Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 2672436, 11 pages
https://doi.org/10.1155/2018/2672436

Research Article

Decentralized Adaptive Double Integral Sliding Mode Controller for Multi-Area Power Systems

Bui Le Ngoc Minh,¹ Van Van Huynh,² Tam Minh Nguyen,¹ and Yao Wen Tsai³

¹Faculty of Electrical and Electronics Engineering, Ho Chi Minh City University of Technology and Education, Ho Chi Minh City, Vietnam
²Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
³Department of Mechanical and Automation Engineering, Da-Yeh University, Changhua 51591, Taiwan

Correspondence should be addressed to Van Van Huynh; huynhvanvan@tdtu.edu.vn

Received 12 February 2018; Revised 31 August 2018; Accepted 3 September 2018; Published 8 October 2018

Academic Editor: Adrian Chmielewski

Copyright © 2018 Bui Le Ngoc Minh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Most of the existing results for load frequency control of multi-area interconnected power systems can only be obtained when the norm of the aggregated uncertainties is bounded by a positive constant. This condition is difficult to achieve in real multi-area interconnected power systems. In this paper, a new load frequency control (LFC) for multi-area interconnected power systems is developed based on a decentralised adaptive double integral sliding mode control technique where the above limitation is eliminated. First, an adaptive gain tuning law is adopted to estimate the unknown upper bound of the aggregated uncertainties. Second, a double integral sliding surface based adaptive sliding mode controller is proposed to improve the transient performance of the closed loop system. Simulation results show that the proposed control law results in shortening the frequency’s transient response, avoiding the overshoot, rejecting disturbance better, maintaining required control quality in the wider operating range, and being more robust to uncertainties as compared to some existing control methods.

1. Introduction

Load-frequency control (LFC) plays an important role in the operation of interconnected power systems to regulate the frequency and the tie line interchanges among different control areas [1]. There are many different control methods, which have been proposed in designing load frequency controllers with better performance to maintain the frequency and to keep tie line power flows within prespecified values during the last two decades [1–18]. The most conventional decentralized control methods for LFC are proportional-integral control [1], proportional-double integral control [2], and PID control [3, 4]. PI control has the advantage of a simple controller structure. But it can yield a long settling time and a large overshoot in transient response [4]. PID controller is an effective LFC when the system is operating in the vicinity of the nominal operating point. However, the operating points could deviate from their nominal values significantly due to wearing out of components, the variations of power consumptions, the variations in synchronizing power coefficients, system uncertainties, and the change of the number of power plants in different control areas. The performance of a PI or PID controller would be significantly degraded due to a large deviation of operating points from their nominal values. In order to overcome the limitations of PI or PID control approaches, many advanced control methods such as optimal control [5–8], the intelligent proportional-integral LFC using genetic algorithms [9–11], internal model control [12], and model predictive control [13] are developed to maintain power quality in a wide range of operation. Recently, application of fuzzy logic controller to LFC has also been presented [14–17]. To achieve a good fuzzy logic controller, the fuzzy rules must be correctly formulated, so the designer must have a thorough knowledge of the problem in formulating the rules. This is one of the most important features determining the quality of this type of control. In addition, several researchers have used state observer and disturbance observer-based controller to deal with the LFC problem. Liu et al. [18] proposed a full-order generalized state observer for load frequency control of multi-area interconnected power system. In [19], an output feedback controller based on universal finite-time observer is designed to regulate the frequency of the hydraulic turbine system. In principle, an exact system model has to be known to design such state observer and disturbance observers, which hardly holds in practice due to uncertain parameters (e.g., variations in synchronizing power coefficients, inertia, and damping parameters) in the system.

Among these presented control methods, sliding mode control (SMC) is recognized as one of the most efficient tools due to its fast response and strong robustness with respect to uncertainties and external disturbances [20]. Recently, there has been an increasing research interest in the sliding mode based load frequency control for power systems with matched and unmatched uncertainties [20–24]. In [20], a full-state sliding mode controller was developed for load frequency control of power systems. In order to improve system dynamic performance in reaching intervals, the decentralized sliding mode controller based on integral switching surface was designed for multi-area interconnected power systems [21]. The scheme of sliding mode control by model order reduction for the LFC problem of micro hydro power plants was addressed in [22]. In [23], the neural-network-based integral sliding mode controller was employed to achieve the LFC problem. In [24], a load frequency control strategy based on sliding mode control theory and disturbance observer was proposed for the single area power system. The above works obtained important results related to the load frequency control of interconnected power systems using SMC theory. As a result, the stability of interconnected power systems was assured under certain conditions. However, the traditional SMC method given in [21, 23, 24] is based on integral sliding surface which may take long settling time and high overshoot.

