Energy and Power 2012, 2(2): 17-23 DOI: 10.5923/j.ep.20120202.03 EP-Based Optimisation for Estimating Synchronising and Damping Torque Coefficients N. A. M. Kamari, I. Musirin*, Z. A. Hamid, M. N. A. Rahim Faculty of Electrical Engineering, Universiti Teknologi Mara, Shah Alam, Selangor, Malaysia Abstract This paper presents Evolutionary Programming (EP) based optimisation technique for estimating synchronising torque coefficients, Ks and damping torque coefficients, Kd of a synchronous machine. These coefficients are used to identify the angle stability of a system. Initially, a Simulink model was utilised to generate the time domain response of rotor angle under various loading conditions. EP was then implemented to optimise the values of Ks and Kd within the same loading conditions. Result obtained from the experiment are very promising and revealed that it outperformed the Least Square (LS) and Artificial Immune System (AIS) methods during the comparative studies. Validation with respect to eigenvalues determination confirmed that the proposed technique is feasible to solve the angle stability problems. Keywords Angle Stability, Synchronising Torque Coefficient, Damping Torque Coefficient, Evolutionary Programming, Artificial Immune System techniques in addressing the stable and unstable phenomena. It has been used as static parameter estimation[8]. However, 1. Introduction several disadvantages have been identified in LS method. Amongst them are the long computation time and the Small signal stability analysis of power systems has requirement for data updating. It also requires monitoring the become more important nowadays. Under small entire period of oscillation. perturbations, this analysis predicts the low frequency Recently, optimisation algorithms such as Evolutionary electromechanical oscillations resulting from poorly damped Programming (EP) and Artificial Intelligent System (AIS) rotor oscillations. The oscillations stability has become a have received much attention in global optimisation very important issue as reported in[4-6]. The operating problems. EP and AIS are heuristic population-based search conditions of the power system are changed with time due to methods that use both random variation and selection. The the dynamic nature, so it is needed to track the system optimal solution search process is based on the natural stability on-line. To track the system, some stability process of biological evolution and is accomplished in a indicators will be estimated from given data and these parallel method in the parameter search space. EP-based indicators will be updated as new data are received. method has been applied in various researches in Synchronising torque coefficients, Ks and damping torque static[12-16] and dynamic system stability[17-19]. On the coefficients, Kd are used as stability indicators. To achieve other hand, AIS optimisation approach is still new in power stable condition, both the Ks and Kd must be positive[1-3]. system compared to the EP. EP and AIS share many Certain techniques have been proposed to estimate the common aspects; EP tries to model the natural evolution values of Ks and Kd involving optimisation technique. Some while AIS tries to benefit from the characteristics of human techniques have been explored by means of frequency immune system[20-22]. response analysis[6,7].[3] decomposed the change in This paper presents an efficient online estimation electromagnetic torque into two orthogonal components in technique of synchronising and damping torque coefficients the frequency domain. The two equations were expressed in in solving angle stability problems. It is based upon the terms of the load angle deviation then solved directly. Static population-based search methods that use both random variation and selection. The method is used to estimate and dynamic time domain estimation methods were also synchronising torque coefficients, K , and damping torque proposed in this study. s coefficients, K , from the machine time responses of the Least Square (LS) method can be one of the possible d change in rotor angle, Δδ(t), the change in rotor speed, Δω(t), and the change in electromechanical torque, ΔTe(t). The goal *Corresponding author: [email protected] (I. Musirin) is to minimise the estimated coefficient error and the time Published online at http://journal.sapub.org/ep consumed. The proposed EP technique is used to find the Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved. best solution of the formulated problem. Results obtained 18 N. A. M. Kamari et al.: EP-Based Optimisation for Estimating Synchronising and Damping Torque Coefficients from the experiment using EP were compared with AIS and well as the excitation levels in the generator. K3 is a function LS methods. Then, the results were verified with of the ratio of impedance. Details of matrix A and matrix B eigenvalues. are explained as follows. EBEq0 K1 = (RT sin δ0 + XTq cosδ0 ) 2. The System Model D (5) E i + B q0 (X − X ′ )(X sin δ − R cosδ ) A simplified block diagram model of the small signal D q d Tq 0 T 0 performance is shown in Figure 1. In this representation, the L dynamic characteristics of the system are expressed in terms K = ads 2 L + L of K constants with linearised single machine infinite bus ads fd (6) (SMIB) system. This model is represented with some R X (X − X ′ ) × T + Tq q d + variables such as electrical torque, rotor speed, rotor angle, Eq0 1iq0 D D and exciter output voltage. K4 iq0 = It cos(δ i + φ) (7) − ∆E fd K ∆ψ fd + + 3 Lads + L fd 1 (8) Σ K2 Σ K1 K3 = − X + 1 + sT3 X d X L Tq 1+ (X d − X d′ ) ∆TE D − ∆ω ∆δ 1 r ω0 ∆T Σ Lads + L fd 1 m T = (9) 2Hs + KD s 3 + ω0 R fd X Tq 1+ (X d − X d′ ) Figure 1. Block diagram model of small signal performance. D L E As in the case of the classical generator model, the = ads B − K 4 (X d X L ) (10) acceleration and the field circuit dynamic equations are: Lads + L fd D ∆ω T − T − K ∆ω × (X Tq sin δ 0 − RT cosδ 0 ) r = m e d r (1) ∆t 2H E (X sinδ − R cosδ ) = B Tq 0 T 0 (11) ∆δ m1 = ω ∆ω (2) D ∆ 0 r t X Tq Lads (12) m2 = ∆ψ fd D (Lads + L fd ) = ω (e − R i ) (3) ∆t 0 fd fd fd E (R sinδ + X cosδ ) n = B T 0 Td 0 (13) where Tm is a mechanical torque, Te is a electromagnetic 1 D torque, efd is a field voltage, Rfd is a rotor circuit resistance, ifd R L is a field circuit current, and ω0=2πf0. n = T ads (14) 2 D (L + L ) From the transfer function block diagram, the following ads fd state-space form is developed. 2 2 X = AX + BU EB = EBd 0 + EBq0 (15) ∆ω r K D K1 K 2 ∆t − − − ∆ω EBd 0 = Et sin δ i − I t [Re sin(δ i +φ)− X e cos(δ i +φ)] (16) 2H 2H 2H r ∆δ = ω0 0 0 ∆δ ∆t E = E cosδ − I [R cos(δ +φ)+ X sin(δ +φ)] (17) ∆ψ K3K 4 1 ∆ψ Bq0 t i t e i e i fd 0 − − fd (4) T T ∆ 3 3 t φ − φ −1 It X qs cos It Ra sin 1 δ = tan (18) i 0 Et + It Ra cosφ + It X qs sinφ 2H ∆Tm + 0 0 ∆ 1 E fd = 0 − X qs K sq X q (19) T3 The system matrix A is a function of the system 2 D = R + X X (20) parameters that depends on the opening conditions. The T Tq Td perturbation matrix B depends on the system parameters only. The interaction among these variables is expressed in terms X Tq = X e + K sd (X q − X L )+ X L (21) of the 4 constants K1, K2, K3, and K4. Constants K1, K2, and K4 are functions of the operating real and reactive loading as X Td = X e + Lads′ + X L (22) Energy and Power 2012, 2(2): 17-23 19 RT = Ra + Re (23) possibility of getting trapped in local minima. Therefore, EP can reach to a global optimal solution. X d − X L + L fd R = (24) (b) EP uses performance index or objective function fd T ′ ω d 0 0 information to guide the search for solution. Therefore, EP (X ′ − X )(X − X ) can easily deal with non-smooth and non-continuous = d L d L (25) L fd objective functions. X d − X d′ (c) EP uses probabilistic transition rules instead of P −1 t (26) non-deterministic rules to make decisions. Moreover, EP is a φ = tan Et It kind of stochastic optimisation algorithm that can search a complicated and uncertain area to find the global minimum. 2 + 2 Pt Qt (27) It = This makes EP more flexible and robust than conventional Et methods. Pt and Qt are terminal active and reactive power, In the EP algorithm, the population has 2n candidate respectively. All related equations are given in[1]. solutions with each candidate solution is an m-dimensional vector, where m is the number of optimised parameters. The EP algorithm can be described as: 3. The System Model Step 1 (Initialisation): Generation counter i is set to 0. n random solutions (x , k=1, …, n) are generated. The kth trial A single machine connected to infinite bus system is k solution x can be written as x =[p ,…, p ], where the lth considered. The system comprises a steam generator k k 1 m optimised parameter p is generated by random value in the connected via a tie line to a large system represented as i range of[p min, p max] with uniform probability.