On-line Load Flow Analysis Using Radial Basis Neural Network

Load flow (LF) study, which is performed to determine the power system static states (voltage magnitudes and voltage angles) at each bus to find the steady state operating condition of a system, is very important and is the most frequently carried out study by power utilities for power system planning, operation and control. In this paper, a radial basis function neural network (RBFN) is proposed to solve load flow problem under different loading/contingency conditions for computing bus voltage magnitudes and angles of the power system. The RBFN has many advantageous features such as optimised system complexity, minimized learning and recall times as compared to multi-layer perceptron model. The composition of the input variables for the proposed neural network has been selected to emulate the solution process of a conventional load flow program. The effectiveness of the proposed RBFN based approach for on-line application is demonstrated by computation of bus voltage magnitudes and voltage angles for different loading conditions and single line-outage contingencies in IEEE 14-bus system. Keywords— Admittance matrix, load flow studies, security analysis, line outage contingency, radial basis function neural network, real loads, reactive loads, bus voltage magnitude, voltage angle, Gaussian function. 1.0 Introduction Load flow or power flow studies are conducted to determine the steady state operating condition of a power system, by solving the static load flow equations (SLFE) that mathematically are represented by a set of non-linear algebraic equations for a given network. The main objective of load flow (LF) studies is to determine the bus voltage magnitude with its angle at all the buses, real and reactive power flows (line flows) in different lines and the transmission losses. It is the most frequently carried out study by power utilities and is required to be performed at almost all the stages of power system planning, optimization, operation and control. Fast security assessment is of paramount importance in a modern power system to provide reliable and secure electricity supply to its consumers. To perform the contingency screening, which is one of the most CPU timeconsuming tasks for on-line security assessment, the computation in a few minutes of many LF scenarios is required simulating the occurrence of several contingencies and different loading conditions [1]. During last four decades, almost all the known methods of numerical analysis for solving a set of non-linear algebraic equations have been applied in developing load flow algorithms [2,3]. One or more desirable features to compare the different LF methods can be the speed of solution, memory storage requirement, accuracy of solution and the reliability of convergence depending on a given situation. Though, robustness or reliability of convergence of the method is required for all types of application, the speed of solution is more important for on-line applications compared to the off-line studies. For contingency selection, fast non-iterative approximate load flow methods such as DC load flow method, linearised AC load flow, decoupled load flow, fast decoupled load flow methods are used, which provide results having significant inaccuracies. Full AC load flow methods are accurate but become unacceptable for on-line implementation due to high computational time requirements. With the advent of artificial intelligence, in recent years, expert systems, pattern recognition, decision tree, neural networks and fuzzy logic methodologies have been applied to the security assessment problem [4-6]. Amongst these approaches, the application of artificial neural networks (ANNs) have shown great promises in power system ♣ Corresponding author, email: [email protected] engineering due to their ability to synthesize complex mappings accurately and rapidly. Most of the published work in this area utilizes multi-layer perceptron (MLP) model based on back propagation (BP) algorithm, which usually suffers from local minima and over-fitting problems [5-7]. Its ability to generalise a pattern depends on the learning rate and the number of units in hidden layer. In reference [8], a neural network load flow using an ANN-based minimisation model is proposed. A separate MLP model based on Levenberg-Marquardt second order training method has been used for computation for bus voltage magnitude and for angle at each bus of power system in reference [9]. As the number of neural networks required to solve load flow problem are large, it may not be applicable to a practical power system having huge number of buses. A radial basis neural network [7,10,11] is proposed in this paper for on-line load flow studies. The RBFN has many advantageous features such as optimised system complexity, minimised learning and recall times. RBF model has an input layer, one hidden layer and output layer. The input variables are directly fed to the hidden units without weights. The effectiveness of the proposed RBFN based approach is demonstrated by computation of bus voltage magnitudes and angles following different single line-outage contingency at different loading conditions in IEEE 14-bus system [12]. 2.0 Methodology Figure1 shows the architecture of the proposed radial basis function neural network. The composition of the input variables for the proposed neural network has been selected to emulate the solution process of a conventional load flow program. Gdiagonal V1 Bdiagonal RADIAL V2 BASIS : Pg (PV buses) FUNCTION : NEURAL : NETWORK : Vg (PV + Slk buses) Vn Figure 1. Proposed RBFN Architecture The input consists of the electric network parameters represented by the diagonal elements of the bus conductance and susceptance matrix, voltage magnitudes Vg of generation and slack buses, the active power generations Pg of PV buses. In order to speed up the neural network training, the conductance and susceptance are normalised between 0.1 and 0.9. For this RBFN based load flow model, the system loads, active and reactive power components are represented like constant admittance and they are included into the diagonal of the bus admittance matrix [Y]=[G]+j[B], where [G] and [B] are the bus conductance and susceptance matrices respectively. 2.1 Radial Basis Function Neural Network The RBF network consists of three layers, the input layer, hidden layer and output layer. The nodes within each layer are fully connected to the previous layer as shown in Figure 2. The input variables are assigned to each node in the input layer and are passed directly to the hidden layer without weights. The hidden nodes (units) contain the radial basis functions, and are analogous to the sigmoidal function commonly used in the BP networks. The radial basis function is similar to the Gaussian density function, which is defined by a centre position and a width parameter. The width of the RBF unit controls the rate of decrease of function. The output of the th i unit ( ) Xp ai in the hidden layer is given by ( ) [ ]       − − = ∑ = r j i ji jp p i x x X a 1 2 2 / exp σ (1) where ji x = centre of th i RBF unit for input variable j σi = width of th i RBF unit r = dimension of input vector The connection between the hidden units and the output units are weighted sums as shown in Figure 2. The output value mi y of the th m output node is given as