models combine modern computer technology with mathematical sciences, and are able to handle high-dimensional nonlinear problems through their powerful learning and processing capabilities, which can improve the prediction accuracy. The common intelligent forecasting models are support vector machines (SVM) and Back Propagation(BP) neural network. BP neural network has a strong nonlinear fitting ability to create the non-linear relationship between the output and a variety of impact factors. With its strong learning ability and forecasting capability, the non-linear output can arbitrarily approximate the actual value. For example, paper [6] presented a short-term icing prediction model based on a three layer BP neural network, and the results showed that the BP forecasting model is accurate for transmission lines of different areas. Paper [7] presented an ice thickness prediction model based on fuzzy neural network, the tested result demonstrated its forecasting abilities of better learning and mapping. However, the single BP model easily falls into the local optimum, and cannot always reach the expected accuracy. For this problem, some scholars adopted optimization algorithms to optimize the parameters of BP neural network, thereby improving the prediction accuracy. For instance, Du and Zheng et al. used Genetic Algorithm (GA) to optimize BP network and built the GA-BP ice thickness prediction model [8], and this model proved that the GA-BP model was more effective than the BP model in ice thickness prediction for transmission lines. Although this combination model could improve the prediction accuracy of BP neural networks, some scholars started to use SVM to build the icing forecasting model for transmission lines due to its slow calculating speed and overall poor performance. SVM can establish the non-linear relationship between various factors and ice thickness, and has better nonlinear mapping ability and generalization ability. In addition, SVM has a strong learning ability and can quickly approach the target value through continuous repetitive learning. Therefore, theSVM model is more widely used in transmission line research. For example, paper [9–11] introduced the icing forecasting model based on the Support Vector Regression learning algorithm, and obtained the ideal results. As is known, the standard SVM uses a Gaussian kernel function to solve the support vector by quadratic programming, and the quadratic programming will involve the calculation of an m order matrix (m is sample number). When m is bigger, the processing ability of the Gaussian function is more unsatisfactory. It will take much computing time, thereby seriously affecting the learning accuracy and predictive accuracy of the algorithm. In transmission line icing forecasting, there are many influencing factors, so a large amount of input data will make the SVM algorithm become unfeasible using the traditional Gaussian kernel for such a large-scale training sample. In view of this problem, this paper will replace the Gaussian kernel function with the wavelet kernel function, and establishes the wavelet support vector machine (w-SVM) for icing forecasting. Using wavelet kernel function in place of the Gaussian kernel is mainly based on the following considerations [12]: (1) the wavelet kernel function Appl. Sci. 2016, 6, 54 3 of 23 has the fine characteristic of progressively describing the data, and using the SVM of the wavelet kernel function can approximate any function with high accuracy while the traditional Gaussian kernel function cannot; (2) the wavelet kernel functions are orthogonal or nearly orthogonal, and the traditional Gaussian kernel functions are relevant, even redundant; (3) the wavelet kernel function has multi-resolution analyzing ability for wavelet signals, so the nonlinear processing capacity of the wavelet kernel function is better than that of Gaussian kernel function, which can improve the generalization ability of the support vector machine regression model. The forecasting performance of the w-SVM model largely depends on the values of its parameters; however, most researchers choose the parameters of SVM only by subjective judgment or experience. Therefore, the parameters values of SVM need to be optimized by meta-heuristic algorithms. Currently, several algorithms have been successfully applied to determine the control parameters of SVM, such as genetic algorithms [13], particle swarm optimization [14], differential evolution [15] and so on. However, those algorithms have the defects of being hard to control, achieving the global optima slowly, etc. In this paper, we use the fireworks optimization algorithm (FA) proposed by Tan and Zhu in 2010 [16] to determine the parameter values of w-SVM. The advantage of using FA over other techniques is that it can be easily realized and be able to reach global optimal with greater convergence speed. Besides, in order to strengthen the optimization ability and obtain better results, this paper also attempts to improve the FA model by using a quantum evolutionary algorithm. The rest of this paper is organized as follows: In Section 2, the quantum fireworks algorithm (QFA) and w-SVM are presented in detail. Also, in this section, a hybrid icing forecasting model (QFA-w-SVM) that combines the QFA and w-SVM models is established; In Section 3, several real-world cases are selected to verify the robustness and feasibility of QFA-w-SVM, and the computation, comparison and discussion of the numerical cases are discussed in detail; Section 4 concludes this paper. 2. Experimental Section 2.1. Quantum Fireworks Algorithm 2.1.1. Fireworks Algorithm A fireworks algorithm (FA) [16] is used to simulate the whole process of the explosion of fireworks. When the fireworks explosion generates a lot of sparks, the sparks can continue to explode to generate new sparks, resulting in beautiful and colorful patterns. In an FA, each firework can be regarded as a feasible solution of the solution space for an optimization problem, and the fireworks explosion process can be seen as a searching process for the optimal solution. In a particular optimization problem, the algorithm needs to take into account the number of sparks of each fireworks explosion, how wide the explosion radius is, and how to select an optimal set of fireworks and sparks for the next explosion (searching process). The most important three components of FA are the explosion operator, mutation operator and selection strategy. (1) Explosion operator. The number of sparks each fireworks explosion generates and explosion radius are calculated based on fitness value of fireworks. For the fireworks x pi  1, 2, , Nq, the calculation formula for the number of sparks S and explosion radius R are: i i y  fpx q # max i S  M  (1) py  fpx qq # max i1 fpx q y # i min R  R  (2) p f px q y q # i min i1 Appl. Sci. 2016, 6, 54 4 of 23 In the above formula, y , y represent the maximum and minimum fitness value of the current max min population, respectively; f px q is the fitness value of fireworks x ; and M is a constant to adjust the i i number of explosive sparks. R is a constant to adjust the size of the fireworks explosion radius. # is the minimum machine value to be used to avoid zero operation. (2) Mutation operator. The purpose of mutation operator is to increase the diversity of the sparks population; the mutation sparks of fireworks algorithm is obtained by Gaussian mutation, namely Gaussian mutation sparks. Select fireworks x to make the Gaussian mutation, and the k-dimensional Gaussian mutation is: x  x  e (3) ik ik Where x ˆ representsk-dimensional mutation fireworks, e represents the Gaussian distribution. ik In the fireworks algorithm, the new generated explosion fireworks and mutation sparks may fall out of the search space, which makes it necessary to map it to a new location, using the following formula: x ˆ  x |x ˆ | % x  x (4) ik LB,k ik UB,k LB,k where x , x are the upper and lower search spaces, respectively; and % denotes the modulo UB,k LB,k operator for floating-point number. (3) Selection strategy. In order to transmit the information to the next generation, it is necessary to select a number of individuals as the next generation. Assume that K individuals are selected; the number of population is N and the best individual is always determined to become the fireworks of next generation. Other N  1 fireworks are randomly chosen using a probabilistic approach. For fireworks x , its probability ppx q of being chosen is i i calculated as follows: ¸ ¸ Rpx q ppx q  ° , Rpx q  d x  x  ||x  x || (5) i i i j i j x PK x PK j j x PK where Rpxq is the sum of the distances between all individuals in the current candidate set. In the candidate set, if the individual has a higher density, that is, the individual is surrounded by other candidates, the probability of the individual selected will be reduced. If Fpxq is the objective function of fireworks algorithm, the steps of the algorithm are shown as follows: (1) Parameters initialized; Randomly select N fireworks and initialize their coordinates. (2) Calculate the fitness value fpx q of each firework, and calculate the blast radius R and i i generated sparks number of each firework. Randomly select dimension z-dimensional coordinates to update coordinates, coordinate updating formula is as follows: x ˆ  x R  U p1, 1q, U(–1,1) ik ik i stands for the uniform distribution on [–1,1]. (3) Generate M Gaussian mutation sparks; randomly select sparks x , use the Gaussian mutation formula to obtain M Gaussian mutation sparks x ˆ , and save those sparks into the Gaussian mutation ik sparks population. (4) Choose N individuals as the fireworks of next generation by using probabilistic formula from fireworks, explosion sparks and Gaussian mutation sparks population. (5) Stop condition. If the stop condition is satisfied, then output the optimal results; if not, return step (2) and continue to cycle. 