Improving Power System Stability Using Transfer Function: A Comparative Analysis

In this paper, a small-signal dynamic model of a singlemachine infinite-bus (SMIB) power system that includes IEEE type-ST1 excitation system and PSS based on transfer function structure is presented. The changes in the operating condition of a power system on dynamic performance have been examined. The dynamic performance of the closed-loop system is analyzed base on its eigenvalues. The effectiveness of the parameters changes on dynamic stability is verified by simulation results. Three types of PSS have been considered for analysis: (a) the derivative PSS, (b) the lead-lag PSS or conventional PSS, and (c) the proportional-integral-derivative PSS. The objective function is formulated to increase the damping ratio of the electromechanical mode eigenvalues. Simulation results show that the PID-PSS performs better for less overshoot and less settling time compared with the CPSS and DPSS under different load operation and the significant system parameter variation conditions. Keywords-power system stabilizer; PID controller; lead-lag controller; stability.


INTRODUCTION
Modern power systems are complex, nonlinear and often exhibit electromechanical oscillations due to inadequate system damping [1,2].Power systems continuously experience changes during abnormal operating conditions due to variations in generation or load and a wide range of disturbances.Power system stability improvements have been considered an important problem for secure system operation over many years [3,4].Low frequency electromechanical oscillations are a characteristic of the power system and they are inevitable.These oscillations can be observed in most power system variables like line current, bus voltage, synchronous generator power and speed.Generally, the damping control methods of power system oscillations can be divided into two broad groups: damping control at generator locations (such as excitation control) and damping control in the transmission path (such as line reactance control).Because of the complexity of the network evolved from the interconnected large transmission systems and heavy generations, the use of the power system stabilizer (PSS) has become common by the utilities today.PSS plays an important role to suppress the electromechanical oscillation, increase the system positive damping and improve the steady-state stability margin and improve the stability in power system [5].Generally, PSS control design methodologies can be categorized as (a) classical method, (b) adaptive and variable structure methods, (c) robust control approaches, (d) intelligent techniques and (e) digital control schemes [6,7].A number of studies have been performed about the PSS parameters design and its applications to improve the dynamic stability of power systems [8,9].An adaptive fuzzy PSS based on robust synergetic control theory and terminal attractor techniques is developed in [10], which fuzzy logic systems are used to approximate the unknown power system dynamic functions without calling upon usual model linearization and simplifications.In [11] a PSS designed using the improved simple adaptive control based on quadratic performance, which this approach can track the reference model and decrease the control increment.A method of designing fixed parameter decentralized PSS for interconnected multi-machine power systems is proposed in [12].A technique for designing fixed parameter decentralized PSSs for interconnected power systems is proposed in [13], which local information available at each machine in the multi-machine environment, is used to tune parameters of PSS.A modified fruit fly optimization algorithm combined with a probabilistic approach to coordinate and optimize the parameters of PSS and SVC damping controller for improving the probabilistic small signal stability of power systems is proposed in [14].A space recursive least square algorithm developed for tuning of PSS parameters on SMIB power system based PID is proposed in [15] to meet the vulnerable conditions.An objective function and algorithm to obtain a set of optimal PSS parameters that include a feedback signal of a remote machine and local and remote input signal ratios for each machine in a multi-machine power system under various operating conditions in proposed in [16].A robust PID based PSS to properly function over a wide range of operating conditions is proposed in [17].
The objective of this paper is to investigate the effects of PSS based PID controllers on power system electromechanical oscillation damping.The synchronous generator is represented by the third-order model.The parameters of PSS are determined based on a linearized model of the power system around a nominal operating point where they can provided good performance.The effectiveness of the proposed PSS in increasing the damping of low-frequency oscillation is demonstrated in a SMIB for different operating conditions of the power system.

II. POWER SYSTEM DESCRIPTION
Power system stabilizer is used to enhance damping of power system oscillations, mainly through excitation control.The IEEE type-ST1 is used for the voltage regulator excitation system [18].The dynamic model in state-space form of the linearized SMIB power system model around an operation point can be expressed as [19]: where the state variables are angle load (), field voltage (E F ), angular velocity ( r ) and voltage proportional to direct axis flux linkages (E' q ).Also, J M is the generator inertia constant, D I is the inherent damping constant, T' do is the d-axis open circuit transient time constant, and  b is the base electrical angular velocity.
The primary inputs to the generating unit are the mechanical torque deviation (TM) and reference terminal voltage deviation (UR), which are supplied from a higher level of control.K1 and K2 are the constant derived from electrical torque, K3 and K4 are the constant derived from field voltage equation, and K5 and K6 are the constant derived from terminal voltage magnitude.The parameters Kl-K6 are constants for a particular operating point (PEO, QEO, UTO) but they are sensitive to power system network parameters and generator operating conditions.Figure 1 shows the functional block diagram of the SMIB power system based on control transfer function (between the output electrical torque and load angle), HQ(s), and the electrical loop transfer function (between exciter input and the output electrical torque), GE(s).Also GM(s) is the transfer function of the dynamic machine.
where  O (s) is the open loop characteristic polynomial in power system.It has four eigenvalues.Therefore, the characteristic equation of the open loop SMIB power system is given by: By varying the operating point, the coefficient parameter values p 0 through p 3 also vary.

