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Solution of reactive power optimisation including interval uncertainty using genetic algorithm

Solution of reactive power optimisation including interval uncertainty using genetic algorithm IntroductionReactive power optimisation (RPO) is used to reduce the real power losses of a power grid by adjusting the voltage of the generators, static volt‐ampere reactive (VAR) compensators, tap positions of the transformers, and output of the shunt capacitors/reactors in the presence of a series of physical and operating constraints. It is a special kind of optimal power flow [1] that is dedicated to optimising the profile of the reactive power and voltage of the power grid, thus reducing the operating costs of the system as well as maintaining a normal voltage level [2].For the conventional formalisation of the RPO, referred to as a deterministic RPO, all input data are determined by a snapshot of the system or several assumptions about the system under study. The RPO as a traditional non‐linear optimisation problem has already been solved adequately by both traditional algorithms [2–7] and artificial algorithms [8–14]. Traditional methods mainly search for the optimal solution based on the gradient direction, which is supposed to be the fastest descending (ascending) direction of differential functions in mathematics. All of the traditional methods demand the differentiability of the solved problem, which is irrational because some variables of the RPO are actually discrete. Therefore, artificial algorithms were proposed to deal with these problems such as the simulated annealing algorithm [8], genetic algorithm (GA) [9], tabu search [10], particle swarm optimisation method [11], heuristic optimisation algorithm [12], seeker optimisation algorithm [13], and fuzzy clustering [14]. These algorithms are mainly based on Monte Carlo simulations (or fuzzy simulations), and they randomly search for the optimal solution in the whole feasible space (including discrete space), so they can solve the RPO by incorporating discrete variables. Although artificial algorithms exhibit better adaptability than traditional methods, a significant amount of time is usually spent on the Monte Carlo (fuzzy simulation) procedure, which results in significant computational consumption.The deterministic RPO only solves a single system state that is representative of the limited set of system conditions corresponding to the data assumptions. However, when the input data are considered to be uncertain, the RPO problem becomes an uncertain programme; this type of problem is very difficult for the deterministic programming approaches to solve because numerous scenarios need to be analysed when dealing with uncertainties. There have been several attempts to solve the uncertain RPO such as using chance‐constrained programming (CCP) [15, 16] and robust programming (RP) [17, 18]. CCP is a kind of stochastic programming that assumes uncertainties are related to special stochastic distributions, and it obtains solution‐satisfying constraints at specific confidence levels. It offers a solution with statistical significance that can be realised with an artificial algorithm (e.g. GA [9]). However, the CCP method takes a significant amount of time to judge the constraints through the Monte Carlo simulation. In addition, the satisfaction of the constraints is not for all the possible scenarios of the uncertainties because the simulation process is conducted just for the extracted scenarios. These problems make the CCP method time‐consuming, so it is mainly used in long‐term optimal reactive power planning. The RP method supplies a robust solution that satisfies the constraints of the problem for the whole range of the uncertainties. The RP method searches for a solution just at the bounds of the uncertainties under the assumption that the uncertainties of the model can be bounded in intervals [17]. The obtained solution meets the constraints for all the values within the bounds of the uncertainties. Since only a few special scenarios are analysed, this method saves more central processing unit (CPU) time than the CCP method. However, the RP method is actually not suitable to solve the uncertain RPO. The RP method requires that the solved problem constraints be linear while the RPO has non‐linear power flow equations. Therefore, the non‐linear balance equations of the RPO must be linearised before using the RP method [17]. The solution given by the RP method satisfies the linearised constraints of the RPO, not the original constraints. The linearisation of the reactive power balance equations is rather inaccurate due to the crude approximation of the cosine function, making usage of the RP method even more infeasible. These drawbacks limit the application of the RP method in solving the uncertain RPO. To overcome these problems, we attempted to obtain conservative solutions of the uncertain RPO that satisfy the original constraints for all values of the uncertainties. Therefore, the following work was conducted:The model of the RPO incorporating interval uncertainty (RPOIU) was constructed. In this model, the uncertainties are assumed to be intervals, variables are divided into controllable variables and state variables, and the value of the state variables are considered as intervals. This model utilises multi‐objective programming because the objective function real power losses is an interval that is determined by its radius and midpoint. The RPOIU model satisfies the constraints perfectly in the presence of the IUs.GA was adopted to solve the single‐objective RPOIU. The midpoint of the real power losses was chosen as the objective function only for simplicity. The reliable power flow calculation (RPFC) was used by GA to acquire the intervals of the state variables to judge the constraints with the intervals. This algorithm is based on affine arithmetic (AA), which is an enhanced model for self‐validated numerical computations [19].We compared the proposed RPOIU method with the CCP method [15], which is a conventional, but practical approach to deal with stochastic uncertainty.We obtained the Pareto front of the multi‐objective RPOIU model using the non‐dominated sorting GA‐II (NSGA‐II) [20] and chose an optimal Pareto optimum solution. The radius and midpoint of the real power losses interval were selected as the two objective functions of the RPOIU model.The rest of this paper is arranged as follows. The formulation procedure of the RPOIU model is introduced in Section 2. Section 3 describes the solution courses of the RPOIU model with GA and NSGA‐II. Section 4 contains two case studies. Finally, the conclusions and contributions of this paper are given in Section 5.Problem formulationThe RPO is mainly used to reduce the steady‐state real power losses under a series of constraints such as the bound limits of the voltage magnitude of the load buses, reactive power generation, output of the shunt capacitors/reactors, and ratios of the transformers. Unlike the optimal real power dispatch [1], which is mainly concerned with the coordination of the real power and frequency in the power system, the RPO aims at obtaining an optimal reactive power and voltage profile of the grid, so the real power generation are assumed to be known as the input data. Other input information is needed such as the load demand, bound information of the related variables, and voltage magnitude at the slack bus.Model of the RPOThe RPO incorporates the real and reactive power balance equations at all buses, as well as some physical and operating constraints. We expressed each nodal voltage as the polar coordinate form V.