Multi-objective non dominated sorting genetic algorithm-II optimized PID controller for automatic voltage regulator systems

Abstract

Automatic Voltage Regulators (AVRs) are widely used to regulate the system voltage. For satisfactory performance of AVR system, optimal design of the Proportional Integral Derivative (PID) controllers is essential. Design tasks of controllers are multiple objective problems since the controller has to satisfy performance measures. The present paper reports the performance study of PID controller with multi-objective Non Dominated Shorting Genetic Algorithm-II (NSGA-II). The design objectives are to enhance the controller performance while minimizing integral error criteria, settling time and maximum overshoot of terminal voltage. Initially a simple AVR system has been considered for checking the superiority of the proposed multi-objective approach. Comparative analysis has been carried out with recently reported techniques. Additionally, the study is extended to an autonomous power system with synchronous and diesel generators. Finally, robustness analysis is also reported by varying the time constant of generator, exciter, amplifier and sensor.

Keywords

Automatic voltage regulator multi objective autonomous power system

1 Introduction

Stability of the voltage level in electrical power system is also identified to be the preliminary control constraint [1]. Maintaining constant voltage is critical as all the equipments connected through the power network will be designed for certain voltage rating. Any violation in the operating voltage level affects the performance and life time of the equipment. Other reason to study voltage control is due to the fact that line losses depend on real and reactive power flow and this reactive power flow depends on terminal voltage (Vt) of the network [2]. To facilitate these mechanism of these controls, Automatic Voltage Regulator (AVR) is adopted with generating units [3]. AVR gains control on Vt by varying the generator excitation voltage. AVR control mechanism is basically a closed loop system which is being compensated with a PID controller which controls V_t to the required value. In view of wide operating ranges, simplicity and high performance PID controllers are been in usage in the instrumentation and process control applications [4 –6]. Proportional, integrating and derivative gains are the three individual control factors of a fundamental PIDcontrol.

Numerous approaches have been reported in the literature to tune these control factors. Traditionally, Ziegler–Nichols method is adopted to compute the control factors of a PID Controller [7, 8]. Recently, various intelligence techniques including neural networks, fuzzy logic, Adaptive neuro-fuzzy systems and several optimization techniques were reported to tune the PID controller factors [7 –9]. Linear-quadratic regulator (LQR) technique is also adopted to design an optimal controller for second order systems [10]. From the comprehensive literature survey it is observed that, modern heuristic optimization techniques have been also applied for optimal designing of the AVR systems with various traditional and modified single objective functions. Meanwhile, several authors reported the same with multiple objective functions. However, in case of these multi-objective studies, several single objective functions have been combined by adopting suitable weights. Nevertheless, in practice identification of these weights will be a challenging aspect and leads scope of the element error or brings the down the actual performance. Additionally, the final solution obtained may be subjected to an inconsistent behavior thus leading to demand of optimization techniques for selection of change in weights.

2 System modeling and proportional integral derivative (PID) controller design

Various transfer functions and the ranges of their control parameters of PID based AVR system shown in Fig. 1 have been tabulated in Table 1 [13].

Fig.1

Transfer function of AVR system with PID controller.

Table 1

Transfer function of individual blocks and the limits of the parameters

Component	Transfer function	Parameter limits
PID controller	$G_{PID} = K_{p} + \frac{K_{i}}{s} + {sK}_{d}$	0.0001 ≤ K_p, K_i, K_d ≤ 1.0
Integral controller	$G_{I} (s) = \frac{K_{ii}}{s}$	0.0001 ≤ K_ii ≤ 1.0
Amplifier	${TF}_{amplifier} = \frac{K_{a}}{1 + {sT}_{a}}$	10 ≤ K_a ≤ 40 ; 0.02s ≤ T_a ≤ 1.0s
Exciter	${TF}_{exciter} = \frac{K_{e}}{1 + {sT}_{e}}$	1≤K_e≤10;0.4s≤T_e≤1.0s
Sensor	${TF}_{sensor} = \frac{K_{s}}{1 + {sT}_{s}}$	0.001s ≤ T_s ≤ 0.06s
Generator	${TF}_{generator} = \frac{K_{g}}{1 + {sT}_{g}}$	K_g depends on load (0.7–1.0); 1.0s≤T_g≤2.0s
Diesel engine generator	${TF}_{deg} = \frac{1}{1 + {sT}_{deg}}$	T_deg = 2.31s
Valve actuator	${TF}_{valveactuator} = \frac{1}{1 + {sT}_{va}}$	T_va = 0.82s

The values of proportional gain (K_p), integral gain (K_i) and derivative gain (K_d) can be computed through the heuristic optimization techniques to minimize the objective function values. There are different performance criteria that are traditionally used in the system design, one among them is Integral of Time-squared Error (ITSE). Improved performance of a system is attributable to objective functions ITSE, Ts, and Mos which often conflicts among them. Hence, the same may be overcome by using a suitable multi-objective technique. In the present work, multi-objective NSGA-II technique has been adopted to generate Pareto solutions and to compute optimal values of K_p, K_i and K_d

Formulation of objective function J is presented below. $J = (J_{1}, J_{2}, J_{3})$ (1)

The transfer function of Fig. 1 is as follows.

