Robust FOPID Controller for Load Frequency Control Using Particle Swarm Optimization Ashu Ahuja Associate Professor, MMEC, Mullana, Haryana, India, email-id:
[email protected]
Shiv Narayan Professor, PEC University of Technology, Chandigarh, India, email-id:
[email protected]
Abstract- In this paper, design of Fractional order proportional – integral – derivative (FOPID) controller is designed for load frequency control (LFC) of power system. The performance of FOPID controller is compared with proportional – integral – derivative (PID) and proportional – integral (PI) controllers. The comparison is made based on various time domain performance indices such as Integral of absolute error (IAE), Integral of square error (ISE), Integral of time absolute error (ITAE), Integral of time square error (ITSE) and integral of square time square error. When load frequency control of power system is to design for open communication network, delay occur in area control error (ACE) signal. Delay issues in damping frequency oscillation w.r.t. change in load variation are also discussed. Particle Swarm Optimization (PSO) technique is being used to tune the parameters of the controllers. It is shown that FOPID controller has better dynamic performance than other controllers. Keywords- LFC; FOPID; PSO; deviation in frequency; error criterions.
I. INTRODUCTION A frequency oscillation when load is suddenly changed is an important issue in power system [1]. Convention PI [25]and PID controller [6-14] have been used to damp out such oscillation in the literature in single area and multi area. Many artificial intelligence (AI) based controllers have also been investigated by the various researchers like decentralized controllers such as sliding mode control[1518], artificial neural network (ANN) controller [19], fuzzy logic (FL) controller [20-22], and neuro-fuzzy controller [23]. Many optimization techniques have also been applied to tune the parameters of the various controllers such as Differential Evolution (DE) [6, 24], Genetic Algorithm [7], and craziness based particle swarm optimization [12]. However, all of these controllers are integer controllers, but many of the physical systems are realized by fractional order differential equations. So, designing the fractional order controllers for these kinds of systems ensure better performance than integer controllers [25, 26]. Therefore, the main focus of the paper is designing the FOPID controller for load frequency control. In this paper, five tuning parameters ( K p , K i , K d , λ , μ ) have been tuned to design FOPID controller. PSO is used for optimization [27, 28] and various objective functions (IAE, ISE, ITAE, ITSE and ISTSE) have been employed for the purpose. To show the effectiveness of FOPID controller the results are compared with PID and PI controller. The effect of delay when power system works in open communication network has also been discussed. The various delay issues discussed in this paper are gain crossover frequency, phase margin, gain margin and robustness. Presentation of this paper is as follows: Section II explains the problem formulation of load frequency control. Section IIIdescribes fractional order system and controllers.Section IV describes PSO technique. Section V is devoted to tuning
Jagdish Kumar Associate Professor, PEC University of Technology, Chandigarh, India, email-id:
[email protected]
of various controllers for LFC and their comparison. The effect of delay is also discussed in this section. Section VI concludes the paper. II. LOAD FREQUENCY CONTROL PROBLEM In a power system, the frequency should remain nearly constant for satisfactory operation. Relatively close control of frequency ensures constancy of speed of induction and synchronous motor which is required for satisfactory performance of generating units. Frequency deviation cause high magnetizing currents in induction motors and transformers. Also, change in frequencyreflects change in active power demand. As there are many generators supplying power into the system, some means must be provided to allocate change in demand to the generators. A speed governor on each generating unit provides the speed control function [1]. The control of generation and frequency is referred as load frequency control (LFC). LFC model is shown in Fig. 1 in detail.
Fig. 1 Load Frequency Control Model Symbols used are given in table 1. Table 1 Symbols used in LFC model Δf
Frequency deviation
ΔPD
Change in load demand
ΔPg
Electromechanical air gap power
ΔPv
Governor valve position
ΔPT
Mechanical power output of turbine-generator
Tg = 0.4
Governor time constant (in sec)
Tt = 0.5
Turbine time constant (in sec)
K PS = 100
Power system gain
TPS = 20
Power system time constant (in sec)
R=3
Governor speed droop
III. FRACTIONAL ORDER SYSTEMS Many real-world physical systems need fractional calculus as they are modeled by fractional-order differential equations, i.e., equations involving non-integer order integrals and derivatives. A. Fractional Calculus Fractional calculus may be explained as the extension of the concept of a derivative operator from integer order ‘n’ to
arbitrary order ‘z’ where z may be a real value or a complex : value or may be a complex valued function dn dz → z n dx dx The most commonly referred definition of fractional derivatives and fractional integrals was given by G. F. B. Riemann and J. Liouville, according to which, the fractional integral of order for a function is given by: x
0
D x− n f ( x ) =
³ c
( x − t ) n −1 f ( t ) dt , ( n − 1)!
