A new teaching-learning-based optimization algorithm for distribution system state estimation

Abstract

Distribution system state estimation (DSSE) is of vital importance to the monitoring and control of recent active distribution networks. This paper proposes a new teaching-learning-based optimization algorithm (TLBO) for estimating state variables of radial distribution systems. In this method, the state variables are estimated by minimizing the sum of weighted squared errors considering either real or pseudo measurements. TLBO is a successful recently-proposed optimization technique that simulates the educational system in a classroom. In this paper, an effective mutation has been incorporated into original TLBO algorithm to evade trapping in local minima and develop search process. Nevertheless, developing the search process does not considerably lessen the speed of the algorithm. So the proposed method is an efficient algorithm for DSSE. Finally, the proposed method is studied on three radial distribution test systems. The numerical results have been depicted to demonstrate the efficiency and accuracy of the method for solving DSSE.

Keywords

Distribution system state estimation teaching-learning-based optimization algorithm (TLBO)single objective optimization algorithm

1 Introduction

Distribution networks have seen major changes throughout the last couple of years, due largely to the advent of various kinds of renewable energy sources and new technologies such as storage devices and plug-in hybrid electric vehicles [1]. Furthermore, recent distribution networks should be compatible with demand side involvement [2]. These changes present new challenges to the system operators for proper monitoring and optimal control of active distribution systems. Reliable information of state variables is needed to conduct many important analyses associated with efficiency and reliability such as security monitoring, volt/var control and feeder reconfiguration. Thus distribution system state estimation (DSSE) is an essential requirement for monitoring and optimal control of modern distribution systems.

Weighted least squares (WLS) is the usual method for power system state estimation [3, 4]. WLS minimizes the error of estimation considering the accuracy of measurement devices. DSSE has to provide a dependable background database for monitoring and optimal control of distribution systems.

Several papers extend WLS methods to be suitable for distribution systems. State variables are estimated based on the raw measurements collected by supervisory control and data acquisition system (SCADA). [5] introduced a method for state estimation of distribution network with the least number of real measurement data. A three phase voltage node state estimator for load data forecasting in distribution systems was suggested in [6]. Distinguishing features of distribution systems such as unbalanced networks and paucity of real-time measurement data were considered in the method. In some papers, branch current components are considered as the state variables. Current based three phase state estimators were proposed in [7, 8] which reduce the computational burden of DSSE in comparison with bus voltage based methods because the three phase branch current equations were solved independently with reasonable accuracy. [9, 10] develop the methods later. An excellent literature review of the DSSE methods is provided in [11, 12].

The collected measurements in distribution systems are insufficient for DSSE. So, distribution systems are often unobservable. The problem is exacerbated whenever some of the real measurements are incorrect. To deal with the shortage of real-time measurement data in distribution systems, load forecasting schemes or load tracking methods can be used to generate pseudo measurements [13, 14]. The details are beyond the scope of this paper.

Another issue in DSSE is the computational efficiency and robustness of the state estimator. In distribution systems, some devices such as static var compensators, voltage control devices and DGs make the objective function of WLS method nonlinear. Therefore evolutionary algorithms can be better choices for solving DSSE in preference to mathematical methods because evolutionary algorithms have the ability to cope with nonlinear and difficult objective functions [15 –17] suggested an evolutionary algorithm for DSSE. A hybrid particle swarm optimization algorithm was employed to estimate output power of loads and DGs as the state variables through minimizing the difference between measured and obtained values. [18] proposed a method based on ant colony optimization for DSSE. In [19], a three phase distribution state estimator was proposed based on Nelder–Mead and particle swarm optimization algorithms. Distribution system equipment which makes the problem nonlinear and discrete, was considered in themethod.

Teaching-learning based optimization algorithm is a new optimization approach that imitates educational mechanism in a classroom [20]. Recently, several researchers have begun using TLBO for their purposes [21 –24]. It has been shown that TLBO algorithm is a powerful and high speed technique to solve difficult and nonlinear optimization problems.