=V∠θ, and then described the RPO model as1Pl=∑i∈SVi∑j∈SVjGijcosθijs.t.2PGi−PLi−Pi=0,i∈S′GQGi−QLi−Qi=0,i∈S′G3−PLi−Pi=0,i∈SLQCi−QLi−Qi=0,i∈SL4QGimin≤QGi≤QGimax,i∈SG5QCimin≤QCi≤QCimax,i∈SC6Vimin≤Vi≤Vimax,i∈SG∪SL7Tlmin≤Tl≤Tlmin,l∈STwhere8Pi=Pi(V,θ,T)=Vi∑j∈SVj(Gijcosθij+Bijsinθij)9Qi=Qi(V,θ,T)=Vi∑j∈SVj(Gijsinθij−Bijcosθij)θij=θi−θj ; S, SL, SG, ST, SC, and SG′ are the sets of system buses, load buses, generator buses, transformer branches, buses with reactive compensators, and non‐balance generator buses, respectively. V, θ, and T are vectors consisting of all the voltage magnitudes, bus angles, and ratios of the transformers, respectively. Equation (1) represents the real power losses. Equations (2) and (3) consist of the balance equations of the RPO. Equation (4) represents the constraints of the reactive power generation and (6) expresses the bound limits of all the bus voltages (not including the slack bus). Equations (5) and (7) are the outputs of the shunt capacitors/reactors and ratios of the transformers, respectively. PLi and QLi represent the active and reactive loads; QCi is the capacitor switched to the reactive power compensation, which belongs to the discrete variables due to its inherent attributes. Bij is the imaginary portion of Yij. Tl is the ratio of transformer l with a discrete value, and the admittance matrix components G={Gij} and B={Bij} are functions of the control variable T [19].We can rewrite the RPO model as10minf(x)s.t.h(x)=0gmin≤g(x)≤gmaxwhere f(x) represents the real power losses, h(x)=0 represents the real and reactive power balance equations, and gmin≤g(x)≤gmax represents the constraints from (2)–(7).Model of the RPOIUAs noted in the introduction, the uncertainty can be expressed as a random number, fuzzy number, and interval. An interval gives a reliable range of the uncertainty despite the distribution and membership function. In addition, the operational security should always be guaranteed in power systems, and intervals supply a relatively conventional result that helps keep the security constraints satisfied. Therefore, the load demand and power generation were regarded as intervals in our RPOIU model. If we denote the active power generation, active power, and reactive power demand as [PGiSL,PGiSU](fori∈SG), [PLiSL,PLiSU], and [QLiSL,QLiSU](fori∈SL), respectively, then the balanced equations incorporating the IUs can be classified and rewritten as follows.For the slack bus11PLi+Pi=PGi,QLi+Qi=QGiFor the generator bus12PLi+Pi=[PGiSL,PGiSU]QLi+Qi=QGiFinally, for the load bus13−Pi=[PLiSL,PLiSU],QCi−Qi=[QLiSL,QLiSU]It should be noted that the active and reactive powers of the slack and generator bus are assumed to be determined values since they are mainly derived from power plants. Pi and Qi are computed by (8) and (9). Actually, if all the data required in the power flow calculation (including the interval information) are given, (11)–(13) together form the RPFC problem, where the power flow results are all intervals, and it can be effectively solved by the AA‐based method [21].After the formulation of the balance equations with the IU, the model of the RPOIU is given by14min{fM,fR}={mid(f(X,u)),rad(f(X,u))}s.t.h(X,u)=[hL,hU]gmin≤g(X,u)≤gmaxwhere [hL,hU] is an interval vector formed from a combination of the items on the right‐hand side of (11)–(13). If there is a real number at the i th equation, then hiL=hiU. We arranged the buses in the following order: slack bus (No. 1), generator bus (No. 2−m), and load bus (No. m  + 1−n); the load bus with the capacitor/reactor (No. m  + 1−m  + r) comes before the others. On the basis of the above order, we have X=[PG1QG1⋯QGmVm+1⋯Vnθ2⋯θn]T and u=[V2⋯VmQCm+1⋯QCm+rT1⋯Tk]T. u is a vector with a real value because the voltage of the generator bus can be fixed by coordinating the excitation system, and the compensation of the capacitor/reactor and ratios of the transformer can be controlled artificially. However, X is a vector with an interval value, since it cannot be regulated under uncertain circumstances. Its value is decided by u through power flow equations. Therefore, X is called a ‘state variable’ and u is called a ‘controllable variable’ [7]. f(X,u) is an interval number because it contains the interval vector X. It is impossible to minimise an interval number directly, so we switched to obtain the midpoint and radius of f(X,u), which are denoted by fM and fR, respectively.The above discussion shows that (11) is a multi‐objective non‐linear integer programme with IU. At present, there is no effective method for solving (11) because it is very difficult to deal with the intervals that are hidden in the objective functions, constraint equations, and inequalities. Relevant research mainly focuses on problems whose intervals can be expressed explicitly [22, 23]. Fortunately, we noted that if the controllable variable u is fixed, then the state variable X can be computed by the RPFC [21], which gives a relatively high accuracy of the interval results. The GA excels in searching solutions in discrete and continuous space despite the continuity and differentiability of the solving problem. Therefore, a method based on the GA and RPFC can be used to solve the RPOIU model, and it is discussed in the next section.Solution of the RPOIUFrom the discussion in Section 2, we know that the RPOIU method is a multi‐objective problem. We decided to acquire its Pareto front, which is a general method for handling multi‐objective programming. However, to provide greater insight, the solution of the single‐objective RPOIU model is introduced first to exhibit the detailed processes of the RPFC and GA. Section 3.1 provides details about the RPFC. The results of the RPFC were employed by the GA and NSGA‐II to solve the single‐objective and multi‐objective RPOIU models, respectively. The solution of the single‐objective is realised through the GA, and it is introduced in Section 3.2. It is followed by a brief introduction of the NSGA‐II in Section 3.3, which was used to obtain the Pareto front of the multi‐objective RPOIU model. These techniques are described and explained in detail in the following sections.RPFC based on AAThe RPFC is a significant method for obtaining the range of the power flow of the power grid incorporating the interval power input data. At present, it can be conducted through both interval mathematics (IA) and AA. IA has been used and tested only in small grids, and there are many problems with its process such as error explosion. AA provides more accurate results because it keeps track of the correlations between the input and computed quantities, and this extra information provides tighter intervals in computation, reducing the likelihood of error explosion problems that occur with IA computations. Therefore, we chose to use AA as the key technique in our RPFC. Since this process is described in Vaccaro's work [21] in detail, we only provide a brief overview of its principles and execution.Suppose that we obtained the conventional power flow results as V0 and θ0, and some IU [−ΔP,ΔP] (i.e. wind power) is exerted on the nodal power of a bus. What are the results of the power flow? In AA, the results of the power flow are V=V0+VPεP and θ=θ0+θPεP, where εP=[−1,1] is the basic noise element and VP and θP are the coefficients sensitive to the nodal power, which can be obtained through the Jacobi matrix of the power flow equation. In other words, the power flow should be changed with [−ΔP,ΔP] at the ‘nominal’ operating point, where the input data are set as the values of the midpoints of the IUs. Without a loss of generality, if all the IUs of the power flow are considered, the voltage magnitude and phase angle can be expressed as15Vi=Vi,0+∑j∈SG∪SLVi,jPεPi+∑j∈SLVi,jQεQi,fori∈SLθi=θi,0+∑j∈SG∪SLθi,jPεPi+∑j∈SLθi,jQεQi,fori∈SG∪SLThe calculations for the coefficients in (15) are performed in Vaccaro's work [21]. To ensure the inclusion of the solution domain, each sensitive coefficient in (15) should be multiplied by an amplification [24].Actually, (15) only gives a rough estimate of the range of the power flow results because the whole process is a linear approximation, but the power flow equations are non‐linear. To contract the range of the interval, a ‘domain‐contraction’‐based method was proposed by Vaccaro et al. [21]. These researchers substituted (15) into (8) and (9), and conducted the AA computation (including the Chebyshev approximation of non‐affine functions) to obtain the affine form of each nodal powers P^i and Q^i. After that, Pi and Qi in the power flow (which contain the interval input data) are replaced by P^i and Q^i. Finally, the interval mathematics (IM) computation is used to form a linear IA problem16Ay=Cwhere y=[εP2,εP3,…,εPn,εQm+1,εQm+2,…,εQn]T, C is an interval vector, and A is a real known matrix. To solve (16), we assume −1≤εPi≤1, −1≤εQi≤1 and set εPi and εQi as the objective functions; we then solve the corresponding linear ‘min’ and ‘max’ programme. By doing this, we obtain the ranges of εPi and εQi, and substitute them into (15), thus acquiring the intervals of the voltage magnitude and phase angle using IA computation. Other power flow variables such as the reactive power generation, active power, and reactive power of the slack bus can be computed by (15) and AA computation. This is the complete process of the RPFC and the steps are shown in Fig. 1. It should be noted that results of the RPFC are always slightly conservative. This is due to the fact that AA, such as IA, yields the ‘worst case’ bounds, which take into account any uncertainties in the input data and all the internal truncation and round‐off errors. Therefore, the ranges of the results obtained are slightly wider than the ‘real’ intervals. In other words, though the AA‐based RPFC is also a linearised model, it can obtain conservative ranges of the solutions, which include the ‘real’ ranges of the interval power flow model. When we use the results provided by the AA‐based methods to judge whether the solution satisfies the constraints of the RPOIU model, we find that the feasible