with the condition that
and
n∈ N 0
D
−n x
(1) f ( x ) are causal
functions. For initial conditions to be zero, the Laplace Transform of is given by L[ 0 D xz ] = s z F ( s ) i.e., for zero initial conditions, the system whose dynamic behavior described by differential equations having fractional derivatives results in transfer functions with fractional orders of s. To simulate fractional order of s in MATLAB, this is to be approximated by usual integer order transfer function having an infinite number of poles and zeroes. It is also possible to logically approximate it with a finite number of poles and zeros. Oustaloup proposed a method of approximation of a function of the form [29]: (2) by a rational function: (3)
The differential equation of a fractional order PIȜDȝ controller can be described as: u (t ) = K p e(t ) + K i Dt− λ e(t ) + K d Dtμ e(t ) (8) and its transfer function can be given as: K ( s) = K p + K i s − λ + K d s μ
(9)
FOPID controller involves selection of five parameters: three parameters [ K p , K i , K d ] (same as PID) and two fractional parameters λ , μ . More flexibility is added in accomplishing control objective by this expansion. So, there are five parameters to tune now:
p = [K p , K i , K d , λ, μ ] C. Objective Function Various time domain error performance indices taken as objective function which are minimized using PSO are: (i) Integral of Absolute error ∞
IAE = ³ e(t ) dt
(10)
0
(ii)
Integral of Square error ∞
ISE = ³ e 2 (t )dt
(11)
0
(iii) Integral of time multiplied absolute error ∞
ITAE = ³ t e(t ) dt
(12)
0
and by applying the following synthesis formulas: (4)
(iv) Integral of time multiplied square error ∞ (13) ITSE = te 2 (t )dt
³ 0
(v) Integral of square time multiplied by square error ∞ (14) ISTSE = t 2 e 2 (t )dt (5) here is geometrical mean of the unit gain frequency and the central frequency of a band of frequencies distributed around it. That is, (6) where, and are the high and low transitional frequencies. B. Designed ControllersPI controller has only two terms to control and is the simplest one. Tuning parameters for PI controller are: p = [ K p , K i ] PID control with its three term functionality offers the simplest solution to many real world control problems. A PID controller with four tuning parameters is usually selected: K Kd s (7) K ( s) = K + i + p
s
τ d s +1
Tuning parameters of the controller are: p = [ K p , K i , K d ,τ d ] . τ d term is introduced to attenuate noise amplification.
³ 0
IV. PARTICLE SWARM OPTIMIZATION The PSO was developed by J. Kenedy and R. Eberhart in 1995. It is an evolutionary optimization technique based on the movement and intelligence of swarm that move around in a search space looking for the best solution. Each particle is treated as a point in N-dimensional space which adjusts its flight according to its own flying experience as well as the flying experience of the other particles. Each particle tries to improve its position using its current position, its current velocity, the distance between its current position and its best position ‘pbest’ and the distance between its current position and global best position ‘gbest’[28]. Proposed algorithm is summarized as follows: (i) Randomly initialize the particles of the population including searching points and velocities for each parameter of the controller. (ii) For each initial particle ‘i’ of the population, calculate the values of the fitness function in (10) to (14).
(iii) Compare each particle’s calculated value with its personal best Ki. The best calculated value among the Ki is denoted as Kg. (iv) Modify the member velocity of each particle ‘i’ by adjusting value of inertia weights. (v) Modify the member position of each particle ‘i’ w.r.t. its best position and global best position. (vi) If the number of iterations reaches the maximum, then go to step (vii), otherwise go to step (ii). (vii) The latest Kg is the optimal controller parameter.
