Abstract
This paper proposes a delay discretization based load frequency control strategy for interconnected power systems. The effect of time delay is considered in the system for the design of stabilizing controller. To improve the tolerable delay margin of the system, a two-term state feedback controller structure is used. The controller requires delayed state information as control input. In the proposed approach, the amount of delay introduced in the state of the system, i.e., artificial delay, for taking control action is assumed to be constant. The approach is based on the discretization of this delay interval. In order to define a simple Lyapunov-Krasovskii (LK) function for each of the discretized interval, a stabilization criterion is developed in such a way that a single one satisfies the requirement of all the intervals. The developed criterion is computationally simple and efficient.
IN a large-scale power system, multiple control areas are connected through tie-lines. For supplying reliable and sufficient power of good quality, one of the most important components of the large-scale power system is the load frequency control (LFC) [
There has been available literature on designing suitable controllers for LFC scheme of an interconnected power system. One of the simplest controller, i.e., a proportional-integral (PI) control, is proposed in [
In this paper, the LFC problem of interconnected power system with delay in ACE is analysed by using delay-discretization approach. performance based delayed state feedback control strategy is proposed by using an artificial delay for tolerable delay margin enhancement of the interconnected power system. The artificial delay is chosen for discretization because it incorporates the delayed state information of interconnected power system into the dynamics of the controller, which is the primary requirement of the proposed control method. The number of decision variables increases with the number of delay intervals in the existing delay discretization approaches [
The contributions of this paper are listed as follows.
1) It deals with the effect of time delay related to ACE on the LFC problem of an interconnected power system.
2) A state feedback controller containing both present and delayed state information is designed to improve tolerable delay margin of the interconnected power system.
3) By using delay-discretization approach, a new stabilization criterion with performance index is derived in terms of LMI based on LK function for interconnected power system with time delay.
4) To compute suitable controller gains and performance index, a constrained LMI optimization problem is developed by formulating a multi-objective function.
5) A study is conducted to show the effect of the number of delay intervals on tolerable delay margin of the interconnected power system.
There are different types of LFC structures in regulated and deregulated power markets. In this paper, charged LFC structure is considered. For simple understanding, a charged LFC structure of two-area interconnected power system is shown in

Fig. 1 Charged LFC structure without bilateral contract.
The LFC model of the

Fig. 2 LFC model of the
The objective of this paper is to design a suitable controller to stabilize the closed loop system, which, at the same time, can ascertain the performance criterion. The performance index is described as:
(1) |
where , is the Laplace operator, is disturbance vector, is the output vector; and is the frequency domain function.
(2) |
where the performance index is the load rejection ratio of the controller. It is required to obtain an controller to minimize , i.e., norm bounded performance measure, in order to have the minimal effect of load variation on the performance of system.
The dynamics of an interconnected power system with control areas for can be described as follows.
The linearized model of the alternator output mechanical power deviation is given by:
(3) |
The linearized model of ACE is given by:
(4) |
The linearized model of the tie-line power deviation is given by:
(5) |
The linearized model of the governor valve position is given by:
(6) |
The linearized model of frequency deviation is given by:
(7) |
where .
The dynamic equations (

