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A sparse recovery model with fast decoupled solution for distribution state estimation and its performance analysis

A sparse recovery model with fast decoupled solution for distribution state estimation and its... J. Mod. Power Syst. Clean Energy (2019) 7(6):1411–1421 https://doi.org/10.1007/s40565-019-0522-9 A sparse recovery model with fast decoupled solution for distribution state estimation and its performance analysis 1 1 2 Junwei YANG , Wenchuan WU , Weiye ZHENG , Yuntao JU Abstract This paper introduces a robust sparse recovery 1 Introduction model for compressing bad data and state estimation (SE), based on a revised multi-stage convex relaxation (R-Cap- State estimation (SE) is a fundamental module in energy ped-L1) model. To improve the calculation efficiency, a management systems (EMSs) and its key task is to provide fast decoupled solution is adopted. The proposed method estimates of state variables which are as accurate as pos- can be used for three-phase unbalanced distribution net- sible. For distribution networks, a robust and efficient works with both phasor measurement unit and remote distribution state estimator assists in integrated operation terminal unit measurements. The robustness and the com- with distributed energy resources, assures power quality putational efficiency of the R-Capped-L1 model with fast levels, and improves the reliability of a power system [1]. decoupled solution are compared with some popular SE A number of distribution system state estimation (DSSE) methods by numerical tests on several three-phase distri- methodologies have been proposed based on different state bution networks. variables, treatments for load data and bad data, and measurements. Weighted least square (WLS) estimators Keywords Distribution system, State estimation, Sparse form the basis of the most popular methods. To suppress recovery, Fast decoupled method the influence of bad data, some robust estimators, such as least median of square (LMS) estimators [2], weighted least absolute value (WLAV) estimators [3, 4], and M-es- timators [5], have been proposed. To mitigate the effect of CrossCheck date: 15 January 2019 leverage points and improve the applicability of WLAV, a Received: 26 September 2018 / Accepted: 15 January 2019 / Published WLAV estimation with optimal transformations (WLAV- online: 9 May 2019 OT) that systematically solves the problem of computing The Author(s) 2019 rotation angles and scaling factors was proposed [6]. Most & Wenchuan WU robust estimators consider the measurement residual as a wuwench@tsinghua.edu.cn whole and minimize the value of a penalty function of Junwei YANG residuals. However, the residuals usually consist of two yang-jw11@mails.tsinghua.edu.cn parts: observation noise and abnormally large measurement Weiye ZHENG errors caused by bad data, which obey different distribu- neo_adonis@139.com tions. The errors are normally sparse. Therefore, based on Yuntao JU the theory of compress sensing (CS), some sparse recovery juyuntao@cau.edu.cn models [7, 8] considering this point have been proposed to Department of Electrical Engineering, Tsinghua University, detect bad data. An L1-relaxation (L1-R) model con- Beijing 100084, China straining the sparse vector by the L1-norm instead of the Department of Electrical and Electronic Engineering, The L0-norm, which is used to denote the number of non-zero University of Hong Kong, Hong Kong, China values in a sparse vector, was proposed [8]. Relaxing the L0-norm problem to the L1-norm problem is a common College of Information and Electrical Engineering, China Agricultural University, Beijing 100083, China 123 1412 Junwei YANG et al. practice in CS [9]. The effectiveness of the L1-R model has formulate the three-phase SE problem with hybrid mea- been proved both mathematically and practically. How- surements. The branch current measurements are formu- ever, it has been reported that the relaxation process often lated as the branch active power and reactive power losses, leads to sub-optimal solutions [10]. To handle this issue allowing them to be incorporated into the FDSE model. mathematically, a multi-stage convex relaxation (Capped- Thus, the contributions of this paper can be summarized as L1) method was proposed [10]. We first introduced this follows: method for SE in a transmission power system in our 1) The Capped-L1 model is first introduced for DSSE, previous work [11], and found that Capped-L1 method has which has a powerful capacity to compress bad data. an advantage in precision but not in computational speed 2) A novel three-phase fast decoupled state estimation because iterations are required and because the optimiza- model with hybrid measurements for DSSE is adopted. tion problem contains both the L1-norm and the L2-norm, 3) A transformation strategy for Capped-L1 to which are relatively nonlinear compared to WLS and R-Capped-L1 is applied and the computational effi- WLAV. Actually, it is easier to solve a pure quadratic ciency is significantly improved. optimization problem than a mixed problem of square values and absolute values [12]. In this paper, efforts are made to transform the original Capped-L1 model to a quadratic optimization model. The efficiency of the revised 2 Proposed state estimation model Capped-L1 (R-Capped-L1) model is improved significantly. This section first reviews the formulation of the sparse Recently, to improve the monitoring of distribution L1-R model for state estimation that has been proposed in system operating conditions, some utilities have started previous work [8]. Based on this formulation, the proposed installing phasor measurement units (PMUs) at the distri- Capped-L1 model and the revised model are introduced. bution level. PMUs provide measurements of voltage phasor and current phasor with high frequency. Combining 2.1 Sparse L1-R model measurements from both remote terminal units (RTUs) and PMUs can promote the ability to perform state forecasting When bad data are presented in the measurements, the for the distribution network [13]. The inclusion of branch relationship between the measurements and the state vari- current and voltage measurements helps to improve the ables can be represented as [7]: precision but also introduces additional components to the z ¼ hðyÞþ o þ e ð1Þ solution procedure [14]. A novel branch current-based SE where z is the raw measurement vector; h(y) is a set of has been proposed [15], with a two-stage solution. In [16], measurement functions; o is the vector of errors corre- the active and reactive power measurements were trans- sponding to bad data; e is the noise vector. formed to linear complex current measurements based on Generally, the elements of e follow random Gaussian estimated phase angle and voltage. References [17] and distributions and are independent in most cases; the error [18] formulated the multi-source measurements SE prob- vector o is a sparse vector with few non-zero values. lem by extending the state variables. Reference [19] Mathematically, the L0-norm of a vector represents the combines the estimates independently obtained from number of its non-zero values. Therefore, the error vector supervisory control and data acquisition (SCADA)-based can be constrained with the L0-norm and the SE model can and PMU-based estimators based on multisensor data be formulated as an optimization problem: fusion theory. They all showed good performance in han- dling branch current measurements. However, owing to the min Jðy; oÞ¼kk z  hðyÞ o ð2Þ repeating factorization of the Jacobian matrix, these s:t: kk o  e methods suffer from a heavy computational burden. To resolve this, a fast decoupled power flow (FDPF) method where e is a positive small number related to the proportion for distribution networks by choosing a complex base of bad data. voltage and adjusting the ratio of R/X was also proposed Given a reasonable e, we can obtain relatively accurate [20, 21]. Its efficiency has been proved in a large number of estimates for state variables by solving (2). However, there case studies. The computational speed of DSSE is expected are two difficulties in solving this problem:  the value of to be faster when introducing this method. parameter e is difficult to specify because the proportion of In this paper, we introduce a robust Capped-L1 model bad data is not normally known; ` L0-norm minimization into DSSE. To reduce the computational burden, we has been proved to be an NP problem that cannot be solved transform the original model to a quadratic form and apply efficiently [9]. a novel fast decoupled state estimation (FDSE) method to 123 A sparse recovery model with fast decoupled solution… 1413 For , according to the duality theory, the solution to (2) corresponds to the solution of the problem (2) for some Lagrange dual variable k  0: min Jðy; oÞ¼kk z  hðyÞ o þkkk o ð3Þ 2 0 For `, according to the theory of CS [9], L1-norm minimization helps obtain sparse solutions. Thereby, the L0-norm in (3) can be relaxed to an L1-norm problem. For this step, the original non-convex sparse mini- mization problem has been evolved into a convex opti- mization problem given by (4), which can be solved efficiently: min Jðy; oÞ¼kk z  hðyÞ o þkkk o ð4Þ 2 1 Fig. 1 Iterative procedure of Capped-L1 The SE model above is exactly the L1-R model process continues until the two error vectors of adjacent previously proposed in [8], which is a typical sparse iteration steps become relatively close. recovery model. However, the Capped-L1 model (5) is an extremely nonlinear model with a fairly heavy computational burden. 