ISSN (Print), ISSN (Online) Volume 1, Number 1, May - June (2010), IAEME

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1 International Journal Journal of Electrical of Electrical Engineering Engineering and Technology (IJEET), and Technology (IJEET), ISSN (Print) ISSN (Online), Volume 1 Number 1, May June (2010), pp IAEME, IJEET I A E M E SMALL SIGNAL STABILITY ANALYSIS USING FUZZY CONTROLLER AND ARTIFICIAL NEURAL NETWOR STABILIZER A.padmaja EEE Dept., GIT, Gitam University Visahapatnam padmaja_a14@yahoo.co.in V.s.vaula EEE Dept., GIT, Gitam University Visahapatnam vaulavs@yahoo.com T.Padmavathi EEE Dept., GIT, Gitam University Visahapatnam tadipadma@gmail.com S.v.Padmavathi EEE Dept., GIT, Gitam University Visahapatnam sv.padmavathi@gmail.com ABSTRACT: Power system Stabilizers are used in order to damp out the low frequency oscillations which are due to disturbances. This paper attempts to investigate the performance of Conventional Power System Stabilizer (CPSS), Fuzzy Logic Power System Stabilizer (FLPSS) and Artificial Neural Networ based Power System Stabilizer (ANNPSS) under Prefault and Postfault conditions for different loadings. The parameters of Conventional Power System Stabilizer (CPSS) is designed using PolePlacement Technique and the parameters of Fuzzy logic Power System Stabilizer (FLPSS) is tuned to their optimal values in order to minimize the overshoot in step response of rotor angle 47

2 deviation using Particle Swarm Optimization (PSO) technique whereas Artificial Neural networ based Power System Stabilizer (ANNPSS) is trained by Linear Optimal Control (LOC) theory. Designed power system stabilizers are applied to a single machine infinite bus system and are tested in four operational conditions; normal load, heavy load, light load and solid 3phase fault occurrence in a transmission line. The simulation study reveals that the performance of Linear Optimal Control (LOC) based Artificial Neural Networ Power System Stabilizer is much improved with Artificial Neural Networ based Power System Stabilizer under different operating conditions. ey words: Power System Stabilizer, Fuzzy Logic Controller, Particle Swarm Optimization, Artificial Neural networ, Pole placement technique, Linear Optimal Control. 1. INTRODUCTION Power Systems experience low frequency oscillations which are in the range of 0.1 to 2.5Hz due to disturbances which may grow and lead to dynamic instability of the system if the applied damping torque is insufficient and may limit the ability to transmit power. Power System Stabilizer (PSS) provides required damping by producing an electrical torque component in phase with rotor speed deviation to enhance the system stability from low frequency oscillations by controlling the excitation using auxiliary stabilizing signals. PSS should be provided with appropriate phase lead circuits to compensate for the phase lag between the exciter input and electrical torque. In this paper, the pole placement technique is considered to determine the parameters of Conventional Power System Stabilizer (CPSS) in order to stabilize a single machine connected infinite bus system. However, conventional PSS which is designed at certain operating point does not provide satisfactory results for all operating conditions due to approximation in modeling and variation of the system topology due to error occurrence. Therefore, in recent years, soft computing methodologies lie fuzzy logic and neural networs have been investigated for designing the PSS. The fuzzy Logic based Power System Stabilizers (FLPSS) have great potential in increasing the damping of generator oscillations. The inputs to the FLPSS are ω, &and ω ω. The first step in the design of FLPSS is to design conventional PID (ProportionalIntegralDerivative) 48