Kp
0.08
0.08
0.0799
Ki
0.19
0.2011
0.3000
Kd
---
0.1825
0.1732
τd
---
0.005
---
λ
---
---
0.9545
μ
---
---
0.7010
F(Fitness function)
4.8010
1.0576
1.2681
Deviation in frequency with ISEas objective function 0.06 PID FOPID 0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1 0
10
20
30
40
50
60
Time (seconds)
Fig.3 Comparison of PI, PID and FOPID based on ISE Table3 Comparative values of parameters using ISE Parameters Settling Time (sec) Overshoot Undershoot Gain Margin Phase Margin Gain Crossover Freuency Phase Crossover Freuency Kp
PI 38.4 0.0449 -0.0807 0.0385 75.8011 1.4858 0.1981 0.08
PID 7.86 0.0128 -0.0544 0.1891 80.960 1.2789 0.2857 0.08
FOPID 4.99 ---0.0545 0.3060 68.7148 1.5390 0.2125 0.0798
0
Ki
0.19
0.3
0.2996
-0.02
Kd
---
0.21
0.2098
-0.04
τd
---
0.0047
---
-0.06
λ
---
---
0.9465
μ
---
---
0.7023
F(Fitness function)
0.1535
0.0381
0.0344
Deviation in frequency with IAEas objective function 0.06 PI PID
0.04
FOPID 0.02
Deltaf
PI 0.04
Delta f
V. SIMULATION AND DISCUSSION The parameters of FOPID, PID and PI controllers have been designed using PSO techniques with the objective functions given by equations (10-14). The simulation has been done using MatLabR2007a. The parameters of the LFC model are given in Table 1.Initialization parameters used for PSO are: population size = 25, maximum number of iterations = 2500, minimum and maximum velocities are 0 and 2, cognitive and social acceleration coefficient ‘C1’ = 2.1, ‘C2’ = 1.3, minimum and maximum inertia weight = 0.6 and 0.9.Various comparative figures are given in Fig. 2-6 and their comparative values are given in Tables 2-6. Figures and tables show that settling time, overshoot, undershoot and phase margin are less and gain margin, gain and phase crossover frequencies are more in case of FOPID controller than PID controller except ITAE where settling time is less in case of PID. Settling time and overshoot are very much high in case of PI controller. Therefore, FOPID controller is best out of these three controllers.
-0.08
-0.1 0
10
20
30
40
50
60
Time (seconds)
Deviation in frequency with ITAEas objective function
Fig.2 Comparison of PI, PID and FOPID based on IAE
0.06 PI PID
0.04
FOPID
Table2 Comparative values of parameters using IAE
0.02
PI 38.4 0.0449 -0.0807 0.0385 75.8010 1.4859
PID 4.97 0.00213 -0.0577 0.2188 76.0044 1.2204
FOPID 4.88 ---0.0576 0.2575 70.0350 1.5287
0.1981
0.2016
0.2138
Delta f
0
Parameters Settling Time (sec) Overshoot Undershoot Gain Margin Phase Margin Gain Crossover Freuency Phase Crossover Freuency
-0.02
-0.04
-0.06
-0.08
-0.1 0
10
20
30 Time (seconds)
40
50
60
;
Fig.4 Comparison of PI, PID and FOPID based on ITAE Deviation in frequency with ISTSEas objective function 0.06
Table4 Comparative values of parameters using ITAE PI 38.4 0.0449 -0.0807 0.0385 75.801 1.4859 0.1981
PID 3.4 0.0011 -0.0581 0.2211 75.568 1.2155 0.1941
FOPID 4.68 0.00046 -0.0568 0.2834 71.1714 1.4281 0.2103
Kp
0.08
0.08
0.0543
Ki
0.19
0.1927
0.3
Kd
---
0.1795
0.1909
τd λ μ
---
0.005
---
---
---
0.9545
---
---
0.6995
F(Fitness function)
46.323
1.4539
10.8552
FOPID
0
-0.02
-0.04
-0.06
-0.08
-0.1 0
10
20
30
40
50
60
Time (seconds)
Fig.6 Comparison of PI, PID and FOPID based on ISTSE Table6 Comparative values of parameters using ISTSE Parameters Settling Time (sec) Overshoot Undershoot Gain Margin Phase Margin Gain Crossover Freuency Phase Crossover Freuency
PI 38.4 0.0452 -0.0807 0.0385 75.801 1.4859
PID 6.11 0.00521 -0.055 0.2228 77.682 1.2123
FOPID 4.86 ---0.0547 0.2606 70.0496 1.5271
0.1981
0.2305
0.2136
0
Kp
0.08
0.08
0.0788
-0.02
Ki
0.19
0.2347
0.3
-0.04
Kd
---
0.21
0.1756
τd λ μ
---
0.0046
---
---
---
0.9545
---
---
0.7
F(Fitness function)
6.7069
0.0713
0.1559
Deviation in frequency with ITSEas objective function 0.06 PI PID
0.04
FOPID 0.02
Delta f
PID
0.02
Delta f
Parameters Settling Time (sec) Overshoot Undershoot Gain Margin Phase Margin Gain Crossover Freuency Phase Crossover Freuency
PI 0.04
-0.06
-0.08
-0.1 0
10
20
30
40
50
60