solution satisfied the constraints perfectly. This is quite different from the RP method [17], which may obey the original constraints after linearising the power flow balance constraints, even though it solves the linearised model on the bounds of the uncertainties.1Fig.Process of the RPFC for obtaining the ranges of the power flow including the IUSolving the single‐objective RPOIU model using the GATo simplify the RPOIU model, we first solved the single‐objective RPOIU model formulated by (14), and set the midpoint of the real power losses fM=mid(f(X,u)) as the objective function. By doing this, we were able to compare it more effectively with other methods (e.g. the CCP method). Since the RPOIU method incorporates interval variables, traditional algorithms cannot solve it. Therefore, an artificial algorithm, i.e. the GA, was employed to solve the single‐objective RPOIU model. The GA is a classical intelligent algorithm, and it can be used to solve all kinds of programming problems that contain features such as convexity, continuity, and differentiability. It seeks the solution in the whole feasible field randomly, which can be conducted in both discrete and continuous space. To solve the single‐objective RPOIU model, the GA searches the optimal controllable variables randomly over the constrained space, while the RPFC gives the range of the state variables and real power losses. The results were used by the GA to judge the inequalities’ constraints and sort the solutions. Here, we present the procedure for the GA in solving the RPOIU method formulated in (14). A detailed explanation of the principles behind the GA can be found in Iba's work [9]. The process of GA is given below.First, set the input data and parameters of the GA. The input data mainly include the intervals of the load demand and active power generation; these parameters consist of the number of the population NP, probability of mutation Pm, probability of crossing Pc, and maximum of the genetic iteration M.Second, initialise the population. The population is composed of controllable variables, denoted as (u1,u2,…,uNP). All the individuals must satisfy the constraints of (14), and the RPFC is needed to obtain the ranges of the state variables and corresponding objective fM.Finally, execute the loop iteration of the GA. It repeats the following steps:Crossing: This involves extracting elements of the current population at a chance of Pc and pairing them randomly. For example, (ui, uj) is a pair, then17ui∗=cui+(1−c)ujand18uj∗=(1−c)ui+cujwhere c is a randomly generated value in the range of (0, 1). If there is at least one of the values from (17) and (18) to satisfy the constraints of (14), for example, ui∗, then we replace ui with ui∗. Otherwise, we need to compute (17) and (18) repeatedly with a new c until the maximum step is reached.ii. Mutation: The current population at probability Pm is sampled, and for each sampled individual ui, the following equation is used:19ui0=ui+Ddwhere d is a random direction and its each dimension is limited in the range of (−1, 1) and D is an amplification. If (19) conforms to the constraints of (14), then ui0 replaces ui or D=D⋅r [r is a random value in the range of (0, 1)] and (19) is repeated until it reaches a maximum step. Here, the RPFC is needed to obtain the ranges of the state variables for judging the constraints of (14).iii. Sorting: We set20f=1/fMas the fitness function, which is the basis for sorting the current population. According to f, the corresponding u is made in ascending order.iv. Selecting: By constructing the evaluation function21q(i)=q(i−1)+fi,1≤i≤NP+1q(0)=0and adopting the Roulette method to filter the current population, a random rq is generated in the range of (0, 1). If rqq(NP)∈[q(i−1),q(i)], then the i th individual ui is selected as a member of the new population. A new population is produced by repeating this operation NP times.After the loop iteration, we obtain an optimal solution u∗ as well as its corresponding objective, fM∗. The whole procedure is presented in Fig. 2. Note that the rounding computation should be used when dealing with the discrete variables in (17)–(19).2Fig.Process of GA in solving the RPOIU modelObtaining the Pareto front of the multi‐objective RPOIU using the NSGA‐IIAs discussed in Section 3.2, the RPOIU model is reduced to a multi‐objective programme (MOP), so this section focuses on the solution of the MOP. Many solution methods have already been proposed to solve an MOP, and they can be classified as methods of switching single‐objective programming and approaches to getting a set of Pareto‐optimal solutions. However, in principle, Pareto‐optimal solutions cover more optimal situations of an MOP than those found by single‐objective programming. Therefore, in the absence of any further information, it is more realistic to obtain the Pareto front of an MOP, which is defined as a hyperplane consisting of a set of Pareto‐optimal solutions.There are many ways to obtain a Pareto front [20, 25–27], and they can be divided into classical optimisation methods and evolutionary algorithms (EAs). However, the classical optimisation methods (including the multi‐criterion decision‐making methods) are known to be time‐consuming because they involve converting the multi‐objective optimisation problem into a single‐objective optimisation problem by emphasising one particular Pareto‐optimal solution at a time. However, the EA can find multiple Pareto‐optimal solutions in a single simulation run, so it saves a lot of CPU time. The EA also includes the NSGA, Pareto‐archived evolution strategy, strength‐Pareto evolution algorithm (SPEA), and NSGA‐II. However, the NSGA‐II outperforms the other algorithms in terms of finding a diverse set of solutions and in converging near the true Pareto‐optimal set. The Pareto solutions obtained by the NSGA‐II have a better distance between each other when compared with the other methods, so it is the best method currently available. As a result, we chose to use the NSGA‐II as the solution algorithm for the MOP in this paper.The NSGA‐II differs from the standard GA in its process of sorting and selection. The sorting process is changed to a non‐dominated sort that orders individuals based on the objectives and crowding distance. The crowding distance is a measure of how close an individual is to its neighbours. A large average crowding distance results in a better diversity in the population. The selection is based on the rank and crowding distance, rather than the objectives. Other procedures such as mutation, crossover, and population initialisation are similar to those found in the standard GA. Details about the NSGA‐II can be found in the work conducted by Deb et al. [20].Simulation resultsThis section discusses the results from the solution of the RPOIU model with the GA, and the IEEE 14 and IEEE 57 test systems are analysed. In the first case, a single‐objective RPOIU model was solved, and a conventional and effective method of solving an uncertain non‐linear programme, i.e. the CCP method, was chosen for comparison. The second case was the solution of a multi‐objective RPOIU model, and the Pareto front and an optimal Pareto optimum solution for (14) were obtained. For convenience, the data descriptions of the IEEE test systems were ignored here, but detailed information can be found in [28]. The bus order in both test cases was rearranged as the slack bus, generator bus, and load bus (a bus with a capacitor/reactor) for easy description.Comparison with the CCP methodThe CCP method is an effective and common approach to solving non‐linear programmes with uncertainty [15]. The model of (14) can be expressed as22minf¯s.t.Pro{f(x,u)<f¯}≥βh(x,u,ξ)=0Pro{g_≤g(x,u)≤g¯}≥αwhere ξ is the uniform distribution random vector that lies in [hL,hU], Pro{⋅} stands for the probability function, and β and α are the confidence levels of the constraints. The CCP model can also be solved by the GA, where the constraints of (22) are judged by the usage of the Monte Carlo technology, which is based on the theory of the law of large numbers. However, the constraints of (22) are not all satisfied in the actual computation, though the confidence levels are set to the maximum, 1. This is because the judgement of the constraints occurs with the Monte Carlo simulation, and it only demands the extracted samples conform to the constraints; other values in [hL,hU] may not.The CCP model is quite different from the RPOIU model in terms of its objective function and constraints, and these differences can be expressed as given below:Objective: The objective function of the CCP model is obtained through the Monte Carlo simulation. For example, suppose we apply a uniform distribution to generate N scenarios (ξ1,ξ2,…,ξN) in [hL,hU] which correspond to a series of real power losses (f1,f2,…,fN). Then, the objective function f¯ is defined as the Nβ th of the smaller values of these values, where ⋅ is the ceiling function. However, the RPOIU method regards the real power losses as an interval, which can be obtained through the RPFC directly. Its midpoint is selected as the single‐objective function of the RPOIU model for simplicity.Constraints: The constraints of the CCP method hold true in the statistical sense, which means the constraints are not affirmatively satisfied. However, the constraints of the RPOIU method are perfectly satisfied because it uses more conservative ranges of the state variables obtained by the RPFC to judge the constraints. It should be noted that all the power flow equation constraints of the CCP model and RPOIU model are satisfied because they are solved by the power flow calculation and RPFC before the judgement of the inequality constraints.From the above discussion, we know that the CCP model and RPOIU model have different meanings for their objective functions and constraints. To minimise these differences in the CCP model, we set α=1 to provide for the low possibility of a constraint violation and made β=0.5 to get the ‘midpoint’ of the real power losses, which corresponds to the midpoint of the interval of the real power losses in the RPOIU model. The GA was also chosen as the solution algorithm for the CCP model to allow for a good comparison and analysis. The parameters of the GA used in the two methods were the same: M=80, NP=30, Pm=0.3, Pc=0.2, and N=1000 was set as the times simulating constraints and objective function of the CCP model. Under these settings, a small test system, i.e. the modified IEEE 14 system, was used to provide for easy analysis. A tolerance of ±20% was assumed in the load demand and active power generation. For the CCP method, the power load and generation were assumed to have uniform distributions in the corresponding intervals. To show the differences between the CCP model and RPOIU model in the constraint holding, limited ranges of the load voltages were modified and narrowed down to [0.97, 1.02].On the basis of the assumed data and parameters, the CCP method and RPOIU method based on the GA were used to solve the uncertain RPO problem. The final solutions obtained by the two approaches were examined by the Monte Carlo method; 5000 simulations were used to obtain the ‘real’ bounds of the state variables. The ranges of the state variables optimised by the three methods are presented in Figs. 3 and 4. Fig. 3 depicts the profiles of the reactive power generation and Fig. 4 illustrates the intervals of the load voltage magnitudes. We observed that the range of the load voltage magnitude at bus 14 obtained by the CCP method was beyond the limitation. This was expected because the number of the extracted samples used in the Monte Carlo simulation was too small. For a better graphic representation of the outcome, the simulation results of bus 14 are illustrated in Fig. 5.3Fig.Intervals of the load voltage magnitudes obtained by the RPOIU method and CCP method4Fig.Interval results of the reactive power generation acquired by the RPOIU method and CCP method5Fig.Simulation results of the voltage at bus 14 obtained by the CCP methodTo show the relationship between the number of samples and satisfaction of the constraints as well as the CPU time of the CCP method, we tested the population in the last GA iteration by using various simulation times for the judgement of the constraints. The corresponding results are listed in Table 1. This table shows that the isolation of the constraints can be avoided by increasing the simulation times, but the CPU time goes up accordingly. Therefore, the CCP method satisfies the constraints at the expense of CPU time. For a fair comparison with the RPOIU approach, N  = 4500 was set as a smaller number needed to satisfy the constraints of the CCP model. The CPU time and objective function of each method are presented in Table 2. We observed that the RPOIU method costs much less CPU time than the CCP method; this is because it does not need the Monte Carlo simulation. However, the objective function values of the two methods are very close to each other.1TableCPU time and constraint holding of the CCP method using different simulation timesSimulation timesConstraint holdingCPU time, s1000no46522000no91073000no14,2544000no18,9134500yes20,8965000yes22,5972TableCPU time and optimal objectives obtained by the RPOIU method and CCP methodMethodCPU time, sObjective, puCCP20,8960.135321RPOIU4690.135810There are three advantages of the RPOIU method over the CCP method:The constraints in the RPOIU method are satisfied definitely, while the CCP method conforms to the constraints in the statistical sense.The CPU time cost by the RPOIU method is much less than that of the CCP method.No assumptions regarding to the probability distribution of the load and generator power variations are required in the RPOIU method.Simulation results of the multi‐objective RPOIUThe radius of the real power losses is a significant variable because it represents the possible degree of variation; a large radius will increase the RPOIU method's operating cost. The midpoint of the real power losses represents the expected value of the operating cost, and it is an important target. To determine an overall solution of the RPOIU method, there is a need to obtain the Pareto front of the multi‐objective RPOIU model, thus balancing the optimisation of the radius and midpoint of the real power losses.The IEEE 57 configuration was analysed to validate the applicability of the proposed method to a larger system. All the voltage magnitude permitted operating ranges were set to 0.9–1.1 pu. The parameters of the NSGA‐II were set as follows: maximum iteration M=200, population NP=200, mutation probability Pm=0.3, crossing probability Pc=0.2, pool size Ppool=200, and tour size Ptour=4. The multi‐objective RPOIU was solved with the NSGA‐II using these data and parameter settings. The Pareto front obtained by the NSGA‐II is exhibited in Fig. 6, and it is constructed using 200 Pareto optimum solutions.6Fig.Pareto front of the RPOIU method obtained with the NSGA‐IIThe ideal point method was employed to select an optimal Pareto optimum solution from the Pareto solution [27]. For convenience, we denote the midpoint and radius of the real power losses as f1 and f2, respectively. The ideal point method first normalises the Pareto optimum solutions to the space [0,1]×[0,1] by using the formulation fi′=(fi−fimin)/(fimax−fimin), i=1,2, where fimin and fimax are the minimum and maximum of the objective function fi of all the Pareto optimum solutions. By doing this, we can remove the influence on the orders of magnitude of the different objective functions. It then becomes obvious that the absolute optimal value (f1min,f2min) corresponds to the point (0,0) in the new space. Therefore, the optimal Pareto optimum solution can be selected as point (f1′,f2′) that is closest to point (0,0) under the new space. The results are exhibited in Fig. 7, and the optimal Pareto solution is located at (f1′,f2′)=0.29872,0.329611, which corresponds to (f1,f2)=(0.29722,0.07965). Therefore, an optimal Pareto solution is obtained, and it balances the midpoint and radius of real power losses simultaneously.7Fig.Optimal Pareto solution obtained by the ideal point methodAs far as the computational requirements are concerned, the process of obtaining a Pareto front of the multi‐objective RPOIU method using the NSGA‐II costs about 20 min.ConclusionThis paper proposes the RPOIU method, where the constraints of the uncertain RPO are satisfied perfectly. To solve this model, the GA is employed for its convenience to deal with the discrete variables. The RPFC is used to obtain the intervals of the state variables that are used by the GA to judge the constraints. Since the RPOIU model is actually an MOP, the NSGA‐II is employed to obtain its Pareto front. An optimal Pareto optimum solution is chosen as a final strategy for its engineering application. The simulation results and analyses demonstrate that the proposed RPOIU method satisfies the operating constraints better than CCP method, which is an excellent method for solving uncertain programmes; the acquired optimal Pareto optimum solution proved to be closest to the ideal operating point, and it can balance the radius and midpoint of the real power losses at the same time.AcknowledgmentsThis work was supported in part by the National Basic Research Program of China (973 Program) (2016YFB0900102) and in part by the China National Funds for Excellent Young Scientists (51322702). 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Solution of reactive power optimisation including interval uncertainty using genetic algorithm

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Abstract