3 power system stabilizer which is based on a linear approximation of a nonlinear power plant around the operating point. The main disadvantage with the classical PID controller is that it cannot successfully control a plant with strong nonlinearities and various operating conditions. During a major disturbance such as a fault, the operating point of a power system drifts; conventional PID controllers do not wor well under such conditions. A nonlinear controller such as proposed Fuzzy controller will be more effective to cover a wider range of operating conditions. Design of Fuzzy Logic Power System Stabilizer is not an easy tas. It is very important to appropriately tune the parameters used in FLPSS. In this paper, Particle Swarm optimization (PSO) is used to tune the parameters of FLPSS. The main advantage of PSO over other Global Search techniques is its algorithmic simplicity as it uses few parameters and easy to implement. Stabilizing method based on Linear Optimal Control (LOC), mae a suitable damping at different loading condition, but also this method needs determination of all system states that seems to be very difficult. In recent years Artificial Neural Networ (ANN) methodology has widely used in power system engineering. In an attempt to cover a wide range of operating conditions, Artificial Neural Networ Power system stabilizer (ANNPSS) have been proposed which is trained by Linear Optimal Control. The efficacy of the stabilizer is tested on a single machine infinite bus bar system under four operational conditions; normal load, heavy load and light load, and in the case of fault occurrence in a transmission line. This paper is organized as follows. The problem is formulated in section 2. A brief explanation about the design of different stabilizers i.e the design of Conventional Power System Stabilizer (CPSS) using Poleplacement technique, the design of Fuzzy Logic Power system stabilizer using Particle Swarm Optimization (PSO) technique and design of Artificial Neural Networ based Power System Stabilizer using Linear Optimal Control (LOCANNPSS) is presented in section 3. Finally, to evaluate the effectiveness of proposed LOCANNPSS, simulation results are provided in section 4 for various operating conditions and conclusions are given in section 5. 49

4 2. PROBLEM FORMULATION Figure 1 shows the power system under study [1] is a Synchronous machine connected to a large power system through a transformer and a double circuit transmission lines. Figure 2 shows the single line schematic of the system including the Conventional Power System Stabilizer (CPSS), Fuzzy Logic Power System Stabilizer (FLPSS) and proposed Linear Optimal Control based Artificial Neural Networ Power System Stabilizer (LOCANNPSS), where they are discussed in the next sections. In this paper, for the analysis and design of control system, the system may be linearized, since the disturbance considered being small. During low frequency oscillations, the current induced in a damper winding is negligibly small; hence damper windings are completely ignored in the system model. The natural oscillating frequency for daxis and qaxis armature windings being extremely high, their eigen modes will not effect the low frequency oscillations. Hence can be ignored. Finally the field winding which is directly connected to excitation system, and has low eigen mode frequency is taen into account in system modeling. The performance of CPSS, FLPSS and LOCANNPSS is evaluated in four operational conditions; normal, heavy, light loads, and in the case of solid 3phase fault occurrence at bus 2 in second circuit. Figure 1 Single machine infinite bus system 50

5 Figure 2 Schematic of the SMIB with different power system stabilizers Figure 3 Linearized model of SMIB with Conventional PSS The linearized state equations for single machine connected to infinite bus are given as: Where x = δ ω E q E x& = A x + Bu y = Cx + Du (1) fd The state matrix can be written as: T 51

6 0 ω M M A = τ do 3τ do τ do 5 E 6 E 1 0 TE TE TE 0 0 B = 0 E T E, C = [ ], D = [1] (2) Constants 1 to 6 represent the system parameters at certain operating condition [2,3]. Analytical expressions for these parameters as function of loading (P, Q) are derived in [4]. System data is given in APPENDIX To cover multioperating conditions of the machine under study normal, heavy and light loading regimes are selected (pu): And also the performance of conventional PSS for prefault and postfault conditions under all above loading conditions is analyzed. The Constants 1 to 6 for multioperating conditions are tabulate in Table.1 Table 1 1 to 6 for multioperating conditions Loading condition 3. DESIGN OF POWER SYSTEM STABILIZERS (PSS) 3.1 Design of Conventional power System Stabilizer (CPSS) using Poleplacement technique Now, the problem is defined as follows: Given system (1), the transfer function of the system can be obtained using transfer matrix Fault Condition Constants Normal Prefault load Postfault Heavy Prefault Load Postfault Light Prefault Load Postfault G( s) = C( si A) B (3) The Linearised incremental model of synchronous machine with an exciter and power system stabilizer is as shown in Fig 3. The technique for selection of stabilizer parameters is explained by considering the (small Signal) transfer function (Fig 5) from the voltage 52

