An Improved Power System State Estimation Using A Dynamically Adapted JAYA Algorithm

State estimation is a central issue for power systems monitoring and control. Applying state estimation schemes ensures the accuracy of system real-time monitoring process. Due to the high nonlinearities and non-smoothness of the dynamic behavior of power systems, state estimation is getting more importance to lessen the error margins. In order to find the best estimate of the various variables of a power system, optimization-based and statistical techniques are applied. Classically, common metering devices are used to measure power system variables. Nevertheless, these devices are associated with errors, especially with the ongoing expansion of the power networks. These errors are linked to many issues related to operation and communication processes besides the errors caused by the metering device itself. The commonly applied technique to solve the power system state estimation problem is the Weighted Least-Squares (WLS) method. In this paper a dynamically adapted algorithm is introduced employing a WLS-based dynamic JAYA optimization technique. The algorithm was validated and applied on the well-known IEEE 14-bus network. The results proved the efficiency and advantages of the proposed algorithm when compared to other methods used for the state estimation problem.

Inaccuracy of measurement is usually a result of device deficiency and the defects associated with data transfer processes. In the state estimation practice, the determination of near optimal power system state is performed statistically using existing measured data [1]. The Weighted Least-Squares (WLS) and the Maximum Likelihood Methods are commonly used to compute the close to optimal state estimation of power systems [2]. The WLS method is performed by minimizing the errors associated to measured data in order to optimally find the estimated variables (states) of the system. Arithmetically, this could be done by the minimization of the Jacobian matrix in which the received measurements are the elements of the matrix. According to the network configuration, the system may have some busses that are not directly coupled, which result in sparse Jacobian matrix. This sparsity of the Jacobian matrix is the most significant disadvantage of the WLS method [3][4][5]. The WLS method is treated using various optimization techniques to overtake its weakness. One of these techniques is the Newton-Raphson method which is extensively applied in this perspective. The use of this method guarantees the satisfaction of the first order optimality condition. In most cases of power system models, however, the high nonlinearities, nonconvexity and nonsmoothness characteristics make it difficult to apply this method effectively. Moreover, ill condition features are developed when the Hessian matrix is inverted. As a result, an inaccurate estimation is generated so that it is essential to apply powerful optimization methods to correct this situation [1]. In the literature, a wide range of many optimization techniques have been utilized to find optimal or close to optimal solutions to various power system operation and control problems [6,7]. Conventionally, these optimization approaches are deterministic methods and calculus-based techniques. On the other hand, the recent ones are heuristic and artificial methods. Non-calculus-based optimization techniques have shown satisfying convergence features when applied to solve nonconvex large-scaled optimization problems that have high non-linearities [8,9]. In order to compute an exact state estimation, the closeto-optimal placement of phasor measurements units is computed using Genetic algorithms [10]. Particles Swarm Optimization (PSO) method is also applied to tackle the state estimation problem in [11]. A hybrid WLS-based dynamic bacterial foraging algorithm is utilized for state estimation problem in [12]. Recently, JAYA algorithm has been introduced to solve various optimization problems [13]. The JAYA algorithm is simple and popularly applied to solve various optimization problems because of the limited number of parameters required. However, the JAYA algorithm is linked to some hitches that must be considered and treated. Divergence to stuck in local minima is the main issue of the JAYA behaviour. This is caused by some issues associated to the diversity preservation of the population samples. To overcome the drawbacks, a WLS-based dynamically adapted JAYA algorithm is introduced in this paper. The proposed WLS-DAJAYA, is presented and applied to determine the closeto-optimal accurate power system estate estimation. The reminder of the paper is organized as follows: Section 2 provides the formulation of the problem using WLS. In Section 3, the JAYA algorithm is described. Simulation results are given in Section 4. The conclusion is drawn in Section 5.

2-State estimation using WLS method
Determination of unknown variables of power system by means of measurements (samples) is a statistics-based estimation technique. In this approach the accessible inaccurate measurements are used to express the near optimal estimate of the unspecified variables [2].
The measured values are given by a number of measuring devices with some errors. These errors can mathematically be expressed as follows: () ii z h x   z i is the i th measurement value. ℎ i is the i th nonlinear function that relates the estimated value with its measurements.
x is a state vector that represents the estimated variables.

