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Sub-Optimal Load Frequency Control for Interconnected Power System
Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto, Japan, 1981
SUB-OPTIMAL LOAD FREQUENCY CONTROL FOR INTERCONNECTED POWER SYSTEM M. Shimizu Technical Research Cent er, The Kansai Electric Power Company, Amagaskz', Hyogo-ken, japan
Abstract. This paper describes the application of infinite terminal time case to the load frequency control of a multiple-area interconnected power system. From the viewpoint of practical application, the linear feedback controller using the instantaneous values of output variables is proved to be effective. A number of different optimal or sub-optimal output feedback controllers can be designed by selecting different estimation functions of weighting factors, Q and R. The proposal sub-optimal output feedback controller is not significantly degraded by using the abbreviation of state covariance equation. And, it is deduced that this controller is promising as a candidate for on-line computer controls, because of the simplicity of the gain solution. Key Words.
Power plants; automatic control; computer applications; load frequency control.
INTRODUCTION
control systems, and verified, the effectiveness of this control method.
The load frequency control of a power system has been expected to adopt a control system of increased efficiency, due to the decline of the ratio of interconnected line capacity to the power system capacity, and the drop of reserve capacity. To cope with this problem, Fosha & Elgerd have converted the whole system into a model expressed with state space representation and adapted the modern control theory in an effective manner. From the practical point of view, various researches on the design of linear regulators used in combination with what is called Kalman filters employing the separation theorem, or with observers, have also been reported. Such design methods, however, are accompanied with problems including the complication of algorithm. According as the size of the power system expand, the materialization of those design methods have become even more difficult. In view of the abovementioned background, this paper describes the newly-developed practical computation algorith'll for the application of the load frequency control (LFC) system by using instantaneous output feedback law which uses only available observation quantity. In the new computation algorithm whicl1 we have developed, it is not necessary to solve the state convariance equation in the parameter optimization problem based on the matrix minimum principle. The algorithm is simple and only requires the solution of costate equation. We carried out simulations on the three area interconnected power system model to study the effects of the application of these
3091
THE OPTIMAL CONTRO e THEORY BASED ON OUTPUT FEEDBACK LAW In the definite system, the plant under consideration can be expressed as: X( t)
AX( t)
X(to)
Xo
+
Bll (t)
(1)
(2)
Its output Y(t) is given as perfect state information, Y(t) CX (t) (3) where x (t) : u (t) : y (t) : A B
C J (u)
n k m n n m
x x x x x x
1 state vector
1 control vector 1 observation vector
n state matrix k contrel matrix n observation matrix
liml rf T[T
(
T~ TEe to x (t)Qx t)
+ uT(t)Ru(t)]dt } where, J(u): Q R
(4)
estimate function. n x n positive semidefinite symmetric state cost weighting matrix k x k positive definite symmetrix control cost weighting matrix
M. Shimizu
3092
C+~CT[CCTl-1
Transpose
T
The optimal control gain GO in the Eq'(s) is obtained as a result of minimizing the estimate function J(u). VO (t) = _GOy (t)
where
UO(t):
The A in the equation (23) is the solution of the following matrix algebraical equation: I ATA-(C+C)ABR-IBT,'I + !\1\- AER- BTi\C+C
(5)
+
Optimal control vector
This problem is exactly gain adjustment, and by applying the matrix minimum principle(l) the following theorem can be obtained: (2)
[Theorem]
+ (C C)ABR
O(t)
=
-G Oy(t)
(6)
- I
T + B Ac C + Q = 0 (14)
Since the problem is stationary, D. L. Kleinman's iterative approximatio n method is adapted. So, we can be presented as the following equations.