$\frac{Δ V_{t} (s)}{Δ V_{r e f} (s)} = \frac{0.1 K_{d} s^{2} + (0.1 K_{p} + 10 K_{d}) s^{2} + (0.1 K_{i} + 10 K_{p}) s + 10 K_{i}}{0.0004 s^{5} + 0.045 s^{4} + 0.555 s^{3} + (1.51 + 10 K_{d}) s^{2} + (1 + 10 K_{p}) s + 10 K_{i}}$ (2)Where $J_{1} = ITSE = \int_{0}^{t_{sim}} {(V_{ref} - V (t))}^{2} \cdot t \cdot dt$ (3) $J_{2} = T_{S}$ (4) $J_{3} = M_{OS}$ (5)

In the above equations, V_ref is reference voltage, V (t) is terminal voltage, t_sim is simulation time, TS and MOS are the settling time and maximum overshoots in the terminal voltage response respectively.

3 Simulation results of multi objective NSGA-II

Authors have reported the implementation details of multi objective NSGA-II in [14]. The basics of the same are also explicitly discussed and reported in [15, 16]. The terminal voltage response of multi objective NSGA-II in comparison to several techniques reported in literature are presented in Fig. 2. The optimal values of K_p, K_i and K_d are 0.7769, 0.7984, and 0.3615 respectively.

Fig.2

Terminal Voltage response of AVR with different approaches.

4 Extension to autonomous power system (APS)

The APS refers to a system which is self sufficient, thus can supply any extra load that is in demand during peak hours or during any contingency. A standard APS model of a typical Diesel Engine Generator (DEG) comprises of a typical speed governor, AVR and PID controller [10 –18] is considered in the present work and is presented in Fig. 3.

Fig.3

Block diagram of the studied autonomous power system.

The blocks shown in Fig. 3 represent the standard mechanical model of a DEG with a speed governor. DEG supplies power when there is a mismatch in generation and demand. The same causes the rotor to accelerate or de-accelerate. The acceleration/de-acceleration is sensed by the inertia and load block and an error signal, which is the function of torque, is generated. The integral controller comes into play and it changes the valve position proportional to the change in power demand. The integral controller eliminates the steady state frequency error. Various transfer functions and the ranges of their control parameters of PID based autonomous power system are presented in Table 1 [13].

5 Design of controller for autonomous power system (APS)

To illustrate the performance of proposed design, initially the controller parameters are optimized employing a single objective optimization technique using single objectives. Minimization of ITSE given by Equation (3) and ITAE value given by Equation (6) are considered as single objectives. Differential Evolution (DE) is identified to optimize the controller parameters. In the present paper the control parameters of DE are adopted from [20 –22]. $J_{4} = ITAE = \int_{0}^{t_{sim}} (| V_{ref} - V (t) |) \cdot t \cdot dt$ (6)

Proposed approach is employed to generate the Pareto solutions to achieve optimal values of various gains. The obtained gains have been used to minimize the objective function values. These objective functions are rewritten for clarity as below: $J_{4} = ITAE = \int_{0}^{t_{sim}} (| V_{ref} - V (t) |) \cdot t \cdot dt$ (7) $J_{5} = T_{S}$ (8) $J_{6} = M_{OS}$ (9)

Figure 4 presents the step response of change in the V(t) for APS using single objective DE for J₄, J₅, J₆ and proposed approach. It is clear from Fig. 4 that the proposed approach out performs the single objective approaches in terms of T_s and M_os. Later, robustness analysis of APS is also performed by changing the time constant of amplifier as shown in Fig. 5.

Fig.4

Objective functions with techniques.

Fig.5

Voltage change curves for – 50% changes in Ta.