The main aim of this paper is to develop a reliable algorithm for DSSE based on TLBO algorithm. In addition, a new wavelet-based mutation has been incorporated into the original TLBO algorithm. This mutation operand develops the search process to find the best estimates of the state variables and help the algorithm evade trapping in local minima. In the proposed approach, it is assumed that active power of variable loads is state variable. The objective function will be the sum of squared deviation between measured and estimated values. In other words, the objective function of the problem will be to minimize the sum of weighted squared errors of either real or pseudo measurements. It is important to remark that the average and standard deviation of active power of inconstant loads are used for generating pseudo measurements.

This paper is organized as follows: Distribution system state estimation is explained in Section 2. The modified teaching learning optimization algorithm (MTLBO) is described in Section 3. In Section 4, the proposed method is implemented for DSSE. Section 5 indicates the results of the proposed algorithm and lastly Section 6 draws the conclusion.

2 Distribution system state estimation

The mathematical representation of DSSE is discussed in this section. According to Fig. ??, host computer in SCADA assembles all information received from the telemetered devices in the distribution system. Since the raw measurements don’t often include all the states of the system and they may also contain errors, DSSE should be performed to achieve more accurate solutions before using this raw data for operation and control purposes such as voltage/current security monitoring, volt/var control and feeder reconfiguration. Since the distribution networks are often unobservable due to limited number of measurement devices, some pseudo measurements have to be generated in order to perform DSSE. Thus DSSE plays a vital role in monitoring and operation of modern distribution networks [12].

The mathematical equations for DSSE can be represented as a single-objective optimization problem with equality and inequality constraints. The objective function of the optimization problem is described as follows:

$M i n f (X) = \sum_{m} = 1_{m}^{N} ω_{m} (h_{m} (X) - z_{m})^{2}$ (1)

$\begin{array}{l} X = [{\bar{P}}_{L}]_{1} \times D \\ {\bar{P}}_{L} = [P_{L}^{1}, P_{L}^{2}, ..., P_{L}^{N}] \end{array}$

flalignX as the state vector, consists of active power of variable loads ( ${\bar{P}}_{L}$ ). z _m is the mth measurement and ω _m is the corresponding coefficient that shows the accuracy of the measurement device. N _m and N _L are the number of measurements and variable loads, respectively. h _m is a nonlinear function which is related to network topology. In this paper, values of ω _m for real measurements and pseudo measurements are 1,000 and 0.1, respectively. It is important to highlight that mean and standard deviation of the active power of variable loads are assumed as pseudo measurements. In this study, D is the dimension of the state vector which is equal to the number of variable loads ( ${\bar{P}}_{L}$ ).

The objective of the above formulation is to find the state variables or estimate the vector X so that the squared difference between h _m (X) and z _m, for all m = 1, 2, …, N _m, reduces to a minimum. In other words, the state variables will be estimated by minimizing the above objective function.

The following constraints make DSSE a constrained optimizing problem:

Tap of transformers

Bus voltage level

Active power values of loads

Distribution load flow

3 Teaching-learning-based optimization algorithm

TLBO is a recently proposed meta-heuristic that imitates a successful and dynamic educational strategy in a classroom [20]. Similar to most evolutionary algorithms, TLBO is a population-based algorithm. The population consists of some students and a teacher. The teacher is the most knowledgeable one in the population.

The main advantage of this algorithm over other evolutionary algorithms is that TLBO has no adjustable parameters, so there is no need to design a tuning mechanism for the parameters. The educational strategy of this algorithm includes both direct and interactive instruction. Actually, not only can the students be affected by their teacher, but also they can affect each other.

TLBO algorithm can be divided into two phases: 1-Teaching phase (direct instruction) 2-Learning phase (interactive instruction). In the teaching phase, the teacher provides information for all the students and the students learn from their teacher, while in the learning phase they can learn from each other and develop their skills. The pseudo code of TLBO algorithm is shown in Fig. 1.

3.1 Teaching phase

The best member is selected as the teacher in each iteration. The teacher trains his/her students. In practice the teacher can only improve the mean of the students’ knowledge. Students’ improvement depends on the students’ aptitude for learning. The knowledge of each student is changed according to the following equations: $Difference_Mean = rand (1, D) \times (T_new - TF \times M)$ (2) where the teaching factor (TF) is randomly selected from either 1 or 2. In each iteration, the best member (the member with the least objective function value) is called the teacher of the class (T _ new). rand (1, D) is a 1-by-D vector with random elements between 0 and 1. M is the mean of the current learners. Afterwards, the new solution (X _new,i) can be generated by the following equation: $X_{new, i} = X_{i} + Difference_mean$ (3) If X _new,i is a better solution in comparison with X _i, it will be accepted and replaced instead of the ith member.

