Adaptive Integral Second-Order Sliding Mode Control Design for Load Frequency Control of Large-Scale Power System with Communication Delays

Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc ang University, Ho Chi Minh City, Vietnam Faculty of Electrical and Electronics Engineering, Ho Chi Minh City University of Technology and Education, Ho Chi Minh City, Vietnam Faculty of Industrial Engineering, Tien Giang University, M~y o, Tien Giang Province 840000, Vietnam Faculty of Automobile Technology, Van Lang University, Ho Chi Minh City, Vietnam


Introduction
In modern power networks, the frequency stability is one of the significant problems related to the large-scale power system (LSPS) with communication delays.e system has been more and more complicated for LFC due to matched or mismatched uncertainties, load variations, and time-delays [1][2][3][4].Communications time-delay is necessary in the LSPS; it cannot ignore and remove in the practical LSPS.So, the time-delay must consider in the area control error (ACE) signal.Normally, the time-delay can appear in the state variables or in the input channel of LSPS.e input control signal is transported to the power plants through communication networks with delay time.e state delay-time is the essential reason to effect on delaying the basic elements of communication or transportation in the LSPS.e deviation of tie-line exchange power and frequency are combined and known as an ACE to confirm the control frequency aim for the LSPS.e balance in both the tie-line and frequency exchange power is assured when an ACE is controlled and limited.In the LSPS model, the time-delay of ACE from regulation station to control area of the system is important [4][5][6].Moreover, the time-delay is known as time lag in the power system model.at is, a big challenge to design the control algorithm to deal with the time-delay in the LSPS.In ACE control signal, the time lag may occur and lead to oscillation and instability in the LSPS [6][7][8][9].erefore, it is absolutely necessary to consider and explore the LFC for the LSPSwCD.e main objectives of LFC for each control area are continuing to keep the nominal value of the system frequency in range of the standard value and to act in accordance with the scheduled active power interchange with neighboring control areas.
In general, the previous LFC scheme such as PID control has been used to maintain tie-line power and frequency at schedule value which is one of the simplest controllers [10][11][12][13].A suitable design of PID controller which is constructed on the direct synthesis method in frequency domain is given in [10].In [11], a fractional-order PID control approach is built for the single-area LFC model using the Kharitonov's theorem to eliminate steady state error.Robust PID controller-based stability boundary locus and Kharitonov's theorem were used for LFC of the LSPS [12].In [13], an ant lion algorithm is combined with PID to optimize the LFC loop parameters for improving the frequency regulation.e above approaches are suitable for application to design LFC of power network with nominal parameter and no uncertainty.However, the practical power network is always influenced or touched by an external factor of the uncertainties such as different load disturbances, which effect on the stability of power network.In order to overcome drawbacks and resolve the LFC problem effectively, advanced control techniques are developed for LFC design consisting of neural network and fuzzy control [14][15][16].In [14], the LFC scheme for the LSPS was established and constructed on the adaptive fuzzy control method.A novel fuzzy PID control strategy with the fractional-order integrator and filtered derivative action was suggested to resolve automatic generation control for the power network [15].In [16], the optimized operation of LFC is proposed for the LSPS with hybrid energy storage system.
On the other hand, as a powerful robust control strategy, SMC has been successfully applied to a wide variety of practical systems [17][18][19].e terminal SMC is designed for LFC with the renewable power networks [20].In [21], the adaptive double integral SMC is proposed for LFC of the LSPS where the time-delay is not considered.e goal of Sarkar was to design an adaptive integral higher order SMC for LFC problems to guarantee the frequency variation [22].In [23], this article indicated nonlinear SMC with parametric uncertainties for LFC utilization in the LSPS to vary the system damping characteristics under uncertainties and step load disturbances.e approach given in [24] developed the strategy of SMC by model order-reduction of the LFC approach of the microhydropower system.A new full order SMC scheme in the paper [25] is utilized to eliminate and avoid the singularity of derivative of terms with fractional power elements.A novel adaptive SMC method has been used and developed for the LFC of the LSPS [26].e proportional and integral switching surface was established for LFC of LSPS where the chattering problem is existed due to first-order SMC used [27].e SMC-based LFC of above approaches can reduce the tie-line power and frequency deviations to ensure the stability of power network.However, the performance of LSPS using control scheme given in [20][21][22][23][24][25][26][27] is achieved without considering the time-delay in the power grids.In addition, the existence of the delays usually leads to the degradation and/or instability of power network performance.In order to solve this problem, the control scheme given in [28][29][30] is proposed to regulate the frequency deviation of the LSPS in the presence of communication delays and sudden load change.In [31], the LFC of the LSPS with nonlinear perturbations and time-varying delays are proposed using SMC.In [32], the sliding mode LFC is designed for the LSPS with time-delay and load disturbance.e sliding mode LFC strategy is suggested for the LSPS with time-delay and significant wind power penetration [33].However, these SMC approaches are developed based on first-order time derivative [31][32][33].In addition, the first-order SMC provides low accuracies due to chattering phenomenon in the control input.erefore, in this paper, the second-order SMC is proposed to solve this problem.Moreover, an adaptive control method is adopted to estimate the unknown upper bound of aggregated uncertainties.To the best of our knowledge, the adaptive integral second-order sliding mode control (AISOSMC) scheme has not been developed for LFC problem in the LSPSwCD so far.e major contributions of this paper are displayed as follows: e novel theory based on AISOSMC approach is offered for the LSPSwCD e continuous control law is developed to deal with the influence of time-delay related to ACE on the LFC problem of the LSPSwCD A state feedback controller containing both present and delayed state information is designed to improve tolerable delay margin of the LSPSwCD A new LMI is established for proving the stability of whole PSPS based on Lyapunov stability theory e proposed LFC based on AISOSMC can not only decrease effectively the frequency variation but also make successfully less in mount of power oscillation/ fluctuation in tie-line exchange