IntroductionReactive power optimisation (RPO) is used to reduce the real power losses of a power grid by adjusting the voltage of the generators, static volt‐ampere reactive (VAR) compensators, tap positions of the transformers, and output of the shunt capacitors/reactors in the presence of a series of physical and operating constraints. It is a special kind of optimal power flow [1] that is dedicated to optimising the profile of the reactive power and voltage of the power grid, thus reducing the operating costs of the system as well as maintaining a normal voltage level [2].For the conventional formalisation of the RPO, referred to as a deterministic RPO, all input data are determined by a snapshot of the system or several assumptions about the system under study. The RPO as a traditional non‐linear optimisation problem has already been solved adequately by both traditional algorithms [2–7] and artificial algorithms [8–14]. Traditional methods mainly search for the optimal solution based on the gradient direction, which is supposed to be the fastest descending (ascending) direction of differential functions in mathematics. All of the traditional methods demand the differentiability of the solved problem, which is irrational because some variables of the RPO are actually discrete. Therefore, artificial algorithms were proposed to deal with these problems such as the simulated annealing algorithm [8], genetic algorithm (GA) [9], tabu search [10], particle swarm optimisation method [11], heuristic optimisation algorithm [12], seeker optimisation algorithm [13], and fuzzy clustering [14]. These algorithms are mainly based on Monte Carlo simulations (or fuzzy simulations), and they randomly search for the optimal solution in the whole feasible space (including discrete space), so they can solve the RPO by incorporating discrete variables. Although artificial algorithms exhibit better adaptability than traditional methods, a significant amount of time is usually spent on the Monte Carlo (fuzzy simulation) procedure, which results in significant computational consumption.The deterministic RPO only solves a single system state that is representative of the limited set of system conditions corresponding to the data assumptions. However, when the input data are considered to be uncertain, the RPO problem becomes an uncertain programme; this type of problem is very difficult for the deterministic programming approaches to solve because numerous scenarios need to be analysed when dealing with uncertainties. There have been several attempts to solve the uncertain RPO such as using chance‐constrained programming (CCP) [15, 16] and robust programming (RP) [17, 18]. CCP is a kind of stochastic programming that assumes uncertainties are related to special stochastic distributions, and it obtains solution‐satisfying constraints at specific confidence levels. It offers a solution with statistical significance that can be realised with an artificial algorithm (e.g. GA [9]). However, the CCP method takes a significant amount of time to judge the constraints through the Monte Carlo simulation. In addition, the satisfaction of the constraints is not for all the possible scenarios of the uncertainties because the simulation process is conducted just for the extracted scenarios. These problems make the CCP method time‐consuming, so it is mainly used in long‐term optimal reactive power planning. The RP method supplies a robust solution that satisfies the constraints of the problem for the whole range of the uncertainties. The RP method searches for a solution just at the bounds of the uncertainties under the assumption that the uncertainties of the model can be bounded in intervals [17]. The obtained solution meets the constraints for all the values within the bounds of the uncertainties. Since only a few special scenarios are analysed, this method saves more central processing unit (CPU) time than the CCP method. However, the RP method is actually not suitable to solve the uncertain RPO. The RP method requires that the solved problem constraints be linear while the RPO has non‐linear power flow equations. Therefore, the non‐linear balance equations of the RPO must be linearised before using the RP method [17]. The solution given by the RP method satisfies the linearised constraints of the RPO, not the original constraints. The linearisation of the reactive power balance equations is rather inaccurate due to the crude approximation of the cosine function, making usage of the RP method even more infeasible. These drawbacks limit the application of the RP method in solving the uncertain RPO. To overcome these problems, we attempted to obtain conservative solutions of the uncertain RPO that satisfy the original constraints for all values of the uncertainties. Therefore, the following work was conducted:The model of the RPO incorporating interval uncertainty (RPOIU) was constructed. In this model, the uncertainties are assumed to be intervals, variables are divided into controllable variables and state variables, and the value of the state variables are considered as intervals. This model utilises multi‐objective programming because the objective function real power losses is an interval that is determined by its radius and midpoint. The RPOIU model satisfies the constraints perfectly in the presence of the IUs.GA was adopted to solve the single‐objective RPOIU. The midpoint of the real power losses was chosen as the objective function only for simplicity. The reliable power flow calculation (RPFC) was used by GA to acquire the intervals of the state variables to judge the constraints with the intervals. This algorithm is based on affine arithmetic (AA), which is an enhanced model for self‐validated numerical computations [19].We compared the proposed RPOIU method with the CCP method [15], which is a conventional, but practical approach to deal with stochastic uncertainty.We obtained the Pareto front of the multi‐objective RPOIU model using the non‐dominated sorting GA‐II (NSGA‐II) [20] and chose an optimal Pareto optimum solution. The radius and midpoint of the real power losses interval were selected as the two objective functions of the RPOIU model.The rest of this paper is arranged as follows. The formulation procedure of the RPOIU model is introduced in Section 2. Section 3 describes the solution courses of the RPOIU model with GA and NSGA‐II. Section 4 contains two case studies. Finally, the conclusions and contributions of this paper are given in Section 5.Problem formulationThe RPO is mainly used to reduce the steady‐state real power losses under a series of constraints such as the bound limits of the voltage magnitude of the load buses, reactive power generation, output of the shunt capacitors/reactors, and ratios of the transformers. Unlike the optimal real power dispatch [1], which is mainly concerned with the coordination of the real power and frequency in the power system, the RPO aims at obtaining an optimal reactive power and voltage profile of the grid, so the real power generation are assumed to be known as the input data. Other input information is needed such as the load demand, bound information of the related variables, and voltage magnitude at the slack bus.Model of the RPOThe RPO incorporates the real and reactive power balance equations at all buses, as well as some physical and operating constraints. We expressed each nodal voltage as the polar coordinate form V.=V∠θ, and then described the RPO model as1Pl=∑i∈SVi∑j∈SVjGijcosθijs.t.2PGi−PLi−Pi=0,i∈S′GQGi−QLi−Qi=0,i∈S′G3−PLi−Pi=0,i∈SLQCi−QLi−Qi=0,i∈SL4QGimin≤QGi≤QGimax,i∈SG5QCimin≤QCi≤QCimax,i∈SC6Vimin≤Vi≤Vimax,i∈SG∪SL7Tlmin≤Tl≤Tlmin,l∈STwhere8Pi=Pi(V,θ,T)=Vi∑j∈SVj(Gijcosθij+Bijsinθij)9Qi=Qi(V,θ,T)=Vi∑j∈SVj(Gijsinθij−Bijcosθij)θij=θi−θj ; S, SL, SG, ST, SC, and SG′ are the sets of system buses, load buses, generator buses, transformer branches, buses with reactive compensators, and non‐balance generator buses, respectively. V, θ, and T are vectors consisting of all the voltage magnitudes, bus angles, and ratios of the transformers, respectively. Equation (1) represents the real power losses. Equations (2) and (3) consist of the balance equations of the RPO. Equation (4) represents the constraints of the reactive power generation and (6) expresses the bound limits of all the bus voltages (not including the slack bus). Equations (5) and (7) are the outputs of the shunt capacitors/reactors and ratios of the transformers, respectively. PLi and QLi represent the active and reactive loads; QCi is the capacitor switched to the reactive power compensation, which belongs to the discrete variables due to its inherent attributes. Bij is the imaginary portion of Yij. Tl is the ratio of transformer l with a discrete value, and the admittance matrix components G={Gij} and B={Bij} are functions of the control variable T [19].We can rewrite the RPO model as10minf(x)s.t.h(x)=0gmin≤g(x)≤gmaxwhere