3.2 Learning phase

As mentioned previously, students can also learn from each other. After learning from the teacher (teaching phase), the students affect their own knowledge by different means such as discussions, conferences and asking each other some questions. In this way, they can gain maximum benefit from the class. Each student’s knowledge can be changed by the new member obtained from Equations (4) or (5). $X_{new, i} = X_{j} + rand (1, D) \times (X_{j} - X_{K})$ (4) $X_{new, i} = X_{j} + rand (1, D) \times (X_{k} - X_{j})$ (5) where X _K and X _j are two different students selected randomly from the learners. If X _new,i is a better solution in comparison with X _i, it will be replaced instead of this member.

3.3 Mutation

Converging to local optimum solutions is one of the common challenges that most evolutionary methods face. Mutation is a successful method to tackle this problem. In order to prevent the algorithm becoming trapped in the local optimal points, a new effective wavelet-based modification is incorporated in this paper. [25] has presented a new particle swarm optimizing algorithm, using wavelet-based modification as the mutation operand. The new wavelet-based mutation is explained below:

Suppose that $X_{i} = [x_{i}^{1}, \dots, x_{i}^{j}, \dots, x_{i}^{D}]$ is the ith member of the population that should be improved by the mutation. Similarly, $X_{new, i} = [x_{new, i}^{1}, \dots, x_{new, i}^{j},$ $\dots, x_{new, i}^{D}]$ is the ith mutated member and $x_{WL, i}^{j}$ is the jth wavelet-based mutated element of the ith member. Let P be the mutation probability. Then, for each element of the member, if a generated random number is smaller than P, the mutation should change that element. We have $x_{new, i}^{j} = {\begin{matrix} x_{i}^{j} & rand \geq P \\ x_{WL, i}^{j} & rand < P \end{matrix}$ (6) and $x_{WL, i}^{j} = {\begin{matrix} x_{i}^{j} + σ \times (x_{\max}^{j} - x_{i}^{j}) & σ > 0 \\ x_{i}^{j} + σ \times (x_{i}^{j} - x_{\min}^{j}) & σ \leq 0 \end{matrix}$ (7) where $x_{max}^{j}$ and $x_{min}^{j}$ are the maximum and minimum limits of jth element, respectively. σ is defined by $σ = \frac{1}{\sqrt{a}} e^{- (\frac{φ}{a})^{2} / 2} \cos (\frac{5 φ}{a})$ (8) In this equation, φ is a random number in the interval of [-2.5a, 2.5a]. Note that the value of parameter a is increased as the iterations elapse: $a = e^{- Ln (g) \times {(1 - \frac{Iter}{Max_iterations})}^{ξ_{ω m}}} + Ln (g)$ (9) where Iter and Max _ iterations are the current iteration and the maximum number of iterations, respectively. In addition, ξ _ωm and g are the shape factor of the above function and the maximum range of a, respectively. More detailed information is available in [25].

In this paper, to improve this mutation function, we generate an improved element ( $x_{mut, i}^{j}$ ), then Equation (7) will be changed to $x_{WL, i}^{j} = {\begin{matrix} x_{mut, i}^{j} + σ \times (x_{\max}^{j} - x_{mut, i}^{j}) & σ > 0 \\ x_{mut, i}^{j} + σ \times (x_{mut, i}^{j} - x_{\min}^{j}) & σ \leq 0 \end{matrix}$ (10) where σ remains the same. $x_{mut, i}^{j}$ is obtained using differential evolution (DE) algorithm [26]. For all i = 1, …, N, let $x_{mut, i} = [x_{mut, i}^{1}, \dots, x_{mut, i}^{j}, \dots, x_{mut, i}^{D}]$ be an improved solution, and we have $x_{mut, i}^{j} = {\begin{matrix} x_{DE}^{j} & If Cr > rand \\ x_{i}^{j} & otherwise \end{matrix}$ (11) where Cr = 0.1 + rand × 0.8. Let $X_{DE} = [x_{DE}^{1},$ $x_{DE}^{2}, \dots, x_{DE}^{j}, \dots, x_{DE}^{D}]$ be a DE mutated solution which is made from three solutions (X _n, X _r, X _s) so that n ≠ r ≠ s. $X_{DE} = X_{n} + Cr \times (X_{r} - X_{s})$ (12) Finally, if the new solution (X _new,i) is a better solution in comparison with the ith learner (X _i), it is acceptable to replace it instead of that learner.