2.1.2. Quantum Evolutionary Algorithm The development of quantum mechanics impels quantum computing to be increasingly applied in various fields. In quantum computing, the expression of a quantum state is a quantum bit, and usually quantum information is expressed by using the 0 and 1 binary method. The basic quantum states are “0” state and “1” state. In addition, the state can be an arbitrary linear superposition state Appl. Sci. 2016, 6, 54 5 of 23 between “0” and “1”. That is to say, the two states can exist at the same time, which challenges the classic bit expression method in classical mechanics to a large extent [17]. The superposition state of quantum state can be presented as shown in Equation (6). 2 2 |y ¡ a|0 ¡ b|1 ¡ , |a| |b|  1 (6) where |0 ¡ and |1 ¡ are the two quantum states a and b are the probability amplitudes. |a| represents the probability at a quantum state of |0 ¡ and |b| represents the probability at a quantum state of |1 ¡. In QFA, the updating proceeds by quantum rotating gate, and the adjustment is: a cospqq sinpqq a (7) b sinpqq cospqq b cospqq sinpqq in which set U  , U is quantum rotating gate, q is quantum rotating angle, sinpqq cospqq and q  arctanpa{bq. A quantum evolutionary algorithm is proposed based on a probabilistic search, as the conception of qubits and quantum superposition means a quantum evolutionary algorithm has many advantages, such as better population diversity, strong global search capability, especially great robustness, and the possibility of combining with other algorithms. 2.1.3. Quantum Fireworks Algorithm Parameters Initialized In the solution space, randomly generate N fireworks, and initialize their coordinates. Here, we use the probability amplitude of quantum bits to encode the current position of fireworks, and the encoding method is used by the following formula: cospq qcospq q cospq q i1 i2 in P  (8) sinpq qsinpq q sinpq q i1 i2 in where q  2prandpq, randpq is a random number between 0 and 1; i  1, 2, , m; j  1, 2, , n; m is ij the number of fireworks, n is the number of solution space. Therefore, the corresponding probability amplitude of individual fireworks for the quantum states |0 ¡ and |1 ¡ are as follows: P  pcospq q, cospq q cospq qq (9) ic i1 i2 in P  psinpq q, sinpq q sinpq qq (10) is i1 i2 in Solution Space Conversion The searching process of a fireworks optimization algorithm is carried out on the actual parameter space ra, bs. Due to the probability amplitude of fireworks location being in the range of r0, 1s, it needs to k 1 be decoded into the actual parameter space ra, bs to search in the fireworks algorithm. Let the Dq th jd j j k1 quantum bit for the individual a is a , b , and its corresponding conversion equations be: jd i i j j j X  rb p1 a q a p1  a qs i f randpq   P (11) i i id ic i i j j j X  rb p1 b q a p1  b qs i f randpq ¥ P (12) i i id is i i 2 Appl. Sci. 2016, 6, 54 6 of 23 where randpq is a random number between r0, 1s; X is the actual parameter value in j th dimension ic position when the quantum state of i th fireworks individual is |0 ¡ , X is the actual parameter value is in j th dimension position when the quantum state of the i th fireworks individual is |1 ¡ . b and a are i i the lower and upper limits. Assuming the FA is searching in two-dimensional space, that means j  1, 2. Initialize the position of population: InitX_ axis; InitY_axis; and the position of individuals can be determined as follows: i f randpq   P : id 1 1 Xpiq  X_axis rb p1 a q a p1  a qs (13) i i i i 2 2 Ypiq  Y_axis rb p1 a q a p1  a qs (14) i i i i i f randpq ¥ P : id 2 2 Xpiq  X_axis rb p1 b q a p1  b qs (15) i i i i 2 2 Ypiq  Y_axis rb p1 b q a p1  b qs (16) i i i i Calculate the Fitness Value fpx q of Each Individual, and Obtain the Explosion Radius and Generated Sparks Number S y  fpx q # max i S  M  (17) py  fpx qq # max i i1 fpx q y # i min R  R  (18) p f px q y q # min i1 Individual Position Updating The individual position update is operated by using a quantum rotating gate using the following equation: k1 k1 k1 a cosq sinq a jd jd jd jd (19) k1 k1 k1 k b sinq cosq b jd jd jd jd k1 k1 where a and b are the probability amplitude of j th fireworks individual in k 1 th iteration for jd jd k1 d dimension space; q is the rotating angle, which can be get from equation: jd k1 k1 k k q  spa , b qDq (20) jd jd jd jd k1 k k where spa , b q determines the rotating angle direction and is the Dq rotating angle increment. jd jd jd k 1 In order to adapt to operation mechanism of fireworks algorithm, we convert the updated a jd k1 and b to a solution space. jd d k1 k1 X  rb p1 a q a p1  a qs i f randpq   P j j id jc jd jd (21) k1 k1 X  rb p1 b q a p1  b qs i f randpq ¥ P j j id js jd jd Then calculate the positional offset amount: d d d d h  R  X , h  R  X (22) i i jc jc js js Appl. Sci. 2016, 6, 54 7 of 23 i f randpq   P , then d  1 id k1 k1 Xpjq  Xpiq h  Xpiq R  rb p1 a q a p1  a qs i j j jc jd jd (23) d k1 k1 Ypjq  Ypiq h  Ypiq R  rb p1 a q a p1  a qs i j i jc jd jd i f randpq ¥ P : then d  2 id d k1 k1 Xpjq  Xpiq h  Xpiq R  rb p1 b q a p1  b qs i j j js jd jd (24) k1 k1 Ypjq  Ypiq h  Ypiq R  rb p1 b q a p1  b qs i j j js jd jd Detection of cross-border. If the generated explosion sparks exceed the possible domain boundary, the position of sparks can be updated with the following equations: Xpjq  X |Xpjq| %pX  X q LB,k UB,k LB,k (25) Ypjq  Y |Ypjq| %pY  Y q LB,k UB,k LB,k Individual Mutation Operation The main reason for the premature convergence and local optimum of the fireworks group is that the diversity of the population is lost in the process of population search. In the quantum fireworks algorithm, in order to increase the diversity of the population, the Gauss mutation in the original algorithm is replaced by a quantum mutation. Randomly select fireworks x , and generate M quantum mutation sparks, and its operation formula is shown as follows: cosp  q q 01 cospq q sinpq q ij ij ij (26) 10 sinpq q cospq q ij ij sinp  q q ij Let the probability of individual be P , and randpq be a random number between r0, 1s; if randpq   P , the mutation can be operated with the above formula and the probability amplitude in quantum bit is changed; finally, the mutated individual can be converted into the solution space and save it to the mutation sparks population. 1 p p X  rb p1 cosp  q qq a p1  cosp  q qqs i f randpq   P j ij j ij id jc 2 2 2 (27) 1 p p X  rb p1 sinp  q qq a p1  sinp  q qqs i f randpq ¥ P j ij j ij id js 2 2 2 i f randpq   P , then d  1 id d d ˆ ˆ ˆ ˆ ˆ XpMq  XpMq X YpMq  YpMq X jc jc i f randpq ¥ P : then d  2 id d d ˆ ˆ ˆ ˆ ˆ XpMq  XpMq X YpMq  YpMq X js js Detection of cross-border. If the generated explosion sparks exceed the possible domain boundary, the position of sparks can be updated by the following equations: ˆ ˆ  ˆ ˆ XpMq  X XpMq %pX  X q LB,k UB,k LB,k (28) ˆ  ˆ ˆ ˆ YpMq  Y YpMq %pY  Y q LB,k UB,k LB,k Appl. Sci. 2016, 6, 54 8 of 23 (6) Choose N individuals as the fireworks of the next generation by using probabilistic formula ppx q from fireworks, explosion sparks and Gaussian mutation sparks population. (7) Stop condition. If the stop condition is satisfied, then output the optimal results; if not, return to step (2) and continue to cycle. 2.2. Wavelet Support Vector Machine 2.2.1. Basic Theory of Support Vector Machine (SVM) A support vector machine, proposed by Vapnik, is a kind of feed forward network [18]; its main purpose is to establish a hyper-plane to make the input vector project into another high-dimensional space. Given a set of data tpx , d qu , where x is the input vector; d is the expected output; it is i i i i1 further assumed that the estimate value of d is y, which is obtained by the projection of a set of nonlinear functions: y  w f pxq  w fpxq (29) j j j0 T T where fpxq  rf pxq, f pxq, , f pxqs , w  rw , w , , w s ; Let f pxq  1; w represent the bias b, 0 1 m 0 1 m 0 0 and the minimization risk function can be described as follows: 1 T Fpw, x, x q  w w C px x q (30) i1 The minimization risk function must satisfy the conditions: d  w fpx q ¤ # x ' i i i & 1 w fpx q d ¤ # x i i (31) x ¥ 0 ' i % 1 x ¥ 0 where i  1, 2, , N, x and x are slack variables; loss function is a #-insensitive loss function, C is a constant. Establish the Lagrange function and obtain: N N ° ° 1 1 1 T T Jpw, x, x , a, a , g, g q  C px x q w w  a rw fpx q d # x s i i i i i i1 i1 (32) N N ° ° 1 1 1 1 a rd  w fpx q # x s pg x g x q i i i i i i i i i1 i1 1 1 1 1 where a and a are Lagrange multiplier; take the partial derivative of variables w, x, x , a, a , g, g and obtain: w  pa  a qfpx q i i (33) i1 1 1 g  C  a ; g  C  a i i i i The above problem can be converted into a dual problem: N N N N ¸ ¸ ¸ ¸ 1 1 1 1 1 maxQ a , a  d pa  a q # pa a q pa  a qpa  a qf px qfpx q i i i i i j i j i i i i j i1 i1 i1 j1 ' pa  a q  0 i1 0 ¤ a ¤ C (34) 0 ¤ a ¤ C i  1, 2, , N Appl. Sci. 2016, 6, 54 9 of 23 Solve the equation and obtain: w  pa  a qfpxq i1 T T T T Then Fpx, wq  w x  pa  a qf pxqfpxq, let Kpx , xq  f pxqfpxq be the kernel function. i1 In this paper, we choose the wavelet kernel function to replace the Gaussian kernel function, and the construction of the wavelet kernel function will be introduced in detail in Section 2.2.2. 