2.2 Capped-L1 model and transformation strategy To increase the calculation efficiency, the following transformation strategy can be adopted: Convex relaxation such as L1-R indeed solves the L0- ðlÞ ðlÞ ðlÞ norm minimization problem efficiently under some condi- ðlÞ ðlÞ ðlÞ ðlÞ > min Jðy ; o Þ¼ z  hðy Þ o þk c a þ b i i i > 2 tions. However, it often leads to a sub-optimal solution in > ðlÞ reality. To obtain better solutions than the L1-R model, a s:t: a  0 new model Capped-L1 was proposed [10] and we introduce > ðlÞ b  0 this model into the DSSE problem to handle the issues with > ðlÞ ðlÞ ðlÞ a  b  o ¼ 0 a sparse error vector. The Capped-L1 model can be for- i i i mulated as: ð6Þ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ min Jðy where a and b represent auxiliary variables sharing the ; o Þ¼ z  hðy Þ o þk c o i i i i ðlÞ same dimension with o . ð5Þ This strategy transforming the optimization problem (5)to ðlþ1Þ  ðlÞ ðlÞ (6) is actually relaxing the absolute value, and the relaxation where c ¼ Ið o  a Þ is a relaxation parameter, IðÞ i i method has been proved to be valid mathematically [22]. is a step function with a value of 0 or 1, l is the number of Solutions to (6) are exactly the same as those to (5); thereby, ðlÞ iterations, a is a threshold value changing with the the precisions of the two Capped-L1 models are identical. ðlÞ  ðlÞ ðlÞ decision variable o . For this problem, if o  a , the i i However, through the relaxation process, the computational ðlþ1Þ speed of the Capped-L1 model will be significantly improved value of c will be 1; otherwise, the value will be 0. because each iteration of (6) is actually a quadratic opti- In the procedure of solving the problem, the iterations mization problem, which is relatively easily solved. The are needed as shown in Fig. 1. iteration procedure for solving (6) is the same as that of (5), but ðlÞ In the iteration procedure, f ðo Þ is a function of each iteration saves a lot of time thanks to the linearization. ðlÞ ðlÞ ðlÞ ðlÞ o whose value meets o  f ðo Þ o . Generally, the max min By adopting this R-Capped-L1 model, a robust SE model function can be chosen as the average value or the median with a lighter computational burden can be developed. ðlÞ ðlÞ of vector o . The threshold a changes in each iteration. e is a small threshold value to determine whether or not the iteration has been convergent. 3 Fast decoupled model for three-phase Observing the iteration procedure, it can be seen that distribution networks each iteration exactly solves a convex optimization prob- ðlÞ lem. Once the new sparse error vector o is given from the The fast decoupled method has been applied in trans- ðlÞ last iteration, the method studies the new sparse vector o mission networks for a long time and its efficiency has ðlþ1Þ been proved by substantial practice. However, resulting to adjust the relaxation parameter c and solve a new from the large R/X in distribution networks, the optimization problem to obtain better solutions. This 123 1414 Junwei YANG et al. ‘‘decoupled’’ idea fails for almost all feeders. Recently, to where j 2 i indicates that j is connected to i. handle this problem, a fast decoupled algorithm via com- plex per unit (pu) normalization for distribution networks 3) Three-phase branch current measurements: has been proposed [20]. To be more readable, a briefly u / / u /u / u /u I I ðcosðh  h Þr þ sinðh  h Þx Þ ij ij ij ij ij ij ij ij review of the complex pu normalization algorithm is given /2fa;b;cg ð13Þ in Appendix A. In this section, we will introduce a novel u u u u u ¼ P þ P þ o þ e ¼ P ij ji P P ji;loss ji;loss ji;loss three-phase fast decoupled model to estimate state vari- u / / u /u / u /u ables for unbalanced three-phase distribution networks. To I I ðcosðh  h Þx  sinðh  h Þr Þ ij ij ij ij ij ij ij ij take the branch current measurements into consideration, /2fa;b;cg ð14Þ the branch active power and reactive power losses are u u u u u ¼ Q þ Q þ o þ e ¼ Q ij ji Q Q ij;loss ij;loss ij;loss employed. Using the fast decoupled method, the calcula- /u tion efficiency of the R-Capped-L1 sparse recovery model where / and u represent the phases; r is the mutual resis- ij will be further improved, enabling good handling of the /u tance of branch ij between phase u and phase /; x is the ij branch current measurements. mutual reactance of branch ij between phaseu and phase/. In the DSSE problem, the measurement vector has to be extended as (7) and the state variable vector can be rep- 4) Three-phase bus voltage and phase angle resented as (18): measurements: hi 2 3 2 3 2 3 2 3 a a abc abc abc abc abc abc abc abc abc a a o e ðU Þ U U U z ¼ U ; h ; P ; P ; Q ; Q ; P ; Q ; I ð7Þ i i i i i i ij ji ij ji i i ij 6 7 6 7 6 7 6 7 b m b 6 7 6 7 o b e b hi 4 ðU Þ 5 ¼ 4 U 5 þ þ ð15Þ U U i i 4 5 4 5 i i abc abc abc abc abc abc y¼ U ; h ; P ; P ; Q ; Q ð8Þ c m c i i ij ji ij ji ðU Þ U c c o e i i U U i i 2 3 2 3 2 3 2 3 where U is the bus voltage; h is the phase angle; P is the a m a a a o e ðh Þ h h h i i i i active power of branches or injections; Q is the reactive 6 7 6 7 6 7 6 7 b b 6 7 6 7 ¼ þ o b þ e b ð16Þ 4 ðh Þ 5 4 h 5 h h power of branches or injections; I is the magnitude of i i 4 5 4 5 i i c c ðh Þ h c c branch current; the superscript abc denotes the three phases o e i i h h i i of the variables; the superscript ij denotes that the variable flows from bus i to bus j; the superscript i denotes bus i. The measurement function h(y) relating z and 5) Pseudo-measurements formulating network y includes: constraints: 1) Three-phase real and reactive power measurements of P ij / u / u/ u / u/ ¼ U ðcosðh  h Þg þ sinðh  h Þb Þ j i j ij i j ij the branch: /2fa;b;cg 2 3 2 3 2 3 2 3 a m a a a X o e ðP Þ P P P ij ij ij ij / u / u/ u / u/ U ðcosðh  h Þg þ sinðh  h Þb Þ 6 7 6 7 6 7 6 7 i i i ij i i ij b m b 6 7 6 7 6 7 6 7 o b e b ðP Þ P ¼ þ þ ð9Þ P P /2fa;b;cg ij ij 4 5 4 5 4 ij 5 4 ij 5 c c ðP Þ P c c ð17Þ o e P P ij ij ij ij 2 3 2 3 2 3 2 3 m Q a a a a ij / u / u/ u / u/ o e ðQ Þ Q Q Q ij ij ij ij ¼ U ðsinðh  h Þg  cosðh  h Þb Þ j i j ij i j ij 6 7 6 7 6 7 6 7 b b i 6 7 6 7 6 7 6 7 /2fa;b;cg o b e b ðQ Þ ¼ Q þ þ ð10Þ Q Q ij ij 4 5 4 5 4 ij 5 4 ij 5 X / u / u/ u / u/ c m c U ðsinðh  h Þg  cosðh  h Þb Þ c c ðQ Þ Q o e i i i ij i i ij ij ij Q Q ij ij /2fa;b;cg where the superscript m denotes the measurement ð18Þ quantities. u/ where g is the mutual conductance of branch ij between ij 2) Three-phase injection power measurements: u/ 2 3 2 3 2 3 2 3 a phase u and phase /; b is the mutual susceptance of a a ij P o e ðP Þ P P ij i i i 6 7 6 7 6 7 branch ij between phase u and phase /. 6 7 m b 6 7 6 7 6 7 ¼¼ P þ o b þ e b ð11Þ 4 ðP Þ 5 P P ij i 4 5 4 i 5 4 i 5 The fast decoupled method can improve the efficiency of m j2i ðP Þ c c P o e P P this DSSE problem. Fast decoupled state estimation depends ij i i 2 3 2 3 2 3 on the PQ decoupled formulation of the measurement 2 3 a m a a Q o e ðQ Þ Q Q ij i i i equations. Clearly, the real and reactive power measurement 6 7 6 7 6 7 6 7 m b 6 7 6 7 6 7 o b e b ¼ Q þ þ ð12Þ (9) and (10), the injection power measurement (11) and (12), 4 ðQ Þ 5 Q Q i ij 4 5 4 i 5 4 i 5 m j2i c c the branch current measurement (13) and (14), and the bus ðQ Þ Q c c o e i Q Q ij i i 123 A sparse recovery model with fast decoupled solution… 1415 2 3 pffiffiffi pffiffiffi voltage and phase angle measurement (15) and (16) can all be 1 3 1 3 aa ab ab ac ac b  b  g  b þ g 6 7 ij ij ij ij ij expressed in a PQ decoupled formulation. 2 2 2 2 6 7 6 7 pffiffiffi pffiffiffi The network constraints (17) and (18) involve both volt- 6 7 1 3 1 3 6 ab ab bb bc bc 7 A ¼ b þ g b  b  g 6 7 age U and phase angle h. As reviewed above, in a distribution ij ij ij ij ij 2 2 2 2 6 7 6 7 pffiffiffi pffiffiffi network, the normalized R/X can be adjusted by choosing an 4 5 1 3 1 3 ac ac bc bc cc appropriate complex base value and thus original R/X value  b  g  b þ g b ij ij ij ij ij 2 2 2 2 is not required to be small to adapt the proposed formulation. ð24Þ Once normalized R/X is small, U has little impact on the active power, and h has little impact on the reactive power Finally, all of the elements in the Jacobian matrix of h(y) are constant. It is important to note that and U  1. Hence, the first-order differential of the pseudo- abc abc abc abc abc abc abc P ; P ; Q ; Q ; P ; Q , and I are irrelevant to measurement equations can be formulated to have PQ ij ji ij ji i i ij the bus voltage U and phase angle h in our model and the decoupled properties as (19) and (20). u Jacobian matrix can be divided into three blocks arranged DP ij u/ u / u/ u / / / ¼ ðb cosðh  h Þ g sinðh  h ÞÞðDh  Dh Þ in diagonal form. Therefore, the calculation procedure of u ij i j ij i j i j /2fa;b;cg the proposed FDSE is mainly composed of the following ð19Þ three steps. Step 1: Estimates of branch active and reactive powers DQ ij u/ u / u/ u / / / ¼ ðg sinðh  h Þ b cosðh  h ÞÞðDU  DU Þ u ij i j ij i j i j are first calculated as: /2fa;b;cg 2 3 abc ðP Þ ð20Þ ij 6 7 abc 6 7 ðP Þ ji 6 7 In addition, in a three-phase distribution system, some 2 3 6 7 m abc abc 6 7 P ðQ Þ other approximations listed in (21) can be made. ij ij 6 7 6 7 8 6 7 m abc abc 6 7 a b o 6 ðQ Þ 7 h  h  120 6 ji 7 > ji 6 7 ¼ C6 7 ð25Þ 6 7 m abc a c o abc 6 7 h  h 120 ð21Þ 6 ðP Þ 7 ij i 4 5 6 7 b c o 6 abc 7 abc h  h 240 ðQ Þ Q 6 7 i ji 6 7 abc m 6 7 ðP Þ ij loss 4 5 As a result, (19) turns to be (22) and (20) is transformed abc m to be (23): ðQ Þ ij loss 2 3 2 3 a Dh i where C is the constant measurement Jacobian matrix for DP ij 6 7 6 a 7 6 7 branch active and reactive power measurements and cur- Dh 6 i 7 6 7 6 7 6 7 rent measurements. 6 7 6 Dh 7 DP ij 6 7 6 7 Then, the estimates of U and h can be obtained by ¼½A  A ð22Þ 6 7 6 7 Dh U j 6 7 6 7 conducting the fast decoupled iteration process. 