4 Solution methodology

This section provides a step by step procedure for the proposed algorithm. Figure 2 summarizes the proposed procedure for DSSE. In the proposed method, the values of variable loads are assumed as the state variables in preference to voltage and angle of the buses in the conventional methods. The subsequent steps should be followed to solve DSSE using the proposed algorithm.

Step 1: Assemble the required information.

Some data associated with distribution network should be available for DSSE. The status of the distribution lines, the values of real and pseudo measurements, the mean and standard deviation of variable loads should be clear.

Step 2: Including the constraints in the objective function.

Actually, DSSE should be solved within equality and inequality constraints. In order to consider the equality and inequality constraints, penalty factors are included in the objective function, thereby DSSE is converted into an unconstrained single objective optimizing problem. The objective function isrewritten as: $\begin{matrix} F (X) & = f (X) + k_{1} \sum_{f = 1}^{N_{eq}} J_{f} (X)^{2} \\ + k_{2} \sum_{h = 1}^{N_{ineq}} [\max (0, - g_{h} (X))]^{2} \end{matrix}$ (13) $\begin{matrix} J_{f} (X) = 0 f = 1, 2, \dots, N_{eq} \\ g_{h} (X) > 0 h = 1, 2, \dots, N_{ineq} \end{matrix}$ f (X) is the value of DSSE problem objective function in Equation (1). J _f (X) and g _h (X) are the equality and inequality constraints, respectively. N _eq and N _ineq is the number of equality inequality constraints, respectively. k ₁ and k ₂ are penalty factors that should be large numbers. As a consequence, any solution which violates these constraints will be eliminated by the minimizing algorithm.

Step 3: Create the preliminary members randomly for TLBO algorithm.

The preliminary population members are created randomly as follows: ${Pop}_{ini} = [\begin{matrix} {\bar{X}}_{1} \\ {\bar{X}}_{2} \\ ⋮ \\ {\bar{X}}_{N} \end{matrix}]$ (14)

${\bar{X}}_{i} = [x_{i}^{1}, \dots, x_{i}^{j}, \dots, x_{i}^{D}],$

$x_{i}^{j} = rand (P_{L, \max}^{j} - P_{L, \min}^{j}) + P_{L, \min}^{j},$

i = 1, 2, …, N, j = 1, 2, …, N _L

$P_{L, \max}^{j}$ and $P_{L, \min}^{j}$ are the upper and lower limits of the jth variable load, respectively.

Step 4: Evaluate objective function for each member of the population.

For each member, the objective function F (X) should be calculated to assess the effectiveness of that member. In order to include constraints, the distribution load flow (DLF) is performed using active power values of variable loads as the state variables. Afterwards the value of f (X) is calculated using Equation (1) and the constraints are evaluated based on the DLF results. Subsequently, the main objective function value can be obtained using Equation (13).

Step 5: Sort the population.

The population members are ranked based on their objective function values F (X).

Step 6: Select the best member as the teacher of the current iteration.

The member with the minimum value of objective function F (X) is the most knowledgeable member assumed as the teacher. Other members are the learners of TLBO algorithm.

Step 7: Learning from the teacher.

The teaching phase of the TLBO algorithm should be performed in this step. The learners’ knowledge is updated by the teacher, using Equations (2) to (3). For all i = 1, 2, …, N, the objective function F (X) of the new solution (X _new,i) should be calculated and if it is a better solution in comparison with the ith learner (X _i), it is necessary to replace it instead of that learner.

Step 8: Learning from other students.

For the sake of maximum efficiency in learning, the students interact with each other and share their knowledge, using Equations (4) and (5). For all i = 1, 2, …, N, the objective function F (X) of the new solution (X _new,i) should be calculated and if it is a better solution in comparison with the ith learner (X _i), it is necessary to replace it instead of that learner.