2.2.2. Construction of Wavelet Kernel Function The kernel function kpx, x q to SVM is the inner product of the image of two input space data points in the spatial characteristic. It has two important features: first, the symmetric function to inner 1 1 product kernel variables is kpx, x q  kpx , xq; second, the sum of the kernel function on the same plane is a constant. In general, only if the kernel function satisfies the following two theorems, can it become the kernel of support vector machine [19]. Mercer Lemma kpx, x q represents a continuous symmetric kernel, which can be expanded into a series as: 1 1 kpx, x q  l g pxqg px q (35) i i i i1 where l is positive, in order to ensure the above expansion is absolutely uniform convergence, the sufficient and necessary condition is: 1 1 1 n kpx, x qgpxqgpxqdxdx ¥ 0, x, x P R (36) gpq  0 For all gpq needs to satisfy the condition: , gpx q stands for the expansion of the g pxqdx   8 characteristic function, l stands for eigenvalue and all are positive, thus the kpx, x q is positive definite. Smola and Scholkopf Lemma If the support vector machine’s kernel function has meet the Mercer Lemma, then it only needs to prove kpx, x q to satisfy the follow formula: n{2 Fpxqpwq  p2pq exppJpw  xqqkpxqdx ¥ 0, x P R (37) Construction of Wavelet Kernel 2 1 ˆ ˆ If the wavelet kernel satisfies the condition: ypxq P L pRq X L pRq and ypxq  0,ypxq is the Fourier transform of ypxq, the ypxq can be defined as [20]: x  m 1{2 y pxq  psq yp q, x P R (38) s,m where s means contraction-expansion factor, and m is horizontal floating coefficient, s ¡ 0, m P R. When the function fpxq, fpxq P L pRq, the wavelet transform to fpxq can be defined as: x  m 1{2 Wps, mq  s fpxqy p qdx (39) 8 Appl. Sci. 2016, 6, 54 10 of 23 where y pxq is the complex conjugation of ypxq. The wavelet transform Wps, mq is reversible and also can be used to reconstruct the original signal, so: » » 8 8 ds fpxq  C Wps, mqy pxq dm (40) s,m 8 8 ³ ypwq ³ j 8 Among them C dw   8, ypwq  ypxqexppJwxqdx. |w| For the above equation, is a constant. The theory of wavelet decomposition is to approximate the function group by the linear combination of the wavelet function. Assuming ypxq is a one-dimensional function, based on tensor theory, the multi-wavelet function can be defined as: lxd d y pxq  P ypx q, x P R , x P R (41) l i i i1 The horizontal floating kernel function can be built as: x  x 1 i kpx, x q  P yp q, s ¡ 0 (42) i1 s In support vector machines, the kernel function should satisfy the Fourier transform; therefore, only when the wavelet kernel function satisfies the Fourier transform can it be used for support vector machines. Thus, the following formula needs to be proved. l{2 Frkspwq  p2pq exppJpwxqqkpxqdx ¥ 0 (43) In order to keep the generality of wavelet kernel function, choose the Morlet mother wavelet as follows: ypxq  cosp1.75xqexppx {2q (44) N N s can be figured out. Where x P R , s, x P R , the multi-dimensional wavelet function is an allowable multidimensional support vector machine functions. #    + 2 2 1 1 ||x  x || ||x  x || i i 1 i i kpx, x q  P cos 1.75p q exp 2 2 i1 2s 2s (45) 1.75X i 2 P cosp qexpp||x || {2s q i1 Proof: exppJwxqkpxqdx # + 1 1 x  x ||x  x || i i i i exppJwxq p P cosr1.75p qsexpp q qdx R 2 i1 s 2s exppj1.75x {sq expp1.75x {sq i i P exppJw x q p q expp||x || {2s qdx i i i i i1 2 #    + 2 2 1 ||x || 1.75j ||x || 1.75j i i P exp  p  jw sq x exp  p  jw sq x dx i i i i i 2 2 i1 2 2s s 2s s # + 2 2 |s| 2p p1.75  w sq p1.75 w sq i i P expp q expp q 2 2 2 i1 2 2 |s| p1.75  w sq p1.75 w sq i i Then Fpxqpwq  P p qpexpp q expp qq i1 2 2 2 |s|  0 ñ Fpxqpwq ¥ 0 ñ wavelet kernel function is an allowable support vector kernel function. Appl. Sci. 2016, 6, 54 11 of 23 2.3. Quantum Fireworks Algorithm for Parameters Selection of Wavelet Support Vector Machine (w-SVM) Model It is extremely important to select the parameters of w-SVM which can affect the fitting and learning ability. In this paper, the constructed quantum fireworks algorithm (QFA) is used for selecting the appropriate parameters of the w-SVM model in order to improve the icing forecasting accuracy. The flowchart of QFA for parameter selection of the fireworks the w-SVM model is shown in Figure 1 and the details of the QFA-w-SVM model are shown as follows: (1) Initialize the parameters. Initialize the number of fireworks N, the explosion radius A1, the number of explosive sparks M, the mutation probability P , the maximum number of iterations id Maxgen, the upper and lower bound of the solution space V and V respectively, and so on. up down (2) Separate the sample date as training samples and test samples, then normalize the sample data. (3) Initialize the solution space. a In the solution space, randomly initialize N positions, that is, N fireworks. Each of the fireworks has two dimensions, that is, C and s. b Use the probability amplitude of quantum bits to encode current position of fireworks according to Equations (9)–(10). c Converse the Solution space according to Equations (11)–(12). d Input the training samples, use w-SVM to carry out a training simulation for each fireworks, and calculate the value of the fitness function corresponding to each of the fireworks. (4) Initialize the global optimal solution by using the above initialized solution space, including the global optimal phase, the global optimal position quantization of fireworks, the global best fireworks, and the global best fitness value. (5) Start iteration and stop the cycle when the maximum number of iterations Maxgen is achieved. a According to the fitness value of each firework, calculate the corresponding explosion radius Rpiq and the number of sparks Spiq generated by each explosion. The purpose of calculating Rpiq and Spiq is to obtain the optimal fitness values, which means that if the fitness value is smaller, the explosion radius is larger, and the number of sparks generated by each explosion is bigger. Therefore, more excellent fireworks can be retained as much as possible. b Generate the explosive sparks. When each fireworks explodes, carry out the solution space conversion for each explosion spark and control their spatial positions through cross border detection. c Generate M Gaussian mutation sparks. Carry out a fireworks mutation operation according to Equations (26)–(27), resulting in M mutation sparks. d Use the initial fireworks, explosive sparks and Gauss mutation sparks to establish a group, and the fitness value of each firework in the new group is calculated by using w-SVM. e Update the global optimal phase, the global optimal position quantization of fireworks, the global best fireworks, and the global best fitness value. f Use roulette algorithm to choose the next generation of fireworks, and obtain the next generation of N fireworks. g Use w-SVM to carry out a training simulation for each fireworks, and update the value of the fitness function corresponding to each of the fireworks. (6) After all iterations, the best fireworks can be obtained, which correspond to the best parameters of C and s. Appl. Sci. 2016, 6, 54 12 of 23 Appl. Sci. 2016, 6, 54  13 of 24  Start Initialize the global Use w-SVM to calculate the optimal solution fitness of each Fireworks in the new group Parameter setting Iteration number<maxGen Construct training Update global optimum samples and test samples YES Calculate the explosion radius Sample data Use Roulette algorithm to select R(i), the number of sparks normalization the next generation N fireworks generated by each explosion S(i) Use w-SVM to update the fitness Initialize N fireworks Generate explosion sparks function value of each fireworks No Generate Gauss mutation Obtain the fireworks with the Fireworks qua ntization sparks optimal fitness encoding Space so lutio n conversion Use optimal parameters to carry out w-SVM s imulation prediction Calculate the fitness function value of each fireworks using w-SVM End Figure 1. The flowchart of the quantum fireworks algorithm (QFA)-w-SVM (wavelet Support Vector Figure 1. The flowchart of the quantum fireworks algorithm (QFA)‐w‐SVM (wavelet Support Vector  Machine) model. Machine) model.  3. Case Study and Results Analysis 3. Case Study and Results Analysis  3.1. Data Selection 3.1. Data Selection  Transmission line icing is affected by many factors, which mainly include wind direction, light Transmission line icing is affected by many factors, which mainly include wind direction, light  intensity, air pressure, altitude, condensation level, terrain, alignments, wires hanging height, wire intensity, air pressure, altitude, condensation level, terrain, alignments, wires hanging height, wire  stiffness, wire diameter, load current and so on. However, the necessary meteorological conditions stiffness, wire diameter, load current and so on. However, the necessary meteorological conditions  are: (1) the relative air humidity must be above 85%; (2) the wind speed should be greater than 1 m/s; are: (1) the relative air humidity must be above 85%; (2) the wind speed should be greater than 1 m/s;  (3) the temperature needs to reach 0 C and below. In addition, the impact factors, which have the (3) the temperature needs to reach 0 °C and below. In addition, the impact factors, which have the  greater correlations with the line icing, mainly include: wind direction, light intensity and air pressure, greater  correlations  with  the  line  icing,  mainly  include:  wind  direction,  light  intensity  and  air  etc. In general, when the wind direction is parallel to the wire or the angle between the wire and the pressure, etc. In general, when the wind direction is parallel to the wire or the angle between the wire  wire is less than 45, the extent of line icing is lighter; when the wind direction is vertical to the wire and the wire is less than 45, the extent of line icing is lighter; when the wind direction is vertical to  or the angle between the wire and the wire is more than 45, the extent of line icing is more severe. the wire or the angle between the wire and the wire is more than 45, the extent of line icing is more  Similarly, the lower the light intensity is, the more severe the line icing is. severe. Similarly, the lower the light intensity is, the more severe the line icing is.  