The optima l output feedback control quantity for the equations (1) through (5) can be expressed as: U
(l3)
+ Q
n = 1,
0,
=
2,
(15)
where
,',
and its control gain matrix GO; k x m is
A
(7)
n-l~A
- BLn-l, n = 1,2,-- (16)
Ln-I~Gn -lC,
n = 1, 2, ---
(17)
-I T T T-I Gn-l~R B An -IC [CC] ,
where A and X are the respective solutions of the following matrix algebraic equations.
n =
2, 3, ---
(18)
1, 2,
(19)
Costate equation; At this time, (A - BGoC)T A + A(A - BGoC)
1) A < An-l ::; An <--- , n
+ Q = 0
+ CTGoTRGoC
2) Hm An = n->=
State covariance equation; (A - BGoC)X
=
---
(8)
-
+ X(A - BGoC)
A
(20)
T
LOAD FREQUENCY Cm;TROL (LFC) i'IODEL
+
I
=
(9)
0
The above two equations (8) and (9) are nonlinear two-point boundary value problems in a finite terminal time, and also are fairly complicated non-linear simultaneous matrix equations i n a stationary problem. Now, let us describe MCLane's suboptimal output feedback gain solution(3) which introduces the following supposition into the state covariance equation (9). Assume that state quantity x(t) is distributed in the following manner. T E {x(t)x (t) }t,Y (t) tE[to,OO],
=
I
(10)
I: unit matrix
In the case where state quantity x(t) in the equations (1) through (5) satisfies the equation(lO) , the output feedback control quantity can be expressed as (ll) So Then, the feedback control gain G : k x m Hill be R-IBTAC+ HOHever,
(12)
The model which we used for simulation is shown in Fig. 1. Major state variables for Area 2 in Fig. 1 are expressed as follows: X2
~
6f 2
Frequency deviation of Are a 2 (Hz) X4 ~ 6 Ptie 1, 2: Power flow deviation of the interconnected line between Area 1 and Area 2 (Pu MVI) Xs 6 Ptie 2, 3: Power flow deviation of the interconnected li ne between Area 2 and Area 3 (Pu MH) X9 602 Deviation in governor position of AFC power plant for Area 2 (Pu ~~ SES) X IO Q 6 PC2 AFC power plant output deviation- of Area 2 (Pu N\{) XII 6 Pf 2 Governor-free power plant out put deviation of Area 2 (PU MH)
A
A
A
These relations are supposed to apply to the major state variables of the other areas. The relationships within the area of Laplace con version can be expressed from Fig.l. In our study, Area 2 adopted a centralized control system, which had the following two targets: 1) To reduce the static frequency deviation in Area 2 to zero 2) To reduce the static interconnected line power flow deviation between Area 2 and the other areas to zero
SUB-OPTIMAL LOAD FREQUENCY CONTROL FOR INTERCONNECTED POWER SYSTEM M. Shimizu Technical Research Cent er, The Kansai Electric Power Company, Amagaskz', Hyogo-ken, japan
Abstract. This paper describes the application of infinite terminal time case to the load frequency control of a multiple-area interconnected power system. From the viewpoint of practical application, the linear feedback controller using the instantaneous values of output variables is proved to be effective. A number of different optimal or sub-optimal output feedback controllers can be designed by selecting different estimation functions of weighting factors, Q and R. The proposal sub-optimal output feedback controller is not significantly degraded by using the abbreviation of state covariance equation. And, it is deduced that this controller is promising as a candidate for on-line computer controls, because of the simplicity of the gain solution. Key Words.
Power plants; automatic control; computer applications; load frequency control.
INTRODUCTION
control systems, and verified, the effectiveness of this control method.