Step 9: Mutation.

In order to improve the algorithm, mutation should change the population as it was explained in Section 3.3. For all i = 1, 2, …, N, the objective function F (X) of the new solution (X _new,i) should be calculated and if it is a better solution in comparison with the ith learner (X _i), it should be accepted that it be replaced instead of that learner.

Step 10: Check the convergence criterion.

If the convergence criterion is met, the algorithm is finished and the member with minimum value of F (X) is the final estimated value for the state variables. Otherwise it is necessary to go back to step 5. This process continues until the convergence criterionis satisfied.

5 Simulation results and analyses

In this section, the performance of the proposed MTLBO is evaluated by studying five classical benchmark functions [27]. Afterwards the proposed method for DSSE is studied on three distribution test systems. Note that to implement the proposed DSSE method, it is assumed that all of the following conditions are available:

The active power values of fixed loads

Network topology

Bus voltage level

Measured quantitiesl

Three phase balance

In addition, standard deviation and average of variable loads are used as pseudo measurements. Power factor of variable loads is also assumed to be constant.

5.1 Experimental results for MTLBO algorithm

In this subsection, the performance of the proposed MTLBO algorithm is compared with improved artificial bee colony algorithm (I-ABC) [28] and improved TLBO algorithm (I-TLBO) [29] using five multimodal benchmark functions. These functions are Rosenblock, Schwefel, Rastrigin, Ackley and Griewank [27].

In this experiment, maximum number of function evaluations is 40000. The dimension of all of these functions is considered 30. The search range of each function is shown in Table 1.

The mean and the standard deviations (Std) of the function values obtained by the MTLBO, I-ABC and I-TLBO algorithms for 30 independent runs are presented in Table 1. According to this Table, the proposed MTLBO algorithm shows very good performance on Rosenbrock, Rastrigin, Ackley and Griewank functions. The performance of all these algorithms is relatively poor on Schwefel which is a highly complex function, especially in high dimensions. From the results presented in Table 1, it can be concluded that the MTLBO algorithm gives competitive performance in comparison with I-ABC andI-TLBO.

5.2 Test system 1

In the following, the results for IEEE 33-bus radial test feeder are expressed. Figure 3 shows the single line diagram of 12.66 kV radial IEEE 33-bus distribution test system. A detailed description of the test system is available in [30].

In this test system, it is assumed that six inconstant loads are connected to buses 2, 3, 6, 18, 22 and 25. In Table 2, the specifications of the inconstant loads are illustrated.

It is assumed that there are five measurement devices with the values obtained by distribution load flow. Furthermore, these devices are wattmeter placed on buses 19, 2, 5, 22 and 18.

In this paper, MTLBO is performed to estimate the state variables of the test system (active power of variable loads). Therefore, the voltages of all buses can be obtained, using distribution system power flow.

The best results of the proposed algorithm for state variables are compared with other methods in Tables 3–4. The maximum range of error between simulation and real values of loads for TLBO algorithm is 0.001 although this range of error for SA, GA, ABC, Bat, Frog and DE algorithms is 0.778, 2.655, 0.17, 0.222, 0.182 and 0.063, respectively. The simulation results demonstrate that the accuracy of the proposed algorithm is preferable to other algorithms.

Execution time is of paramount importance to online programs such as DSSE. Table 5 shows the required function evaluations, execution time, average and standard deviation for 30 independent runs of different algorithms. For the considered test system with 33 buses, the proposed method needs an execution time of about 15 (s) on an Intel i5 2.50-GHz CPU system (Windows 7 OS) with MATLAB R2011b.

The execution time of the proposed method is changed in the range of 15∼18 seconds and this is less in comparison with other evolutionary algorithms except SA algorithm. The execution time of the SA method is about 12∼15 seconds as SA has only one member to converge into the optimal solution. Although the execution time of SA is less than the proposed method, the proposed algorithm is more accurate and efficient than SA.

According to Table 5, the average value of error for the proposed algorithm is 3.809e-28 which is much less than the average error of other evolutionary algorithms. So the proposed method is much more accurate. In addition, the standard deviation of MTLBO algorithm, 8% , is less than others, therefore it can be concluded that the proposed method is more efficient in comparison with other evolutionary algorithms.