Fig. 3 Two-area LFC model.
Define a state vector as , where , . The dynamic equations (
(8) |
(9) |
where is the load disturbance vector. For and , the following matrices are defined as:
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
It can be noted that the local PI controller is considered as an integral part of the model (3)-(7). As the margin of delay increases, the local PI controller fails to stabilize the system in conventional LFC scheme. For such situation, the conventional PI controller may not improve the performance of the system [
In [
1) To solve the LFC problem of time-delay power system, a two-term controller of the following form is proposed as:
(22) |
where and are the controller gain matrices with satisfactory dimension; is a known finite delay intentionally introduced in the controller by the designer (artificial delay or controller delay). Assume that is a constant delay satisfying , where is the upper bound of . The control signal generated for the system is a function of present and delayed state of the system.
2) To obtain the stabilization criterion using LK function in LMI framework, a discretization approach is proposed. Systems (8) and (9) are considered to validate the proposed control algorithm in this paper.
Using a controller of the form (22), the closed-loop system can be represented as:
(23) |
(24) |
where and .
To derive the main stabilization criterion, an existing result is given in the form of Lemma which is discussed as follows.
Lemma 1 (Jensen’s Inequality [
(25) |
where . The right-hand side of the above inequality is nonconvex in . To approximate a convex criterion involving the uncertain parameter , an equivalent representation can be obtained using the free matrix variable. The approximated representation is as follows:
(26) |
where and are free weighted matrices with appropriate dimensions. Note that, with the choice in (26), we can obtain (25).
The following theorem presents an LMI-based criterion for designing the controller of form (22) while ascertaining the performance criterion (2).
Theorem 1: system (8) with controller (22) for known , , and satisfies the performance (2) if there exists , , , , for , and arbitrary matrices , , , and for , satisfying the following LMI:
(27) |
where , , , , , , , , , , , , , , , , , , , , , , , , , , ,
Proof: considering the
(28) |
(29) |
(30) |
Differentiating with respect to time along the state trajectory of (23) yields:
(31) |
(32) |
(33) |
where and .
Instead of replacing by directly using (23) in (31), we consider in this paper a zero valued quadratic formulation of the system dynamics (23) as:
(34) |
where , are arbitrary matrices of appropriate dimensions. This will incorporate the information regarding the coupling of some important states with the system dynamics. As is not replaced from (23) to (31), it is an important requirement for the analysis to incorporate the information regarding the system dynamics. Therefore, the above zero term (34) can be used in the analysis. This term can easily fulfill the requirement of involving states of system dynamics coupled with some important states while modifying the stabilization requirement. The following inequality is used [
(35) |
where , and .
Following (25) of Lemma 1, two integral terms of and the first integral of are approximated. The last integral term of in (31) may be written as:
(36) |
The above term (36) can be approximated by following (26) of Lemma 1. After the approximation of all integral terms in (31), we can write the stability condition as:
(37) |
where , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
Therefore, the stability requirement of the
(38) |
Next, to ascertain the performance criterion, the performance criterion from (2) can be obtained as:
(39) |
Note that if , the system (23) satisfies the condition (2). Thus, to design an performance based two-term controller with performance index, the performance criterion is adopted. For zero initial condition, i.e., , and since , (39) can be re-written as:
(40) |
Substituting (38) into (40), the following inequality can be obtained:
(41) |
where , and .
Therefore, is satisfied if . The above is a polytope of matrices on , and it is always negative definite if two of its vertices are also negative definite. Then, (38) can be equivalently written as:
(42) |
Note that and it is maximum in the
(43) |
Taking Schur complement for the last term in (43), we can obtain:
(44) |
where , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
The derived inequality (44) is not an LMI because it has some nonlinear terms in . The presence of five arbitrary matrices , , , and in makes the inequality (44) nonlinear. Thus, the inequality (44) can be converted into LMI by restricting the presence of five arbitrary matrices into only one arbitrary matrix (i.e., S1). For this reason, four parameters such as , , and are chosen by the control designer. , , and can be represented in terms of by using , , , and as , , , and , respectively.
The LMI (27) can be obtained by substituting , , , and into (44), pre- and post-multiplying (44) by and its transpose, respectively. Finally, we can change the following variables:, , , , , , and . The proof is completed.
The controller gains can be obtained by using and from the feasible solution of (27) with a suitable value of . To obtain the suitable value of , we have to optimize in (27). Thus, an optimal controller is yielded by defining and minimizing to obtain a solution of (27). The optimal controller gives optimized value of , but provides high value of controller gains. These high controller gains are not practically implementable [
(45) |
where , , and are the norms of the matrices , , and , respectively. By minimizing the above objective function (45), the performance (2) can be achieved. The stabilizing controller gains can also be obtained from the minimization. The number of decision variables and size of the LMI in Theorem 1 does not change with number of division of the delay interval . This is the most important advantage of the proposed approach. No approximation is used to obtain the stability condition (43) from (42). But, the gap in approximating the first integral term of (31) increases with , and increases with . So, the stabilization criterion is indeed ultimately constrained. This limitation arises due to the choice of LK function and the corresponding results may be influenced by the approximations of the first integral term. However, it is easy to search over to obtain the maximum tolerable .