The load frequency control of a power system has been expected to adopt a control system of increased efficiency, due to the decline of the ratio of interconnected line capacity to the power system capacity, and the drop of reserve capacity. To cope with this problem, Fosha & Elgerd have converted the whole system into a model expressed with state space representation and adapted the modern control theory in an effective manner. From the practical point of view, various researches on the design of linear regulators used in combination with what is called Kalman filters employing the separation theorem, or with observers, have also been reported. Such design methods, however, are accompanied with problems including the complication of algorithm. According as the size of the power system expand, the materialization of those design methods have become even more difficult. In view of the abovementioned background, this paper describes the newly-developed practical computation algorith'll for the application of the load frequency control (LFC) system by using instantaneous output feedback law which uses only available observation quantity. In the new computation algorithm whicl1 we have developed, it is not necessary to solve the state convariance equation in the parameter optimization problem based on the matrix minimum principle. The algorithm is simple and only requires the solution of costate equation. We carried out simulations on the three area interconnected power system model to study the effects of the application of these
3091
THE OPTIMAL CONTRO e THEORY BASED ON OUTPUT FEEDBACK LAW In the definite system, the plant under consideration can be expressed as: X( t)
AX( t)
X(to)
Xo
+
Bll (t)
(1)
(2)
Its output Y(t) is given as perfect state information, Y(t) CX (t) (3) where x (t) : u (t) : y (t) : A B
C J (u)
n k m n n m
x x x x x x
1 state vector
1 control vector 1 observation vector
n state matrix k contrel matrix n observation matrix
liml rf T[T
(
T~ TEe to x (t)Qx t)
+ uT(t)Ru(t)]dt } where, J(u): Q R
(4)
estimate function. n x n positive semidefinite symmetric state cost weighting matrix k x k positive definite symmetrix control cost weighting matrix
M. Shimizu
3092
C+~CT[CCTl-1
Transpose
T
The optimal control gain GO in the Eq'(s) is obtained as a result of minimizing the estimate function J(u). VO (t) = _GOy (t)
where
UO(t):
The A in the equation (23) is the solution of the following matrix algebraical equation: I ATA-(C+C)ABR-IBT,'I + !\1\- AER- BTi\C+C
(5)
+
Optimal control vector
This problem is exactly gain adjustment, and by applying the matrix minimum principle(l) the following theorem can be obtained: (2)
[Theorem]
+ (C C)ABR
O(t)
=
-G Oy(t)
(6)
- I
T + B Ac C + Q = 0 (14)
Since the problem is stationary, D. L. Kleinman's iterative approximatio n method is adapted. So, we can be presented as the following equations.
The optima l output feedback control quantity for the equations (1) through (5) can be expressed as: U
(l3)
+ Q
n = 1,
0,
=
2,
(15)
where
,',
and its control gain matrix GO; k x m is
A
(7)
n-l~A
- BLn-l, n = 1,2,-- (16)
Ln-I~Gn -lC,
n = 1, 2, ---
(17)
-I T T T-I Gn-l~R B An -IC [CC] ,
where A and X are the respective solutions of the following matrix algebraic equations.
n =
2, 3, ---
(18)
1, 2,
(19)
Costate equation; At this time, (A - BGoC)T A + A(A - BGoC)
1) A < An-l ::; An <--- , n
+ Q = 0
+ CTGoTRGoC
2) Hm An = n->=
State covariance equation; (A - BGoC)X
=
---
(8)
-
+ X(A - BGoC)
A
(20)
T
LOAD FREQUENCY Cm;TROL (LFC) i'IODEL
+
I
=
(9)
0
The above two equations (8) and (9) are nonlinear two-point boundary value problems in a finite terminal time, and also are fairly complicated non-linear simultaneous matrix equations i n a stationary problem. Now, let us describe MCLane's suboptimal output feedback gain solution(3) which introduces the following supposition into the state covariance equation (9). Assume that state quantity x(t) is distributed in the following manner. T E {x(t)x (t) }t,Y (t) tE[to,OO],
=
I
(10)
I: unit matrix
In the case where state quantity x(t) in the equations (1) through (5) satisfies the equation(lO) , the output feedback control quantity can be expressed as (ll) So Then, the feedback control gain G : k x m Hill be R-IBTAC+ HOHever,
(12)
The model which we used for simulation is shown in Fig. 1. Major state variables for Area 2 in Fig. 1 are expressed as follows: X2
~
6f 2
Frequency deviation of Are a 2 (Hz) X4 ~ 6 Ptie 1, 2: Power flow deviation of the interconnected line between Area 1 and Area 2 (Pu MVI) Xs 6 Ptie 2, 3: Power flow deviation of the interconnected li ne between Area 2 and Area 3 (Pu MH) X9 602 Deviation in governor position of AFC power plant for Area 2 (Pu ~~ SES) X IO Q 6 PC2 AFC power plant output deviation- of Area 2 (Pu N\{) XII 6 Pf 2 Governor-free power plant out put deviation of Area 2 (PU MH)
A
A
A
These relations are supposed to apply to the major state variables of the other areas. The relationships within the area of Laplace con version can be expressed from Fig.l. In our study, Area 2 adopted a centralized control system, which had the following two targets: 1) To reduce the static frequency deviation in Area 2 to zero 2) To reduce the static interconnected line power flow deviation between Area 2 and the other areas to zero