7 regulator reference of the machine where the stabilizer is to be applied to the speed deviation of that machine. Such a transfer function can be obtained from the linearised state space equations of single machine infinite bus system. V ref G(s) ω Figure 4 Bloc diagram of system transfer function with out PSS The state space equations include a reasonable representation of excitation system dynamics. The poles of G(s) are exactly the eigen values of the linearized single machine system without the present stabilizer. The effect of adding a stabilizer with transfer function H(S) to the system can be seen by considering the bloc diagram shown in Figure 5. Figure 5 Bloc diagram of system transfer function with PSS The modified transfer function now becomes the closed loop transfer function G( s) Gc ( s) = (4) 1 G( s) H ( s) The Eigen values of the system including the Stabilizers on the machine are the poles of this closed loop transfer function, and satisfy the closed loop characteristic equation: 1 G ( s ) H ( s ) = 0 (5) If we now specify a pair of desired complex eigen values (complex conjugates of each other), we can substitute one of these into above equation which, upon separation into real and imaginary parts, will yield two equations in the three unnown stabilizer parameters pss, 1 V ref + T and T2 in H(s). G(s) H(S) These equations can then be solved to determine the stabilizer parameters. Since there are two equations with three unnown parameters, the additional degree of freedom can be used to control to some extent the locations of the eigen values other than the ω 53

8 primary desired complex pair. The objective is to design a single stage PSS in the form of ( ) G s V ref = ω (6) st w 1+ st H ( s) = pss 1+ stw 1+ st 1 2 This stabilizes the system by placing the poles at desired location so as to stabilize the system at various operating conditions. The washout time constant (7) T w is taen as 10sec and the Time constant T2 is taen as 0.05 sec. using the poleplacement technique the desired poles are j and j The variables are calculated and tabulated in Table.2. Table 2 Parameters of Conventional power System Stabilizer pss and T 1 pss T 1(sec) T 2 (sec) The Linearized model of the Single Machine Infinite Bus system with Conventional power System Stabilizer is shown in Figure Design of Fuzzy Logic power System Stabilizer (FLPSS) using PSO Fuzzy Logic Controller (FLC) The change in load causes the variation of the generator dynamic characteristics so that the different operating conditions are obtained. Power system stabilizer must be capable of providing appropriate stabilization signals over a broad range of operating conditions and disturbances. Traditional power system stabilizers rely on linear design methods. On the other hand, a conventional PSS is designed for a linear model representing the generator at a certain operating point and it does not cover wide range of operating conditions effectively. During a major disturbance such as a fault, the operating point of a power system drifts; conventional controllers do not wor well under such conditions whereas a nonlinear controller can perform well under such situations. Hence a Fuzzy Logic Controller (FLC), which has nonlinear structure, is proposed in order improve the system stability by minimizing the maximum overshoot. A simple Fuzzy 54

9 logic Controller can be depicted using the bloc diagram shown in Figure 6. The steps followed in designing Fuzzy Logic Controller (FLC) are summarized [12] below: 1. Identify the variables (inputs, states and output) of the plant. 2. Partition the universe of discourse or the interval spanned by each variable into a number of fuzzy subsets, assigning a linguistic label. 3. Determine or assign a membership function for each fuzzy subset. 4. Form the rule base i.e., assigning the fuzzy relationships between the inputs or states fuzzy subsets on the one hand and the outputs fuzzy subsets on the other hand. 5. Choose appropriate scaling factors for the input and output variables in order to normalize the variables to the [0, 1] or the [1, 1] interval. 6. Fuzzify the inputs to the controller 7. Use fuzzy approximate reasoning to infer the output contributed from each rule. 8. Aggregate the fuzzy outputs recommended by each rule. 9. Apply defuzzification to form a crisp output. Figure 6 Structure of Fuzzy Logic Controller The implementation of fuzzy controller in a PSS structure is shown in Fig.6 and its illustrations can be explained as the following steps [5]: Step 1: In the scheme, ω, & ω and output signal of the controller. The coefficients ω are used as the input signals and Vs as p, i and d are used to eep the signals within allowable limit. These coefficients are nown as scaling factors which 55