Motivated by the aforementioned analysis, this paper proposes a new load frequency control for a more general structure of multi-area interconnected power systems based on the decentralised adaptive double integral sliding mode control technique. The main contributions of this paper are as follows.

(i)A double integral sliding surface based adaptive sliding mode controller is proposed to alleviate the steady-state errors and improve the transient performance of the closed loop system.(ii)An adaptive gain tuning law is adopted in the proposed double integral sliding mode controller to estimate the unknown upper bound of the system uncertainty.(iii)The two major limitations in [21, 23] (the lumped uncertainties are bounded by a positive constant and steady-state errors will be existent in the frequency deviation) are both eliminated.(iv)The proposed control law results in shortening the frequency’s transient response, avoiding the overshoot, rejecting disturbance better, maintaining required control quality in the wider operating range, and being more robust to uncertainties as compared to some existing control methods.

2. A Multi-Area Interconnected Power System Model

Without loss of generality, a multi-area interconnected power system same as [21, 23, 24] is considered in this paper; see Figures 1 and 2. Although a power system is nonlinear and dynamic, the use of the linearized model is permissible in the load-frequency control problem because only small changes in load are expected during its normal operation [21, 23, 24].

Figure 1: Multi-area interconnected power system.

Figure 2: Block diagram of the ith area of a multi-area power system.

The dynamic equations of the ith area of a multi-area power system are as follows:

where and is the number of areas. The matrix form of the dynamic equations (1)-(5) can be written as

where

and is the states of the ith area subsystem, is the states of interconnected subsystems and , is the control vector, and is the vector of load disturbance. Variables , and are the changes of frequency, power output, governor valve position, integral control, and rotor angle deviation. , and are the time constants of governor, turbine, and power system. , and are power system gain, speed regulation coefficient, integral control gain, and frequency bias factor. is the interconnection gain between areas and . The dimensions of system matrices in (6) are , , , . If there is no power exchange between and , .

Furthermore, since it is very difficult to find the exact values of the system parameters , , , and because of nonlinear and dynamic multi-area interconnected power system, the dynamic equation (6), the following general model can be drawn:

where , , and are the nominal values of , , and ; the unknown matrices , , and denote the parameter uncertainties and the modeling errors; and is called the aggregated uncertainties and defined as

The aggregated uncertainties are assumed to be bounded and satisfy the following condition:

where and , are unknown positive constants. The positive integer is determined by the designer in accordance with the knowledge about the order of the lumped uncertainty. For example, if the lumped uncertainty in ith area contains a term such as , then one may choose . In practice, the bounds of the system uncertainty are often unknown in advance. So adaptive tuning laws given in (20) and (21) are proposed to estimate .

Remark 1. The current sliding mode control approaches for load frequency control of the multi-area interconnected power system are achieved under assumption that the norm of the lumped uncertainty is bounded by a positive constant [21, 23]. That is where is a positive constant. This condition is quite restrictive.

Remark 2. In this approach, the aggregated uncertainties are bounded by more general function with order of state variable . Notably, the knowledge of the upper bounds on the uncertainties is not a required prerequisite for designing the decentralised adaptive double integral sliding mode controller. Therefore, (10) is a positive function, and it is just an extension of the condition given in Remark 1.

3. Decentralised Adaptive Double Integral Sliding Mode Controller Design

First, let error in which is the desired value. Then, a traditional integral sliding surface is given as below:

where is positive constant; is any full rank matrix such that is invertible. Then, by the time derivative of and using (8), we have

The stability of (12) is assured if the traditional integral sliding mode control effort is given as [25]

where is positive constant and

Because the aggregated uncertainties of the multi-area interconnected power system are usually unknown in advance in load frequency control approaches, the required control parameter to keep the system state within the boundary layer is hard to choose. Thus, an adaptive gain tuning law is adopted in the proposed double integral sliding mode controller to estimate the unknown upper bound of the aggregated uncertainties and to improve the steady-state control performance. The double integral sliding surface is given as below:

where and are positive constants; is any full rank matrix such that is invertible. By the time derivative of and using (8), we can achieve

In order to achieve the stability of the multi-area power system represented by (8), the decentralised adaptive double integral sliding mode control law is designed as below:

where

and

and the proposed adaptive controller for tackling the system lumped uncertainty is designed as

where

in which is the positive constant.

Remark 3. Equations (13), (17), and (18) show that the integral term is only reflected in the proposed decentralised adaptive double integral sliding mode control law (17). Therefore, the control law (17) with I control feature results in improved steady-state error performance compared with the traditional integral sliding mode control (13).