Mathematical Model of a Large-Scale Power System with Communication Delays (LSPSwCD)
In this part, we model the LSPSwCD.Figure 1 is the block diagram of the i th area of the LSPS with communication delay [28][29][30].e model of the i th area system is composed of a governor, a nonreheat turbine, and a generator.e output of the generator is the frequency error.e tie-line power error is linearly proportional to the integration of frequency error.e linear combination of frequency error and the tie-line power error is the area control error (ACE).
In addition, the time-delay of ACE signal is considered in the power network.Nevertheless, when the load variation with minor change occurs during its conventional process, then the 2 Complexity mathematical model of power network can be linearized near the stable operating point.erefore, the dynamic equations of the above i th area system are expressed as follows: with i � 1 to N and N is denoted as the number of areas, where Δf i (t) and Δf j (t) are the frequency variation of the i th area and the j th area, ΔP mi (t) is the variation in governor output command, ΔX gi (t) is the governor valve position of each area, Δδ i (t) and Δδ j (t) are the changes of rotor angle deviations of the i th area and the j th area, ΔP di (t) is the incremental change in local load of each area, T s,ij is the tieline power coefficient between the i th area and the j th area, T gi is the time constant of governor, T ti is the turbine time constant, and T pi is the time constant in power network, respectively.K pi , R i , K Ei , and K Bi are power network gain, droop coefficient of individual area, speed regulation coefficient, and frequency bias factor.ΔE i (t − τ i ) is the area control error with the time-delay and u i (t) is the control input.
In state-space form, the system state variables are used as ( So, the LSPSwCD described by Figure 1 can be written and expressed in state-space representation as follows: where i � 1, 2, . . ., N and N is denoted as the number of areas.e state-space matrix A i , B i , D i , F i , H ij in mathematical model is given as follows: Complexity 3 4 Complexity In applied LSPSwCD, the operating point fluctuates continually induced by the fluctuating resource and load disturbance.In addition, by considering CDs element, the dynamic model of LSPSwCD with the uncertainties and parameter variations in equation ( 3) are further redefined as where A i , D i , B i , and H ij are the system matrices with nominal value, ΔA i (z i , t), ΔH ij (z j , t), and ΔD i (z i , t − τ i ) are the parameter uncertainties, and B i ξ i (z i , t) is the disturbance input signal.e lumped uncertainty w i (z i , t) is defined as follows: Assumption 1. e lumped uncertainties w i (z i , t) and the differential of _ w i (z i , t) are bounded, i.e., there exist known scalars c i and z i such that ‖w i (z i , t)‖ ≤ c i and ‖ _ w i (z i , t)‖ ≤ z i , where ‖.‖ is the matrix norm.