A. Conventional Lead-Lag PSS
The conventional lead-lag PSS (CPSS) transfer function is given by the following [20]: where T W is the washout time constant and K C is the PSS pure gain.T D is the lead time constant and T G is the lag time constant.The selection of the T w value depends upon the type of mode under study [21].Figure 3

C. Proportional-Integral-Derivative PSS
A PID controller is commonly used by industrial utilities.It can be represented in transfer function form as [22,23]:

D. Close-Loop Transfer Function
The linearized model of the close-loop in a SMIB power system has six eigenvalues.The transfer functions in close loop system (with PSS) are given by the following: To increase the system damping, the eigenvalue-based objective function is considered as follows: where  i is the i th electromechanical mode eigenvalue.In the optimization process, it is aimed to minimize J in order to shift the poorly damped eigenvalues to the left in s-plane.

E. Characteristic Equations
The characteristic equation of the close loop power system with DPSS s defined as: where the coefficients a 0 through a 6 are given by: According to this equation, all coefficients are depending on T d , but a 3 only depend on the K d .The characteristic equation of the close loop SMIB power system equipped with PID-PSS is given by: where the coefficients d 0 through d 4 are given by:  The characteristic equation of the close loop SMIB power system equipped with CPSS is given by: where the coefficients m 0 through m 5 are given by the next equation according to which, the time constant T 1 only effected on m 3 and gain K P only effected on m 3 and m 2 .

IV. SIMULATION RESULTS
The small signal stability analysis of a SMIB power system is examined by the eigenvalues of the state matrix.To assess the effectiveness of the proposed controllers, three different loading conditions nominal, light and heavy as shown in Table I are considered for eigenvalue analysis.The data of the system is given in Table II.The constants K 1 to K 6 for the three operating points considered are given in Table III.Note that the constant K 5 is only positive for light loading.The system modes and damping ratio for electromechanical mode without PSS are given in Table IV.Note that the system without PSS is slightly damped only in light loading.The system response without applying any PSS is more oscillatory in heavy load condition.The maximum phase lagging of the open loop system in the H Q (s) is approximately -100 degree and in the G E (s) is approximately -140 degree at 10 rad/s, respectively.Therefore, one first-order blocks will be used to achieve the desired phase compensation.Also, the proportional and integral gains of PID controller are positive.The undamped natural angular frequency ( n ) is  n =5.4741 rad/s.The PSS parameters time-constants T W , T D , T G and gain K C are to be optimized.T W =10s and T G =0.05s are chosen.The required phase-lead can be obtained by choosing the value of time constant T D .The transfer function G E (s) in the j n is 1.2516-62.9522.Therefore, T D =0.8049s is obtained which provides the desired phase-lead of 62.9522.The optimal parameters of the PSS base on phase compensation design are shown in Table V.The system eigenvalues with the stabilizer for three different operating conditions are given in Tables VI, VII and VIII.The damping ratio of the electrometrical mode eigenvalue for different loading of the power system without PSS and PSS are shown in Table IX.A comparative between the conventional PSS and DPSS with PIDPSS in damping power system oscillation under normal operating conditions is shown in Figures 8, 9 and 10.The step response of the angular speed deviation under heavy operating conditions has been shown in Figure 11.The results show the superiorly of PIDPSS and CPSS over DPSS in increasing the damping of low frequency oscillations.Table X shows the summary of the system dynamic characteristics such as settling time (t s ), peak time (t p ) and percent overshoot (M P ).It is seen from these simulation studies that the PID-PSS is more effective than the CPSS and DPSS in damping the electromechanical oscillations under various loading conditions and its damping speed is much faster.Step response of the electrical torque deviation   The main function of power system stabilizer is suppressing the low-frequency oscillation of the power system to improve its dynamic stability.The dynamic response of a SMIB power system with PSS based on transfer function at various operating conditions has been investigated in this paper.Three types of PSS have been considered for analysis.The PID-PSS performs better for less overshoot and less settling time compared with the CPSS and DPSS under different load operation and the significant system parameter variation conditions.

Fig. 1 .Fig. 2 .
Fig. 1.Block diagram of the open loop SMIB power system based on control and electrical loop transfer functions

Fig. 3 .
Fig. 3.Phase frequency response characteristics of the CPSS

Fig. 4 .
Fig. 4. The structures of a DPSS Where K P represents the proportional gain, K I represents the integral gain, and K D represents the derivative gain.The phase angle diagram of the PID controllers for different values of gains are shown in Figure.6.The PID-PSS as shown in Figure 7 with rotor deviation as input have the following transfer function: ( ) ( )( ) 1

Fig. 8 .
Fig. 8.Step response of the angular speed deviation

Fig. 9 .
Fig. 9.Step response of the electrical torque deviation

TABLE IV .
SYSTEM MODES IN POWER SYSTEM WITHOUT PSS SYSTEM MODES FOR POWER SYSTEM WITH PSS UNDER HEAVY LOADING

TABLE IX .
DAMPING RATIO OF ELECTROMECHANICAL MODE FOR POWER SYSTEM UNDER DIFFERENT OPERATION CONDITIONS