Theorem 4. Considering the multi-area power system represented by (8), if the proposed decentralised adaptive double integral sliding mode controller given in (17) includes a robust controller given in (18), an adaptive controller given in (20) with adaptive gain tuning law (21), and a switching controller designed as (19), then the asymptotic stability of the multi-area power system is guaranteed.

Proof. Constructing a Lyapunov function

where . The derivative of is as

Substituting (8) into (23) gives

Then, substituting (17)-(19) into (24), we achieve

It is clear that . This implies that

and therefore , , and are bounded. In addition, by combining (22) and (25), it can be obtained that

where and is the upper bound of . Then, it yields

Using (28), the second derivative of can be obtained such that

Taking that the limit approaches infinity on both sides of (29), it can be shown that

It is clear that is also bounded. Hence, is uniformly continuous. It can be seen from (25), (26), and (30) that is bounded, is negative semi-definite, and is uniformly continuous. By applying Barbalat’s lemma (or the Lyapunov-like lemma) [25] to (25), (26), and (30) we achieve

According to (25), we obtain

It can be seen that . Thus, by standard linear control arguments, is obtained and the asymptotic stability of the multi-area power system can be assured.

Remark 5. To clarify the differences and improvements of the control approaches including traditional integral sliding mode control and decentralised adaptive double integral sliding mode control, the block diagram of the above control approaches is given in Figures 3 and 4. First, the sliding mode control using traditional integral sliding surface is given in Figure 3. From (13) and Figure 3, the sliding mode control using traditional integral sliding surface results in with fixed control gains. However, the bound of the aggregated uncertainties of the multi-area interconnected power system is usually unknown in advance in load frequency control approaches. Thus, the required control parameter to keep the system state within the boundary layer is hard to choose. For this reason, an adaptive gain tuning law is adopted in the proposed double integral sliding mode controller to estimate the unknown upper bound of the system uncertainties and to improve the steady-state control performance. The proposed decentralised adaptive double integral sliding mode control combines the merits of the integral sliding mode control and adaptive control. Moreover, the proposed decentralised adaptive double integral sliding mode control law can be easily applied to the multi-area interconnected power systems with a general structure given in (8).

Figure 3: Configuration of traditional integral sliding mode control.

Figure 4: Configuration of the proposed decentralized adaptive double integral sliding mode control.

Remark 6. The adaptive control (20) with adaptation law (21) offers the advantage that no a priori knowledge about the bounds of is required as it adaptively estimates the bounds of and also ensures that the adaptive gain does not get overestimated. Therefore the adaptive control (20) with adaptation law (21) will reduce the saturated control effort.

Remark 7. In this approach, lumped uncertainties are bounded by more general structure than the one in [21, 23]; the adaptive gain tuning law (20) and (21) is adopted to estimate the unknown upper bound of the aggregated uncertainties . In addition, the steady-state error of frequency deviation is alleviated by using the proposed decentralised double integral sliding mode control law (17). Therefore, the two major limitations in [21, 23] (the aggregated uncertainties are bounded by a positive constant and steady-state errors will be existent in the frequency deviation) are both eliminated.

4. Application Results

To test the effectiveness and superiority of proposed double integral sliding mode control approach, and to compare the results with the recent applied sliding mode control techniques [21], three-area interconnected power system networks are considered as the test system with its parameters given in [21] (see Table 1).

Table 1: Parameters of three-area interconnected power system [21].

Area

1
20
120
0.3
0.08
2.4
10
0.41
0.55

2
25
112.5
0.33
0.072
2.7
9
0.37
0.65

3
20
115
0.35
0.07
2.5
7.1
0.4
0.545

The parameters and in the proposed decentralised adaptive double integral sliding mode control law (17)-(18) can be reflected as the P control and I control gains and , respectively, and will affect the control performance significantly. If is made large to get adequately small steady-state error, the damping may be much too low for satisfactory transient response. The integral gain can be selected purely to provide an acceptable dynamic response; however, typically it will cause instability if raised sufficiently high. In addition, the constant in (21) determines the convergence rate of the estimated bounds . Practically, any constant can be used to estimate the disturbance but a large value only is used for faster estimation of disturbance resulting in larger band of the bounded region and vice versa. Therefore, all the control parameters should be further adjusted manually considering the trade-off between control performance and system stability. As a result, the control parameters of the proposed decentralised adaptive double integral sliding mode control law are chosen as follows: , , , , , , , , , , , , . The sliding surface parameters are selected as , , and .

Case 1. In this base case, nominal parameters of the multi-area power system are used. No disturbances are assumed to be acting on the system; that is, .

