Decentralized Disturbance Observer-Based Sliding Mode Load Frequency Control in Multiarea Interconnected Power Systems

The load frequency control (LFC) problem in interconnected multiarea power systems is facing more challenges due to increasing uncertainties caused by the penetration of intermittent renewable energy resources, random changes in load patterns, uncertainties in system parameters and unmodeled system dynamics, leading to a compromised reliability of power systems and increasing the risk of power outages. In responding to this problem, this paper proposes a decentralized disturbance observer-based sliding mode LFC scheme for multiarea interlinked power systems with external disturbances. First, a reduced power system order is constructed by lumping disturbances from tie-line power deviations, load variations and the output power from renewable energy resources. The disturbance observer is then designed to estimate the lumped disturbance, which is further utilized to construct a novel integral-based sliding surface. The necessary and sufficient conditions to determine the tuning parameters of the sliding surface are then formulated in terms of linear matrix inequalities (LMIs), thus guaranteeing that the resultant sliding mode dynamics meet the ${H_\infty }$ performance requirements. The sliding mode controller is then synthesized to drive the system trajectories onto the predesigned sliding surface in finite time in the presence of a lumped disturbance. From a practical perspective, the merit of the proposed control method is to minimize the impact of the lumped disturbance on the system frequency, which has not been considered to date in sliding mode LFC design. Numerical simulations are illustrated to validate the effectiveness of the proposed LFC strategy and verify its advantages over other approaches.


I. INTRODUCTION
In recent years, LFC in multiarea interconnected power sys-21 tems has experienced more challenges due to the penetration 22 of intermittent renewable energy resources and changes in 23 electricity consumption patterns [1]. In addition to uncertain-24 ties in system parameters, the fluctuating output power of 25 renewable generators and random load changes have intro-26 duced more uncertainties to power systems; these uncertain-27 ties may cause significant frequency deviations [2]. In these 28 The associate editor coordinating the review of this manuscript and approving it for publication was Padmanabh Thakur . environments, the main purpose of an LFC system is to 29 maintain the power balance between the real and scheduled 30 generation quantities for each area and to minimize tie-line 31 power deviations between interlinked neighbouring areas [3]. 32 A wide range of energy management strategies and control 33 techniques have been introduced to deal with the recent LFC 34 problem. Energy storage devices with fast response time such 35 as batteries and supercapacitors have been utilized to improve 36 grid frequency performance [4]. However, the application 37 of these storage devices is limited to a small range due to 38 high installation and maintenance costs [5]. Additionally, 39 the optimal sizing and placement of these energy storage 40 uncertainties were lumped together, but under a restrictive 96 assumption on the rank of the lumped disturbance matrix 97 to retain the robustness of traditional SMC against mis-98 matched disturbances. In addition, no systematic approach 99 was employed to determine the unknown parameters of the 100 sliding mode controller to guarantee the desired response. 101 The authors of [6] designed a hybrid fuzzy logic nonlinear 102 SMC in which the controller gains were tuned by deriving 103 a novel imperialistic competitive algorithm and a continuous 104 approximated function was suggested to handle the chattering 105 problem. 106 As a practical alternative approach, several authors have 107 considered disturbance observer-based SMC approaches for 108 LFC problems to reduce the chattering problem and main-109 tain nominal frequency controller performance. For instance, 110 a discrete-time sliding mode load frequency controller based 111 on a disturbance observer was introduced in [52], but the 112 results showed that the chattering problem was not fully 113 resolved because the estimation of aggregate disturbances 114 was not directly involved in the sliding surface design. The 115 proposed approach in [58] applied a disturbance observer 116 in the controller design and introduced an optimal slid-117 ing manifold based on an LQR algorithm; this approach 118 was robust against mismatched uncertainties and unmod-119 eled dynamics. However, the approach was conservative, 120 as the disturbance observer design was limited to tracking 121 slow-varying step load disturbances, where the effective-122 ness of the approach against random disturbances was not 123 investigated. The authors of [51] and [59] employed dis-124 turbance observer-based SMC frequency controllers, but no 125 optimal method was utilized to determine unknown param-126 eters of the sliding manifolds. Furthermore, the aforemen-127 tioned approaches have not considered an upper bound limit 128 for lumped disturbance estimation error, and thus do not 129 systematically prove the disturbance estimation error to be 130 sufficiently close to zero. The proposed disturbance observers 131 in [51], [53], and [58] have been developed based on a 132 conservative assumption that the first derivative of lumped 133 disturbance is zero, thus degrading the dynamic performance 134 of the system in tracking non-constant random lumped distur-135 bance caused by intermittent renewable energy resources and 136 complex load patterns. To our knowledge, none of the above-137 mentioned sliding mode frequency controllers have straight-138 forwardly incorporated the estimation of disturbances in the 139 design of the sliding surface to compensate for mismatched 140 disturbances. This approach helps maintain the nominal per-141 formance of a system and alleviates chattering problems. 142 Although reports of the aforementioned approaches have 143 claimed that the effect of lumped disturbance on the fre-144 quency deviations has been reduced, the impact of minimiz-145 ing disturbances on system frequency has not been addressed 146 in control objectives. From a practical point of view in 147 relation to LFC problems, it is important to minimize the 148 effects of lumped disturbances on system frequency devia-149 tions straightforwardly.  proposed controller has the advantage of low computa-192 tional burden for practical implementations.