f(x) represents the real power losses, h(x)=0 represents the real and reactive power balance equations, and gmin≤g(x)≤gmax represents the constraints from (2)–(7).Model of the RPOIUAs noted in the introduction, the uncertainty can be expressed as a random number, fuzzy number, and interval. An interval gives a reliable range of the uncertainty despite the distribution and membership function. In addition, the operational security should always be guaranteed in power systems, and intervals supply a relatively conventional result that helps keep the security constraints satisfied. Therefore, the load demand and power generation were regarded as intervals in our RPOIU model. If we denote the active power generation, active power, and reactive power demand as [PGiSL,PGiSU](fori∈SG), [PLiSL,PLiSU], and [QLiSL,QLiSU](fori∈SL), respectively, then the balanced equations incorporating the IUs can be classified and rewritten as follows.For the slack bus11PLi+Pi=PGi,QLi+Qi=QGiFor the generator bus12PLi+Pi=[PGiSL,PGiSU]QLi+Qi=QGiFinally, for the load bus13−Pi=[PLiSL,PLiSU],QCi−Qi=[QLiSL,QLiSU]It should be noted that the active and reactive powers of the slack and generator bus are assumed to be determined values since they are mainly derived from power plants. Pi and Qi are computed by (8) and (9). Actually, if all the data required in the power flow calculation (including the interval information) are given, (11)–(13) together form the RPFC problem, where the power flow results are all intervals, and it can be effectively solved by the AA‐based method [21].After the formulation of the balance equations with the IU, the model of the RPOIU is given by14min{fM,fR}={mid(f(X,u)),rad(f(X,u))}s.t.h(X,u)=[hL,hU]gmin≤g(X,u)≤gmaxwhere [hL,hU] is an interval vector formed from a combination of the items on the right‐hand side of (11)–(13). If there is a real number at the i th equation, then hiL=hiU. We arranged the buses in the following order: slack bus (No. 1), generator bus (No. 2−m), and load bus (No. m  + 1−n); the load bus with the capacitor/reactor (No. m  + 1−m  + r) comes before the others. On the basis of the above order, we have X=[PG1QG1⋯QGmVm+1⋯Vnθ2⋯θn]T and u=[V2⋯VmQCm+1⋯QCm+rT1⋯Tk]T. u is a vector with a real value because the voltage of the generator bus can be fixed by coordinating the excitation system, and the compensation of the capacitor/reactor and ratios of the transformer can be controlled artificially. However, X is a vector with an interval value, since it cannot be regulated under uncertain circumstances. Its value is decided by u through power flow equations. Therefore, X is called a ‘state variable’ and u is called a ‘controllable variable’ [7]. f(X,u) is an interval number because it contains the interval vector X. It is impossible to minimise an interval number directly, so we switched to obtain the midpoint and radius of f(X,u), which are denoted by fM and fR, respectively.The above discussion shows that (11) is a multi‐objective non‐linear integer programme with IU. At present, there is no effective method for solving (11) because it is very difficult to deal with the intervals that are hidden in the objective functions, constraint equations, and inequalities. Relevant research mainly focuses on problems whose intervals can be expressed explicitly [22, 23]. Fortunately, we noted that if the controllable variable u is fixed, then the state variable X can be computed by the RPFC [21], which gives a relatively high accuracy of the interval results. The GA excels in searching solutions in discrete and continuous space despite the continuity and differentiability of the solving problem. Therefore, a method based on the GA and RPFC can be used to solve the RPOIU model, and it is discussed in the next section.Solution of the RPOIUFrom the discussion in Section 2, we know that the RPOIU method is a multi‐objective problem. We decided to acquire its Pareto front, which is a general method for handling multi‐objective programming. However, to provide greater insight, the solution of the single‐objective RPOIU model is introduced first to exhibit the detailed processes of the RPFC and GA. Section 3.1 provides details about the RPFC. The results of the RPFC were employed by the GA and NSGA‐II to solve the single‐objective and multi‐objective RPOIU models, respectively. The solution of the single‐objective is realised through the GA, and it is introduced in Section 3.2. It is followed by a brief introduction of the NSGA‐II in Section 3.3, which was used to obtain the Pareto front of the multi‐objective RPOIU model. These techniques are described and explained in detail in the following sections.RPFC based on AAThe RPFC is a significant method for obtaining the range of the power flow of the power grid incorporating the interval power input data. At present, it can be conducted through both interval mathematics (IA) and AA. IA has been used and tested only in small grids, and there are many problems with its process such as error explosion. AA provides more accurate results because it keeps track of the correlations between the input and computed quantities, and this extra information provides tighter intervals in computation, reducing the likelihood of error explosion problems that occur with IA computations. Therefore, we chose to use AA as the key technique in our RPFC. Since this process is described in Vaccaro's work [21] in detail, we only provide a brief overview of its principles and execution.Suppose that we obtained the conventional power flow results as V0 and θ0, and some IU [−ΔP,ΔP] (i.e. wind power) is exerted on the nodal power of a bus. What are the results of the power flow? In AA, the results of the power flow are V=V0+VPεP and θ=θ0+θPεP, where εP=[−1,1] is the basic noise element and VP and θP are the coefficients sensitive to the nodal power, which can be obtained through the Jacobi matrix of the power flow equation. In other words, the power flow should be changed with [−ΔP,ΔP] at the ‘nominal’ operating point, where the input data are set as the values of the midpoints of the IUs. Without a loss of generality, if all the IUs of the power flow are considered, the voltage magnitude and phase angle can be expressed as15Vi=Vi,0+∑j∈SG∪SLVi,jPεPi+∑j∈SLVi,jQεQi,fori∈SLθi=θi,0+∑j∈SG∪SLθi,jPεPi+∑j∈SLθi,jQεQi,fori∈SG∪SLThe calculations for the coefficients in (15) are performed in Vaccaro's work [21]. To ensure the inclusion of the solution domain, each sensitive coefficient in (15) should be multiplied by an amplification [24].Actually, (15) only gives a rough estimate of the range of the power flow results because the whole process is a linear approximation, but the power flow equations are non‐linear. To contract the range of the interval, a ‘domain‐contraction’‐based method was proposed by Vaccaro et al. [21]. These researchers substituted (15) into (8) and (9), and conducted the AA computation (including the Chebyshev approximation of non‐affine functions) to obtain the affine form of each nodal powers P^i and Q^i. After that, Pi and Qi in the power flow (which contain the interval input data) are replaced by P^i and Q^i. Finally, the interval mathematics (IM) computation is used to form a linear IA problem16Ay=Cwhere y=[εP2,εP3,…,εPn,εQm+1,εQm+2,…,εQn]T, C is an interval vector, and A is a real known matrix. To solve (16), we assume −1≤εPi≤1, −1≤εQi≤1 and set εPi and εQi as the objective functions; we then solve the corresponding linear ‘min’ and ‘max’ programme. By doing this, we obtain the ranges of εPi and εQi, and substitute them into (15), thus acquiring the intervals of the voltage magnitude and phase angle using IA computation. Other power flow variables such as the reactive power generation, active power, and reactive power of the slack bus can be computed by (15) and AA computation. This is the complete process of the RPFC and the steps are shown in Fig. 1. It should be noted that results of the RPFC are always slightly conservative. This is due to the fact that AA, such as IA, yields the ‘worst case’ bounds, which take into account any uncertainties in the input data and all the internal truncation and round‐off errors. Therefore, the ranges of the results obtained are slightly wider than the ‘real’ intervals. In other words, though the AA‐based RPFC is also a linearised model, it can obtain conservative ranges of the solutions, which include the ‘real’ ranges of the interval power flow model. When we use the results provided by the AA‐based methods to judge whether the solution satisfies the constraints of the RPOIU model, we find that the feasible solution satisfied the constraints perfectly. This is quite different from the RP method [17], which may obey the original constraints after linearising the power flow balance constraints, even though it solves the linearised model on the bounds of the uncertainties.1Fig.Process of the RPFC for obtaining