Table 2

Objective function values of various evolutionary algorithms

Technique/objective function	ITSE	Settling Time (s)	Maximum overshoot (p.u.)
ABC [27]	0.0183	0.9200	1.2500
PSO [27]	0.0248	1.0000	1.3006
DEA [27]	0.0226	0.9500	1.3286
MOL [28]	0.0066	0.7300	1.1140
Proposed method	0.01	0.27	1.05

6 Conclusion

In this study, a multi-objective NSGA-II has been presented for optimal design of a PID controller in the AVR system for improving the step response of V(t). Minimization of an integral based error criteria, settling time and maximum overshoot in terminal voltage response for a unit step input are formulated as multiple objectives and NSGA-II is employed to search the optimal PID controller parameters. The optimized step response of terminal voltage using the NSGA-II method is improved compared to some recently proposed single objective optimization approaches such as ABC, DE, PSO and MOL. Robustness analysis is performed to show the effectiveness of designed controllers. The proposed approach is also implemented to an APS and the superiority of proposed multi-objective NSGA-II approach over single objective DE approaches is demonstrated.

References

Kundur

, Power system stability and control, New York: McGraw-Hill, 1994.

Elgerd

, Electric energy systems theory: An introduction. New Delhi: Tata McGraw-Hill, 1983.

Gozde

and Taplamacioglu

M.C.

, Comparative performance analysis of artificial bee colony algorithm for automatic voltage regulator (AVR) system, Int J Franklin Ins348(8) (2011), 1927–1946.

Astrom

K.J.

and Hagglund

, The future of PID control, Control Eng Practice9(11) (2001), 1163–1175.

Visioli

, Tuning of PID controllers with fuzzy logic, IEE Proc Control Theory Appl148(1) (2001), 1–8.

, Ang

K.H.

and Chong

G.C.Y.

, PID control system analysis and design, problems, remedies, and future directions, IEEE Control Syst Mag26(1) (2006), 32–41.

S.R.

, Wang

D.F.

, Han

and Li

Y.H.

, Grey prediction based RBF neural network self-tuning PID control for turning process, Proc Int Conf Mach Learning Cybern2 (2004), 802–805.

Rubaai , Castro-Sitiriche

M.J.

and Ofoli

A.R.

, Design and implementation of parallel fuzzy PID controller for high-performance brushless motor drives: An integrated environment for rapid control prototyping, IEEE Trans Ind Appl44(4) (2008), 1090–1098.

Rubaai

Y.P.

, EKF-based PI-/PD-like fuzzy-neural-network controller for brushless drives, IEEE Trans Ind Appl47(6) (2011), 2391–2401.

10.

and Hwang

R.C.

, Optimal PID speed control of brushless DC motors using LQR approach, Proc Int Conf Mach Learning Cybern2004, pp. 473–478.

11.

Bingquan

, Buckley

and Kechadi

T.M.

, Multi-objective feature selection by using NSGA-II for customer churn prediction in telecommunications, Expert Systems with Applications37(5) (2013), 3638–3646.

12.

Panda

, Multi-objective evolutionary algorithm for SSSC-based controller design, Electr Power Syst Res79(6) (2009), 937–944.

13.

Panda

, Sahu

B.K.

and Mohanty

P.K.

, Design and performance analysis of PID controller for an automatic voltage regulator system using simplified particle swarm optimisation, J of the Franklin Ins349(8) (2012), 2609–2625.

14.

Panda

and Yegireddy

N.K.

, Automatic generation control of multi area power system using multi objective non dominated sorting genetic algorithm-II, Electr Power Syst Res53 (2013), 54–63.

15.

Srinivas

and Deb

, Multi-objective optimization using non-dominated sorting in genetic algorithms, IEEE Trans Evolut Comput2(3) (1994), 221–248.

16.

Deb

, Pratap

, Agarwal

and Meyarivan

, A fast elitist multi-objective genetic algorithm: NSGA-II, IEEE Trans Evolut Comput6(2) (2002), 182–197.

17.

Milano

, Power System Modeling and Scripting, Springer, 2010.

18.

Banerjee

M.V.

and Ghoshal

S.P.

, Seeker optimization algorithm for load tracking performance of an autonomous power system, Energy Sys43(1) (2012), 1162–1170.

19.

Mukherjee

and Ghoshal

S.P.

, Intelligent particle swarm optimized Fuzzy PID controller for AVR system, Electr Power Syst Res72(12) (2007), 1689–1698.

20.

Fan

and Joo

, 4th IEEE Conference on Ind Electron Appl., May 2009, Xi’an, China.

21.

Vural

K.A.M.

, Transient stability enhancement of the power system interconnected with wind farm using generalized unified power flow controller with simplex optimized self-tuning fuzzy damping scheme, Int Rev Electr Eng7 (2012), 5091–5107.

22.

Mohanty

P.S.

and Hota

P.K.

, Controller parameters tuning of differential evolution algorithm and its application to load frequency control of multi-source power system, Int J Electr Power Energy Syst54 (2014), 77–85.