In this paper, three power transmission lines, named “Qianpingxian-95”, “Fusha-I-xian” and In this paper, three power transmission lines, named “Qianpingxian‐95”, ”Fusha‐Ӏ‐xian” and  “Yangtongxian” in PingXi, ChangSha and ZhaoYang of Hunan province, respectively, are selected as the “Yangtongxian” in PingXi, ChangSha and ZhaoYang of Hunan province, respectively, are selected  case studies to demonstrate the effectiveness, feasibility and robustness of the proposed method. The as  the  case  studies  to  demonstrate  the  effectiveness,  feasibility  and  robustness  of  the  proposed  data from the above mentioned three transmission lines are provided by Key Laboratory of Disaster method.  The  data  from  the  above  mentioned  three  transmission  lines  are  provided  by  Key  Prevention and Mitigation of Power Transmission and Transformation Equipment (Changsha, China). Laboratory  of  Disaster  Prevention  and  Mitigation  of  Power  Transmission  and  Transformation  As is known, a large freezing disaster occurred in the south of China, which caused huge damage Equipment (Changsha, China).  to the power grid system. Hunan province, located in the southern part of China, was one of the As  is  known,  a  large  freezing  disaster  occurred  in  the  south  of  China,  which  caused  huge  most serious areas affected by this disaster. During the disaster period, one third of 500 and 200 kV damage to the power grid system. Hunan province, located in the southern part of China, was one of  substations were out of action in the Hunan power grid. According to the statistics, there were 481 line the most serious areas affected by this disaster. During the disaster period, one third of 500 and 200  breakages of 500 kV transmission lines, 673 line breakages of 200 kV transmission lines, 142 tower kV substations were out of action in the Hunan power grid. According to the statistics, there were  Appl. Sci. 2016, 6, 54  14 of 24  Appl. Sci. 2016, 6, 54  14 of 24  Appl. Sci. 2016, 6, 54 13 of 23 481 line breakages of 500 kV transmission lines, 673 line breakages of 200 kV transmission lines, 142  481 line breakages of 500 kV transmission lines, 673 line breakages of 200 kV transmission lines, 142  tower collapses of 500 kV AC and DC transmission lines, 633 tower collapses of 220 kV transmission  collapses of 500 kV AC and DC transmission lines, 633 tower collapses of 220 kV transmission lines, tower collapses of 500 kV AC and DC transmission lines, 633 tower collapses of 220 kV transmission  lines, and 1203 tower collapses of 110 kV transmission lines. Moreover, in the Hunan area, which  and 1203 tower collapses of 110 kV transmission lines. Moreover, in the Hunan area, which suffered lines, and 1203 tower collapses of 110 kV transmission lines. Moreover, in the Hunan area, which  suffered  from  the  influence  of  topography,  it  is  easy  to  form  the  stationary  front  due  to  the  from the influence of topography, it is easy to form the stationary front due to the mountains block, suffered  from  the  influence  of  topography,  it  is  easy  to  form  the  stationary  front  due  to  the  mountains block, when cold air enters the area. Therefore, it is due to their certain typicality that we  when cold air enters the area. Therefore, it is due to their certain typicality that we select those mountains block, when cold air enters the area. Therefore, it is due to their certain typicality that we  select those three transmission lines in Hunan province as cases to verify the validity and robustness  three transmission lines in Hunan province as cases to verify the validity and robustness of the select those three transmission lines in Hunan province as cases to verify the validity and robustness  of the proposed method.  proposed method. of the proposed method.  Case 1: the data on “Qianpingxian‐95” are from 10 January 2008 to 15 February 2008, which has  Case 1: the data on “Qianpingxian-95” are from 10 January 2008 to 15 February 2008, which has Case 1: the data on “Qianpingxian‐95” are from 10 January 2008 to 15 February 2008, which has  221 data groups for the training and testing in total. The former 180 data groups are regarded as a  221 data groups for the training and testing in total. The former 180 data groups are regarded as a 221 data groups for the training and testing in total. The former 180 data groups are regarded as a  training set and the last 41 groups of data are a testing set. The input vectors of SVM are average  training set and the last 41 groups of data are a testing set. The input vectors of SVM are average training set and the last 41 groups of data are a testing set. The input vectors of SVM are average  temperature, relative humidity, wind speed, wind direction, and sunlight intensity, and the output  temperature, relative humidity, wind speed, wind direction, and sunlight intensity, and the output temperature, relative humidity, wind speed, wind direction, and sunlight intensity, and the output  vector is ice thickness. The sample data are shown in Figure 2.  vector is ice thickness. The sample data are shown in Figure 2. vector is ice thickness. The sample data are shown in Figure 2.  Case 2: the data of ”Fusha‐Ӏ‐xian” are from 12 January 2008 to 25 February 2008, which has 287  Case 2: the data of “Fusha-I-xian” are from 12 January 2008 to 25 February 2008, which has Case 2: the data of ”Fusha‐Ӏ‐xian” are from 12 January 2008 to 25 February 2008, which has 287  data groups. The former 240 groups are taken as a training set and the remaining 47 groups as a  287 data groups. The former 240 groups are taken as a training set and the remaining 47 groups as a data groups. The former 240 groups are taken as a training set and the remaining 47 groups as a  testing set. The input vectors are same as that of Case 1. The sample data are shown in Figure 3.  testing set. The input vectors are same as that of Case 1. The sample data are shown in Figure 3. testing set. The input vectors are same as that of Case 1. The sample data are shown in Figure 3.  Case 3: the data of “Yangtongxian” are from 8 January 2008 to 24 February 2008, whose data  Case 3: the data of “Yangtongxian” are from 8 January 2008 to 24 February 2008, whose data Case 3: the data of “Yangtongxian” are from 8 January 2008 to 24 February 2008, whose data  groups  total  up  to  329.  The  former  289  groups  are  taken  as  a  training  set  and  the  remaining   groups total up to 329. The former 289 groups are taken as a training set and the remaining 40 groups groups  total  up  to  329.  The  former  289  groups  are  taken  as  a  training  set  and  the  remaining   40 groups as a testing set. The input vectors are still same as that of Case 1. The sample data are  as a testing set. The input vectors are still same as that of Case 1. The sample data are shown in 40 groups as a testing set. The input vectors are still same as that of Case 1. The sample data are  shown in Figure 4.  Figure 4. shown in Figure 4.  Figure 2. The original sample data charts.  Figure 2. The original sample data charts. Figure 2. The original sample data charts.  Figure Figure 3. 3. Origi Original nal data data of of “Fusha “Fusha-I-xian”. ‐Ӏ‐xian”.  Figure 3. Original data of “Fusha‐Ӏ‐xian”.  Appl. Sci. 2016, 6, 54 14 of 23 Appl. Sci. 2016, 6, 54  15 of 24  Figure 4. Original data charts of “Yangtongxian”.  Figure 4. Original data charts of “Yangtongxian”. 3.2. Data Pre‐Treatment  3.2. Data Pre-Treatment Before the calculation, the data must be screened and normalized to put them in the range of 0  Before the calculation, the data must be screened and normalized to put them in the range of 0 to to 1 using the following formula:  1 using the following formula: y  y i min y  y Z  z  i  1, 2 , 3 ,..., n i min (46)  Z  tz u  i  1, 2, 3, ..., n (46) i y  y max min y  y   max min where    and    are the maximum and minimum value of sample data, respectively. The  y y max min where y and y are the maximum and minimum value of sample data, respectively. The values max min values of each data are in the range [0,1] for eliminating the dimension influence.  of each data are in the range [0,1] for eliminating the dimension influence. Furthermore,  this  paper  will  use  a  quantum  fireworks  algorithm  (QFA)  to  optimize  the  Furthermore, this paper will use a quantum fireworks algorithm (QFA) to optimize the parameters parameters  C,   of the wavelet support vector machine to find optimal parameters to improve the  C, s of the wavelet support vector machine to find optimal parameters to improve the prediction prediction accuracy. In the parameters optimization, we adopt the mean square error (MSE) as the  accuracy. In the parameters optimization, we adopt the mean square error (MSE) as the fitness function fitness function to realize the process of QFA, and the formula of MSE is as follows:  to realize the process of QFA, and the formula of MSE is as follows: ° y  y  i i (47)  y  y i 1 i function (i )  i1 n f unctionpiq  (47) where    is the actual value;  y   is the prediction value; and  n  is the sample number.  