6 7 6 7 c b 4 DP 5 6 7 Dh ij Step 2: The phase angles are first corrected by: 4 5 2 3 abc i Dh j Dðh Þ 6 7 abc 2 3 abc 2 3 4 5 ¼ B Dh ð26Þ DP 1 a DU i ij DQ ij abc 6 7 b U 6 a 7 DU 6 7 6 i 7 6 7 6 7 6 7 6 7 DU Step 3: The voltages are corrected by: DQ 6 7 ij 6 7 ¼½AA6 7 ð23Þ 2 3 6 7 b m abc 6 DU 7 U j 6 7 DðU Þ 6 7 i 6 7 c 6 b 7 6 7 abc 4 DQ 5 abc DU ij 4 5 ¼ B DU ð27Þ j 4 DQ 5 2 ij abc i DU where B and B are the constant measurement Jacobian where A in (22) is a constant matrix of size 3  3: 1 2 matrixes for pseudo-active power measurements. This procedure continues to convergence. The detail of the iteration procedure is similar to that reported previously [1]. In step 1, the constant Jacobian matrix C relates the active and reactive power state variables to active and 123 1416 Junwei YANG et al. reactive power measurements. The elements of C are To perform sufficient analysis on the R-Capped-L1 determined by (9)–(14). C is relatively independent and the model and the fast decoupled method for the three-phase relationship between measurements and state variables is distribution network, we conducted case studies from four linear. Thus, there is no need to calculate iteratively. aspects: abc abc abc abc abc abc abc P ; P ; Q ; Q ; P ; Q and I can be estimated ij ji ij ji i i ij 1) Impact of the dual parameter k on the accuracy of the directly in one calculation. R-Capped-L1 model. In Step 2, the constant Jacobian matrix B relates the 2) Efficiency comparison of FDSE with Newton base SE phase angle state variables to phase angle measurements in [16] and R-Capped-L1 with Capped-L1. and pseudo-active power measurements. B is formulated 3) Performance comparison with the traditional WLS by the Jacobian matrix ½A; A as shown in (22) repre- model, robust WLAV model, and sparse recovery senting the relationship between the incremental of pseudo- model L1-R. active power measurements and incremental of phase angle 4) Impact of imbalance in the three-phase loads. state variables, and unit matrixes representing the rela- In these cases, the WLAV model for comparison is a tionship between the incremental of phase angle measure- traditional robust model that performs superiorly in sup- ments and the incremental of phase angle state variables. pressing bad data. We compared our sparse recovery In Step 3, the constant Jacobian matrix is B relates the Capped-L1 model with this widely accepted robust model voltage state variables to voltage measurements and to verify the efficiency and precision of our model. The pseudo-reactive power measurements. B is formulated by formulation of the WLAV model is as follows: the Jacobian matrix ½A; A as shown in (23) representing the relationship between the incremental of pseudo-reac- min JðyÞ¼ jj w ðz  hðy ÞÞ ð30Þ i i i tive power measurements and incremental of phase angle state variables, and unit matrixes representing the rela- th where w is the weight associated with i measurement. tionship between the incremental of voltage measurements For convenience, we simply considered the unweighted and incremental of voltage state variables. situation and ignored the differences in precision between measurements. Thus, the WLAV model can be simply described as : 4 Numerical tests min JðxÞ¼kk y  hðxÞ ð31Þ To verify the effectiveness of the R-Capped-L1 SE Similarly, the weights in the WLS model were ignored model with fast decoupled solutions in a three-phase dis- and the unweighted model was adopted. tribution network, the method was programmed in Comparisons between these two unweighted and sparse MATLAB and tested on three distribution networks: an recovery models will be discussed next. IEEE 33-bus distribution network, an IEEE 123-bus dis- tribution network, and a 615-bus distribution network 4.1 Impact of dual parameter on accuracy of R- spliced by 5 IEEE 123-bus systems. Capped-L1 model The standard deviation of the errors corresponding to bad data o is 50 times that of the noises e, and the standard Similar to other Lagrangian optimization problems, k is deviation of noises e is given by: a crucial parameter that largely influences the precision of r ¼ 0:001 jj z ð28Þ the solutions. To maximize the precision of the sparse mean recovery SE models and detect the impact of the dual where z is the mean of the per-unit measurements. mean parameter, we scanned k across a range for the L1-R and We measure the estimate errors by resistance of the R-Capped-L1 models (identical to Capped-L1). The opti- branch R ¼kk y ^ y , where y ^ is the estimated value and true mal value of k for different test cases is provided by case ðlÞ y is the true value, and assign a in Fig. 1 to be (29), true study, which is important for the application of sparse which is relatively reasonable for the simulation method. recovery models. Table 1 shows the optimal value of k for ðlÞ ðlÞ ðlÞ a ¼ f ðo Þ¼ meanðoÞð29Þ the three test feeders in cases of bad data proportions (BD) of 0.06 and 0.1. To maintain a situation close to the practical scenario, Figures 2, 3 and 4 show the estimation errors against k 30% of nodes and branches were selected randomly for for the R-Capped-L1 model, the L1-R model, and the installation by PMU, and voltages, phase angles, and robust WLAV model. As a result of the independence of currents on these nodes and branches are part of the WLAV from k, the WLAV trendline is horizontal. How- measurement vector. ever, both the R-Capped-L1 and the L1-R models are 123 A sparse recovery model with fast decoupled solution… 1417 Table 1 Optimal value of k for tested cases 0.25 Case BD = 0.06 BD = 0.10 0.02 L1-R R-Capped-L1 L1-R R-Capped-L1 0.15 IEEE 33-bus 0.36 0.42 0.22 0.36 0.01 IEEE 123-bus 0.14 0.14 0.14 0.28 0.05 615-bus 0.06 0.09 0.06 0.13 0 0.05 0.10 0.15 0.20 0.012 (a) BD=0.06 0.011 0.010 0.07 0.009 0.06 0.008 0.05 0.007 0.04 0.006 0.005 0.03 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.02 0.01 (a) BD=0.06 00 0.05 .10 0.15 0.20 0.06 0.05 (b) BD=0.10 0.04 R-Capped-L1; L1-R; WLAV 0.03 Fig. 4 Estimation error versus k in 615-bus system 0.02 0.01 Table 2 Performance of two SE models using NB or FD methods in 0.15 0.20 0.25 0.30 0.35 0.40 0.45 IEEE 33-bus system (b) BD=0.10 SE method Iteration CPU time (s) Error R-Capped-L1; L1-R; WLAV Capped-L1 with NBSE 13 2.1893 0.0075 Fig. 2 Estimation error versus k in IEEE 33-bus system R-Capped-L1 with NBSE 13 1.9164 0.0075 Capped-L1 with FDSE 24 0.5235 0.0075 R-Capped-L1 with FDSE 24 0.4500 0.0075 0.030 0.025 0.020 sensitive to dual parameter k and the trendlines have 0.015 extreme values. At extreme points, the SE precision of the 0.010 R-Capped-L1 model is highest, and the WLAV model is 0.005 the most inaccurate. Thereby, in the following case studies, the optimal values of k were applied to test the perfor- 0 0.1 0.2 0.3 0.4 mance of the sparse recovery SE models. (a) BD=0.06 4.2 Efficiency comparison 0.045 To reduce the calculation time of the Capped-L1 model, 0.030 the fast decoupled method for three-phase DSSE was adopted and a transformation strategy was applied. To verify 0.015 the efficiency of the FDSE method on the R-Capped-L1 model, the test results comparing the computational speed of 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 the four SE methods are shown in Tables 2, 3 and 4. Bad data proportions were 0.06 for all simulations. The calculation (b) BD=0.10 times were measured by CPU time. All results in the R-Capped-L1; L1-R; WLAV Fig. 3 Estimation error versus k in IEEE 123-bus system R R R R R 1418 Junwei YANG et al. Table 3 Performance of two SE models using NB or FD methods in Table 5 Performance of four SE models in IEEE 33-bus system IEEE123 bus system SE model BD Iteration CPU time (s) Error SE method Iteration CPU time (s) Error WLS 0.06 5 0.7562 0.0610 Capped-L1 with NBSE 12 4.1487 0.0076 0.10 6 0.7073 0.1475 R-Capped-L1 with NBSE 12 2.9389 0.0076 WLAV 0.06 4 0.6162 0.0092 Capped-L1 with FDSE 16 0.9003 0.0076 0.10 4 0.5928 0.0108 R-Capped-L1 with FDSE 17 0.5671 0.0076 L1-R 0.06 4 0.7823 0.0084 0.10 4 0.7139 0.0103 R-Capped-L1 0.06 22 0.4228 0.0075 Table 4 Performance of two SE models using NB or FD methods in 0.10 18 0.3215 0.0080 615-bus system SE method Iteration CPU time (s) Error Table 6 performance of four SE models for IEEE 123-bus system SE model BD Iteration CPU time (s) Error Capped-L1 with NBSE 12 19.848 0.0084 R-Capped-L1 with NBSE 12 13.089 0.0084 WLS 0.06 4 0.9298 0.1125 Capped-L1 with FDSE 18 4.4106 0.0084 0.10 4 0.8961 0.1679 R-Capped-L1 with FDSE 18 2.3406 0.0084 WLAV 0.06 4 0.9008 0.0093 0.10 4 0.8890 0.0106 L1-R 0.06 4 1.1236 0.0090 tables are the average of 20 trials in which the locations of 0.10 4 1.1354 0.0105 bad data, the errors, and the noises were randomized. R-Capped-L1 0.06 19 0.6100 0.0083 From the results, we can conclude that the proposed FDSE 0.10 18 0.5304 0.0090 is more efficient. On one hand, although the Jacobian matrix of the FDSE was enlarged by introducing additional state Table 7 Performance of four SE models in 615-bus system abc abc abc abc variables ðP ; P ; Q ; Q Þ, this matrix is very sparse ij ji ij ji SE model BD Iteration CPU time (s) Error and the function between these variables and measurements WLS 0.06 5 7.8428 0.2266 is linear with no need to iterate in the calculation process. On 0.10 4 8.0753 0.3200 the other hand, despite the fewer iterations required for WLAV 0.06 4 7.3726 0.0122 NBSE, the Jacobian matrix has to be reformulated at every 0.10 4 6.9826 0.0143 iteration, whereas it only needs to be formulated initially for L1-R 0.06 4 8.3318 0.0119 FDSE. Thus, the FDSE method maintains its efficiency. Additionally, the validity of the transformation strategy 0.10 4 8.2323 0.0141 in (6) is proved. The process transforming the original R-Capped-L1 0.06 18 2.5835 0.0074 Capped-L1 to a formulation without absolute value helps 0.10 18 2.8137 0.0131 decrease the computational burden. It is important to note that the calculation errors for the four methods are very close because all the methods met Table 8 Sum of recovered sparse errors and noises in IEEE 33-bus system compared with different pre-set values the same constraints and were solved by the same sparse abc abc recovery model (the solution for Capped-L1 is the same as SE model U h that for R-Capped-L1). Pre-set 0.0265 0.0371 WLS 0.0228 0.0322 4.3 Performance comparison with WLS, WLAV WLAV 0.0262 0.0302 and L1-R models L1-R 0.0259 0.0364 R-Capped-L1 0.0263 0.0371 In this section, test results comparing the performance of the R-Capped-L1 model (with fast decoupled solutions) with the traditional WLS model, the WLAV model, and the that compares the recovered sparse errors of bad data via L1-R model are listed. The dual parameter k was chosen to the proposed R-Capped-L1 model, the traditional WLS be optimal as shown in Table 1 and all results are the model, the WLAV model and the L1-R model with the pre- average for 20 trials. Tables 5, 6 and 7 show the estimation set bad data values. Please note that since the WLS model errors, iterations, and CPU times of each model for the and the WLAV model cannot distinguish between noises three distribution systems. Table 8 lists a specific example and errors, for convenience of comparison, the values listed 123 A sparse recovery model with fast decoupled solution… 1419 Table 9 Iterations and CPU times for R-Capped-L1 model of three- 5 Conclusion phase imbalances in IEEE 33-bus system In this paper, Capped-L1 was first introduced into DSSE Load ratio FDSE NBSE (a:b:c) and a revision, denoted R-Capped-L1, was proposed to Average CPU time Iteration CPU time improve the computational efficiency. In addition, a novel iteration (s) (s) three-phase FDSE model for distribution networks was 1:1:1 23.9 0.4167 13.6 2.0194 adopted, which could also accelerate the solution procedure 1:0.8:1.2 23.4 0.4479 13.6 2.8714 for R-Capped-L1. Numerical tests were conducted to 1:0.6:1.4 22.3 0.4602 13.6 2.4223 analyze