Assumption 2.
e time-delay state vector must satisfy the condition ‖z i (t − τ i )‖ ≤ z i max , z i max � max‖z i max ‖, where ‖.‖ is the matrix norm.
In order to prove the system stability, we recall some lemmas.
Lemma 1 (see [34]).Let X and Y be actual matrices with appropriate dimension, then, for any scalar μ > 0, the sequent matrix inequality obtains Lemma 2 (see [35]).e following matrix inequality where Lemma 3 (see [35]).Assume that and N is the positive definite matrix.en, the inequality holds for all ε > 0.

Adaptive Integral Second-Order Sliding Mode Control (AISOSMC) Design for LFC of Large-Scale Power System with Communication Delays (LSPSwCD)
In this section, the AISOSMC method is developed for LFC of the LSPSwCD under mismatched parameter uncertainties and load disturbances.To solve this problem, we work step by step to design and implement the new controller approach.Firstly, the integral sliding surface (ISS) is represented for LSPSwCD to assure that the whole system is asymptotically stable.Secondly, the decentralized adaptive integral second-order sliding mode control law (DAISOSMCL) is designed to force the system trajectories to the sliding manifold and keep it there for after.

Stability Analysis of a LSPSwCD in Sliding Mode
Dynamics.In detail, we first begin to propose and build an ISS for a LSPS: where E i is the constant matrix and T i is the design matrix, matrix E i is designed to guarantee that matrix E i B i is nonsingular and matrix T i is chosen via pole assignment such that the eigenvalues of matrix (A i − B i T i ) are always less than zero.
If we recognize and differentiate σ i [z i (t)] with respect to time combined with (3), then Complexity 5 So, the setting Substituting u i (t) with u eq i (t) into the LSPSwCD yields the sliding motion: e introduction of the following theorem makes a condition that the AISOSMC dynamic equation ( 11) is asymptotically stable.

Theorem 1.
e sliding motion ( 13) is asymptotically stable if and only if there includes symmetric positive definite matrix Q i , P i , i � 1, 2, . . ., N, and positive scalars q, φ i .and β j such that the following LMIs hold: where Proof.To study and analyze stability of the sliding motion (13), we use the Lyapunov function as follows: where Q i , P i > 0 satisfies (14).en, taking the time derivative of (15) and using equation ( 13), we obtain To apply Lemma 1 in equation ( 16), we obtain Using Lemma 3 and equation ( 17), we get where is the semipositive definite.Since the z i (t) for i � 1, 2, . . ., N are independent of each other.

□ Remark 1.
e adaptive integral second-order sliding mode control design is composed of the hitting phase and the sliding phase.e proposed controller is used to force the system state trajectories to sliding phase and keep the system state trajectories on it thereafter.If the disturbance and uncertainty satisfy the matching condition rank[B i , w i (z i , t)] � rank[B i ], the system in the sliding mode is invariant to disturbance and uncertainty.e stability of system under matched condition is easier than mismatched condition.
e proposed controller can compensate for disturbance and uncertainty directly under matched condition.erefore, the stability of the system in the sliding mode under mismatched condition rank[B i , w i i , t)] ≠ rank[B i ] has been considered and proved using LMI technique based on Lyapunov stability theory.

3.2.
First-Order SMC Design.In order to guarantee the reachability of state variables to the ISS (10), the decentralized first-order integral SMC law is designed as follows [18]: where  ε i > 0 and Remark 2. e first-order SMC can be used to study LFC of power system under matched uncertainties.However, the parametric uncertainties not usually satisfy the matched condition in real power network.Consequently, some main constraints are necessary to design the first-order SMC to compensate the uncertainties, which can guarantee the convergence in nominal frequency and the system stability but the system trajectories cannot reach to origin point.erefore, the second-order ISS has been used as the following part to force the system trajectory to equivalent point and to make better the transient performance.