where y is the actual value; y is the prediction value; and n is the sample number. 3.3. Case Study 1  3.3. Case Study 1 In this case, “Qianpingxian‐95”, which is a 220kV high voltage line, is selected to performed the  In this case, “Qianpingxian-95”, which is a 220kV high voltage line, is selected to performed simulation. After the above preparation, apply the constructed model to verify its feasibility and  the simulation. After the above preparation, apply the constructed model to verify its feasibility robustness.  and robustness. Firstly, initialize the parameters of QFA. Let the maximum iteration number  Maxgen  200 ,  Firstly, initialize the parameters of QFA. Let the maximum iteration number Maxgen  200, PopNum 40 M  100 population  number  ,  Sparks  number  determination  constant  ,  explosion  population number PopNum  40, Sparks number determination constant M  100, explosion radius 510 ˆ 5 10 determination radius determconstant ination con R  stant 150  ,R the bor 150 der , the of parameter border of pa C ra bemr2eter 2 s,  the be  [2 border 2 of ] , parameter the border sof  5 5 55 be r2 2 s, the upper and lower limits of searching space of fireworks individual be V  512 and up parameter    be  [2 2 ] , the upper and lower limits of searching space of fireworks individual  V  512, respectively, and mutation rate P  0.05. Then use the steps of QFA to optimize the down id be  V  512  and  , respectively, and mutation rate  P  0.05 . Then use the steps  V  512 up down id parameters of w-SVM and obtain C  18.3516, s  0.031402. Finally, we predict the icing thickness of of  QFA  to  optimize  the  parameters  of  w‐SVM  and  obtain  C 18.3516,    0.031402 .  Finally,  we  the testing sample after putting the optimal parameters into the w-SVM regression model. predict the icing thickness of the testing sample after putting the optimal parameters into the w‐SVM  Figure 5 shows the optimization process of the quantum fireworks algorithm (QFA). As we can regression model.  see, the proposed model obtains the optimal value when the iteration number is 35, and the optimal Figure 5 shows the optimization process of the quantum fireworks algorithm (QFA). As we can  value is 0.11; this illustrates that the proposed algorithm can obtain the global optimum with a fast see, the proposed model obtains the optimal value when the iteration number is 35, and the optimal  convergence speed. Figure 6 shows the forecasting results of the proposed method. This paper also value is 0.11; this illustrates that the proposed algorithm can obtain the global optimum with a fast  convergence speed. Figure 6 shows the forecasting results of the proposed method. This paper also  Appl. Sci. 2016, 6, 54  16 of 24  Appl. Sci. 2016, 6, 54 15 of 23 Appl. Appl. Sci.  Sci. 2016  2016, 6, ,6 54 , 54    1616 of of 24 24    selects w‐SVM optimized by a regular fireworks algorithm (FA‐w‐SVM), SVM optimized by particle  selects selects w w‐SV ‐SVMM optimized  optimized by by a a reg  reguular lar firework  fireworkss al algo gorit rithhmm (FA  (FA‐w‐w‐SV ‐SVM), M), SV  SVMM opti  optimi mizzeedd by by part  particle icle   selects w-SVM optimized by a regular fireworks algorithm (FA-w-SVM), SVM optimized by particle swarm  optimization  algorithm  (PSO‐SVM),  SVM  model,  and  multiple  linear  regression  model  swarm  optimization  algorithm  (PSO‐SVM),  SVM  model,  and  multiple  linear  regression  model  swarm  optimization  algorithm  (PSO‐SVM),  SVM  model,  and  multiple  linear  regression  model  swarm (MLR) optimization to make a coalgorithm mparison,  (PSO-SVM), and the convergence SVM model,  curves and multiple of FA‐wlinear ‐SVM ran egrdession  PSO‐SV model M ar(MLR) e also  (ML (MLRR) )to to ma  makkee a a co compa mparis risoonn, ,and  and the  the convergence  convergence curves  curves of of FA  FA‐w‐w‐SV ‐SVMM an  andd PSO  PSO‐SV ‐SVMM ar aree al also so   to make a comparison, and the convergence curves of FA-w-SVM and PSO-SVM are also shown in shown in Figure 5. The forecasting results of four models are shown in Figure 7.  shown in Figure 5. The forecasting results of four models are shown in Figure 7.  shown in Figure 5. The forecasting results of four models are shown in Figure 7.  Figure 5. The forecasting results of four models are shown in Figure 7.    Figure 5. The iteration process of QFA‐w‐SVM. PSO‐SVM: particle swarm optimization algorithm.  Figure Figure 5. 5. The  The it iteration eration process  process of of QFA  QFA‐w‐w‐SV ‐SVM. M. PSO  PSO‐SVM ‐SVM: :particle  particle swarm  swarm optimization  optimization al algori gorithm. thm.   Figure 5. The iteration process of QFA-w-SVM. PSO-SVM: particle swarm optimization algorithm. Actual value A QF A ctua cA tua l- v w- l v aS lu aV lu eM e QFA-w-SVM QFA-w-SVM 1 6 11 16 21 26 31 36 41 1 6 11 16 21 26 31 36 41 1 6 11 16 21 26 31 36 41 Figure 6. Forecasting value of QFA‐w‐SVM.  Figure 6. Forecasting value of QFA-w-SVM. Figure 6. Forecasting value of QFA‐w‐SVM.  Figure 6. Forecasting value of QFA‐w‐SVM.  8 Actual value 88 Ac Ac tuta ula l 7.5 FA-w-SVM value value 7.5 7.5 7 FA-w-SVM FA-w-SVM PSO-SVM 6.5 PS PS O-SVM O-SVM 6.5 6.5 5.5 5.5 5.5 4.5 4.5 4.5 3.5 3.5 3.5 1 6 11 16 21 26 31 36 41 1 6 11 16 21 26 31 36 41   1 6 11 16 21 26 31 36 41 Figure 7. Forecasting value of each method. MLR: multiple linear regression model. Figure 7. Forecasting value of each method. MLR: multiple linear regression model.  Figure 7. Forecasting value of each method. MLR: multiple linear regression model.  Figure 7. Forecasting value of each method. MLR: multiple linear regression model.  Forecasting results/mm Forecasting results/mm Forecasting results/mm Forecasting results/mm Forecasting results/mm Forecasting results/mm Appl. Sci. 2016, 6, x Appl. Sci. 2016, 6, 54 16 of 23 For further analysis, this paper will use the relative error (RE), the mean absolute percentage error (MAPE), mean square error (MSE) and average absolute error (AAE) to evaluate the results of prediction. For further analysis, this paper will use the relative error (RE), the mean absolute percentage error (MAPE), mean square error (MSE) and average absolute error (AAE) to evaluate the results Y − Y ′ 1 Y − Y ′ i i i i RE = ×100% MAPE = × 100% of prediction. Y n Y i i 1 1 Y  Y 1 Y  Y i i i i (48) RE   100% MAPE   100% n n n 1 1 Y n Y i i 1 ′ ′ MSE = () Y −Y AAE = ( Y −Y ) ( Y )   i i i i i (48) n n n n n i=1  i== 11   i ° ° ° 1 1 1 1 1 MSE  Y  Y AAE  p Y  Y q{p Y q i i i i i n n n Y Y ′ i1 i1 i1 where is the actual value of power load; is the forecasting value of power load; i i i = 1,2, , n where Y is the actual value of power load; Y is the forecasting value of power load; i  1, 2, , n. The relative errors (RE) value of the QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR The relative errors (RE) value of the QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR models models are shown in Figure 8. It can be clearly seen that the RE curve of QFA-w-SVM is the lowest are shown in Figure 8. It can be clearly seen that the RE curve of QFA-w-SVM is the lowest among among the other four models, which demonstrates that the accuracy of the proposed algorithm is the other four models, which demonstrates that the accuracy of the proposed algorithm is much much higher than other mentioned algorithms. The RE curve of the MLR model is the highest, and higher than other mentioned algorithms. The RE curve of the MLR model is the highest, and this this demonstrates the forecasting results based on the MLR model are not satisfied here. demonstrates the forecasting results based on the MLR model are not satisfied here. 18% QFA-w-SVM FA-w-SVM PSO-SVM SVM 15% MLR 12% 9% 6% 3% 0% 1 4 7 101316 1922 252831 34 3740 Figure 8. Forecasting errors of each model. Figure 8. Forecasting errors of each model. The relative error range [−3%,+3%] is always regarded as a standard to evaluate the The relative error range [3%,+3%] is always regarded as a standard to evaluate the performance performance of a forecasting model. As we can see from Figure 7, 35 points of QFA-w-SVM means of a forecasting model. As we can see from Figure 7, 35 points of QFA-w-SVM means 85% forcasting 85% forcasting points are in the range [−3%,+3%], and only 6 forecasting points are out of this scope. points are in the range [3%,+3%], and only 6 forecasting points are out of this scope. In the model of In the model of FA-w-SVM, only 8 out of 41 points are in the range [−3%,+3%], which means that FA-w-SVM, only 8 out of 41 points are in the range [3%,+3%], which means that 78% of forecasting 78% of forecasting points are out of the scope. In addition, only 5 forecasting points of PSO-SVM and points are out of the scope. In addition, only 5 forecasting points of PSO-SVM and one forecasting one forecasting point of SVM are in the scope of [−3%,+3%]; none of the forecasting points of the point of SVM are in the scope of [3%,+3%]; none of the forecasting points of the MLR model are in MLR model are in the scope. These results demonstrate that the QFA-w-SVM model has a better the scope. These results demonstrate that the QFA-w-SVM model has a better performance in icing performance in icing forecasting when compared with other models. In addition, the maximum and forecasting when compared with other models. In addition, the maximum and minimum relative minimum relative errors (MaxRE and MinRE) can also