Power system static state estimation using JADE-adaptive differential evolution technique

Vedik Basetti; Ashwani Kumar Chandel; K. B. V. S. R. Subramanyam

doi:10.1007/s00500-017-2715-3

Soft Computing

pp 1–20 | Cite as

Power system static state estimation using JADE-adaptive differential evolution technique

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Vedik BasettiEmail author
Ashwani Kumar Chandel
K. B. V. S. R. Subramanyam

Methodologies and Application

First Online: 11 July 2017

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Abstract

Power system state estimation is the important module in monitoring the power system network. The state estimator provides states of the power system by processing the measurements placed optimally across the power system network. However, temporary un-observability may occur due to unexpected loss of measurement devices or communication links. In the present paper, the reliability of the static state estimation has been tested under three different scenarios, viz. over-determined, critically determined, and under-determined systems. The state estimation problem has been solved using JADE-adaptive differential evolution algorithm utilizing both the conventional and the synchronized phasor measurements. From the test results it has been determined that the proposed technique determines the solution even when the power system is partially observable and also detects the un-observable buses present in the network. The results thus obtained using JADE have been compared with conventional weighted least square and improved self-adaptive particle swarm optimization-based SE techniques. On the bases of various performance indices, the results show the effectiveness, reliability, and accuracy of the proposed algorithm when compared to other state estimation techniques.

Keywords

State estimation Power system Hybrid state estimator Phasor measurement unit Differential evolution

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Compliance with Ethical Standards

Conflicts of interest

The authors declare that there is no conflict of interest.

Human participants or Animals performed

This article does not contain any studies with human participants or animals performed by any of the authors.

Appendix A

Differential evolution is a stochastic optimization technique, and the solution obtained is dependent on the set of variables generated randomly by searching over different promising areas (Das and Suganthan 2011; Brest et al. 2006; Wang et al. 2011a). The process of estimating the states of un-observable buses has been explained with an example of IEEE 14-bus system as given below:

JADE uses a population P of size $N_P $ that consists of floating point encrypted chromosomes. Each chromosome $p_j $ is a vector that contains D decision variables. Similar to all other evolutionary algorithms (EAs), JADE begins with randomly generating chromosomes of size $N_P $ with D decision parameters within the promising area constrained by the specified minimum and maximum parameters limits. For example, let us consider the initial value of the jth parameter in the ith chromosome is generated according to (11). Here the size of decision parameters for IEEE 14-bus system is 27 (14 voltage magnitudes and 13 voltage angles) with boundary constraints given in (4) (Das and Suganthan 2011; Wang et al. 2011a). Once the initial population is generated, the fitness of each individual chromosome is calculated according to (2). This population evolves over G generations to attain an optimal solution. Further, a parent population from the current generation is termed as a target vector.

$$\begin{aligned} p_{j,i} =p_j^{\min } +\alpha _j \times \left( {p_j^{\max } -p_j^{\min } } \right) \quad j=1,2,\ldots ,D,\nonumber \\ \end{aligned}$$

(11)

where $p_j^{\min } \hbox { and }p_j^{\max }$ are minimum and maximum values of state variable $p_j $, and $\alpha _j $ is uniformly distributed random number in the range of 0–1.

After initialization, JADE is carried out with three simple cycles of stages, viz. differential vector-based mutation, crossover, and selection.

Difference vector mutation

Another population known as donor vector $v_{i,G} $ population is obtained by performing difference vector mutation on each individual (i) of the current population $\{x_{i,G} |i=1,2,\ldots ,N_P \}$. In order to perform difference vector mutation, for each chromosome in the current population three other chromosomes are selected. These three chromosomes are selected as follows. First to improve the convergence characteristic, best chromosome $x_\mathrm{best,G}^p $ is selected from current population, second chromosome $\tilde{x}_{r2,G} $ is selected from historical data archive A which consists of recently explored inferior solutions to provide the information of progress direction and for improving the diversity of the population, and the third chromosome $x_{r1,G} $ is selected randomly from the current population (Zhang and Sanderson 2009). The mutation vector strategy with external archive is achieved in the following manner:

$$\begin{aligned} v_{i,G}= & {} x_{i,G} +F_i .\, \left( x_\mathrm{best,G}^p -x_{i,G}\right) \nonumber \\&+F_i .\,(x_{r1,G} -\tilde{x}_{r2,G} ), \end{aligned}$$

(12)

where the indices $r_1 \hbox { and }r_2 $ are two randomly selected distinct integers and $F_i $ is the scaling factor.