Decentralized Adaptive Integral Second-Order Sliding
Mode Control Law (DAISOSMCL) Design.In this step, the DAISOSMCL is developed for the LSPSwCD to reduce the frequency deviation.
e main purpose of the proposed control scheme is to effect on the second-order derivative of the sliding variables σ i [z i (t)].By using the discontinuous control signal _ u i (t), it is simple to make σ i [z i (t)] and _ σ i [z i (t)] converge to zero.So, the input control signal u i (t) of LSPSwCD can get by integrating the discontinuous signal _ u i (t) to make continuous signal u i (t).
erefore, the DAISOSMCL approach removes some undesired frequency oscillations in the control signal of LSPSwCD.
We define and establish the sliding manifold (SMd) where ε i > 0 is a positive constant, according to equation (30); the equation ( 31) can be redefined as

Complexity
Based on the definition of sliding surface and SMd, the continuous DAISOSMCL for LFC of a LSPSwCD is given as follows: where where in which p i and p i are the positive constants.en, we have the main result which is presented as follows.
Theorem 2. Consider the closed loop of the power systems with the DAISOSMCL (33).en, every solution trajectory of system state is directed towards the SMd G i [z i (t)], and once the trajectory hits the SMd G i [z i (t)], it remains on the sliding manifold thereafter. Proof.
e Lyapunov function is introduced as follows: where  z i (t) � z i −  z i (t).So, taking the derivative of V(t), we have According to equation (37) and property ‖AB‖ ≤ ‖A‖‖B‖, we have 10 Complexity Using Assumption 1, we achieve Using the DAISOSMCL (33) yields e above inequality implies that the system trajectories of the LSPSwCD (3) reach the SMd G i [z i (t)] and keep it for later.

□ Remark 3.
e SMC is capable to reach the bounded system stability and is able to maintain within a range to converge zero, and then the power system is notably stable.Equation Complexity11 (10) indicates that the integral term dτ is only reflected in the proposed DAISOSMCL (33).
erefore, the control law (33) is to improve the performance of steady state error in comparison with the traditional integral SMC.

Results and Discussion
Based on the interconnected time-delay power network, the four cases are offered to prove the strength of the suggested AISOSMC approach under the required conditions such as the different load disturbances and the parameter variations.A LSPSwCD is analyzed to explain the robustness and effectiveness of the suggested AISOSMC approach.e parameters of LSPSwCD were given in [22] as shown Table 1.