193
The remaining sections of this paper are structured as 194 follows. Section 2 describes the dynamic model of a multi-195 area power system for the LFC problem. Section 3 demon-196 strates the design of the disturbance observer. The slid-197 ing mode frequency controller is synthesized in Section 4. 198 Section 5 presents simulation results for different scenarios 199 to verify the feasibility and effectiveness of the proposed 200 approach for the LFC problem. Conclusions and discussion 201 of possible future work can be found in Section 6.

II. DYNAMIC MODEL OF A MULTIAREA POWER SYSTEM 203
It has been well established that a multiarea power system is a 204 coupled nonlinear system that is exposed to parametric uncer-205 tainties and exogenous disturbances. However, a linearized 206 power system model can be used for studying LFC problems 207 due to slow changes in loads and resources during normal 208 operations [3], [7]. A conventional decentralized LFC model 209 for the ith area of a multiarea power system is depicted in 210 Fig. 1.

211
The governing equations of the system dynamics for the 212 ith area are given as follows: for the ith area [58]; this disturbance can be added to the constant. Based on this notion, a new reduced-order model of 231 the system dynamics can be derived by inserting the integral 232 of (4) into (1) and substituting the result into (2) and (3), 233 which is represented in the following state space form: 236 where x i ∈ R n is the system state vector as gate disturbance is D i (t) and the controlled output is y i (t).

239
The system matrices A i (t), B i (t), H i (t) and C i are real matrices 240 with appropriate dimensions, which are given as follows: The system model expressed in (6) can be rearranged as  (7) can be rewritten as 252  2) ∀d i (t) = 0 , t ∈ [ 0, ∞) and under zero initial 261 condition, The objective of this paper is to design a disturbance 264 observer-based SMC for the LFC system given by (8) in such 265 a way that 1) the reachability of the closed-loop system onto 266 the proposed sliding surface is satisfied and 2) the resulting 267 sliding mode dynamics meet the H ∞ performance require-268 ments defined in Definition 1.

269
Remark 1: Note that the control objective is to stabilize the 270 state trajectories of the controlled plant (8), meaning that both 271 the state vectors x i (t) and the control input u i (t) are bounded, 272 and since D i (t) is assumed to be bounded, d i (t) is concluded 273 to be a bounded disturbance.

274
Remark 2: One of the practical limitations for the LFC 275 problem is inability to measure all the external and internal 276 disturbances. A realistic solution is to estimate all the distur-277 bances by utilizing the defined lumped disturbance term.

279
In this section, a disturbance observer is designed to estimate 280 the lumped disturbance in (8). The dynamic equation of the 281 disturbance observer is expressed as follows: where p i (t) is the state vector of the observer,d i (t) is the esti-285 mate of d i (t) in (8), and i is a Hurwitz matrix. Combining (8)   286 and (9) gives

288
Defining the estimation error asd 291 Accordingly, we have the following lemma regarding the

299
We can use the bound e i t ≤ α i e λ max ( i )t to estimate the 300 solution by In accordance with Assumption 1, ḋ i (t) ḋ * i . Therefore, 307 λ i is bounded, and the proof is completed. for the problem of frequency deviation in multiarea power 318 systems as described by (8). The objective is to analyze 319 sliding mode dynamics with bounded L 2 -gain performance.

320
In this section, an integral sliding surface is proposed based 321 on the disturbance estimation d i (t). A sliding mode control 322 law is then constructed to reject the lumped disturbance so 323 that the reachability of the sliding surface is guaranteed.

324
Finally, an H ∞ -based control strategy is employed to deter-325 mine the controller and disturbance observer gains. The struc-326 ture of the proposed controller is shown in Fig. 2.