10 transform the real value scale to required value in decision limit. The output signal ( V s ) is injected at the summing point of the exciter as the supplementary signal. Step 2: Identical membership functions have been chosen for each of input and out put variables in order to achieve normalization on the physical variables. Normalization allows the controller to associate equitable weight to each of the rules and to calculate the stabilizing signal correctly. Step 3: Each of the input fuzzy variables x i is assigned three linguistic fuzzy subsets such as Negative (N), Zero (Z), Positive (P) and output fuzzy variables, is assigned seven linguistic fuzzy subsets such as Large Negative (LN), Medium Negative (MN), Small Negative (SN), Zero (Z), Small Positive (SP), Medium Positive (MP) and Large Positive (LP). Symmetrical and normalized triangular membership functions for input and output fuzzy variables are shown in Fig.4.14 and 4.15 Figure 7 Membership functions for input variables Figure 8 Membership functions for output variables. Step 4: The range of each fuzzy variable is normalized between 1 to 1 by introducing a scaling factor to represent the actual signal. Step 5: Formation of rule base i.e., the set of decision rules which gives all possible combinations of inputs and outputs is based on the previous experience in the 56

11 field. The decision table made up of (3 X 3 X 3 =) 27 rules expressed using the same linguistic labels as those of the inputs is shown in Table.3. Table.3 Rule base for three input Fuzzy Controller 1 N N N PL 2 N N Z P 3 N N P P 4 N Z N P 5 N Z Z P 6 N Z P Z 7 N P N P 8 N P Z Z 9 N P P N 10 Z N N P 11 Z N Z P 12 Z N P Z 13 Z Z N P 14 Z Z Z Z 15 Z Z P N 16 Z P N Z 17 Z P Z N 18 Z P P N 19 P N N P 20 P N Z Z 21 P N P N 22 P Z N Z 23 P Z Z N 24 P Z P N 25 P P N N 26 P P Z N 27 P P P NL Step 6: Centroid method of defuzzification is applied to convert fuzzy output to crisp out put signal supplementary signal. Vs which is added at the summing junction of the exciter as the Figure 9 depicts the implementation of fuzzy controller in a Linearized model of SMIB 57

12 Overview of PSO Figure 9 Linearized model of SMIB with FLPSS Particle swarm optimization (PSO) is an evolutionary computation technique developed by Dr. Eberhart and Dr.ennedy in 1995, inspired by social behavior of bird flocing or fish schooling. PSO is a population based optimization tool. As shown in Fig.10, the system is initialized with a population of random solutions and searches for optima by updating generations. All the particles have fitness values, which are evaluated by the fitness function to be optimized, and have velocities, which direct the flying of the particles. The particles are flown through the problem space by following the current optimum particles. PSO is initialized with a group of random particles (solutions) and then searches for optima by updating generations. In every iteration, each particle is updated by following two best values. The first one is the best solution (fitness) it has achieved so far. (The fitness value is also stored.) This value is called pbest. Another best value that is traced by the particle swarm optimizer is the best value, obtained so far by any particle in the population. This best value is a global best and called gbest. When a particle taes part of the population as its topological neighbors, the nbest value is a local best and is called lbest. After finding the two best values, the particle updates its velocity and positions with following equations: V = W * V + c1* rand () * ( P X ) + c 2 * rand () * ( P X ) (8) id id id id id id 58

