2013 International Conference on Power, Energy and Control (ICPEC)
Iterative Linear Matrix Inequality Algorithm Based Decentralized Controller for Load Frequency Control of Two-Area Thermal Power Systems S. K. Pandey skp1111.1969@rediff mail.com
S R. Mohanty
[email protected] m
Nand Kishor
[email protected] o.in
Rajendra P. Payasi payasirp@rediffmail
Department of Electrical Engineering, Motilal Nehru National Institute of Technology Allahabad, India Abstract-In this paper, iterative linear matrix inequalities (ILMI) algorithm based design of decentralized controller for load frequency control (LFC) is proposed. The ILMI algorithm is used to tune the control parameters of the proportional-integralderivative (PID) controller subject to H∞ constraints in terms of ILMI. Hence, the control design is called IPIDH∞ controller. The proposed controller is tested on a two-area interconnected power system considering reheat type turbine in both area with different load scenarios. The results of the proposed controller are compared with the H∞ -controller. From the simulation results, the proposed controller is superior to H∞ controller.
Keywords-- Interconnected power systems, iterative linear matrix inequalities, load frequency control, robust controller, reheat turbine.
I. INTRODUCTION The main goal of LFC is to maintain the power balance in the system such that the frequency deviates from its nominal value to within specified bounds and according to practically acceptable dynamic performance of the system. To ensure the quality of the power supply, it is necessary to regulate the generator loads depending on the optimal frequency value with a proper LFC design. Simple PI/PID controller which is tuned based on experiences/trial-error methods is difficult. The different methods of tuning of PI/PID controllers are given in [1-2]. A method for PID control design which uses a combination of H∞-control is proposed in [3]-[4]. Many other robust control methods have been applied to load frequency control problem, for example, the LFC using genetic algorithms and linear matrix inequalities is described in [5], and design of
multivariable PID controller based on ILMI is discussed in [6]. The LFC using robust PI controller based on ILMI algorithm is given in [7]. In this paper, the LFC problem is formulated as a H∞-static output feedback (SOF) control problem to obtain a desired PID controller. An ILMI algorithm is developed to compute the PID gains. The proposed strategy is applied to a two-area interconnected power systems consist reheated steam turbine. Simulation results show that the robustness of the proposed controller is much superior to that of the H∞ controller against various load changes. This paper is organized as follows: The proposed control strategy for LFC is given in section II. Section III presents the proposed dynamic model and design of novel robust controller. The two types of robust load frequency controllers, which are based on the conventional H∞ control and IPIDH∞, are tested on two area interconnected power system with different scenarios of load disturbances, and their robust performance with different load scenario demonstrated in section in section IV. II. PROPOSED CONTROL STRATEGY This section deals the brief overview of robust H∞ control design via LMI approach. The design of robust iterative PIDH∞ control via LMI approach is also described in this section. A. Design of H∞ Controller via LMI Approach The classical closed-loop system via robust H∞ control is represented as Fig. 1, in which P(s) represents a linear-invariant system and K∞(s) represents robust H∞ controller [7]. The objective of H∞ control theory is to design the control law u on the basis of the measured variable y , so that the
978-1-4673-6030-2/13/$31.00 ©2013 IEEE 431
2013 International Conference on Power, Energy and Control (ICPEC) effect of the disturbance ω on the control variable Z ∞ , expressed in terms of the infinity norm of the transfer function from Z ∞ to ω, ( Tz∞ w ) does not exceed a specified limit γ defined as guaranteed robust performance.