Now, let us consider a chromosome $x_{32,300} $, which represents the 32nd chromosome of 300th generation in target population of size, say, 100. For performing difference vector mutation, corresponding to chromosome $x_{32,300} $, three other chromosomes were selected, viz. best chromosome $x_\mathrm{best,300}^p$ from current population; second chromosome $\tilde{x}_{116,300} $ which denotes 116th chromosome of 300th generation, randomly selected from the union of current population and historical data archive $(P\cup A)$; and third chromosome $x_{66,300} $ which is 66th chromosome of 300th generation, randomly selected from current population. These three chromosomes appear as shown below:

$$\begin{aligned}&x_{32,300} = [1.0213 1.0455 1.0104 1.0136 1.0170 1.0702 \\&\quad 1.0461 1.0364 1.0307 1.0298 1.0679 1.0535 1.0467 1.0196 \\&\quad -0.0864 -0.2207 -0.1779 -0.1521 -0.2518 \\&\quad -0.2303 -0.0579 -0.2578 -0.2614 \\&\quad -0.0008 -0.2667 -0.2672 -0.2801]; \\&\quad x_\mathrm{best,300}^p = [1.0251 1.0453 1.0103 1.0135 \\&\quad 1.0169 1.0698 1.0460 1.0368 1.0307 \\&\quad 1.0724 1.0670 1.0532 1.0464 1.0193 \\&\quad -0.0869 -0.2212 -0.1785 -0.1527 \\&\quad -0.2518 -0.2308 -0.0570 -0.2582 \\&\quad -0.2532 -0.0012 -0.2666 -0.2672 -0.2800]; \\&\quad x_{66,300} = [1.0234 1.0452 1.0101 1.0133 1.0167 \\&\quad 1.0703 1.0458 1.0360 1.0304 1.0718 \\&\quad 1.0670 1.0536 1.0469 1.0197 -0.0866 \\&\quad -0.2209 -0.1782 -0.1524 -0.2518 -0.2306 \\&\quad -0.0598 -0.2580 -0.2387 -0.0013 \\&\quad -0.2667 -0.2672 -0.2800], \\&\quad \tilde{x}_{116,300} = [1.0232 1.0456 1.0106 1.01381.0172 \\&\quad 1.0700 1.0462 1.0362 1.0307 1.0732 \\&\quad 1.0687 1.0533 1.0466 1.0196 -0.0866 \\&\quad -0.2209 -0.1781 \\&\quad -0.1523 -0.2519 -0.2303 -0.0562 \\&\quad -0.2578 -0.2306 -0.0011 \\&\quad -0.2666 -0.2673 -0.2801]. \end{aligned}$$

Now difference vector mutation with a scaling factor F of 0.604909433 is performed according to (12). The resultant 32nd chromosome $v_{32,300} $ of donor population for 300th generation is as given below.

$$\begin{aligned} v_{32,300}= & {} [1.0237 1.0451 1.0100 1.0132 1.0166 \\&1.0701 1.0458 1.0365 1.0305 1.0547 \\&1.0663 1.0535 1.0467 1.0195 \\&-0.0867 -0.2210 -0.1783 -0.1525 -0.2517 \\&-0.2308 -0.0595 -0.2582 -0.2613 \\&-0.0012 -0.2667 -0.2671 -0.2800]. \end{aligned}$$

Binomial crossover

The binomial crossover operation is performed on donor population to enhance the exploration capability of the present technique. This is accomplished as follows. For each decision variable in the population, a random number is generated between 0 and 1. If the generated random number is less than or equal to crossover rate (CR) then the decision variable is inherited from the donor vector $v_{32,300}$, else decision variable is inherited from the target vector $x_{32,300} $ and is defined as (Das and Suganthan 2011; Liu and Wang 2009; Brest et al. 2006; Wang et al. 2011b):

$$\begin{aligned} u_{j,i,G} =\left\{ {\begin{array}{ll} v_{j,i,G} ,&{} \quad \mathrm{if}\,rand(0,1)\le \mathrm{CR}\, \mathrm{or}\, j=q \\ x_{j,i,G} ,&{} \quad \mathrm{otherwise}, \\ \end{array}} \right. \end{aligned}$$