13 X i d = X i d + V i d (9) V id is the particle velocity; X id is the current particle (solution). P id and P gd are pbest and gbest. rand ( ) is a random number between (0, 1). c 1, c 2 are learning factors. Now, the problem is defined as follows: Given system (1), the objective is to design a single stage PSS in the form of i V s = U G c ( s ) ( p + d s + ) ω s I G c ( s ) = ( P + D s + ) s Where, P p, D and i, d and (10) I are the proportional, derivative and Integral gains, U are the input and out put gains or scaling factors Figure 10 Flow Chart for PSO 59

14 The input gains, p i, and output gain are tuned to their optimal values d by solving the minmax optimization problem defined in (11) using Particle Swarm optimization Technique. Minimize J = ( t ) max max δ δ ss δ ss Where J represents worst overshoot over selected regimes, U selected regimes (11) ( t) max δ and δ ss represents respectively the maximum and steady state values of torque angle deviation. Table.4 gives the PSO Optimized parameters for FLPSS Table 4 PSO Optimized parameters of FLPSS P I D Design of Artificial Neural Networ Power System Stabilizer (ANNPSS) The use of Artificial Neural Networs (ANNs) is the most powerful approach in Artificial Intelligence. ANNs are information processing structures which emulate the architecture and operational mode of the biological nervous tissue. Any ANN is a system made up of several basic entities (named neurons) which are interconnected and operate in parallel transmitting signals to one another in order to achieve a certain processing tas. One of the most outstanding features of ANNs is their capability to simulate the learning process. They are supplied with pairs of input and output signals from which general rules are automatically derived so that the ANN will be capable of generating the correct output for a signal that was not previously used Linear Optimal Control (LOC) Stabilizing method based on linear optimal control [6], mae a suitable damping at different loading condition, but also this method needs determination of all system states that seems to be very difficult. Using of ANN to self tuning of stabilizers parameters is introduced. In the proposed PSS designed in this wor, the ANN is trained by LOCPSS. This stabilizer uses the linear optimal control theory to design a linear state feedbac control loop. U 60

15 There are two important steps in the successful design of such a controller. One is to have a precise mathematical model of the controlled system. Another is to select a suitable weighting matrix in the performance index. The stabilization problem is to design a stabilizer which provides a supplementary stabilizing signal to increase the damping torque of the system. The design linear optimal stabilizer is based on the theory of linear optimal regulator. In order to formulate the problem of stabilization using linear optimal control theory, a set of state variables must be first selected. Then the state equation for the system is written in the vector matrix differential equation form x & = Ax + Bu (12) Where x is the state n vector, u is the control m vector, A and B are constant matrices of dimensions n x n and n x m respectively. Suppose that the performance index is to minimize the integral J = 1 T T ( x Qx + u Ru )dt (13) 2 0 Where Q is the positive semidefinite matrix and R is a positive definite matrix, then the optimal control for the system with performance index is given by u = x (14) Where 1 T = R B P (15) and P is the solution of the algebraic matrix Riccati equation PA + A T P PBR 1 B T P + Q = 0 (16) Riccati equation is the ey to the design of the linear optimal control systems. Once the matrices Q and R are nown, the matrix P can be obtained by solving above equation and then the optimal control signal U is calculated. The traditional way to select Q, R matrices is to use trial and error method. Many simulation studies have to be done in time domain with different weighting matrices to choose the ones that provide desired performance. Because of the complexity, the matrices Q and R are commonly chosen as diagonal matrices. The Single Machine Connected to Infinite Bus with Linear Optimal Controlled ANNPSS is shown in Figure