y = C2 x
(1)
The PID controller defined as:
y = C2 + D21 w + D22 u
t
The state space controller model is assumed as follows: ξ = A ξ + B y k
u = Ck ξ + Dk y
(2)
Combining equations (1) and (2), we get closed loop system as: w P (s)
u
u = K1 y + K 2 ∫ ydt + K3 0
Where, K1 , K and K3 matrices to be designed (PID 2 gains). The output feedback H∞-control problem is to find a controller of the form (8) u = Ky
y
( Tz∞ w ) < γ
t , and z = ⎡ z1 ⎤ , the variable z = ⎢z ⎥ , 2 ∫ ydt 0 ⎣ 2⎦ that can be viewed as the state vector of new system whose dynamics are represented by z1 = x = Az1 + B1 w + B2 u (10) z2 = y = C2 z1
(3)
z∞ = Ccl xcl + Dcl w Where, ⎡ x⎤ ⎡ A+ B2Ck C2 B2Ck ⎤ xcl = ⎢ ⎥ Acl = ⎢ Ak ⎥⎦ ⎣ξ ⎦ , ⎣ Bk C2 ,
D12Ck ]
,
The problem of a PID is reduced to a static output feedback (SOF) control system as
z = Az + B1 w + B 2 u
Dcl = [ D11 + D12 Dk D22 ]
z ∞ = C1 z + D12u
The closed-loop RMS gain T∞ ( s ) or H ∞ norm of the
y = C 2 z + D 21 w
transfer
u=Ky
does
not
exceed
performance index γ , if and only if there exists a symmetric matrix X ∞ [7] such that
X∞ > 0
(11)
z∞ = C1 x + D12 u
⎡ B + B2 Ck D22 ⎤ C = C + D D C Bcl = ⎢ 1 12 k 2 ⎥ cl [ 1 ⎣ Bk D22 ⎦,
function ( Tz∞ w ) ,
(9)
Let z1 = x
xcl = Acl xcl + Bcl w
−I Dcl
(7)
such that the infinite-norm of the closed-loop transfer function from z∞ to w
Fig.1. Close-loop system via H∞ control.
Bcl
dy dt
z∞
K∞ ( s)
⎡ Acl X ∞ + X ∞ AclT ⎢ BclT ⎢ ⎢ Ccl X ∞ ⎣
(6)
z∞ = C1 x + D12 u
x = Ax + B1w + B2u
k
B. Design of Iterative PIDH-infinity (IPIDH∞) Controller via LMI Approach Consider the system model is defined as:
x = Ax + B1w + B2u
The state space model of the system is given by:
z∞ = C1 x + D11 w + D12 u
matrix inequalities (4) and (5). In order to solve the H∞-SOF an iterative LMI algorithm has been used.
X ∞ CclT ⎤ ⎥ DclT ⎥ < 0 −γ 2 I ⎥⎦
(4)
Where, A = ⎡ A ⎢C ⎣ 2
C1 = [C1 0] ,
(5)
Hence, the optimal H ∞ control is achieved by minimizing the performance index γ , subject to the
0 ⎤ , B = ⎡ B1 ⎤ , B = ⎡ B2 ⎤ , 1 2 ⎢0⎥ ⎢0⎥ ⎣ ⎦ ⎣ ⎦ 0 ⎥⎦ ,
C 2 = ⎡⎣C 21 C 22
D 1 2 = D1 2 D 21 = [ 0 ,
and K = ⎡ K 1 ⎣
K
2
0
K 3 ⎤⎦
C 23 ⎤⎦
C 2 B1 ]
T
T
(12) (13)
Once K is found, the original PID gains can be obtained from
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2013 International Conference on Power, Energy and Control (ICPEC)
(
)
−1
K1 = ( I − K3C2 B2 ) K 1
Step 7: If obtained solution K satisfies the gain constant, it is desirable, otherwise change constant weights (ni ) , Q and γ and go to step 1.
equation (1) then compute A , B1 , B2 , C1 , and C 2 as equation (12) and select the performance index γ .
III. PROPOSED DYNAMIC MODEL AND DESIGN OF NOVEL ROBUST CONTROLLER Consider the state space model of proposed power system as shown in Fig. 2 is as follows: x = Ax + B1 w + B2 u
K 3 = K 3 I + C2 B2 K 3
, K 2 = ( I − K 3C2 B2 ) K 2 ,
(14) The algorithm of an iterative PIDH∞ via LMI approach for the optimization problem mentioned in Eq. (9) is as follows: Step 1: Obtain A , B1 , B2 , C1 , C2 , and D12 as
Step 2: Select Q > 0 and solve P for the Riccati T
T
equation A P + PA− PB2 B2 P + Q = 0 , P > 0
y = C2 x x = ⎡ΔXE1 ΔPt1 ΔPr1 Δf1 ΔXE2 ΔPt2 ΔPr2 Δf2 ΔPtie ⎣
w = [ w1
Step 3: Solve the following optimization problem for Pi , K and ai . Optimization 1: Minimize ai subject to the following LMI constraints Pi B 1
( C 1 + D 12 K C 2 ) T
−γ
0
0
−I
0
0
( B Pi + K C 2 ) T ⎤ ⎥ ⎥ 0 ⎥