the performance of the proposed R-Capped-L1 1:0.4:1.6 21.9 0.4470 13.3 2.7062 model and it was shown to have advantages in both com- 1:0.2:1.8 22.0 0.4837 13.7 2.7539 putational efficiency and compressing bad data. Furthermore, the transformation strategy shown in (6) can be applied to the L1-R model to accelerate the calcu- in the table are the sums of them and since the numbers of lation process, and the R-Capped-L1 model with fast sparse errors for the tested systems are very large even if decoupled solutions can also be used in solving the SE bad data proportion is 0.06, only the recovered errors and problem for transmission networks. The optimal placement noises of bus voltage and phase angle measurements with of PMUs to enhance accuracy of state estimation is another larger amplitude are listed in the table. interesting topic but probably beyond the scope within this It can be discerned from the results that the R-Capped- single paper. Also, spatial decomposition into multi-area L1 model has the highest precision and efficiency of all SE merits further investigation. These issues will be three cases, but it also has the largest number of iterations regarded as future works. because the process of solving an R-Capped-L1 model is a multi-stage issue, with each stage exactly solving a convex Acknowledgements This work was supported in part by the National optimization problem (each stage is similar to the L1-R Key Research and Development Plan of China (No. 2018YFB0904200) and in part by the National Natural Science model). Although the number of iterations for solving the Foundation of China (No. 51725703). R-Capped-L1 model is larger, its computational speed is still the fastest because of the application of the FDSE Open Access This article is distributed under the terms of the method and the relaxation step in (6). Furthermore, as the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted BD increases from 0.06 to 0.1, all SE models except for use, distribution, and reproduction in any medium, provided you give WLS show good performance in compressing bad data. appropriate credit to the original author(s) and the source, provide a The R-Capped-L1 model was more robust than the other link to the Creative Commons license, and indicate if changes were models. Please note that the errors here are different made. because different models are adopted. 4.4 Impact of imbalance in three-phase loads Appendix A: Review of complex per unit Finally, we consider the impact of imbalance in the normalization three-phase loads. On the basis of the original IEEE 33-bus system, the ratio among the load of each phase was mul- To adjust the R/X in distribution networks, the param- eters and variables are normalized on a complex per unit tiplied by a coefficient to explore the effect of three-phase basis. The main idea is that by adjusting the phase angle in imbalance on the iteration times of the FDSE method. the base of complex impedance, the R and X in SE cal- Table 9 lists the number of iterations and the CPU time of culation can be normalized into a small R/X ratio. The FDSE and NBSE for the R-Capped-L1 models. The bad complex volt-ampere base is adopted, that is [20]: data rate was 0.1. As shown in Table 9, the iteration of both FDSE and j/ _  _  base S ¼ S e ðA1Þ base base NBSE for R-Capped-L1 are almost constant as the load ratio changes. The imbalance in the three-phase distribu- where S is the complex power base; / is the base base base tion system has little impact on the iteration process. angle. The voltage base is given by: Actually, a similar conclusion has already been discussed elsewhere [20] and it was concluded that the iterative times j0 _ _ V ¼ V e ¼ V ðA2Þ base base base for FDPF depend on R/X and the power factor to some degree, but not the imbalance in the loads. where V is the complex voltage base. base 123 1420 Junwei YANG et al. decision and control and European control conference, Orlando, According to (A1) and (A2), the impendence base will USA, 12–15 December 2011, 6 pp also be complex: [9] Davenport MA, Duarte MF, Eldar YC et al. (2011) Introduction V to compressed sensing. http://www.cems.uvm.edu/*gmirchan/ base j/ _  _  base Z ¼ ¼ Z e ðA3Þ base base classes/EE275/2015/Projects/CompressedSensing/Comp_ base Smpling_Rice.pdf [10] Zhang T (2010) Analysis of multi-stage convex relaxation for The complex normalized value of impendences can be sparse regularization. J Mach Learn Res 11:1081–1107 determined by: [11] Yang J, Wu W, Zheng W et al (2016) Performance analysis of sparse recovery models for bad data detection and state esti- jh R þ jX Z e X X jð/ þhÞ mation in electric power networks. In: Proceedings of IEEE PES _ _ base Z ¼ ¼ ¼ Z e ðA4Þ cpu pu j/ _ _ base general meeting, Boston, USA, 17–21 July 2016, 5 pp Z Z e base base [12] Stephen B, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge Then, the ratio of X/R can be expressed as (A5), which [13] Pau M, Pegoraro PA, Sulis S (2013) Efficient branch-current- can be adjusted by changing / . base based distribution system state estimation including synchro- cpu nized measurements. IEEE Trans Instrum Meas ¼ tanð/ þ hÞðA5Þ base 62(9):2419–2429 cpu [14] Chakrabarti S, Kyriakides E, Ledwich G et al (2010) Inclusion of PMU current phasor measurements in a power system state The determination of the base angle is achieved by: estimator. IET Gener Transm Distrib 4(10):1104–1115 a þ c p avg avg [15] Baran ME, Kelley AW (1995) A branch-current-based state / ¼  ð1 þ eÞðA6Þ base estimation for distribution systems. IEEE Trans Power Syst 2 2 10(1):483–491 [16] Lu CN, Teng JH, Liu WHE (1995) Distribution system state where a ¼ arctanðX =R Þ n; n is the total avg i i estimation. IEEE Trans Power Syst 10(1):229–240 i¼1 [17] Wang H, Schulz NN (2004) A revised branch current-based number of branches. distribution system state estimation algorithm and meter placement impact. IEEE Trans Power Syst 19(1):207–213 c ¼½maxðarctanðX=RÞÞ þ minðarctanðX=RÞÞ=2; avg [18] Wu W, Ju Y, Zhang B et al (2012) A distribution system state estimator accommodating large number of ampere measure- ments. Int J Electr Power Energy Syst 43(1):839–848 e ¼ 1  cosðarctanðQ =P ÞÞ=l: i i [19] Costa AS, Albuquerque A, Bez D (2013) An estimation fusion i¼1 method for including phasor measurements into power system real-time modeling. IEEE Trans Power Syst 28(2):1910–1920 With the complex per unit normalization, the fast [20] Tortelli OL, Lourenc¸o EM, Garcia AV et al (2015) Fast decoupled algorithm can be used for state estimation for decoupled power flow to emerging distribution systems via distribution networks with a high R/X ratio. complex pu normalization. IEEE Trans Power Syst 30(3):1351–1358 [21] Portelinha RK, Tortelli OL (2015) Three phase fast decoupled power flow for emerging distribution systems. In: Proceedings References of IEEE PES innovative smart grid technologies Latin America Montevideo, Uruguay, 5–7 October 2015, 6 pp [1] Primadianto A, Lu CN (2017) A review on distribution system [22] Luenberger DG, Ye Y (2015) Basic properties of linear pro- state estimation. IEEE Trans Power Syst 32(5):3875–3883 grams. Linear Nonlinear Program 116:11–31 [2] Rousseeuw PJ, Leroy AM (2005) Robust regression and outlier detection. Wiley, Hoboken Junwei YANG received the M.Sc degree in Tsinghua University, [3] Kotiuga WW, Vidyasagar M (1982) Bad data rejection prop- Beijing, China, in 2017. Her research interests state estimation and erties of Weughted least absolute value techniques applied to security and stability analysis of power grid. static state estimation. IEEE Trans Power Appar Syst 101(4):844–853 [4] Go¨l M, Abur A (2014) LAV based robust state estimation for Wenchuan WU received the B.S., M.S., and Ph.D. degrees from the systems measured by PMUs. IEEE Trans Smart Grid Electrical Engineering Department, Tsinghua University, Beijing, 5(4):1808–1814 China. He is currently a Professor in the Department of Electrical [5] Chilard O, Grenard S (2013) Detection of measurements errors Engineering of Tsinghua University. His research interests include with a distribution network state estimation function. In: Pro- Energy Management System, active distribution system operation and nd ceedings of 22 international conference and exhibition on control, and EMTP-TSA hybrid real-time simulation. Prof. Wu is an electricity distribution, Stockholm, Sweden, 10–13 June 2013, 4 IET Fellow, Distinguished Young Scholar supported by the National pp Science Fund, associate editor of IET Generation, Transmission and [6] Chen Y, Liu F, Mei S et al (2015) A robust WLAV state esti- Distribution, Electric Power Components and Systems. mation using optimal transformations. IEEE Trans Power Syst 30(4):2190–2191 Weiye ZHENG received the B.S. and Ph.D. degrees from the [7] Candes EJ, Randall PA (2008) Highly robust error correction by Department of Electrical Engineering, Tsinghua University, Beijing, convex programming. IEEE Trans Inf Theory 54(7):2829–2840 China. He received the excellent graduate with thesis award from [8] Xu W, Wang M, Tang A (2011) On state estimation with bad Tsinghua University in 2013 and Outstanding Ph.D. of Beijing City in th data detection. In: Proceedings of 50 IEEE conference on 123 A sparse recovery model with fast decoupled solution… 1421 2018, respectively. His research interests lie in the areas of distributed to University of Toronto for a year. In 2015, Dr. Yuntao Ju joined optimization in smart grids and renewable energy management. china electric power research institute as a research fellow. He is now an associate professor in College of Information and Electrical Engineering, China Agricultural University (CAU), Beijing, China. Yuntao JU received his B.Sc in mechanical engineering in 2008 and His research interests include hybrid energy system modeling, high his Ph.D. degree in electrical engineering in 2013, all from Tsinghua speed dynamic simulation, large scale system parameter identifica- University. He was an awardee of excellent graduates from Tsinghua tion, state estimation and uncertainty optimization. University in 2008. In November 2013, He has been a visiting scholar http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Modern Power Systems and Clean Energy Springer Journals

A sparse recovery model with fast decoupled solution for distribution state estimation and its performance analysis

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Springer Journals