the ranges of the power flow including the IUSolving the single‐objective RPOIU model using the GATo simplify the RPOIU model, we first solved the single‐objective RPOIU model formulated by (14), and set the midpoint of the real power losses fM=mid(f(X,u)) as the objective function. By doing this, we were able to compare it more effectively with other methods (e.g. the CCP method). Since the RPOIU method incorporates interval variables, traditional algorithms cannot solve it. Therefore, an artificial algorithm, i.e. the GA, was employed to solve the single‐objective RPOIU model. The GA is a classical intelligent algorithm, and it can be used to solve all kinds of programming problems that contain features such as convexity, continuity, and differentiability. It seeks the solution in the whole feasible field randomly, which can be conducted in both discrete and continuous space. To solve the single‐objective RPOIU model, the GA searches the optimal controllable variables randomly over the constrained space, while the RPFC gives the range of the state variables and real power losses. The results were used by the GA to judge the inequalities’ constraints and sort the solutions. Here, we present the procedure for the GA in solving the RPOIU method formulated in (14). A detailed explanation of the principles behind the GA can be found in Iba's work [9]. The process of GA is given below.First, set the input data and parameters of the GA. The input data mainly include the intervals of the load demand and active power generation; these parameters consist of the number of the population NP, probability of mutation Pm, probability of crossing Pc, and maximum of the genetic iteration M.Second, initialise the population. The population is composed of controllable variables, denoted as (u1,u2,…,uNP). All the individuals must satisfy the constraints of (14), and the RPFC is needed to obtain the ranges of the state variables and corresponding objective fM.Finally, execute the loop iteration of the GA. It repeats the following steps:Crossing: This involves extracting elements of the current population at a chance of Pc and pairing them randomly. For example, (ui, uj) is a pair, then17ui∗=cui+(1−c)ujand18uj∗=(1−c)ui+cujwhere c is a randomly generated value in the range of (0, 1). If there is at least one of the values from (17) and (18) to satisfy the constraints of (14), for example, ui∗, then we replace ui with ui∗. Otherwise, we need to compute (17) and (18) repeatedly with a new c until the maximum step is reached.ii. Mutation: The current population at probability Pm is sampled, and for each sampled individual ui, the following equation is used:19ui0=ui+Ddwhere d is a random direction and its each dimension is limited in the range of (−1, 1) and D is an amplification. If (19) conforms to the constraints of (14), then ui0 replaces ui or D=D⋅r [r is a random value in the range of (0, 1)] and (19) is repeated until it reaches a maximum step. Here, the RPFC is needed to obtain the ranges of the state variables for judging the constraints of (14).iii. Sorting: We set20f=1/fMas the fitness function, which is the basis for sorting the current population. According to f, the corresponding u is made in ascending order.iv. Selecting: By constructing the evaluation function21q(i)=q(i−1)+fi,1≤i≤NP+1q(0)=0and adopting the Roulette method to filter the current population, a random rq is generated in the range of (0, 1). If rqq(NP)∈[q(i−1),q(i)], then the i th individual ui is selected as a member of the new population. A new population is produced by repeating this operation NP times.After the loop iteration, we obtain an optimal solution u∗ as well as its corresponding objective, fM∗. The whole procedure is presented in Fig. 2. Note that the rounding computation should be used when dealing with the discrete variables in (17)–(19).2Fig.Process of GA in solving the RPOIU modelObtaining the Pareto front of the multi‐objective RPOIU using the NSGA‐IIAs discussed in Section 3.2, the RPOIU model is reduced to a multi‐objective programme (MOP), so this section focuses on the solution of the MOP. Many solution methods have already been proposed to solve an MOP, and they can be classified as methods of switching single‐objective programming and approaches to getting a set of Pareto‐optimal solutions. However, in principle, Pareto‐optimal solutions cover more optimal situations of an MOP than those found by single‐objective programming. Therefore, in the absence of any further information, it is more realistic to obtain the Pareto front of an MOP, which is defined as a hyperplane consisting of a set of Pareto‐optimal solutions.There are many ways to obtain a Pareto front [20, 25–27], and they can be divided into classical optimisation methods and evolutionary algorithms (EAs). However, the classical optimisation methods (including the multi‐criterion decision‐making methods) are known to be time‐consuming because they involve converting the multi‐objective optimisation problem into a single‐objective optimisation problem by emphasising one particular Pareto‐optimal solution at a time. However, the EA can find multiple Pareto‐optimal solutions in a single simulation run, so it saves a lot of CPU time. The EA also includes the NSGA, Pareto‐archived evolution strategy, strength‐Pareto evolution algorithm (SPEA), and NSGA‐II. However, the NSGA‐II outperforms the other algorithms in terms of finding a diverse set of solutions and in converging near the true Pareto‐optimal set. The Pareto solutions obtained by the NSGA‐II have a better distance between each other when compared with the other methods, so it is the best method currently available. As a result, we chose to use the NSGA‐II as the solution algorithm for the MOP in this paper.The NSGA‐II differs from the standard GA in its process of sorting and selection. The sorting process is changed to a non‐dominated sort that orders individuals based on the objectives and crowding distance. The crowding distance is a measure of how close an individual is to its neighbours. A large average crowding distance results in a better diversity in the population. The selection is based on the rank and crowding distance, rather than the objectives. Other procedures such as mutation, crossover, and population initialisation are similar to those found in the standard GA. Details about the NSGA‐II can be found in the work conducted by Deb et al. [20].Simulation resultsThis section discusses the results from the solution of the RPOIU model with the GA, and the IEEE 14 and IEEE 57 test systems are analysed. In the first case, a single‐objective RPOIU model was solved, and a conventional and effective method of solving an uncertain non‐linear programme, i.e. the CCP method, was chosen for comparison. The second case was the solution of a multi‐objective RPOIU model, and the Pareto front and an optimal Pareto optimum solution for (14) were obtained. For convenience, the data descriptions of the IEEE test systems were ignored here, but detailed information can be found in [28]. The bus order in both test cases was rearranged as the slack bus, generator bus, and load bus (a bus with a capacitor/reactor) for easy description.Comparison with the CCP methodThe CCP method is an effective and common approach to solving non‐linear programmes with uncertainty [15]. The model of (14) can be expressed as22minf¯s.t.Pro{f(x,u)<f¯}≥βh(x,u,ξ)=0Pro{g_≤g(x,u)≤g¯}≥αwhere ξ is the uniform distribution random vector that lies in [hL,hU], Pro{⋅} stands for the probability function, and β and α are the confidence levels of the constraints. The CCP model can also be solved by the GA, where the constraints of (22) are judged by the usage of the Monte Carlo technology, which is based on the theory of the law of large numbers. However, the constraints of (22) are not all satisfied in the actual computation, though the confidence levels are set to the maximum, 1. This is because the judgement of the constraints occurs with the Monte Carlo simulation, and it only demands the extracted samples conform to the constraints; other values in [hL,hU] may not.The CCP model is quite different from the RPOIU model in terms of its objective function and constraints, and these differences can be expressed as given below:Objective: The objective function of the CCP model is obtained through the Monte Carlo simulation. For example, suppose we apply a uniform distribution to generate N scenarios (ξ1,ξ2,…,ξN) in [hL,hU] which correspond to a series of real power losses (f1,f2,…,fN). Then, the objective function f¯ is defined as the Nβ th of the smaller values of these values, where ⋅ is the ceiling function. However, the RPOIU method regards the real power losses as an interval, which can be obtained through the RPFC directly. Its midpoint is selected as the single‐objective function of the RPOIU model for