Formulation of the state estimation problem
The state estimation problem is formulated as an optimization problem where the cost function to be minimized is the summation of the residual errors. If the number of available measurements is denoted as m and the number of unknown variables is n, then this problem can be formulated as follows [2]: variance for the i measurement, and ( ) measurement residual The expression given in Equation (2) is known as the weighted least-squares estimator which is the maximum likelihood estimator where errors are modelled as random numbers with normal distribution characteristics. The above minimization function can be expressed in a vector form as follows: where [H]x is an m by n matrix and its elements are the coefficients of the linear function h i (x). The measurements are expressed in a column vector as: Then Equation (2) can be expressed in a compact matrix notation as: Where [R] is known as the covariance matrix of measurement errors and defined as follows: The formulation in Equation (6) can be extended to find a general minimization form expressed in the following equation [2]: 11 11 min

Constraints
The cost function given above is subject to a number of constraints including equality and inequality ones that have to be well satisfied. The upper and lower limits that represent the problem boundaries are specified as [1]: where λ is the penalty factor and N is the number of variables. text

3-The Basic and Proposed JAYA Algorithms
The proposed DAJAYA algorithm is an improved version of the basic JAYA. In this section both basic and dynamically adapted JAYA algorithms are investigated.

The basic JAYA Algorithm
The original JAYA algorithm is a non-deterministic optimization method that was introduced by R. Venkata [13]. The name JAYA is a Sanskrit word which means victory. This algorithm was effectively utilized to solve practical optimization problems with nonconvex objective functions. The principal advantage of this method is the relatively small number of control parameters needed to run the algorithm. A brief explanation of the application of the JAYA algorithm is given here: 1-The size of population, total number of variables (states), decision and stopping criteria are specified and the implementation is initialized. 2-Population of size P (candidate solution) × q (decision variables) is produced. 3-A starting solution vector is determined. 4-Best and worst solutions (X k i, best , X k i, worst ) are determined.

5-
The solution vector is updated as follows:

6-Comparison of solution candidates is performed to find out if the updated solution is
better than the previous one. The update is only accepted if the new candidate is better. 7-The algorithm's stopping criterion is checked out so that the process is terminated when satisfied and else step 2 is to be updated. It is obvious that according to Equation (11), the candidate gets closer to the best solution and drives away from the worst one. The JAYA algorithm keeps specifying the maximum iteration as well as the population size.

The DAJAYA Algorithm
The disadvantages of the basic JAYA algorithm, are treated by introducing a modified updating pattern for the size of population. This adjustment is proposed to deal with large scaled non-linear cost functions with non-convex characteristics.
In order to achieve the best algorithm performance for the power system state estimation problem, the size of population is updated adaptively and dynamically. If the initial population size is designated as P i , then the updated size is calculated using the following equation [14]: where; s is a random variable such that -1 < s < 1.
To apply Equation (12), in the proposed dynamically adapted algorithm (DAJAYA), the population are entirely updated to the next population providing that the updated size is larger than the previous one and the close-to-optimal solution will be appointed to the residual candidates. Instead, the best solution will only be updated to the following population when the old size is larger than the new one. Understandably, no need for any actions when there is no change in the population size.

4-Simulation results
The proposed WLS-based DAMBFA was applied to find the optimal or near optimal "best" accurate state estimation of the recognized IEEE 14-bus standard power network, which is given in [14] and illustrated in figure 1. The algorithm was coded and executed in MATLAB on an Intel-Core i7-8750-H 2.20-GHz PC with 8-GB-RAM. To guarantee consistency and stability, 45 independent runs were performed using fresh random initial solution for every new run. The data line and bus are found in [15] and shown in tables 1, 2 respectively. The existing measurements are offered in [16].   A benchmark three-buss small power system was employed to apply the proposed algorithm and show its effectiveness. Afterwards, the proposed WLS-based DAMBFA was applied to the well-known IEEE 14-bus system. The algorithm was implemented for the test system with its upper and lower boundaries of the system variables were properly specified. The algorithm's parameters are pre-set appropriately as well as its stopping criteria. The obtained results were compared to those acheived by applying the WLS-Newton-Raphson, WLS and the Particle Swarm Optimization PSO-WLS [16]. They were also compared to those given in [14] where the WLS-base dynamic bacterial foraging algorithm WLS-DBFA was applied for the power system state estimation problem. The results as well as the comparison are demonstrated in table 3. Figure 3 shows the absolute percent error associated with the algorithm.  Figure 2 illustrates the absolute percent error for the bus voltage magnitude and angle when the proposed WLS-based DAMBFA was applied. It can be seen that the error margin is within the pre-set tolerated edges which demonstrates the effectiveness of the proposed algorithm.

5-CONCLUSION
In this paper a WLS-based dynamically adapted JAYA algorithm was introduced and applied to determine the close-to-optimal and "best" accurate state estimation of a designated power system. The well-known IEEE 14-bus system was employed to validate the algorithm. The statistical weighted least squares method enhanced by the JAYA algorithm was formulated, applied, and accomplished for the power system state estimation problem. The WLS-based DAJAYA was effectively verified to be successful when compared to other techniques. The absolute percent error was computed for the estimated voltage magnitude and angle and found to be within the tolerable and reasonable boundaries.