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Copyright © 2019 by The Author(s)
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Energy; Energy Systems; Renewable and Green Energy; Power Electronics, Electrical Machines and Networks
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10.1007/s40565-019-0522-9
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J. Mod. Power Syst. Clean Energy (2019) 7(6):1411–1421 https://doi.org/10.1007/s40565-019-0522-9 A sparse recovery model with fast decoupled solution for distribution state estimation and its performance analysis 1 1 2 Junwei YANG , Wenchuan WU , Weiye ZHENG , Yuntao JU Abstract This paper introduces a robust sparse recovery 1 Introduction model for compressing bad data and state estimation (SE), based on a revised multi-stage convex relaxation (R-Cap- State estimation (SE) is a fundamental module in energy ped-L1) model. To improve the calculation efficiency, a management systems (EMSs) and its key task is to provide fast decoupled solution is adopted. The proposed method estimates of state variables which are as accurate as pos- can be used for three-phase unbalanced distribution net- sible. For distribution networks, a robust and efficient works with both phasor measurement unit and remote distribution state estimator assists in integrated operation terminal unit measurements. The robustness and the com- with distributed energy resources, assures power quality putational efficiency of the R-Capped-L1 model with fast levels, and improves the reliability of a power system [1]. decoupled solution are compared with some popular SE A number of distribution system state estimation (DSSE) methods by numerical tests on several three-phase distri- methodologies have been proposed based on different state bution networks. variables, treatments for load data and bad data, and measurements. Weighted least square (WLS) estimators Keywords Distribution system, State estimation, Sparse form the basis of the most popular methods. To suppress recovery, Fast decoupled method the influence of bad data, some robust estimators, such as least median of square (LMS) estimators [2], weighted least absolute value (WLAV) estimators [3, 4], and M-es- timators [5], have been proposed. To mitigate the effect of CrossCheck date: 15 January 2019 leverage points and improve the applicability of WLAV, a Received: 26 September 2018 / Accepted: 15 January 2019 / Published WLAV estimation with optimal transformations (WLAV- online: 9 May 2019 OT) that systematically solves the problem of computing The Author(s) 2019 rotation angles and scaling factors was proposed [6]. Most & Wenchuan WU robust estimators consider the measurement residual as a wuwench@tsinghua.edu.cn whole and minimize the value of a penalty function of Junwei YANG residuals. However, the residuals usually consist of two yang-jw11@mails.tsinghua.edu.cn parts: observation noise and abnormally large measurement Weiye ZHENG errors caused by bad data, which obey different distribu- neo_adonis@139.com tions. The errors are normally sparse. Therefore, based on Yuntao JU the theory of compress sensing (CS), some sparse recovery juyuntao@cau.edu.cn models [7, 8] considering this point have been proposed to Department of Electrical Engineering, Tsinghua University, detect bad data. An L1-relaxation (L1-R) model con- Beijing 100084, China straining the sparse vector by the L1-norm instead of the Department of Electrical and Electronic Engineering, The L0-norm, which is used to denote the number of non-zero University of Hong Kong, Hong Kong, China values in a sparse vector, was proposed [8]. Relaxing the L0-norm problem to the L1-norm problem is a common College of Information and Electrical Engineering, China Agricultural University, Beijing 100083, China 123 1412 Junwei YANG et al. practice in CS [9]. The effectiveness of the L1-R model has formulate the three-phase SE problem with hybrid mea- been proved both mathematically and practically. How- surements. The branch current measurements are formu- ever, it has been reported that the relaxation process often lated as the branch active power and reactive power losses, leads to sub-optimal solutions [10]. To handle this issue allowing them to be incorporated into the FDSE model. mathematically, a multi-stage convex relaxation (Capped- Thus, the contributions of this paper can be summarized as L1) method was proposed [10]. We first introduced this follows: method for SE in a transmission power system in our 1) The Capped-L1 model is first introduced for DSSE, previous work [11], and found that Capped-L1 method has which has a powerful capacity to compress bad data. an advantage in precision but not in computational speed 2) A novel three-phase fast decoupled state estimation because iterations are required and because the optimiza- model with hybrid measurements for DSSE is adopted. tion problem contains both the L1-norm and the L2-norm, 3) A transformation strategy for Capped-L1 to which are relatively nonlinear compared to WLS and R-Capped-L1 is applied and the computational effi- WLAV. Actually, it is easier to solve a pure quadratic ciency is significantly improved. optimization problem than a mixed problem of square values and absolute values [12]. In this paper, efforts are made to transform the original Capped-L1 model to a quadratic optimization model. The efficiency of the revised 2 Proposed state estimation model Capped-L1 (R-Capped-L1) model is improved significantly. This section first reviews the formulation of the sparse Recently, to improve the monitoring of distribution L1-R model for state estimation that has been proposed in system operating conditions, some utilities have started previous work [8]. Based on this formulation, the proposed installing phasor measurement units (PMUs) at the distri- Capped-L1 model and the revised model are introduced. bution level. PMUs provide measurements of voltage phasor and current phasor with high frequency. Combining 2.1 Sparse L1-R model measurements from both remote terminal units (RTUs) and PMUs can promote the ability to perform state forecasting When bad data are presented in the measurements, the for the distribution network [13]. The inclusion of branch relationship between the measurements and the state vari- current and voltage measurements helps to improve the ables can be represented as [7]: precision but also introduces additional components to the z ¼ hðyÞþ o þ e ð1Þ solution procedure [14]. A novel branch current-based SE where z is the raw measurement vector; h(y) is a set of has been proposed [15], with a two-stage solution. In [16], measurement functions; o is the vector of errors corre- the active and reactive power measurements were trans- sponding to bad data; e is the noise vector. formed to linear complex current measurements based on Generally, the elements of e follow random Gaussian estimated phase angle and voltage. References [17] and distributions and are independent in most cases; the error [18] formulated the multi-source measurements SE prob- vector o is a sparse vector with few non-zero values. lem by extending the state variables. Reference [19] Mathematically, the L0-norm of a vector represents the combines the estimates independently obtained from number of its non-zero values. Therefore, the error vector supervisory control and data acquisition (SCADA)-based can be constrained with the L0-norm and the SE model can and PMU-based estimators based on multisensor data be formulated as an optimization problem: fusion theory. They all showed good performance in han- dling branch current measurements. However, owing to the min Jðy; oÞ¼kk z  hðyÞ o ð2Þ repeating factorization of the Jacobian matrix, these s:t: kk o  e methods suffer from a heavy computational burden. To resolve this, a fast decoupled power flow (FDPF) method where e is a positive small number related to the proportion for distribution networks by choosing a complex base of bad data. voltage and adjusting the ratio of R/X was also proposed Given a reasonable e, we can obtain relatively accurate [20, 21]. Its efficiency has been proved in a large number of estimates for state variables by solving (2). However, there case studies. The computational speed of DSSE is expected are two difficulties in solving this problem:  the value of to be faster when introducing this method. parameter e is difficult to specify because the proportion of In this paper, we introduce a robust Capped-L1 model bad data is not normally known; ` L0-norm minimization into DSSE. To reduce the computational burden, we has been proved to be an NP problem that cannot be solved transform the original model to a quadratic form and apply efficiently [9]. a novel fast decoupled state estimation (FDSE) method to 123 A sparse recovery model with fast decoupled solution… 1413 For , according to the duality theory, the solution to (2) corresponds to the solution of the problem (2) for some Lagrange dual variable k  0: min Jðy; oÞ¼kk z  hðyÞ o þkkk o ð3Þ 2 0 For `, according to the theory of CS [9], L1-norm minimization helps obtain sparse solutions. Thereby, the L0-norm in (3) can be relaxed to an L1-norm problem. For this step, the original non-convex sparse mini- mization problem has been evolved into a convex opti- mization problem given by (4), which can be solved efficiently: min Jðy; oÞ¼kk z  hðyÞ o þkkk o ð4Þ 2 1 Fig. 1 Iterative procedure of Capped-L1 The SE model above is exactly the L1-R model process continues until the two error vectors of adjacent previously proposed in [8], which is a typical sparse iteration steps become relatively close. recovery model. However, the Capped-L1 model (5) is an extremely nonlinear model with a fairly heavy computational burden. 