reflect the forecasting accuracy of icing errors (MaxRE and MinRE) can also reflect the forecasting accuracy of icing forecasting models. Firstly, forecasting models. Firstly, both the MaxRE and MinRE values of combined models are smaller than both the MaxRE and MinRE values of combined models are smaller than those of single models, those of single models, which proves the optimization algorithm can facilitate improving accuracy of which proves the optimization algorithm can facilitate improving accuracy of w-SVM through finding w-SVM through finding optimal parameters. Secondly, the MaxRE and MinRE values of QFA-w-SVM optimal parameters. Secondly, the MaxRE and MinRE values of QFA-w-SVM model are 3.61% and model are 3.61% and 0.89%, respectively, both of which are the smallest among the five icing 0.89%, respectively, both of which are the smallest among the five icing forecasting models, which forecasting models, which illustrates that QFA-w-SVM has a better nonlinear fitting ability in illustrates that QFA-w-SVM has a better nonlinear fitting ability in sample training. Thirdly, in the sample training. Thirdly, in the FA-w-SVM model, the MaxRE and MinRE values are 5.91% and FA-w-SVM model, the MaxRE and MinRE values are 5.91% and 1.02%, respectively, both of which are 1.02%, respectively, both of which are smaller than that of PSO-SVM model(6.75% and 1.82%), which smaller than that of PSO-SVM model(6.75% and 1.82%), which means FA-w-SVM has better forecasting means FA-w-SVM has better forecasting accuracy. Finally, both the MaxRE and MinRE values of accuracy. Finally, both the MaxRE and MinRE values of SVM are 8.73% and 2.73%, respectively, both of SVM are 8.73% and 2.73%, respectively, both of which are smaller than that of MLR, that mean the 17 Appl. Sci. 2016, 6, 54  18 of 24  nonlinear processing ability is better than that of MLR; in other words, the robustness of SVM is  much Appl.  stronge Sci. 2016 r ,tha 6, 54n that of the MLR model.  17 of 23 The MAPE, MSE and AAE values of the above mentioned five models are shown in Figure 9.  The experimental results show that the forecasting effect of the SVM model obviously precedes the  which are smaller than that of MLR, that mean the nonlinear processing ability is better than that of multiple  linear  regression  model;  this illustrates  that  the  intelligent  forecasting  model  has strong  MLR; in other words, the robustness of SVM is much stronger than that of the MLR model. learning ability and nonlinear mapping ability which the multiple linear regression model cannot  The MAPE, MSE and AAE values of the above mentioned five models are shown in Figure 9. match.  In  addition,  the  MAPE  value  of  single  SVM  is  4.988%,  which  is  much  higher  than  that  The experimental results show that the forecasting effect of the SVM model obviously precedes the multiple linear regression model; this illustrates that the intelligent forecasting model has strong obtained by hybrid QFA‐w‐SVM, FA‐w‐SVM and PSO‐SVM models (which are 1.56%, 2.83% and  learning ability and nonlinear mapping ability which the multiple linear regression model cannot 2.776%, respectively); this result proves that the optimization algorithm can help single SVM to find  match. In addition, the MAPE value of single SVM is 4.988%, which is much higher than that obtained the optimal parameters, and heighten the learning capacity and forecasting accuracy. Furthermore,  by hybrid QFA-w-SVM, FA-w-SVM and PSO-SVM models (which are 1.56%, 2.83% and 2.776%, the MAPEs of QFA‐w‐SVM and FA‐w‐SVM are 1.56% and 2.83%, respectively; this shows that the  respectively); this result proves that the optimization algorithm can help single SVM to find the QFA model can greatly improve the optimization performance of regular FA model through the  optimal parameters, and heighten the learning capacity and forecasting accuracy. Furthermore, the combination  of  quantum  optimization  algorithm,  which  makes  the  algorithm  can  easily  get  the  MAPEs of QFA-w-SVM and FA-w-SVM are 1.56% and 2.83%, respectively; this shows that the QFA optimal results. Meanwhile, the MAPE value of PSO‐SVM model is 2.776%, which is higher than that  model can greatly improve the optimization performance of regular FA model through the combination of  QFA‐w‐SVM;  this  illustrates  that  QFA‐w‐SVM  model  has  a  better  global  convergence  of quantum optimization algorithm, which makes the algorithm can easily get the optimal results. performance compared with PSO‐SVM, and also proves that w‐SVM has better nonlinear mapping  Meanwhile, the MAPE value of PSO-SVM model is 2.776%, which is higher than that of QFA-w-SVM; capability compared with SVM.  this illustrates that QFA-w-SVM model has a better global convergence performance compared with PSO-SVM, and also proves that w-SVM has better nonlinear mapping capability compared with SVM. 40% MAPE MSE AAE 30% 20% 10% 0% QFA-w-SVM FA-w-SVM SVM MLR PSO-SVM Figure 9. Values of mean absolute percentage error (MAPE), mean square error (MSE) and average Figure 9. Values of mean absolute percentage error (MAPE), mean square error (MSE) and average  absolute error (AAE). absolute error (AAE).  Furthermore, the MSE values of QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR models are Furthermore, the MSE values of QFA‐w‐SVM, FA‐w‐SVM, PSO‐SVM, SVM and MLR models  0.0174, 0.0466, 0.0590, 0.1094 and 0.3581, respectively, and the AAE values of those five models are 0.024, 0.0389, 0.0439, 0.0602 and 0.1092, respectively. As we know, when the value of MSE and AAE is are 0.0174, 0.0466, 0.0590, 0.1094 and 0.3581, respectively, and the AAE values of those five models  smaller, the prediction effect of model is more ideal. It can be clearly seen that the MSE value and AAE are 0.024, 0.0389, 0.0439, 0.0602 and 0.1092, respectively. As we know, when the value of MSE and  value of QFA-w-SVM is the smallest among other models, which directly demonstrates the feasibility AAE is smaller, the prediction effect of model is more ideal. It can be clearly seen that the MSE value  and effectiveness of QFA-w-SVM model, and QFA can improve the optimization performance of the and AAE value of QFA‐w‐SVM is the smallest among other models, which directly demonstrates the  regular FA to help SVM find the optimal parameters to improve the forecasting accuracy. feasibility  and  effectiveness  of  QFA‐w‐SVM  model,  and  QFA  can  improve  the  optimization  performance of the regular FA to help SVM find the optimal parameters to improve the forecasting  3.4. Case Study 2 accuracy.  “Fusha-I-xian”, which is a representative 500 kV transmission line between Fuxing and Shapin of Hunan province, is chosen to prove the robustness and stability of the proposed QFA-w-SVM icing 3.4. Case Study 2  forecasting model. The sample data of “Fusha-I-xian” are also predicted by the five models and the results of five models are also used to make a comparison. ”Fusha‐Ӏ‐xian”, which is a representative 500kV transmission line between Fuxing and Shapin  Figure 10 shows the iteration trend of the QFA-w-SVM searching of optimization parameters. of Hunan province, is chosen to prove the robustness and stability of the proposed QFA‐w‐SVM  As can be seen from Figure 3, the convergence can be seen in generation 39 with the optimal MSE icing forecasting model. The sample data of ”Fusha‐Ӏ‐xian” are also predicted by the five models and  value of 0.1124, and the parameters of w-SVM are obtained with C  23.68, s  0.1526, respectively. the results of five models are also used to make a comparison.  Figure 10 shows the iteration trend of the QFA‐w‐SVM searching of optimization parameters.   As can be seen from Figure 3, the convergence can be seen in generation 39 with the optimal MSE  Appl. Sci. 2016, 6, 54  19 of 24  Appl. Sci. 2016, 6, 54  19 of 24    0.1526 value  of  0.1124,  and  the  parameters  of  w‐SVM  are  obtained  with  C  23.68 ,  ,  respectively. In FA‐w‐SVM model, the convergence can be seen in generation 53 with the optimal    0.1526 value  of  0.1124,  and  the  parameters  of  w‐SVM  are  obtained  with  C  23.68 ,  ,  C  31.58   0.0496 MSE value of 0.1307, and the parameters of w‐SVM are obtained with  ,  . In the  respectively. Appl. Sci. 2016 In  ,FA 6, 54‐w‐SVM model, the convergence can be seen in generation 53 with18 the of 23 optimal  PSO‐SVM  model,  the  convergence  can  be  seen  in  generation  58  with  the  optimal  MSE  value  of  C  31.58   0.0496 MSE value of 0.1307, and the parameters of w‐SVM are obtained with  ,  . In the  C  19.48   0.5962 0.1324, and the parameters of SVM are obtained with  ,  . This proves that the  PSO‐SVM  model,  the  convergence  can  be  seen  in  generation  58  with  the  optimal  MSE  value  of  In FA-w-SVM model, the convergence can be seen in generation 53 with the optimal MSE value of proposed  QFA‐w‐SVM  model  can  find  the  global  optimal  value  with  the  faster  convergence  0.1324, 0.1307,  and the and  the para parameters meters of of SV w-SVM M are are obtained obtained with  witCh C31.58  19. , s 48, 0.0496   0. . 59 In 62 the. PSO-SVM This proves model,  that the  compared with FA‐w‐SVM and PSO‐SVM models. This also validates the stability of the proposed  the convergence can be seen in generation 58 with the optimal MSE value of 0.1324, and the parameters proposed  QFA‐w‐SVM  model  can  find  the  global  optimal  value  with  the  faster  convergence  icing forecasting model.  