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

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 DucThang University, Ho Chi Minh City, Vietnam 3Department of Mechanical and Automation Engineering, Da-Yeh University, Changhua 51591, Taiwan


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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].The most conventional decentralized control methods for LFC are proportionalintegral 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][6][7][8], the intelligent proportional-integral LFC using genetic algorithms [9][10][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][15][16][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 Mathematical Problems in Engineering 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][21][22][23][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.

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].The dynamic equations of the ith area of a multi-area power system are as follows: where  = 1, 2, . . .,  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  = 1, 2, . . .,  and  ̸ = ,   () ∈    is the control vector, and Δ   () ∈    is the vector of load disturbance.Variables Δ  (), Δ   (), Δ   (), Δ  (), and Δ  () 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 ,   = 0. 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 Δ  (  , ), Δ  (  , ), and Δ  (  , ) 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   ,  = 1, 2, . . .,  and  = 0, 1, 2, . . ., , 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  3  (), then one may choose  = 3.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 ‖  (  , )‖ ≤ ℎ  where ℎ  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.

Decentralised Adaptive Double Integral Sliding Mode Controller Design
First, let error   () =   () − x (),  = 1, 2, . . . in which x () 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 multiarea 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 steadystate control performance.The double integral sliding surface is given as below: where   and ε 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 ∫  0   () 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.
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).
Remark 6.The adaptive control (20) with adaptation law (21) offers the advantage that no a priori knowledge about the bounds of δ is required as it adaptively estimates the bounds of δ 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.

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).The parameters   and ε in the proposed decentralised adaptive double integral sliding mode control law ( 17)-( 18) can be reflected as the P control and I control gains K P and K I , 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 ε 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 δ .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 Case 1.In this base case, nominal parameters of the multiarea power system are used.No disturbances are assumed to be acting on the system; that is,   (  , ) = 0.
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.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 Δ  1 () = 0.01 pu, Δ  2 () = 0.015 pu, and Δ  3 () = 0.02 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].
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 Δ 23 = Δ 31 = Δ 12 .The rank[  , Δ  ] ̸ = rank[  ] and rank[  , Δ  ] ̸ = rank[  ] 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 Δ   () is taken to be the function of frequency deviation as Δ   () = 0.015 + 0.015Δ  () + 0.015Δ 2  () 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 ‖ 1 ( 1 , )‖ ≤ 1+ 1 + 2 1 , ‖ 2 ( 2 , )‖ 1+ 2 + 2 2 , and ‖ 3 ( 3 , )‖ ≤ 1+ 3 + 2 3 , 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.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 ( 2  1 ,  2 1 , and  2 3 ).Therefore, the method given in [21,23] can not applied for this kind of lumped uncertainty.

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

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

Figure 3 :
Figure 3: Configuration of traditional integral sliding mode control.

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

Figure 6 :
Figure 6: Tie line power deviation for Case 1.

Figure 8 :
Figure 8: Tie line power deviation for Case 2.

Figure 10 :
Figure 10: Tie line power deviation for Case 3.