2.2 Capped-L1 model and transformation strategy To increase the calculation efficiency, the following transformation strategy can be adopted: Convex relaxation such as L1-R indeed solves the L0- ðlÞ ðlÞ ðlÞ norm minimization problem efficiently under some condi- ðlÞ ðlÞ ðlÞ ðlÞ > min Jðy ; o Þ¼ z  hðy Þ o þk c a þ b i i i > 2 tions. However, it often leads to a sub-optimal solution in > ðlÞ reality. To obtain better solutions than the L1-R model, a s:t: a  0 new model Capped-L1 was proposed [10] and we introduce > ðlÞ b  0 this model into the DSSE problem to handle the issues with > ðlÞ ðlÞ ðlÞ a  b  o ¼ 0 a sparse error vector. The Capped-L1 model can be for- i i i mulated as: ð6Þ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ min Jðy where a and b represent auxiliary variables sharing the ; o Þ¼ z  hðy Þ o þk c o i i i i ðlÞ same dimension with o . ð5Þ This strategy transforming the optimization problem (5)to ðlþ1Þ  ðlÞ ðlÞ (6) is actually relaxing the absolute value, and the relaxation where c ¼ Ið o  a Þ is a relaxation parameter, IðÞ i i method has been proved to be valid mathematically [22]. is a step function with a value of 0 or 1, l is the number of Solutions to (6) are exactly the same as those to (5); thereby, ðlÞ iterations, a is a threshold value changing with the the precisions of the two Capped-L1 models are identical. ðlÞ  ðlÞ ðlÞ decision variable o . For this problem, if o  a , the i i However, through the relaxation process, the computational ðlþ1Þ speed of the Capped-L1 model will be significantly improved value of c will be 1; otherwise, the value will be 0. because each iteration of (6) is actually a quadratic opti- In the procedure of solving the problem, the iterations mization problem, which is relatively easily solved. The are needed as shown in Fig. 1. iteration procedure for solving (6) is the same as that of (5), but ðlÞ In the iteration procedure, f ðo Þ is a function of each iteration saves a lot of time thanks to the linearization. ðlÞ ðlÞ ðlÞ ðlÞ o whose value meets o  f ðo Þ o . Generally, the max min By adopting this R-Capped-L1 model, a robust SE model function can be chosen as the average value or the median with a lighter computational burden can be developed. ðlÞ ðlÞ of vector o . The threshold a changes in each iteration. e is a small threshold value to determine whether or not the iteration has been convergent. 3 Fast decoupled model for three-phase Observing the iteration procedure, it can be seen that distribution networks each iteration exactly solves a convex optimization prob- ðlÞ lem. Once the new sparse error vector o is given from the The fast decoupled method has been applied in trans- ðlÞ last iteration, the method studies the new sparse vector o mission networks for a long time and its efficiency has ðlþ1Þ been proved by substantial practice. However, resulting to adjust the relaxation parameter c and solve a new from the large R/X in distribution networks, the optimization problem to obtain better solutions. This 123 1414 Junwei YANG et al. ‘‘decoupled’’ idea fails for almost all feeders. Recently, to where j 2 i indicates that j is connected to i. handle this problem, a fast decoupled algorithm via com- plex per unit (pu) normalization for distribution networks 3) Three-phase branch current measurements: has been proposed [20]. To be more readable, a briefly u / / u /u / u /u I I ðcosðh  h Þr þ sinðh  h Þx Þ ij ij ij ij ij ij ij ij review of the complex pu normalization algorithm is given /2fa;b;cg ð13Þ in Appendix A. In this section, we will introduce a novel u u u u u ¼ P þ P þ o þ e ¼ P ij ji P P ji;loss ji;loss ji;loss three-phase fast decoupled model to estimate state vari- u / / u /u / u /u ables for unbalanced three-phase distribution networks. To I I ðcosðh  h Þx  sinðh  h Þr Þ ij ij ij ij ij ij ij ij take the branch current measurements into consideration, /2fa;b;cg ð14Þ the branch active power and reactive power losses are u u u u u ¼ Q þ Q þ o þ e ¼ Q ij ji Q Q ij;loss ij;loss ij;loss employed. Using the fast decoupled method, the calcula- /u tion efficiency of the R-Capped-L1 sparse recovery model where / and u represent the phases; r is the mutual resis- ij will be further improved, enabling good handling of the /u tance of branch ij between phase u and phase /; x is the ij branch current measurements. mutual reactance of branch ij between phaseu and phase/. In the DSSE problem, the measurement vector has to be extended as (7) and the state variable vector can be rep- 4) Three-phase bus voltage and phase angle resented as (18): measurements: hi 2 3 2 3 2 3 2 3 a a abc abc abc abc abc abc abc abc abc a a o e ðU Þ U U U z ¼ U ; h ; P ; P ; Q ; Q ; P ; Q ; I ð7Þ i i i i i i ij ji ij ji i i ij 6 7 6 7 6 7 6 7 b m b 6 7 6 7 o b e b hi 4 ðU Þ 5 ¼ 4 U 5 þ þ ð15Þ U U i i 4 5 4 5 i i abc abc abc abc abc abc y¼ U ; h ; P ; P ; Q ; Q ð8Þ c m c i i ij ji ij ji ðU Þ U c c o e i i U U i i 2 3 2 3 2 3 2 3 where U is the bus voltage; h is the phase angle; P is the a m a a a o e ðh Þ h h h i i i i active power of branches or injections; Q is the reactive 6 7 6 7 6 7 6 7 b b 6 7 6 7 ¼ þ o b þ e b ð16Þ 4 ðh Þ 5 4 h 5 h h power of branches or injections; I is the magnitude of i i 4 5 4 5 i i c c ðh Þ h c c branch current; the superscript abc denotes the three phases o e i i h h i i of the variables; the superscript ij denotes that the variable flows from bus i to bus j; the superscript i denotes bus i. The measurement function h(y) relating z and 5) Pseudo-measurements formulating network y includes: constraints: 1) Three-phase real and reactive power measurements of P ij / u / u/ u / u/ ¼ U ðcosðh  h Þg þ sinðh  h Þb Þ j i j ij i j ij the branch: /2fa;b;cg 2 3 2 3 2 3 2 3 a m a a a X o e ðP Þ P P P ij ij ij ij / u / u/ u / u/ U ðcosðh  h Þg þ sinðh  h Þb Þ 6 7 6 7 6 7 6 7 i i i ij i i ij b m b 6 7 6 7 6 7 6 7 o b e b ðP Þ P ¼ þ þ ð9Þ P P /2fa;b;cg ij ij 4 5 4 5 4 ij 5 4 ij 5 c c ðP Þ P c c ð17Þ o e P P ij ij ij ij 2 3 2 3 2 3 2 3 m Q a a a a ij / u / u/ u / u/ o e ðQ Þ Q Q Q ij ij ij ij ¼ U ðsinðh  h Þg  cosðh  h Þb Þ j i j ij i j ij 6 7 6 7 6 7 6 7 b b i 6 7 6 7 6 7 6 7 /2fa;b;cg o b e b ðQ Þ ¼ Q þ þ ð10Þ Q Q ij ij 4 5 4 5 4 ij 5 4 ij 5 X / u / u/ u / u/ c m c U ðsinðh  h Þg  cosðh  h Þb Þ c c ðQ Þ Q o e i i i ij i i ij ij ij Q Q ij ij /2fa;b;cg where the superscript m denotes the measurement ð18Þ quantities. u/ where g is the mutual conductance of branch ij between ij 2) Three-phase injection power measurements: u/ 2 3 2 3 2 3 2 3 a phase u and phase /; b is the mutual susceptance of a a ij P o e ðP Þ P P ij i i i 6 7 6 7 6 7 branch ij between phase u and phase /. 6 7 m b 6 7 6 7 6 7 ¼¼ P þ o b þ e b ð11Þ 4 ðP Þ 5 P P ij i 4 5 4 i 5 4 i 5 The fast decoupled method can improve the efficiency of m j2i ðP Þ c c P o e P P this DSSE problem. Fast decoupled state estimation depends ij i i 2 3 2 3 2 3 on the PQ decoupled formulation of the measurement 2 3 a m a a Q o e ðQ Þ Q Q ij i i i equations. Clearly, the real and reactive power measurement 6 7 6 7 6 7 6 7 m b 6 7 6 7 6 7 o b e b ¼ Q þ þ ð12Þ (9) and (10), the injection power measurement (11) and (12), 4 ðQ Þ 5 Q Q i ij 4 5 4 i 5 4 i 5 m j2i c c the branch current measurement (13) and (14), and the bus ðQ Þ Q c c o e i Q Q ij i i 123 A sparse recovery model with fast decoupled solution… 1415 2 3 pffiffiffi pffiffiffi voltage and phase angle measurement (15) and (16) can all be 1 3 1 3 aa ab ab ac ac b  b  g  b þ g 6 7 ij ij ij ij ij expressed in a PQ decoupled formulation. 2 2 2 2 6 7 6 7 pffiffiffi pffiffiffi The network constraints (17) and (18) involve both volt- 6 7 1 3 1 3 6 ab ab bb bc bc 7 A ¼ b þ g b  b  g 6 7 age U and phase angle h. As reviewed above, in a distribution ij ij ij ij ij 2 2 2 2 6 7 6 7 pffiffiffi pffiffiffi network, the normalized R/X can be adjusted by choosing an 4 5 1 3 1 3 ac ac bc bc cc appropriate complex base value and thus original R/X value  b  g  b þ g b ij ij ij ij ij 2 2 2 2 is not required to be small to adapt the proposed formulation. ð24Þ Once normalized R/X is small, U has little impact on the active power, and h has little impact on the reactive power Finally, all of the elements in the Jacobian matrix of h(y) are constant. It is important to note that and U  1. Hence, the first-order differential of the pseudo- abc abc abc abc abc abc abc P ; P ; Q ; Q ; P ; Q , and I are irrelevant to measurement equations can be formulated to have PQ ij ji ij ji i i ij the bus voltage U and phase angle h in our model and the decoupled properties as (19) and (20). u Jacobian matrix can be divided into three blocks arranged DP ij u/ u / u/ u / / / ¼ ðb cosðh  h Þ g sinðh  h ÞÞðDh  Dh Þ in diagonal form. Therefore, the calculation procedure of u ij i j ij i j i j /2fa;b;cg the proposed FDSE is mainly composed of the following ð19Þ three steps. Step 1: Estimates of branch active and reactive powers DQ ij u/ u / u/ u / / / ¼ ðg sinðh  h Þ b cosðh  h ÞÞðDU  DU Þ u ij i j ij i j i j are first calculated as: /2fa;b;cg 2 3 abc ðP Þ ð20Þ ij 6 7 abc 6 7 ðP Þ ji 6 7 In addition, in a three-phase distribution system, some 2 3 6 7 m abc abc 6 7 P ðQ Þ other approximations listed in (21) can be made. ij ij 6 7 6 7 8 6 7 m abc abc 6 7 a b o 6 ðQ Þ 7 h  h  120 6 ji 7 > ji 6 7 ¼ C6 7 ð25Þ 6 7 m abc a c o abc 6 7 h  h 120 ð21Þ 6 ðP Þ 7 ij i 4 5 6 7 b c o 6 abc 7 abc h  h 240 ðQ Þ Q 6 7 i ji 6 7 abc m 6 7 ðP Þ ij loss 4 5 As a result, (19) turns to be (22) and (20) is transformed abc m to be (23): ðQ Þ ij loss 2 3 2 3 a Dh i where C is the constant measurement Jacobian matrix for DP ij 6 7 6 a 7 6 7 branch active and reactive power measurements and cur- Dh 6 i 7 6 7 6 7 6 7 rent measurements. 6 7 6 Dh 7 DP ij 6 7 6 7 Then, the estimates of U and h can be obtained by ¼½A  A ð22Þ 6 7 6 7 Dh U j 6 7 6 7 conducting the fast decoupled iteration process. 