of SVM are obtained with C  19.48, s  0.5962. This proves that the proposed QFA-w-SVM model can compared with FA‐w‐SVM and PSO‐SVM models. This also validates the stability of the proposed  find the global optimal value with the faster convergence compared with FA-w-SVM and PSO-SVM icing forecasting model.  models. This also validates the stability of the proposed icing forecasting model. Figure 10. The convergence curves of algorithms.  Figure 11 shows the forecasting results of the five models. The forecasting errors of the five  Figure 10. The convergence curves of algorithms. Figure 10. The convergence curves of algorithms.  models are shown in Figure 12. It can be seen from the comparison of the forecasting curve and the  actual value that the forecasting results of ice thickness of those five models have approximation to  Figure 11 shows the forecasting results of the five models. The forecasting errors of the five models Figure 11 shows the forecasting results of the five models. The forecasting errors of the five  actual  curve.  Among  them,  the  proposed  QFA‐w‐SVM  has  the  best  fitting  learning  and  fitting  are shown in Figure 12. It can be seen from the comparison of the forecasting curve and the actual models are shown in Figure 12. It can be seen from the comparison of the forecasting curve and the  ability, and 44 forecasting points mean that almost 94% of points are in the scope of [−3,+3]; only  value that the forecasting results of ice thickness of those five models have approximation to actual actual value that the forecasting results of ice thickness of those five models have approximation to  threecurve.  forecaAmong sting points them, are the out proposed  of this QF sc A-w-SVM ope. In FA has‐wthe ‐SVbest M model, fitting learning 27 forecand astin fitting g points ability  mea , and ns that  actual  curve.  Among  them,  the  proposed  QFA‐w‐SVM  has  the  best  fitting  learning  and  fitting  44 forecasting points mean that almost 94% of points are in the scope of [3,+3]; only three forecasting nearly 57% points are in the range [−3,+3], and 20 points fall out this scope. In PSO‐SVM model, 20  ability, points  and ar 44 e out forec ofa this stin scope. g points In F A-w-SVM mean tha model, t almo 27stfor  94 ecasting % of points points are means  in that the nearly scope57%  of [points −3,+3]; only  forecasting points means that nearly 43% of points are in the scope, and 27 points are out of the  are in the range [3,+3], and 20 points fall out this scope. In PSO-SVM model, 20 forecasting points three forecasting points are out of this scope. In FA‐w‐SVM model, 27 forecasting points means that  range. In SVM model, only 2 points are in this range, and the rest of the forecasting points are not in  means that nearly 43% of points are in the scope, and 27 points are out of the range. In SVM model, nearly 57% points are in the range [−3,+3], and 20 points fall out this scope. In PSO‐SVM model, 20  this scope. In MLR model, none of points are in this scope. This reveals that QFA‐w‐SVM model has  only 2 points are in this range, and the rest of the forecasting points are not in this scope. In MLR forec a higher asting forec  poiansttsing  mea  accnus rathcyat tha  nearly n other  43%  mentioned  of points mo  are de lin s,  an thed  it sco alpsoe,  has and stron  27 po gerin robustness ts are out  and of the     model, none of points are in this scope. This reveals that QFA-w-SVM model has a higher forecasting range nonlinear . In SVM  fitting  model,  abilit only y.   2 points are in this range, and the rest of the forecasting points are not in  accuracy than other mentioned models, and it also has stronger robustness and nonlinear fitting ability. this scope. In MLR model, none of points are in this scope. This reveals that QFA‐w‐SVM model has  a higher forecasting accuracy than other mentioned models, and it also has stronger robustness and  Original data QFA-w-SVM FA-w-SVM nonlinear fitting ability.  PSO-SVM SVM MLR Original data QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR 1 6 11 16 21 26 31 36 41 46 Figure 11. Forecasting results of the five models. Figure 11. Forecasting results of the five models.  1 6 11 16 21 26 31 36 41 46 Figure 11. Forecasting results of the five models.  Foreasting results/mm Foreasting results/mm Appl. Sci. 2016, 6, 54  20 of 24  Appl. Sci. 2016, 6, 54  20 of 24  Appl. Sci. 2016, 6, 54 19 of 23 10% QFA-w-SVM FA-w-SVM PSO-SVM 10% SVM MLR QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR 8% 8% 6% 6% 4% 4% 2% 2% 0% 0% 1 6 11 16 21 26 31 36 41 46 1 6 11 16 21 26 31 36 41 46 Figure 12. Forecasting errors of each model.  Figure 12. Forecasting errors of each model. Figure 12. Forecasting errors of each model.  The values of MAPE, MSE, AAE of the five models in icing forecasting of ”Fusha‐Ӏ‐xian” are  The values of MAPE, MSE, AAE of the five models in icing forecasting of “Fusha-I-xian” are The values of MAPE, MSE, AAE of the five models in icing forecasting of ”Fusha‐Ӏ‐xian” are  shown in Figure 13. It can be seen that the proposed QFA‐w‐SVM model still has the smallest MAPE,  shown in Figure 13. It can be seen that the proposed QFA-w-SVM model still has the smallest MAPE, shown in Figure 13. It can be seen that the proposed QFA‐w‐SVM model still has the smallest MAPE,  MSE  and  AAE  values,  which  are  1.9%,  0.026  and  0.018,  respectively.  This  again  reveals  that  the  MSE and AAE values, which are 1.9%, 0.026 and 0.018, respectively. This again reveals that the MSE  and  AAE  values,  which  are  1.9%,  0.026  and  0.018,  respectively.  This  again  reveals  that  the  proposed QFA‐w‐SVM model has the best performance in the icing forecasting results. In addition,  proposed QFA-w-SVM model has the best performance in the icing forecasting results. In addition, the proposed QFA‐w‐SVM model has the best performance in the icing forecasting results. In addition,  the MAPE, MSE and AAE values of FA‐w‐SVM are 3.01%, 0.063 and 0.028, respectively, which are  MAPE, MSE and AAE values of FA-w-SVM are 3.01%, 0.063 and 0.028, respectively, which are both the MAPE, MSE and AAE values of FA‐w‐SVM are 3.01%, 0.063 and 0.028, respectively, which are  both lar large gerr than  than that  tha oft QF of A-w-SVM, QFA‐w‐SV bM, ut smaller  but smaller than the  tha rest n the of thr  rest ee  models. of threeIn models. PSO-SVM,  In PSO the MAPE, ‐SVM, the  both larger than that of QFA‐w‐SVM, but smaller than the rest of three models. In PSO‐SVM, the  MSE and AAE values are 3.45%, 0.099and 0.034, respectively, which are smaller than those of SVM and MAPE, MSE and AAE values are 3.45%, 0.099and 0.034, respectively, which are smaller than those of  MAPE, MSE and AAE values are 3.45%, 0.099and 0.034, respectively, which are smaller than those of  MLR models. That proves that the combined algorithms have better forecasting performance and the SVM  and  MLR  models.  That  proves  that  the  combined  algorithms  have  better  forecasting  optimization algorithms can help single regression model to achieve a better accuracy through finding SVM  and  MLR  models.  That  proves  that  the  combined  algorithms  have  better  forecasting  performance and the optimization algorithms can help single regression model to achieve a better  better parameters. This result agrees with the one presented in Section 3.3. performance and the optimization algorithms can help single regression model to achieve a better  accuracy through finding better parameters. This result agrees with the one presented in Section 3.3.  accuracy through finding better parameters. This result agrees with the one presented in Section 3.3.  30% 30% MAPE MSE AAE MAPE MSE AAE 20% 20% 10% 10% 0% QFA-w-SVM FA-SVM PSO-SVM SVM MLR 0% QFA-w-SVM FA-SVM PSO-SVM SVM MLR   Figure 13. Values of MAPE, MSE, AAE. Figure 13. Values of MAPE, MSE, AAE.  Figure 13. Values of MAPE, MSE, AAE.  3.5. Case Study 3 3.5. Case Study 3  In this case, the icing data from a 110 kV transmission line called “Yangtongxian” is selected to 3.5. Case Study 3  In this case, the icing data from a 110 kV transmission line called “Yangtongxian” is selected to  validate the robustness and stability of the proposed QFA-w-SVM icing forecasting model. Similarly, In this case, the icing data from a 110 kV transmission line called “Yangtongxian” is selected to  validate  the  robustness  and  stability  of  the  proposed  QFA‐w‐SVM  icing  forecasting  model.  the five models are still used to make a comparison for prediction results. validate  the  robustness  and  stability  of  the  proposed  QFA‐w‐SVM  icing  forecasting  model.  Similarly, the five models are still used to make a comparison for prediction results.  Similarly, the five models are still used to make a comparison for prediction results.  Figure 14 shows the convergence plots of QFA‐w‐SVM, FA‐w‐SVM and PSO‐SVM. As shown  Figure 14 shows the convergence plots of QFA‐w‐SVM, FA‐w‐SVM and PSO‐SVM. As shown  in  the  figure,  QFA‐w‐SVM  obtains  the  optimal  value  after  40  iterations  while  FA‐w‐SVM  and  in  the  figure,  QFA‐w‐SVM  obtains  the  optimal  value  after  40  iterations  while  FA‐w‐SVM  and  PSO‐SVM  models  converge  to  global  solutions  after  53  and  57  iterations,  respectively.  In  other  PSO‐SVM  models  converge  to  global  solutions  after  53  and  57  iterations,  respectively.  In  other  Appl. Sci. 2016, 6, 54 20 of 23 Appl. Sci. 2016, 6, 54  21 of 24  Appl. Sci. 2016, 6, 54  21 of 24  Figure 14 shows the convergence plots of QFA-w-SVM, FA-w-SVM and PSO-SVM. As shown in words,  the  QFA‐w‐SVM still  has  the fastest  convergence  speed and  stronger global  optimization  the figure, QFA-w-SVM obtains the optimal value after 40 iterations while FA-w-SVM and PSO-SVM ability compared with other models.  