Time (seconds)
Fig.5 Comparison of PI, PID and FOPID based on ITSE Table5 Comparative values of parameters using ITSE Parameters Settling Time (sec) Overshoot Undershoot Gain Margin Phase Margin Gain Crossover Freuency Phase Crossover Freuency
Kp
PI 38.4 0.0452 -0.0807 0.0385 75.801 1.4859 0.1981 0.08
PID 5.83 0.0082 -0.0547 0.2091 78.9379 1.2366 0.2517 0.08
FOPID 4.88 ---0.0545 0.3052 69.463 1.5397 0.2123 0.0799
Fig. 7 LFC model with delay Effect of delay with IAEas objective function 0.08 without delay 0.06
0.5 s delay 1.5 s delay
0.04
Ki
0.19
0.2594
0.2999
Kd
---
0.21
0.2096
τd λ μ
---
0.0006
---
---
---
0.8544
-0.06
---
---
0.7000
-0.08
F(Fitness function)
0.6860
0.0468
0.0421
-0.1
2.5 s delay
0.02
Delta f
0 -0.02 -0.04
-0.12 0
5
10
15
20 Time (seconds)
25
30
35
40
Fig. 8 Deviation in frequency in LFC with different Communication delays Table7 Comparative values of parameters with FOPID having different delays using IAE Parameters
No delay
0.5 s delay
1.5 s delay
2.5 s delay
Settling Time (sec)
4.88
15.9
21.8
25.3
Overshoot Undershoot Gain Margin Phase Margin
--0.0576 0.2575 70.035
0.02 -0.0839 0.1034 73.962
0.0697 -0.1050 0.0643 89.280
0.0463 -0.116 0.0810 89.326
Gain Crossover Freuency Phase Crossover Freuency
1.5287
1.5476
1.2254
1.1578
0.2138
0.1621
0.1943
0.1424
Kp
0.0799
0.05
0.05
0.05
Ki
0.3000
0.2309
0.2999
0.2231
Kd
0.1732
0.0975
0.0900
0.0900
λ
0.9545
0.9545
0.9545
0.9545
μ
0.7010
0.7100
0.7100
0.7100
F(Fitness function)
1.2681
2.3669
4.3542
4.7201
Fig. 7 and table 7 discuss the effect of communication delay in LFC. As the delay increases, the performance of the system degrades. Settling time and overshoot increases. Gain margin and crossover frequency also decreases. As a result of all issues, significant dead time is a significant source of instability for closed loop response. CONCLUSION VI. In this paper, a comparison of PI, PID and FOPID controllers for LFC has been investigated. The comparison involves time domain performance criterion. In order to examine the effect of cost function, different error criterions (IAE, ISE, ITAE, ITSE and ISTSE) are examined. The performance of PI, PID and FOPID is compared using PSO and it is observer that in each case, FOPID controller has better performance characteristics than PID and PI controllers. The effect of time delay is also studied and it is observed that as time delay increases ultimate gain and crossover frequencies decreases. [1] [2] [3]
[4]
REFERENCES P. Kundur, Power System Stability and Control, New York, McGraw Hill, 1994. H. Glavitsch and J. Stoffel, “Automatic generation control,” Electrical Power and Energy Systems, vol. 2, no. 1, January 1980, pp. 21-28. H. Shayeghi, H. A. Shayanfar, A. Jalili, “Load frequency control strategies: A state-of-the-art survey for the researcher,” Energy Conversion and Management, vol. 50, no. 2009, pp. 344–353. S. K. Pandey, S. R.Mohanty, and N. Kishor, “A literature survey on load–frequency control for conventional and distribution generation power
[5]
[6]
[7]
[8] [9]
[10] [11]
[12]
[13]
[14]
[15]
[16]
[17]
systems,” RenewableandSustainableEnergyReviews, vol. 25, 2013, pp. 318–334. Ibraheem, P. Kumar, and D. P. Kothari, “Recent Philosophies of Automatic Generation Control Strategies in Power Systems,” IEEE Transactions on Power Systems, vol. 20, no. 1, February 2005, pp. 346-357. U. K. Rout, R. K. Sahu, and S. Panda, “Design and analysis of differential evolution algorithm based automatic generation control for interconnected power system,” Ain Shams Engineering Journal,vol. 4, issue 3, pp. 409-421, 2013. S. Panda and N. K.Yegireddy, “Automatic generation control of multi-area power system using multi-objective non-dominated sorting genetic algorithm-II,” Electrical Power and Energy Systems, vol. 43, pp. 54-63, 2013. A. K. Mahalanabis and G. Ray, “Modeling large power systems for efficient load frequency control,” Mathematical Modelling, vol. 7, pp. 259-272, 1986. E. Cam and I. Kocaarslan, “Load frequency