simplicity.Constraints: The constraints of the CCP method hold true in the statistical sense, which means the constraints are not affirmatively satisfied. However, the constraints of the RPOIU method are perfectly satisfied because it uses more conservative ranges of the state variables obtained by the RPFC to judge the constraints. It should be noted that all the power flow equation constraints of the CCP model and RPOIU model are satisfied because they are solved by the power flow calculation and RPFC before the judgement of the inequality constraints.From the above discussion, we know that the CCP model and RPOIU model have different meanings for their objective functions and constraints. To minimise these differences in the CCP model, we set α=1 to provide for the low possibility of a constraint violation and made β=0.5 to get the ‘midpoint’ of the real power losses, which corresponds to the midpoint of the interval of the real power losses in the RPOIU model. The GA was also chosen as the solution algorithm for the CCP model to allow for a good comparison and analysis. The parameters of the GA used in the two methods were the same: M=80, NP=30, Pm=0.3, Pc=0.2, and N=1000 was set as the times simulating constraints and objective function of the CCP model. Under these settings, a small test system, i.e. the modified IEEE 14 system, was used to provide for easy analysis. A tolerance of ±20% was assumed in the load demand and active power generation. For the CCP method, the power load and generation were assumed to have uniform distributions in the corresponding intervals. To show the differences between the CCP model and RPOIU model in the constraint holding, limited ranges of the load voltages were modified and narrowed down to [0.97, 1.02].On the basis of the assumed data and parameters, the CCP method and RPOIU method based on the GA were used to solve the uncertain RPO problem. The final solutions obtained by the two approaches were examined by the Monte Carlo method; 5000 simulations were used to obtain the ‘real’ bounds of the state variables. The ranges of the state variables optimised by the three methods are presented in Figs. 3 and 4. Fig. 3 depicts the profiles of the reactive power generation and Fig. 4 illustrates the intervals of the load voltage magnitudes. We observed that the range of the load voltage magnitude at bus 14 obtained by the CCP method was beyond the limitation. This was expected because the number of the extracted samples used in the Monte Carlo simulation was too small. For a better graphic representation of the outcome, the simulation results of bus 14 are illustrated in Fig. 5.3Fig.Intervals of the load voltage magnitudes obtained by the RPOIU method and CCP method4Fig.Interval results of the reactive power generation acquired by the RPOIU method and CCP method5Fig.Simulation results of the voltage at bus 14 obtained by the CCP methodTo show the relationship between the number of samples and satisfaction of the constraints as well as the CPU time of the CCP method, we tested the population in the last GA iteration by using various simulation times for the judgement of the constraints. The corresponding results are listed in Table 1. This table shows that the isolation of the constraints can be avoided by increasing the simulation times, but the CPU time goes up accordingly. Therefore, the CCP method satisfies the constraints at the expense of CPU time. For a fair comparison with the RPOIU approach, N  = 4500 was set as a smaller number needed to satisfy the constraints of the CCP model. The CPU time and objective function of each method are presented in Table 2. We observed that the RPOIU method costs much less CPU time than the CCP method; this is because it does not need the Monte Carlo simulation. However, the objective function values of the two methods are very close to each other.1TableCPU time and constraint holding of the CCP method using different simulation timesSimulation timesConstraint holdingCPU time, s1000no46522000no91073000no14,2544000no18,9134500yes20,8965000yes22,5972TableCPU time and optimal objectives obtained by the RPOIU method and CCP methodMethodCPU time, sObjective, puCCP20,8960.135321RPOIU4690.135810There are three advantages of the RPOIU method over the CCP method:The constraints in the RPOIU method are satisfied definitely, while the CCP method conforms to the constraints in the statistical sense.The CPU time cost by the RPOIU method is much less than that of the CCP method.No assumptions regarding to the probability distribution of the load and generator power variations are required in the RPOIU method.Simulation results of the multi‐objective RPOIUThe radius of the real power losses is a significant variable because it represents the possible degree of variation; a large radius will increase the RPOIU method's operating cost. The midpoint of the real power losses represents the expected value of the operating cost, and it is an important target. To determine an overall solution of the RPOIU method, there is a need to obtain the Pareto front of the multi‐objective RPOIU model, thus balancing the optimisation of the radius and midpoint of the real power losses.The IEEE 57 configuration was analysed to validate the applicability of the proposed method to a larger system. All the voltage magnitude permitted operating ranges were set to 0.9–1.1 pu. The parameters of the NSGA‐II were set as follows: maximum iteration M=200, population NP=200, mutation probability Pm=0.3, crossing probability Pc=0.2, pool size Ppool=200, and tour size Ptour=4. The multi‐objective RPOIU was solved with the NSGA‐II using these data and parameter settings. The Pareto front obtained by the NSGA‐II is exhibited in Fig. 6, and it is constructed using 200 Pareto optimum solutions.6Fig.Pareto front of the RPOIU method obtained with the NSGA‐IIThe ideal point method was employed to select an optimal Pareto optimum solution from the Pareto solution [27]. For convenience, we denote the midpoint and radius of the real power losses as f1 and f2, respectively. The ideal point method first normalises the Pareto optimum solutions to the space [0,1]×[0,1] by using the formulation fi′=(fi−fimin)/(fimax−fimin), i=1,2, where fimin and fimax are the minimum and maximum of the objective function fi of all the Pareto optimum solutions. By doing this, we can remove the influence on the orders of magnitude of the different objective functions. It then becomes obvious that the absolute optimal value (f1min,f2min) corresponds to the point (0,0) in the new space. Therefore, the optimal Pareto optimum solution can be selected as point (f1′,f2′) that is closest to point (0,0) under the new space. The results are exhibited in Fig. 7, and the optimal Pareto solution is located at (f1′,f2′)=0.29872,0.329611, which corresponds to (f1,f2)=(0.29722,0.07965). Therefore, an optimal Pareto solution is obtained, and it balances the midpoint and radius of real power losses simultaneously.7Fig.Optimal Pareto solution obtained by the ideal point methodAs far as the computational requirements are concerned, the process of obtaining a Pareto front of the multi‐objective RPOIU method using the NSGA‐II costs about 20 min.ConclusionThis paper proposes the RPOIU method, where the constraints of the uncertain RPO are satisfied perfectly. To solve this model, the GA is employed for its convenience to deal with the discrete variables. The RPFC is used to obtain the intervals of the state variables that are used by the GA to judge the constraints. Since the RPOIU model is actually an MOP, the NSGA‐II is employed to obtain its Pareto front. An optimal Pareto optimum solution is chosen as a final strategy for its engineering application. The simulation results and analyses demonstrate that the proposed RPOIU method satisfies the operating constraints better than CCP method, which is an excellent method for solving uncertain programmes; the acquired optimal Pareto optimum solution proved to be closest to the ideal operating point, and it can balance the radius and midpoint of the real power losses at the same time.AcknowledgmentsThis work was supported in part by the National Basic Research Program of China (973 Program) (2016YFB0900102) and in part by the China National Funds for Excellent Young Scientists (51322702). 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Journal

IET Generation Transmission & DistributionWiley

Published: Oct 1, 2017

Keywords: reactive power; genetic algorithms; voltage control; load flow; Pareto optimisation; RPO; interval uncertainty; genetic algorithm; reactive power optimisation; optimal profile; power systems; steady state; deterministic sets; demand loads; generation values; voltage control; uncertain nonlinear programme; power flow calculation; Pareto front; chance‐constrained programming method

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