(13)

where $i=1,\ldots ,N_{p}, j=1,\ldots ,D$, and q is a randomly chosen index $\in \left\{ {1,\ldots ,N_p } \right\} $. The obtained trial chromosome $u_{32,300} $ is given as follows:

$$\begin{aligned} u_{32,300}= & {} [1.0237 1.0451 1.0100 1.0132 \\&1.0166 1.0702 1.0461 1.0365 1.0307 \\&1.0298 1.0663 1.0535 1.0467 \\&1.0196 -0.0867 -0.2210 \\&-0.1779 -0.1521 -0.2517 \\&-0.2303 -0.0595 -0.2578 -0.2614 \\&-0.0012 -0.2667 -0.2671 -0.2801]. \end{aligned}$$

Now if both target chromosome $x_{32,300} $ and the trial chromosome $u_{32,300} $ obtained after binary crossover are compared with the true values then the error between $x_{32,300} $ and true values comes out to be 0.5402. This is high compared to error obtained using trial chromosome $u_{32,300} $ which is 0.5319.

Selection

Finally, after performing the binomial crossover operation the selection process is performed between the target vector and the trial vector chromosomes depending upon the fitness of the each individual in order to select better individual to the next generation. Selection operation is described as (Das and Suganthan 2011; Liu and Wang 2009; Brest et al. 2006; Wang et al. 2011b):

$$\begin{aligned} x_{i,G+1} =\left\{ {\begin{array}{ll} u_{i,G} , &{} \quad \hbox {if}\, f(u_{i,G} )<f(x_{i,G} ) \\ x_{i,G} , &{} \quad \hbox {otherwise.} \\ \end{array}} \right. \end{aligned}$$

(14)

By using selection operation, trial chromosome $u_{32,300} $ has been found to be fittest chromosome and is then selected for the next generation.

At each generation, the control parameters CR and F for each target vector are updated by utilizing th e normal and the Cauchy distributions, respectively (Zhang and Sanderson 2009). The crossover rate $\text {CR}_i $ of each chromosome $x_i $ is achieved according to (15), and next, the value of $\text {CR}_i $ is trimmed to $\left[ {0,1} \right] $.

$$\begin{aligned} \text {CR}_i = randn_i (\mu _\mathrm{CR}, 0.1). \end{aligned}$$

(15)

The mean $\mu _\mathrm{CR} $ of normal distribution is initially initialized to 0.5, and this value is then updated for each iteration according to (16).

$$\begin{aligned} \mu _\mathrm{CR} =(1-c){\cdot }\mu _\mathrm{CR} + c{\cdot }mean_A (S_\mathrm{CR} ), \end{aligned}$$

(16)

where $S_\mathrm{CR} $ represents the set of all effective crossover rates $\text {CR}_i$’s at each generation G, c indicates a positive constant value between 0 and 1, and $\hbox {mean}_A (\cdot )$ shows the arithmetic mean (Zhang and Sanderson 2009).

Similarly, the scaling factor $F_i $ of each chromosome $x_i $ is generated independently using (17) at every generation G and next the value $F_i $ is trimmed to 1 if $F_i \ge 1$ or restored if $F_i \le 0$.

$$\begin{aligned} F_i =randc_i (\mu _F ,0.1). \end{aligned}$$

(17)

The location parameter $\mu _F $ is initially set to 0.5, and at the end of each generation the parameter is updated as:

$$\begin{aligned} \mu _F =(1-c){\cdot }\mu _F +c{\cdot }\hbox {mean}_L (S_F ), \end{aligned}$$

(18)

where $\hbox {mean}_L (\cdot )$ is the Lehmer mean and $S_F $ denotes the set of all successful scaling factors in generation G (Zhang and Sanderson 2009).

$$\begin{aligned} \hbox {mean}_L (S_F )=\frac{\sum \limits _{F\in S_F } {F^{2}} }{\sum \limits _{F\in S_F } F }. \end{aligned}$$

(19)

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Authors and Affiliations

Vedik Basetti
- 1
Email author
Ashwani Kumar Chandel
- 2
K. B. V. S. R. Subramanyam
- 3

1.Electrical Engineering DepartmentIMS Engineering CollegeGhaziabadIndia
2.Electrical Engineering DepartmentNIT-HamirpurHamirpurIndia
3.Electrical and Electronics Engineering DepartmentSR Engineering CollegeWarangalIndia

Power system static state estimation using JADE-adaptive differential evolution technique

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Compliance with Ethical Standards

Conflicts of interest

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