control in two area power systems using fuzzy logic controller,” Energy Conversion and Management, vol. 46, pp. 233-243, 2005. W. Tan, “Unified tuning of PID load frequency controller for power systems via IMC,” IEEE Transactions on Power Systems, vol. 25, no. 1, 2010. M. Farahani and S. Ganjefar, “Solving LFC problem in an interconnected power system using superconducting magnetic energy storage, Physica C, vol. 487, pp. 60-66, 2013. H. Gozde and M. C. Taplamacioglu, “Automatic generation control application with craziness based particle swarm optimization in a thermal power system,” Electrical Power and Energy Systems, vol. 33, pp. 8-16, 2011. A. M. Stankovic, G. Tadmor, and T. A. Sakharuk, “On robust control analysis and design for load frequency regulation,” IEEE Transactions on Power Systems, vol. 13, no. 2, 1998. H. Shabani, B. Vahidi, and M. Ebrahimpour, “A robust PID controller based on imperialist competitive algorithm for load-frequency control of power systems,” ISA Transactions, vol. 52, pp. 8895, 2013. T. C. Yang, Z. T. Ding, and H. Yu, “Decentralized Power System load frequency control beyond the limit of diagonal dominance,” Electrical Power and Energy Systems, vol. 24, pp. 173-184, 2002. M. Zribi, M. Al-Rashed, and M. Alrifai, “Adaptive decentralized load frequency control of multi-area power systems,” Electrical Power and Energy Systems, vol. 27, pp. 575-583, 2005. K.R. Sudha and R. VijayaSanthi, “Robust decentralized load frequency control of interconnected power system with Generation Rate Constraint using Type-2 fuzzy approach,” Electrical
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27] [28] [29]
Power and Energy Systems, vol. 33, pp. 699-707, 2011. M. T. Alrifai , M. F. Hassan, and M.Zribi, “Decentralized load frequency controller for a multiarea interconnected power system,” Electrical Power and Energy Systems, vol. 33, pp. 198-209, 2011. H. Shayeghi and H.A. Shayanfar, “Application of ANN technique based on l-synthesis to load frequency control of interconnected power system,” Electrical Power and Energy Systems, vol. 28, pp. 503-511, 2011. E. Cam andI.Kocaarslan, “Application of fuzzy logic for load frequency control of hydro-electrical power plants,” Energy Conversion and Management, vol. 46, pp. 233-243, 2005. S.Pothiya, I.Ngamroo, “Optimal fuzzy logic-based PID controller for load–frequency control including superconducting magnetic energy storage units,” Energy Conversion and Management, vol. 49, pp. 2833-2838, 2008. K.R. Sudha, R. VijayaSanthi, “Load Frequency Control of an Interconnected Reheat Thermal system using Type-2 fuzzy system including SMES units,” Electrical Power and Energy Systems, vol. 43, pp. 1383–1392, 2012. R. Farhangi, M. Boroushaki, and S. H.Hosseini, “Load–frequency control of interconnectedpower system using emotional learning-based intelligent controller,” Electrical Power and Energy Systems, vol. 36, pp. 76-83, 2012. R. K.Sahu, S. Panda, U. K. Rout, “DE optimized parallel 2-DOF PID controller for load frequency control of power system with governor dead-band nonlinearity”, Electrical Power and Energy Systems, vol. 49, pp. 19–33, 2013. S. Debbarma, L. C.Saikia, and N. Sinha, “AGC of a multi-area thermal system under deregulated environment using a non-integer controller,” Electric Power Systems Research, vol. 95, pp. 175-183, 2013. A. Ahuja and S. K. Aggarwal, “Design of fractional order PID controller for DC motor using evolutionary optimization techniques,” WSEAS transactions on system and control, vol. 9, pp. 171-182, 2014. M. Clerc, Particle Swarm Optimization,ISTE Ltd., 2006. K. E. Parsopoulos and M. N. Vrahatis, “Particle swarm optimization and intelligence: Advances and applications,” Information Sciences, 2010. Gao, Zhe, Liao, Xiaozhong, “Improved Oustaloup approximation of fractional order operator using adaptive chaotic particle swarm optimization,” Journal of System Engineering Electronics, vol. 23, issue 1, Feb. 2012, pp. 145-153.