16 Figure 11 Linearized model of SMIB with LOCANNPSS Design of Artificial Neural Networ Power System Stabilizer (ANNPSS) using Linear Optimal Control (LOC) In the proposed PSS, the ANN is trained by using Linear Optimal Control theory to mae a suitable damping at different loading condition. Taing active power (P), reactive power (Q),Terminal Voltage(V t ) and the line reactance (X e ) as input layers and the optimal feedbac gain matrix = [ ] as output layer the training data is obtained for 180 operating conditions. Using this data a neural networ is trained with different algorithms and found that the cascaded forward feed bac propagation learning algorithm is capable of giving the best results with least error. The ANN is trained using Neural Networs tool box GUI/MATLAB environment and the trained networ is shown in Figure

17 Loadi ng condit ion Light Norm al Heav y Figure 12 Trained Neural networ The same networ has been simulated to obtain state feedbac gains for prefault and postfault conditions under multioperating conditions i.e., heavy, normal and light loads. ANNPSS is designed using the feedbac gains from all states and stabilizes the Single Machine Infinite Bus System as shown in Figure 11. Table 5 shows the PSS parameters computed offline and with ANN for different operating conditions. Fault Condi Inputs(p.u) State feedbac gains computed with LOC State feedbac gains computed with ANN tion P Q V X e Prefau lt 0. 7 Postfa ult 0. Prefau lt Postfa ult 7 Prefau lt 1. 2 Postfa ult Table 5 PSS parameters computed offline and with ANN for different operating conditions

18 4. RESULTS & DISCUSSIONS The performance of CPSS, FLPSS and LOCANNPSS for Prefault and Postfault conditions is investigated under three different loading conditions [3] shown in Table 6 at a unit step disturbance of. ± 5 %. Table 6 Loading Conditions Loading Conditions P Q Normal Heavy Light The Constants 1 to 6 constants for multioperating conditions tabulated in Table 7. Loading condition Table 7 1 to 6 constants for multioperating conditions Fault Constants Condition Normal load Prefault Postfault Heavy Load Prefault Postfault Light Load Prefault Postfault The performance of CPSS, FLPSS and LOCANNPSS is evaluated in MATLAB/SIMULIN for Prefault i.e., before the occurrence of a solid 3phase fault at bus 2 and postfault i.e., after the clearance of fault by line outage. The efficacy of PSS can be studied only after the clearance of fault by opening the circuit breaers at both ends simultaneously. These simulation studies are carried out under above loading conditions. Figure 13 to Figure 18 shows the rotor angle deviation response curves for a disturbance of +5% in T under Prefault and Postfault conditions for a solid threephase m fault at bus 2 when the system is considered at multioperating conditions. It can be observed that Without Power System Stabilizer (WOPSS), the system in unstable and the overshoot is minimized to negligible value and also settling time is significantly reduced with Artificial Neural Networ Power System Stabilizer (ANNPSS) where as Conventional Power System Stabilizer (CPSS) and Fuzzy Logic Power System Stabilizer (FLPSS) are stabilizing the system with small oscillations and overshoot. 64

19 Table 8 and Table 9 shows the comparison between WOPSS, CPSS, FLPSS and LOCANNPSS in view of percentage pea over shoot and settling time respectively for multioperating conditions. Table 8 Comparision between CPSS, FLPSS and LOCANNPSS (Percentage Pea Overshoot) Loading Fault condition Pea Overshoot (%) Light Heavy Normal CPSS FLPSS LOCANNPSS Prefault Postfault Prefault Postfault Prefault Postfault Table.9 Comparision between CPSS, FLPSS and LOCANNPSS (Settling Time) Loading Fault condition Settling Time (in sec.) CPSS FLPSS LOCANNPSS Light Prefault Postfault Heavy Prefault Postfault Normal Prefault Postfault From the comparision table, it can be observed that during postfault condition, the electrical power transfer is more than the prefault condition and therefore the pea over shoot is high compared to prefault condition. The aim of the proposed wor is to minimize the percentage pea over shoot and settling time of the step response of generator rotor angle by optimally tuning the parameters of ANNPSS using LOC. The performance of LOCANNPSS is compared with CPSS and FLPSS. It can be observed from the presented results that the LOC ANNPSS minimizes the percentage pea overshoot and settling time under multioperating conditions compared with CPSS and FLPSS. The eigen values of the system without any stabilizer, with CPSS and with Linear Optimal Controller are tabulated in Table