To simplify Theorem 1 by eliminating the number of variables, the following corollary is proposed.
Corollary 1: system (8) with controller (22) for known , , and satisfies the performance (2) if there exists , , , , for , and arbitrary matrices , , , and for , satisfying the following LMI:
(46) |
where , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
Proof: since the last term in (43) is positive definite, one can derive the stability criterion in the form of a single matrix inequality as:
(47) |
Following Lemma 1, by substituting the free matrix variables as in (47) and following the linearization technique adopted in Theorem 1, (46) is obtained. The proof is completed.
Although the above stability criterion derived in Corollary 1 is conservative compared to Theorem 1 due to the approximations incorporated, the bounding gap decreases with the decrease of integral limit , i.e., the increase of the number of delay intervals (), in both criteria. Hence, the criterion developed in Corollary 1 is more useful for large , since it involves less number of free variables. By modifying the single objective problem to a multi-objective one, the gains of the designed controller will be within the practical range. They can be easily implemented in real time.
The proposed Corollary 1 can be used to solve the optimization problem with objective function (45) by replacing the constraint (27) by (46). This causes the reduction of computation complexity with conservative results.
A well-known numerical example is presented below to validate the developed control approach for two-area interconnected power system.
An example of a well-known two-area interconnected power system [
For the simulation study of the two-area power system (8) with controller (22) in MATLAB, the delay in the ACE signal of area 1 and the delay in the ACE signal of area 2 are set to be fixed. To design the controller (22), the gains of the stabilizing controller and are required. These controller gains with minimized can be obtained from the minimization of objective function (45). The LMI optimization problem containing the objective function (45) can be solved by using mincx solver of LMI control Toolbox in MATLAB. However, it is unable to find the solution of the optimization problem using mincx solver alone, because mincx solver can not get the values of the four parameters , , , and chosen by the control designer as explained in Theorem 1. Now, it is a challenge for the control designer to select suitable values for such unknown parameters, i.e., , , , and . These four unknown parameters can be obtained suitably by using fminsearch routine of MATLAB Toolbox. Therefore, the LMI optimization problem is solved by using both mincx solver and fminsearch routine. The fminsearch routine takes four input values at the time of invoking which are treated as initial values for the parameters, then searches the suitable values of , , and , and finally gives the suitable values of these unknown parameters to the mincx solver. Then mincx solver solves the LMI optimization problem and gives the values of , , along with , , and . Next, the controller (22) is designed by using and .
The maximum tolerable delay margin of the closed loop system (23) can be verified by checking tolerability of delay as well as minimizing . The proposed discretization approach gives an opportunity to study the effect of maximum tolerable delay margin with number of delay intervals . A study has been made by obtaining using Theorem 1 with respect to change in and presented in
It can be observed from
The major concern in the delay discretization approaches proposed in [
Using the above conditions for simulation, the maximum is also obtained using Corollary 1 by changing the value . The analysis has been made and presented in
Some variable approximations in Corollary 1 make the criterion conservative. Though the criterion is conservative, the number of variable involved in the criterion is less than that of Theorem 1. From the above study, it is confirmed that the maximum tolerable delay margin can be obtained by setting . A comparative analysis is made in
To evaluate the performance of Corollary 1 and Theorem 1 with respect to existing approaches in [
(48) |
(49) |
The deviations in frequency ( and ) and the mechanical power output of the turbines ( and ) for both the areas modeled in (3)-(7) can be studied with random step load disturbances ( and ). For simulation, the random step load disturbances of two areas are generated for 200 s as shown in Figs.

Fig. 4 Change in load disturbance of area 1.

Fig. 5 Change in load disturbance of area 2.
The simulation results , , and at maximum tolerable delay margin () are presented in Figs.

Fig. 6 Change in frequency of area 1.

Fig. 7 Change in frequency of area 2.

Fig. 8 Deviation in mechanical power output of area 1.

Fig. 9 Deviation in mechanical power output of area 2.
In this paper, a delay discretization approach is proposed to improve the tolerable delay margin of the interconnected power system. To validate the approach, a well-known existing example is considered. From the demonstration, it is observed that the designed performance based two-term controller is able to withstand random load disturbances. The proposed approach is simple and less conservative than some existing results. Though the approach is simple and less conservative, it is time-consuming and tedious to search the tuning parameters , , and for the minimum . This opens up a new direction for research to avoid the above tuning parameters required during linearization. One may also analyze the tolerable delay margin improvement capability using dynamic state feedback controller in place of a two-term one.
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