Case 1.
e presented LFC based on AISOSMC is used to compare with the traditional LFC approach under the same condition in [22].e proposed LFC based on AISOSMC has been examined with different step load disturbance effect on the thermal power plant with nominal parameter conditions.We have considered the system parameters of matched uncertainties.In this case, the load disturbances are ΔP d1 � 0.02, ΔP d2 � 0.015 , and ΔP d3 � 0.01 (p.u.MW) at t 1 � 0 in three areas of the LSPS.e frequency deviations of Δf 1 , Δf 2 , and Δf 3 for the LSPS with delay-time as τ i � 0.1 s (i � 1, 2, 3) are displayed in Figures 2-4 which are the tie-line power deviation and control input signal for three-area power networks.It is easy to realize that the transient responses achieved in the proposed LFC based on AISOSMC is faster in the settling time, but it has less magnitude of overshoot percentage with the proposed recent controller in [22].2. In particular, the report of results can show in an effective comparison while the time-delay communication is considered for the largescale power network.erefore, the system performance of the suggested AISOSMC is well balanced, and frequency variation is zero after 1s.
Case 2. As the same matched parameter uncertainty of the three areas is used in [22], the cosine function around the nominal operation point is used to verify the usefulness and robustness of the suggested controller to load disturbance.
Figure 5 indicates that the proposed LFC based on AISOSMC to three areas with CDs, τ i � 0.1; the frequency deviation reaching to zero is about 2 s much smaller than the comparative LFC used in [22] within 10 s. Figure 6 plots the control signal of the system under matched uncertainty.e tie-line power variations of three areas with LFC based on  12 Complexity AISOSMC always are kept with maximum value as 4.7 × 10 −3 p.u.MW in Figure 7.
Remark 5.In this configuration, the influents of time-delay signals are considered to compare with the simulation result in [22], the DAISOSMCL based on the offered switching surface can not only make better in the response speed but also upgrade the transient performance to decrease the overshoot percentage.So, the designed control method is powerful and strong enough to regulate and control the matched parameter uncertainties of multiarea interconnected time-delay grids.
Case 3. In this case, the load disturbances areΔP d1 � 0.01 p.u.MW at t 1 � 0 s, ΔP d2 � 0.02 p.u.MW at t 2 � 0, and ΔP d3 � 0.03 p.u.MW at t 3 � 0 in area 1, area 2, and area 3,   respectively.We discuss about the impact of mismatch parameter uncertainty in the state matrix and values of the time-delay as τ i � 0.2 s (i � 1, 2, 3) for all the subsystems of LSPSwCD which are designed as follows: e mismatch interconnected between subsystems is designed as follows: ΔH 12 � 0 0 0 0 0.178 cos(t) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.296 sin(t) 0 0 0 0 0 e frequency variation in LSPSwCD is displayed in Figure 8, the control input signal is presented in Figure 9, and tie-line power variation is displayed in Figure 10.We observe the system performance, the transient response makes achievement by the proposed control scheme which reduces the amplitude of over/undershoot percentage.It is     Complexity to be clear that the input control signal is to converge quickly and drive the tie-line power variation and frequency variation to zero.erefore, the suggested controller carried out with better design, both in terms of reducing the over/undershoots and minimizing the settling time in comparison to [21,22].
Remark 6.In this case, though considering to the influence of the mismatched uncertainty of LSPSwCD, the performance of proposed controller still is kept powerful and stable.erefore, the proposed ISOSMC is proved to be suitable for LSPSwCD.e approach given in [21,22] cannot be applied for the LSPSwCD of this case.
Case 4. In this case, the nominal parameters of LSPSwCD are in Table 1 and the same parameters with previous cases.However, in the practical power system, the parameters of LSPSwCD are always varied due to different operation conditions.erefore, the parameters of LSPSwCD are selected to vary by ±20% synchronously from their normal values.e delay time is 0.2 s and load variation is as shown in Figure 11.e tie-line power and frequency deviation decay quickly as observed in Figures 12 and 13. e control signal has been presented in Figure 14.It is to be clear that the proposed control scheme is able to handle with the timedelay and random load disturbances.It is proved that the proposed DAISOSMCL has the good quality and achievement to reject disturbance with small control signal.
Remark 7. In this approach, the structure of LSPSwCD is more general than the approach given in [28][29][30][31][32][33].Also, the system responses in terms of convergence of frequency variation and power exchange error are improved by using the proposed DAISOSMCL.erefore, the proposed design controller is suitable for LFC of LSPSwCD.

Conclusions
In this paper, the LFC based on decentralized adaptive integral second-order sliding mode control (DAISOSMC) has been developed for the large-scale power system with communication delays (LSPSwCD), load variations, and parameter uncertainties.It is shown that the proposed DAISOSMCL ensures the finite time reachability of the system states and moreover the dynamics of LSPSwCD in the sliding mode is asymptotically stable under certain conditions.To the best of our knowledge, this is the first time the DAISOSMC approach is designed for LFC of the LSPSwCD.e report of simulation results indicates that the proposed DAISOSMC approach can adequately make less the deviation of the tie-line power and frequency variation of LSPSwCD.e suggested control scheme is therefore proved to be more effective for the LSPSwCD implementation.

Figure 1 :
Figure 1: e structure of i th area of the LSPSwCD.

Remark 4 .
Due to Figures2 to 4, the testing simulation results in this part are appointed in Table

Figure 2 :Figure 3 :
Figure 2: Frequency variation of the system.

Figure 4 :
Figure 4: Control signal of the system.

Figure 5 :
Figure 5: Frequency deviation of the system under matched uncertainty.

Figure 6 :
Figure 6: Control signal of the system under matched uncertainty.

Figure 11 :Figure 12 :
Figure 11: Load variations of three-area power network.

Table 1 :
e parameters of three-area power system.