20 Table 10 Comparison of the Eigen values without stabilizer, with CPSS and with Linear Optimal Controller Fault Without stabilizer With CPSS condition ( pss =10.67, Loading Conditi on Normal (P=1.0, Q=0.0) Light (P=0.7, Q=0.3) Heavy (P=1.2, Q=0.2) Prefault ( X e =0.475) Post fault ( X e =0.65) Prefault ( X e =0.475) Post fault ( X e =0.65) Prefault ( X e =0.475) Post fault ( X e =0.65) 5. CONCLUSIONS i i , i i , i i , i i , i i , i i , T 1=0.485sec, T =0.05sec) i i , i i , i i , i i , i i , i i , With Linear Optimal Controller (LOC) i i , i i , i i , i i , i i , i i , From the results listed in Table 8, Table 9 and Table.10, it is evident that a Conventional Power System Stabilizer (CPSS) designed for certain operating point does not cover all operating conditions effectively. Under this situation, a nonlinear controller such as Fuzzy Logic Power System Stabilizer (FLPSS) can control the plant to some extent but with small percentage overshoot in its response whereas Linear optimal Control (LOC) based Artificial Neural Networ Power System Stabilizer (LOC ANNPSS) can perform very well with least overshoot and negligible settling time. 66

21 The simulation studies reveal that the proposed Linear Optimal Control based Artificial Neural Networ Power System Stabilizer (LOCANNPSS) is very effective to fault conditions and minimizes the overshoot and settling time. From the Fig 13 to 18, it is evident that the dynamic performance of the proposed Linear Optimal Control based Artificial Neural Networ Power System Stabilizer (LOC ANNPSS) is also superior to that of CPSS and FLPSS WOPSS CPSS FLPSS LOCANNPSS Rotor Angle Deviation > Time, t in sec > Figure 13 Rotor Angle Deviation for a disturbance of +5% in for prefault condition T under normal loading m WOPSS CPSS FLPSS LOCANNPSS Rotor Angle Deviation > Time t in sec > Figure 14 Rotor Angle Deviation for a disturbance of +5% in for postfault condition T under normal loading m 67

22 WOPSS CPSS FLPSS LOCANNPSS Rotor Angle Deviation > Time, t in sec > Figure 15 Rotor Angle Deviation for a disturbance of +5% in prefault condition T under light loading for m WOPSS CPSS FLPSS LOCANNPSS 0.08 Rotor Angle Deviation > Time, t in sec > Figure 16 Rotor Angle Deviation for a disturbance of +5% in postfault condition T under light loading for m 68

23 0.08 WOPSS CPSS 0.07 FLPSS LOCANNPSS 0.06 Rotor Angle Deviation > Time,t in sec > Figure 17 Rotor Angle Deviation for a disturbance of +5% in prefault condition T under heavy for m 0.1 WOPSS 0.09 CPSS FLPSS LOCANNPSS Rotor Angle Deviation > Time,t in sec > Figure 17 Rotor Angle Deviation for a disturbance of +5% in postfault condition APPENDIX T under heavy for m The System under study is a thermal generating station consisting of four 555MVA, 24V, 60Hz units. The networ reactances shown in Figure 1 are in p.u. on 2220MVA, 24V base(referred to LT side of stepup transformer). Resistances are assumed to be negligible. Equivalent generator parameters in p.u: X d = 1.81, X d = 0.3, X q = 1.76, T do = 8sec, H = 3.5MJ/MVA, V t = 1.0 Exciter: E = 25, T E = 0.05Sec. 69

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