Figure 4 aims to appraise the performance of six different methods for estimating the state variables of the first test system. Each method is evaluated 30 times and then an average is obtained for comparison. It shows that MTLBO algorithm is more accurate and faster than other algorithms due to its improved search aptitude.

After 100 iterations, the error of estimation of MTLBO algorithm considerably reduces to approximately 1 × 10^-28 while Frog, Bat, DE, ABC and original TLBO algorithms reach around 1 × 10^-7, 1 × 10^-12, 1 × 10^-10, 1 × 10^-7 and 1 × 10^-16, respectively. Moreover, MTLBO algorithm reaches its final value in less than 100 iterations (about 60 iterations). In other words, its convergence speed is relatively high to find the global minimum.

5.3 Test system 2

In order to demonstrate the accuracy and efficiency of the proposed method, another distribution system is studied in this subsection. The second distribution test system is a three phase 11.4 kV distribution network with 83 buses and 11 feeders [31]. Single line diagram of the second test system is illustrated in Fig. 5.

Seventeen inconstant loads with constant power factors are assumed for this system connected to buses 6, 15, 21, 28, 33, 37, 40, 46, 51, 55, 61, 65, 69, 72, 76, 80 and 83. The necessary data for the inconstant loads has been shown in Table 6. In addition, it is assumed that thirteen measurement devices are available in this test system. The measurement devices are watt meters and voltmeters. The watt meters are installed on buses 18, 26, 34, 44, 49, 57, 70, 73 and 78 while the voltmeters are connected to buses 5, 12, 53 and 67.

Similar to the previous section, the best results of the proposed algorithm for state variables are compared with other algorithms in Table 7. Since the power factors of the variable loads are assumed to be constant, their estimated reactive power values can be calculated easily by using the estimated active power values.

From Table 7 it can be understood that SA and GA algorithms are trapped in local minimum in minimizing the objective function of the state estimation problem. In other words, the estimation contains a glaring error. Frog, DE, ABC and Bat algorithms put slightly more accurate estimates on state variables in comparison with SA and GA. However their performances are not acceptable for DSSE, because their results also have noticeable errors. The average error of each algorithm is shown in Table 8. On the other hand, MTLBO is more successful in estimating the state variables and can provide more accurate data for the state variables. From Table 7 we can conclude that the proposed method for DSSE contains minor error in comparison with other algorithms.

As it can be seen in Table 8, the average value of error for the proposed algorithm is 6.516e-10 which is less than the average error of other evolutionary algorithms. So the proposed method is more accurate in this test system. In addition, the standard deviation of the MTLBO, SA, GA, ABC, Bat, Frog and DE is 10, 30, 25, 20, 15, 15 and 15 percent, respectively. The standard deviation of MTLBO algorithm is less than others. So it can be concluded that the proposed method is more reliable. The evaluation results show that the proposed method is a powerful tool to put an accurate and reliable estimation on state variables of distribution systems.

5.4 Test system 3

In this subsection, a 167-bus radial test system is assessed to show the good convergence of the proposed method for larger test systems. This test system consists of two similar systems in Fig. 5. with one feeder. Therefore the number of state variables in this test system is twice as many as in the test system 2. The information of each network is the same as the system in Fig. 5.

According to Table 9, when the number of function evaluation is assumed to be 50000, the execution time is less than 70,000 and 100,000 number of function evaluations but the average of the error is more in comparison to them. So solving the problem with more number of function evaluations obtains more accurate solution. The simulation results presented in Table 9 indicate that the proposed method keeps its good performance in dealing with larger systems.

6 Conclusions

In this paper, a new method for distribution system state estimation was proposed based on TLBO algorithm. A new wave-let mutation had been incorporated into original TLBO algorithm to improve the performance of the algorithm. In this method, the active power of variable loads was considered as state variables instead of voltage and angle of the buses in the conventional methods. The proposed MTLBO algorithm is evaluated by studying five classical benchmark functions. Then the proposed method for DSSE was tested on three radial distribution test systems. The evaluation results show that the method is not only a confident and accurate tool but also a relatively high speed program for DSSE problem.

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