6 7 6 7 c b 4 DP 5 6 7 Dh ij Step 2: The phase angles are first corrected by: 4 5 2 3 abc i Dh j Dðh Þ 6 7 abc 2 3 abc 2 3 4 5 ¼ B Dh ð26Þ DP 1 a DU i ij DQ ij abc 6 7 b U 6 a 7 DU 6 7 6 i 7 6 7 6 7 6 7 6 7 DU Step 3: The voltages are corrected by: DQ 6 7 ij 6 7 ¼½AA6 7 ð23Þ 2 3 6 7 b m abc 6 DU 7 U j 6 7 DðU Þ 6 7 i 6 7 c 6 b 7 6 7 abc 4 DQ 5 abc DU ij 4 5 ¼ B DU ð27Þ j 4 DQ 5 2 ij abc i DU where B and B are the constant measurement Jacobian where A in (22) is a constant matrix of size 3  3: 1 2 matrixes for pseudo-active power measurements. This procedure continues to convergence. The detail of the iteration procedure is similar to that reported previously [1]. In step 1, the constant Jacobian matrix C relates the active and reactive power state variables to active and 123 1416 Junwei YANG et al. reactive power measurements. The elements of C are To perform sufficient analysis on the R-Capped-L1 determined by (9)–(14). C is relatively independent and the model and the fast decoupled method for the three-phase relationship between measurements and state variables is distribution network, we conducted case studies from four linear. Thus, there is no need to calculate iteratively. aspects: abc abc abc abc abc abc abc P ; P ; Q ; Q ; P ; Q and I can be estimated ij ji ij ji i i ij 1) Impact of the dual parameter k on the accuracy of the directly in one calculation. R-Capped-L1 model. In Step 2, the constant Jacobian matrix B relates the 2) Efficiency comparison of FDSE with Newton base SE phase angle state variables to phase angle measurements in [16] and R-Capped-L1 with Capped-L1. and pseudo-active power measurements. B is formulated 3) Performance comparison with the traditional WLS by the Jacobian matrix ½A; A as shown in (22) repre- model, robust WLAV model, and sparse recovery senting the relationship between the incremental of pseudo- model L1-R. active power measurements and incremental of phase angle 4) Impact of imbalance in the three-phase loads. state variables, and unit matrixes representing the rela- In these cases, the WLAV model for comparison is a tionship between the incremental of phase angle measure- traditional robust model that performs superiorly in sup- ments and the incremental of phase angle state variables. pressing bad data. We compared our sparse recovery In Step 3, the constant Jacobian matrix is B relates the Capped-L1 model with this widely accepted robust model voltage state variables to voltage measurements and to verify the efficiency and precision of our model. The pseudo-reactive power measurements. B is formulated by formulation of the WLAV model is as follows: the Jacobian matrix ½A; A as shown in (23) representing the relationship between the incremental of pseudo-reac- min JðyÞ¼ jj w ðz  hðy ÞÞ ð30Þ i i i tive power measurements and incremental of phase angle state variables, and unit matrixes representing the rela- th where w is the weight associated with i measurement. tionship between the incremental of voltage measurements For convenience, we simply considered the unweighted and incremental of voltage state variables. situation and ignored the differences in precision between measurements. Thus, the WLAV model can be simply described as : 4 Numerical tests min JðxÞ¼kk y  hðxÞ ð31Þ To verify the effectiveness of the R-Capped-L1 SE Similarly, the weights in the WLS model were ignored model with fast decoupled solutions in a three-phase dis- and the unweighted model was adopted. tribution network, the method was programmed in Comparisons between these two unweighted and sparse MATLAB and tested on three distribution networks: an recovery models will be discussed next. IEEE 33-bus distribution network, an IEEE 123-bus dis- tribution network, and a 615-bus distribution network 4.1 Impact of dual parameter on accuracy of R- spliced by 5 IEEE 123-bus systems. Capped-L1 model The standard deviation of the errors corresponding to bad data o is 50 times that of the noises e, and the standard Similar to other Lagrangian optimization problems, k is deviation of noises e is given by: a crucial parameter that largely influences the precision of r ¼ 0:001 jj z ð28Þ the solutions. To maximize the precision of the sparse mean recovery SE models and detect the impact of the dual where z is the mean of the per-unit measurements. mean parameter, we scanned k across a range for the L1-R and We measure the estimate errors by resistance of the R-Capped-L1 models (identical to Capped-L1). The opti- branch R ¼kk y ^ y , where y ^ is the estimated value and true mal value of k for different test cases is provided by case ðlÞ y is the true value, and assign a in Fig. 1 to be (29), true study, which is important for the application of sparse which is relatively reasonable for the simulation method. recovery models. Table 1 shows the optimal value of k for ðlÞ ðlÞ ðlÞ a ¼ f ðo Þ¼ meanðoÞð29Þ the three test feeders in cases of bad data proportions (BD) of 0.06 and 0.1. To maintain a situation close to the practical scenario, Figures 2, 3 and 4 show the estimation errors against k 30% of nodes and branches were selected randomly for for the R-Capped-L1 model, the L1-R model, and the installation by PMU, and voltages, phase angles, and robust WLAV model. As a result of the independence of currents on these nodes and branches are part of the WLAV from k, the WLAV trendline is horizontal. How- measurement vector. ever, both the R-Capped-L1 and the L1-R models are 123 A sparse recovery model with fast decoupled solution… 1417 Table 1 Optimal value of k for tested cases 0.25 Case BD = 0.06 BD = 0.10 0.02 L1-R R-Capped-L1 L1-R R-Capped-L1 0.15 IEEE 33-bus 0.36 0.42 0.22 0.36 0.01 IEEE 123-bus 0.14 0.14 0.14 0.28 0.05 615-bus 0.06 0.09 0.06 0.13 0 0.05 0.10 0.15 0.20 0.012 (a) BD=0.06 0.011 0.010 0.07 0.009 0.06 0.008 0.05 0.007 0.04 0.006 0.005 0.03 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.02 0.01 (a) BD=0.06 00 0.05 .10 0.15 0.20 0.06 0.05 (b) BD=0.10 0.04 R-Capped-L1; L1-R; WLAV 0.03 Fig. 4 Estimation error versus k in 615-bus system 0.02 0.01 Table 2 Performance of two SE models using NB or FD methods in 0.15 0.20 0.25 0.30 0.35 0.40 0.45 IEEE 33-bus system (b) BD=0.10 SE method Iteration CPU time (s) Error R-Capped-L1; L1-R; WLAV Capped-L1 with NBSE 13 2.1893 0.0075 Fig. 2 Estimation error versus k in IEEE 33-bus system R-Capped-L1 with NBSE 13 1.9164 0.0075 Capped-L1 with FDSE 24 0.5235 0.0075 R-Capped-L1 with FDSE 24 0.4500 0.0075 0.030 0.025 0.020 sensitive to dual parameter k and the trendlines have 0.015 extreme values. At extreme points, the SE precision of the 0.010 R-Capped-L1 model is highest, and the WLAV model is 0.005 the most inaccurate. Thereby, in the following case studies, the optimal values of k were applied to test the perfor- 0 0.1 0.2 0.3 0.4 mance of the sparse recovery SE models. (a) BD=0.06 4.2 Efficiency comparison 0.045 To reduce the calculation time of the Capped-L1 model, 0.030 the fast decoupled method for three-phase DSSE was adopted and a transformation strategy was applied. To verify 0.015 the efficiency of the FDSE method on the R-Capped-L1 model, the test results comparing the computational speed of 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 the four SE methods are shown in Tables 2, 3 and 4. Bad data proportions were 0.06 for all simulations. The calculation (b) BD=0.10 times were measured by CPU time. All results in the R-Capped-L1; L1-R; WLAV Fig. 3 Estimation error versus k in IEEE 123-bus system R R R R R 1418 Junwei YANG et al. Table 3 Performance of two SE models using NB or FD methods in Table 5 Performance of four SE models in IEEE 33-bus system IEEE123 bus system SE model BD Iteration CPU time (s) Error SE method Iteration CPU time (s) Error WLS 0.06 5 0.7562 0.0610 Capped-L1 with NBSE 12 4.1487 0.0076 0.10 6 0.7073 0.1475 R-Capped-L1 with NBSE 12 2.9389 0.0076 WLAV 0.06 4 0.6162 0.0092 Capped-L1 with FDSE 16 0.9003 0.0076 0.10 4 0.5928 0.0108 R-Capped-L1 with FDSE 17 0.5671 0.0076 L1-R 0.06 4 0.7823 0.0084 0.10 4 0.7139 0.0103 R-Capped-L1 0.06 22 0.4228 0.0075 Table 4 Performance of two SE models using NB or FD methods in 0.10 18 0.3215 0.0080 615-bus system SE method Iteration CPU time (s) Error Table 6 performance of four SE models for IEEE 123-bus system SE model BD Iteration CPU time (s) Error Capped-L1 with NBSE 12 19.848 0.0084 R-Capped-L1 with NBSE 12 13.089 0.0084 WLS 0.06 4 0.9298 0.1125 Capped-L1 with FDSE 18 4.4106 0.0084 0.10 4 0.8961 0.1679 R-Capped-L1 with FDSE 18 2.3406 0.0084 WLAV 0.06 4 0.9008 0.0093 0.10 4 0.8890 0.0106 L1-R 0.06 4 1.1236 0.0090 tables are the average of 20 trials in which the locations of 0.10 4 1.1354 0.0105 bad data, the errors, and the noises were randomized. R-Capped-L1 0.06 19 0.6100 0.0083 From the results, we can conclude that the proposed FDSE 0.10 18 0.5304 0.0090 is more efficient. On one hand, although the Jacobian matrix of the FDSE was enlarged by introducing additional state Table 7 Performance of four SE models in 615-bus system abc abc abc abc variables ðP ; P ; Q ; Q Þ, this matrix is very sparse ij ji ij ji SE model BD Iteration CPU time (s) Error and the function between these variables and measurements WLS 0.06 5 7.8428 0.2266 is linear with no need to iterate in the calculation process. On 0.10 4 8.0753 0.3200 the other hand, despite the fewer iterations required for WLAV 0.06 4 7.3726 0.0122 NBSE, the Jacobian matrix has to be reformulated at every 0.10 4 6.9826 0.0143 iteration, whereas it only needs to be formulated initially for L1-R 0.06 4 8.3318 0.0119 FDSE. Thus, the FDSE method maintains its efficiency. Additionally, the validity of the transformation strategy 0.10 4 8.2323 0.0141 in (6) is proved. The process transforming the original R-Capped-L1 0.06 18 2.5835 0.0074 Capped-L1 to a formulation without absolute value helps 0.10 18 2.8137 0.0131 decrease the computational burden. It is important to note that the calculation errors for the four methods are very close because all the methods met Table 8 Sum of recovered sparse errors and noises in IEEE 33-bus system compared with different pre-set values the same constraints and were solved by the same sparse abc abc recovery model (the solution for Capped-L1 is the same as SE model U h that for R-Capped-L1). Pre-set 0.0265 0.0371 WLS 0.0228 0.0322 4.3 Performance comparison with WLS, WLAV WLAV 0.0262 0.0302 and L1-R models L1-R 0.0259 0.0364 R-Capped-L1 0.0263 0.0371 In this section, test results comparing the performance of the R-Capped-L1 model (with fast decoupled solutions) with the traditional WLS model, the WLAV model, and the that compares the recovered sparse errors of bad data via L1-R model are listed. The dual parameter k was chosen to the proposed R-Capped-L1 model, the traditional WLS be optimal as shown in Table 1 and all results are the model, the WLAV model and the L1-R model with the pre- average for 20 trials. Tables 5, 6 and 7 show the estimation set bad data values. Please note that since the WLS model errors, iterations, and CPU times of each model for the and the WLAV model cannot distinguish between noises three distribution systems. Table 8 lists a specific example and errors, for convenience of comparison, the values listed 123 A sparse recovery model with fast decoupled solution… 1419 Table 9 Iterations and CPU times for R-Capped-L1 model of three- 5 Conclusion phase imbalances in IEEE 33-bus system In this paper, Capped-L1 was first introduced into DSSE Load ratio FDSE NBSE (a:b:c) and a revision, denoted R-Capped-L1, was proposed to Average CPU time Iteration CPU time improve the computational efficiency. In addition, a novel iteration (s) (s) three-phase FDSE model for distribution networks was 1:1:1 23.9 0.4167 13.6 2.0194 adopted, which could also accelerate the solution procedure 