words,  the  QFA‐w‐SVM still  has  the fastest  convergence  speed and  stronger global  optimization  models converge to global solutions after 53 and 57 iterations, respectively. In other words, the ability compared with other models.  QFA-w-SVM still has the fastest convergence speed and stronger global optimization ability compared with other models. Figure 14. The convergence curves of algorithms.  Figure 14. The convergence curves of algorithms. Figure 14. The convergence curves of algorithms.  The forecasting results and relative errors are shown in Figures 15 and 16, respectively. From  The forecasting results and relative errors are shown in Figures 15 and 16 respectively. From the The forecasting results and relative errors are shown in Figures 15 and 16, respectively. From  the  figures,  it  is  obvious  that  the  proposed  QFA‐w‐SVM  model  still  has  better  accuracy  in  icing  figures, it is obvious that the proposed QFA-w-SVM model still has better accuracy in icing forecasting, the  figures,  it  is  obvious  that  the  proposed  QFA‐w‐SVM  model  still  has  better  accuracy  in  icing  forecasting,  which  demonstrates  that  the  proposed  model  is  superior  for  solving  the  icing  which demonstrates that the proposed model is superior for solving the icing forecasting problem forecasting,  which  demonstrates  that  the  proposed  model  is  superior  for  solving  the  icing  forecasting problem compared to other algorithms, because it can achieve global optima in fewer  compared to other algorithms, because it can achieve global optima in fewer iterations and is a critical forecasting problem compared to other algorithms, because it can achieve global optima in fewer  iterations and is a critical factor in convergence process of algorithms.  factor in convergence process of algorithms. iterations and is a critical factor in convergence process of algorithms.  5 5 Original data QFA-w-SVM FA-w-SVM Original data QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR PSO-SVM SVM MLR 1 6 11 16 21 26 31 36 1 6 11 16 21 26 31 36 Figure 15. Forecasting results of each model.  Figure 15. Forecasting results of each model. Figure 15. Forecasting results of each model.  Forecasting results/mm Forecasting results/mm Appl. Sci. 2016, 6, 54  22 of 24  Appl. Sci. 2016, 6, 54 21 of 23 Appl. Sci. 2016, 6, 54  22 of 24  QFA-w-SVM FA-w-SVM PSO-SVM 12% QFA-w-SVM FA-w-SVM PSO-SVM 12% SVM MLR SVM MLR 10% 10% 8% 8% 6% 6% 4% 4% 2% 2% 0% 0% 1 6 11 16 21 26 31 36 1 6 11 16 21 26 31 36 Figure 16. Forecasting errors of each model.  Figure 16. Forecasting errors of each model. Figure 16. Forecasting errors of each model.  Figure 17 shows the MAPE, MSE, and AAE values of the five models; it is clear that the values  Figure 17 shows the MAPE, MSE, and AAE values of the five models; it is clear that the values of Figure 17 shows the MAPE, MSE, and AAE values of the five models; it is clear that the values  of MAPE, MSE, AAE of the proposed QFA‐w‐SVM model are the smallest among the five models,  MAPE, MSE, AAE of the proposed QFA-w-SVM model are the smallest among the five models, which of MAPE, MSE, AAE of the proposed QFA‐w‐SVM model are the smallest among the five models,  which again demonstrates that the proposed model performs with great robustness and stability.  again demonstrates that the proposed model performs with great robustness and stability. which again demonstrates that the proposed model performs with great robustness and stability.  20% 20% MAPE MSE AAE MAPE MSE AAE 16% 16% 12% 12% 8% 8% 4% 4% 0% QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR 0% QFA-w-SVM FA-w-SVM PSO-SVM SVM MLR Figure 17. MAPE, MSE, AAE values of each model. Figure 17. MAPE, MSE, AAE values of each model.  Figure 17. MAPE, MSE, AAE values of each model.  In summary, the proposed QFA-w-SVM model can greatly close the gap between forecasting In summary, the proposed QFA‐w‐SVM model can greatly close the gap between forecasting  values and original data, which means it outperforms the FA-w-SVM, PSO-SVM, SVM, and MLR values and original data, which means it outperforms the FA‐w‐SVM, PSO‐SVM, SVM, and MLR  In summary, the proposed QFA‐w‐SVM model can greatly close the gap between forecasting  models in icing forecasting with different voltage transmission lines. Moreover, the QFA-w-SVM models in icing forecasting with different voltage transmission lines. Moreover, the QFA‐w‐SVM  values and original data, which means it outperforms the FA‐w‐SVM, PSO‐SVM, SVM, and MLR  which uses QFA to select the parameters of the SVM model, can effectively make a great improvement which  uses  QFA  to  select  the  parameters  of  the  SVM  model,  can  effectively  make  a  great  models in icing forecasting with different voltage transmission lines. Moreover, the QFA‐w‐SVM  for icing forecasting accuracy. It might be a promising alternative for icing thickness forecasting. improvement for icing forecasting accuracy. It might be a promising alternative for icing thickness  which  uses  QFA  to  select  the  parameters  of  the  SVM  model,  can  effectively  make  a  great  forecasting.  4. Conclusions improvement for icing forecasting accuracy. It might be a promising alternative for icing thickness  4. Conclusions  forecasting.  For better forecasting accuracy for icing thickness, this paper proposes an intelligent method for a wavelet support vector machine (w-SVM) based on a quantum fireworks optimization algorithm For better forecasting accuracy for icing thickness, this paper proposes an intelligent method for  4. Conclusions  a wavelet support vector machine (w‐SVM) based on a quantum fireworks optimization algorithm  For better forecasting accuracy for icing thickness, this paper proposes an intelligent method for  a wavelet support vector machine (w‐SVM) based on a quantum fireworks optimization algorithm  Appl. Sci. 2016, 6, 54 22 of 23 (QFA). Firstly, the regular fireworks optimization algorithm is improved through combination with a quantum optimization algorithm and proposes the quantum fireworks optimization algorithm (QFA); the steps of QFA are listed in detail. Secondly, in order to make use of the advantages of kernel function of SVM, the wavelet kernel function is applied to SVM instead of a Gaussian kernel function. Finally, this paper uses the proposed QFA model to optimize the parameters of w-SVM, and builds the icing thickness forecasting model of QFA-w-SVM. During the application of icing thickness forecasting, this paper also gives full consideration to the impact factors and selects the temperature, humidity, wind speed, wind direction and sunlight as the main impact factors. Through the numerical calculations of a self-written program, the empirical results show that the proposed forecasting model of QFA-w-SVM has great robustness and stability in icing thickness forecasting and is effective and feasible. Author Contributions: Drafting of the manuscript: Tiannan Ma and Dongxiao Niu; Implementation of numerical simulations and preparations of figures: Tiannan Ma and Ming Fu; Finalizing the manuscript: Tiannan Ma. Planning and supervision of the study: Dongxiao Niu and Tiannan Ma. Acknowledgments: This work is supported by the Natural Science Foundation of China (Project Nos. 71471059). Conflicts of Interest: The authors declare no conflict of interest. Reference 1. Yin, S.; Lu, Y.; Wang, Z.; Li, P.; Xu, K.J. Icing Thickness forecasting of overhead transmission line under rough weather based on CACA-WNN. Electr. Power Sci. Eng. 2012, 28, 28–31. 2. Jia-Zheng, L.U.; Peng, J.W.; Zhang, H.X.; Li, B.; Fang, Z. Icing meteorological genetic analysis of hunan power grid in 2008. Electr. Power Constr. 2009, 30, 29–32. 3. Goodwin, E.J.I.; Mozer, J.D.; Digioia, A.M.J.; Power, B.A. Predicting ice and snow loads for transmission line design. 1983, 83, 267–276. 4. Makkonen, L. Modeling power line icing in freezing precipitation. Atmos. Res. 1998, 46, 131–142. [CrossRef] 5. Liao, Y.F.; Duan, L.J. Study on estimation model of wire icing thickness in hunan province. Trans. Atmos. Sci. 2010, 33, 395–400. 6. Sheng, C.; Dong, D.; Xiaotin, H.; Muxia, S. Short-term Prediction for Transmission Lines Icing Based on BP Neural Network. In Proceedings of the IEEE 2012 Asia-Pacific Power and Energy Engineering Conference (APPEEC), Sanya, China, 31 December 2012–2 January 2013; pp. 1–5. 7. Liu, J.; Li, A.-J.; Zhao, L.-P. A prediction model of ice thickness based on T-S fuzzy neural networks. Hunan Electr. Power 2012, 32, 1–4. 8. Du, X.; Zheng, Z.; Tan, S.; Wang, J. The study on the prediction method of ice thickness of transmission line based on the combination of GA and BP neural network. In Proceedings of the 2010 International Conference on IEEE E-Product E-Service and E-Entertainment (ICEEE), Henan, China, 7–9 November 2010; pp. 1–4. 9. 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Appl. Energy 2012, 94, 65–70. [CrossRef] 19. Dai, H.; Zhang, B.; Wang, W. A multiwavelet support vector regression method for efficient reliability assessment. Reliab. Eng. Syst. Saf. 2015, 136, 132–139. [CrossRef] 20. Kang, J.; Tang, L.W.; Zuo, X.Z.; Li, H.; Zhang, X.H. Data prediction and fusion in a sensor network based on grey wavelet kernel partial least squares. J. Vib. Shock 2011, 30, 144–149. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Applied SciencesMultidisciplinary Digital Publishing Institute

Published: Feb 17, 2016

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