Simulation results of the frequency deviations of the three-area interconnected power system for Case 1 using the proposed double integral sliding mode controller are shown in Figure 5. It is observed from Figure 4 that the frequency deviations converge to zero in about 6 seconds. Figure 6 shows that the tie line power deviation reaches zero with the proposed double integral sliding mode controller. In comparing the simulation results with the results given by [21], the proposed double integral sliding mode controller (17)-(21) can assure not only a fast response but also the smaller overshoot.

Figure 5: The frequency deviations (Hz) for Case 1: (dashed), (solid), (dotted).

Figure 6: Tie line power deviation for Case 1.

Case 2. The main goal of designing a controller is its ability to work well under uncertain environment. In this case, the system performance with the proposed double integral sliding mode controller is test under matched parameter uncertainties and load disturbances.

Load disturbances of pu, pu, and pu are assumed to occur in areas 1, 2, and 3, respectively. The matched parameter uncertainties in the three areas are given in [21]

and .

The closed-loop responses for each control area using the proposed double integral sliding mode controller are shown in Figures 7 and 8. It is observed from Figures 7 and 8 that the system responses are better, in terms of overshoots and settling time, compared to the one proposed in [21].

Figure 7: The frequency deviations (Hz) for Case 2: (dashed), (solid), (dotted).

Figure 8: Tie line power deviation for Case 2.

Case 3. In the previous case, the proposed double integral sliding mode controller has its ability to work well under matched parameter uncertainties and load disturbances. However, in reality, there exists mismatched parameter uncertainties in the state matrix (due to wearing out of components or variation of operating points), mismatched interconnections among subsystems (due to unknown or variations in synchronizing power coefficients), and unknown disturbances. Therefore, the robustness against those aggregated uncertainties needs to be tested. The mismatched parameter uncertainties in the state matrix in three areas are chosen as follows:

and . The mismatched interconnections among subsystems are assumed as

and . The and for i = 1, 2, 3; therefore parameter uncertainties in this case are mismatched uncertainties. The aggregated uncertainties of interconnected power systems include mismatched parameter uncertainties, nonlinear terms, and load disturbances. The load disturbances can be approximated by two parts, that is, nonfrequency-sensitive load change and frequency-sensitive load change. In this case, the load disturbance for the ith area subsystem is taken to be the function of frequency deviation as for i = 1, 2, 3. Then, the aggregated uncertainties of the first area, the second area, and the third area are assumed to be bounded by , , and , respectively.
Figures 9 and 10 show the plots of the frequency deviation and the tie line power deviation for the three-area interconnected power system when the mismatched parameter uncertainties are used and input disturbances are assumed to be acting on the system. It is observed that the frequency deviation and the tie line power deviation converge to zero in about 15 seconds.

Figure 9: The frequency deviations (Hz) for Case 3: (dashed), (solid), (dotted).

Figure 10: Tie line power deviation for Case 3.

Even under mismatched parameter uncertainties and load disturbances, the proposed double integral sliding mode controller successfully preserves system stability. Accordingly, the proposed design becomes a good choice to cope with mismatched parameter uncertainties in the state matrix, mismatched interconnections among subsystems, and load disturbances.

Remark 8. Comparing the simulation results for the three cases, the proposed double integral sliding mode controller is robust to disturbances acting on the system associated with variations of the matched and mismatched parameter uncertainties. Noticeably, the performance of the proposed double integral sliding mode controller can strongly outperforms that of the sliding mode controller proposed in [21].

Remark 9. In this simulation, the norm of lumped uncertainty of the three-area interconnected power system is bounded by unknown function of system states with the second order , and . Therefore, the method given in [21, 23] can not applied for this kind of lumped uncertainty.

5. Conclusions

This paper presents a new load frequency controller for multi-area power system where the aggregated uncertainties are bounded by unknown function of state variables, which is more general structure. An adaptive gain tuning law is adopted to estimate the unknown upper bound of the aggregated uncertainties. A double integral sliding surface based adaptive sliding mode controller is proposed to alleviate the steady-state errors and improve the transient performance of the closed loop system. Simulation results show that the proposed double integral sliding mode controller successfully preserves system stability and is robust to disturbances acting on the system associated with variations of the matched and mismatched parameter uncertainties. Moreover, comparing the simulation results with the conventional sliding mode control, the proposed decentralized adaptive double integral sliding mode controller can assure not only a fast response but also the smaller overshoot. Therefore, the proposed double integral sliding mode controller becomes a good choice to cope with matched and mismatched parameter uncertainties and load disturbances in multi-area interconnected power systems.

Appendix

Barbalat’s Lemma (or the Lyapunov-Like Lemma) [25]

If a scalar function satisfies the conditions that is lower bounded, is negative semi-definite, and is uniformly continuous, then as [25].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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