1:0.8:1.2 23.4 0.4479 13.6 2.8714 for R-Capped-L1. Numerical tests were conducted to 1:0.6:1.4 22.3 0.4602 13.6 2.4223 analyze the performance of the proposed R-Capped-L1 1:0.4:1.6 21.9 0.4470 13.3 2.7062 model and it was shown to have advantages in both com- 1:0.2:1.8 22.0 0.4837 13.7 2.7539 putational efficiency and compressing bad data. Furthermore, the transformation strategy shown in (6) can be applied to the L1-R model to accelerate the calcu- in the table are the sums of them and since the numbers of lation process, and the R-Capped-L1 model with fast sparse errors for the tested systems are very large even if decoupled solutions can also be used in solving the SE bad data proportion is 0.06, only the recovered errors and problem for transmission networks. The optimal placement noises of bus voltage and phase angle measurements with of PMUs to enhance accuracy of state estimation is another larger amplitude are listed in the table. interesting topic but probably beyond the scope within this It can be discerned from the results that the R-Capped- single paper. Also, spatial decomposition into multi-area L1 model has the highest precision and efficiency of all SE merits further investigation. These issues will be three cases, but it also has the largest number of iterations regarded as future works. because the process of solving an R-Capped-L1 model is a multi-stage issue, with each stage exactly solving a convex Acknowledgements This work was supported in part by the National optimization problem (each stage is similar to the L1-R Key Research and Development Plan of China (No. 2018YFB0904200) and in part by the National Natural Science model). Although the number of iterations for solving the Foundation of China (No. 51725703). R-Capped-L1 model is larger, its computational speed is still the fastest because of the application of the FDSE Open Access This article is distributed under the terms of the method and the relaxation step in (6). Furthermore, as the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted BD increases from 0.06 to 0.1, all SE models except for use, distribution, and reproduction in any medium, provided you give WLS show good performance in compressing bad data. appropriate credit to the original author(s) and the source, provide a The R-Capped-L1 model was more robust than the other link to the Creative Commons license, and indicate if changes were models. Please note that the errors here are different made. because different models are adopted. 4.4 Impact of imbalance in three-phase loads Appendix A: Review of complex per unit Finally, we consider the impact of imbalance in the normalization three-phase loads. On the basis of the original IEEE 33-bus system, the ratio among the load of each phase was mul- To adjust the R/X in distribution networks, the param- eters and variables are normalized on a complex per unit tiplied by a coefficient to explore the effect of three-phase basis. The main idea is that by adjusting the phase angle in imbalance on the iteration times of the FDSE method. the base of complex impedance, the R and X in SE cal- Table 9 lists the number of iterations and the CPU time of culation can be normalized into a small R/X ratio. The FDSE and NBSE for the R-Capped-L1 models. The bad complex volt-ampere base is adopted, that is [20]: data rate was 0.1. As shown in Table 9, the iteration of both FDSE and j/ _  _  base S ¼ S e ðA1Þ base base NBSE for R-Capped-L1 are almost constant as the load ratio changes. The imbalance in the three-phase distribu- where S is the complex power base; / is the base base base tion system has little impact on the iteration process. angle. The voltage base is given by: Actually, a similar conclusion has already been discussed elsewhere [20] and it was concluded that the iterative times j0 _ _ V ¼ V e ¼ V ðA2Þ base base base for FDPF depend on R/X and the power factor to some degree, but not the imbalance in the loads. where V is the complex voltage base. base 123 1420 Junwei YANG et al. decision and control and European control conference, Orlando, According to (A1) and (A2), the impendence base will USA, 12–15 December 2011, 6 pp also be complex: [9] Davenport MA, Duarte MF, Eldar YC et al. (2011) Introduction V to compressed sensing. http://www.cems.uvm.edu/*gmirchan/ base j/ _  _  base Z ¼ ¼ Z e ðA3Þ base base classes/EE275/2015/Projects/CompressedSensing/Comp_ base Smpling_Rice.pdf [10] Zhang T (2010) Analysis of multi-stage convex relaxation for The complex normalized value of impendences can be sparse regularization. J Mach Learn Res 11:1081–1107 determined by: [11] Yang J, Wu W, Zheng W et al (2016) Performance analysis of sparse recovery models for bad data detection and state esti- jh R þ jX Z e X X jð/ þhÞ mation in electric power networks. In: Proceedings of IEEE PES _ _ base Z ¼ ¼ ¼ Z e ðA4Þ cpu pu j/ _ _ base general meeting, Boston, USA, 17–21 July 2016, 5 pp Z Z e base base [12] Stephen B, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge Then, the ratio of X/R can be expressed as (A5), which [13] Pau M, Pegoraro PA, Sulis S (2013) Efficient branch-current- can be adjusted by changing / . base based distribution system state estimation including synchro- cpu nized measurements. IEEE Trans Instrum Meas ¼ tanð/ þ hÞðA5Þ base 62(9):2419–2429 cpu [14] Chakrabarti S, Kyriakides E, Ledwich G et al (2010) Inclusion of PMU current phasor measurements in a power system state The determination of the base angle is achieved by: estimator. IET Gener Transm Distrib 4(10):1104–1115 a þ c p avg avg [15] Baran ME, Kelley AW (1995) A branch-current-based state / ¼  ð1 þ eÞðA6Þ base estimation for distribution systems. IEEE Trans Power Syst 2 2 10(1):483–491 [16] Lu CN, Teng JH, Liu WHE (1995) Distribution system state where a ¼ arctanðX =R Þ n; n is the total avg i i estimation. IEEE Trans Power Syst 10(1):229–240 i¼1 [17] Wang H, Schulz NN (2004) A revised branch current-based number of branches. distribution system state estimation algorithm and meter placement impact. IEEE Trans Power Syst 19(1):207–213 c ¼½maxðarctanðX=RÞÞ þ minðarctanðX=RÞÞ=2; avg [18] Wu W, Ju Y, Zhang B et al (2012) A distribution system state estimator accommodating large number of ampere measure- ments. Int J Electr Power Energy Syst 43(1):839–848 e ¼ 1  cosðarctanðQ =P ÞÞ=l: i i [19] Costa AS, Albuquerque A, Bez D (2013) An estimation fusion i¼1 method for including phasor measurements into power system real-time modeling. IEEE Trans Power Syst 28(2):1910–1920 With the complex per unit normalization, the fast [20] Tortelli OL, Lourenc¸o EM, Garcia AV et al (2015) Fast decoupled algorithm can be used for state estimation for decoupled power flow to emerging distribution systems via distribution networks with a high R/X ratio. complex pu normalization. IEEE Trans Power Syst 30(3):1351–1358 [21] Portelinha RK, Tortelli OL (2015) Three phase fast decoupled power flow for emerging distribution systems. In: Proceedings References of IEEE PES innovative smart grid technologies Latin America Montevideo, Uruguay, 5–7 October 2015, 6 pp [1] Primadianto A, Lu CN (2017) A review on distribution system [22] Luenberger DG, Ye Y (2015) Basic properties of linear pro- state estimation. IEEE Trans Power Syst 32(5):3875–3883 grams. Linear Nonlinear Program 116:11–31 [2] Rousseeuw PJ, Leroy AM (2005) Robust regression and outlier detection. Wiley, Hoboken Junwei YANG received the M.Sc degree in Tsinghua University, [3] Kotiuga WW, Vidyasagar M (1982) Bad data rejection prop- Beijing, China, in 2017. Her research interests state estimation and erties of Weughted least absolute value techniques applied to security and stability analysis of power grid. static state estimation. IEEE Trans Power Appar Syst 101(4):844–853 [4] Go¨l M, Abur A (2014) LAV based robust state estimation for Wenchuan WU received the B.S., M.S., and Ph.D. degrees from the systems measured by PMUs. IEEE Trans Smart Grid Electrical Engineering Department, Tsinghua University, Beijing, 5(4):1808–1814 China. He is currently a Professor in the Department of Electrical [5] Chilard O, Grenard S (2013) Detection of measurements errors Engineering of Tsinghua University. His research interests include with a distribution network state estimation function. In: Pro- Energy Management System, active distribution system operation and nd ceedings of 22 international conference and exhibition on control, and EMTP-TSA hybrid real-time simulation. Prof. Wu is an electricity distribution, Stockholm, Sweden, 10–13 June 2013, 4 IET Fellow, Distinguished Young Scholar supported by the National pp Science Fund, associate editor of IET Generation, Transmission and [6] Chen Y, Liu F, Mei S et al (2015) A robust WLAV state esti- Distribution, Electric Power Components and Systems. mation using optimal transformations. IEEE Trans Power Syst 30(4):2190–2191 Weiye ZHENG received the B.S. and Ph.D. degrees from the [7] Candes EJ, Randall PA (2008) Highly robust error correction by Department of Electrical Engineering, Tsinghua University, Beijing, convex programming. IEEE Trans Inf Theory 54(7):2829–2840 China. He received the excellent graduate with thesis award from [8] Xu W, Wang M, Tang A (2011) On state estimation with bad Tsinghua University in 2013 and Outstanding Ph.D. of Beijing City in th data detection. In: Proceedings of 50 IEEE conference on 123 A sparse recovery model with fast decoupled solution… 1421 2018, respectively. His research interests lie in the areas of distributed to University of Toronto for a year. In 2015, Dr. Yuntao Ju joined optimization in smart grids and renewable energy management. china electric power research institute as a research fellow. He is now an associate professor in College of Information and Electrical Engineering, China Agricultural University (CAU), Beijing, China. Yuntao JU received his B.Sc in mechanical engineering in 2008 and His research interests include hybrid energy system modeling, high his Ph.D. degree in electrical engineering in 2013, all from Tsinghua speed dynamic simulation, large scale system parameter identifica- University. He was an awardee of excellent graduates from Tsinghua tion, state estimation and uncertainty optimization. University in 2008. In November 2013, He has been a visiting scholar

Journal

Journal of Modern Power Systems and Clean Energy – Springer Journals

Published: May 9, 2019

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