327
Consider the following disturbance observer-based integral 328 sliding surface: where G i ∈ R m×n is to be selected to satisfy G i B i = I n . 332 Since B i is of full column rank, G i can be chosen as B i † , 333 and u N i (t) = −K i x i (t) − G idi (t), in which K i is designed 334 such that the sliding mode dynamics meet the required control 335 objectives defined later.

336
Remark 4: The advantage of the proposed sliding sur-337 face (13) over traditional sliding surfaces is the use of 338 d i (t) to actively reduce the impacts of the unknown 339 disturbance d i (t).

340
Let us assume that K i is a known parameter. The next 341 theorem shows that the state trajectories of System (8) reach 342 the sliding surface S i (t) = 0 by applying the control law as Proof: as a Lyapunov func-349 tion candidate, we obtain Substituting u i (t) from (14) into (15), we obtain Therefore,V i (t) 0, and the reachability condition is 359 achieved in finite time. This completes the proof.

360
Implementing the function sgn(G i T S i (t)) may cause an 361 undesirable chattering phenomenon. A practical solution 362 is to replace this function with either a smooth func-363 tion tanh(G i T S i (t)) or a high slope saturation function 364 sat(G i T S i (t)) [70].

365
In the next step, an equivalent control law is utilized to 366 determine the unknown parameter K i of the sliding sur-367 face (13). By solvingṠ i (t) = 0, the equivalent control law is 368 obtained as u i eq (t) = u N i (t) − G idi (t). By substituting u i eq (t) 369 in (8), it can be verified that   Step load disturbance.
Considering that B i G i = I n − B i ⊥ G i ⊥T , the sliding mode 373 dynamics are obtained as follows: positive-definite matrix P i such that the following LMI holds: Note that by changing the second and 391 third columns of (19) and in turn the corresponding second 392 and third rows, the following equivalent condition can be 393 obtained:

Lemma 3 [66]:
Consider matrices , and , and sup-397 pose = T . The following two statements are equivalent: 398 The Schur complement verifies that the above inequality is If we choose the following matrix variables it can be readily obtained that ⊥T ⊥ 424 VOLUME 10, 2022    invertible matrix. Hence, the parameter K i is obtained as This completes the proof.

432
Remark 5: Note that condition (24) has the advantage of 433 introducing the slack variables X i and Q i ; therefore, the con-

437
To demonstrate the effectiveness of the proposed con-438 trol scheme, three simulation scenarios are studied for a 439 three-area interconnected thermal power system under dif-440 ferent load perturbations, parameter uncertainties and renew-441 able output power fluctuations. The nominal values for the 442 three-area power system are given in Table 1. The tun-443 ing parameter of the disturbance observer for all areas is 444 chosen as The tuning parameter ρ of the proposed controller (14)     parameter of the sliding surface K i in each area. A summary 451 of the results is presented in Table 2. areas are compared with the conventional PI and SMLFC 460 methods in [53], as shown in Fig. 4.

461
From the simulation results, the proposed approach out-462 performs the aforementioned approaches. Compared with 463 the other two strategies, the scheme exhibits a better tran-464 sient response in relation to overshoot, settling time and 465 nominal performance recovery for all three controlled areas. 466 We highlight that compared to [53], the proposed algorithm 467 demonstrates better transient performance in accommodating 468 relatively large disturbances. This is an important practical 469 enhancement in ensuring that the stability of the power sys-470 tem is maintained following the occurrence of large load 471 perturbations. It is also evident in this work that the proposed 472 LFC method demonstrates a better steady-state response 473 by minimizing the impact of the lumped disturbance on 474 the controlled output f i , as considered in the controller 475 design. Fig. 5 illustrates the response curve of the disturbance 476 VOLUME 10, 2022 FIGURE 10. Thermal power plant with GRC and GBD nonlinearities [53].   The sliding surface (13) and control effort (14) for all 479 control areas are shown in Figs. 6 and 7, which show that 480 the system trajectories are driven to the sliding surfaces in all 481 control areas.

482
Scenario 2: To examine the robustness of the proposed 483 method under real operating conditions, the system perfor-484 mance is evaluated in each controlled area with a lumped 485 disturbance, consisting of varying step load disturbance and 486 system parameter uncertainties. Fig. 8 represents the varying 487 step load disturbance patterns in all three areas. The variations 488 of the system parameters are modeled by the cosine functions, 489 varying ±50% around the nominal values of the system, 490 as listed in Table 1. Fig. 9 shows the dynamic response 491 of system frequency deviations for all control areas. The 492 parameter uncertainties have a minor impact on the frequency 493    Fig. 12; the disturbance includes random load variations 513 and wind turbine output power modeled in [69]. The dynamic 514 response f i of the proposed controller is compared with that 515 of the SMLFC in [53] and the conventional PI and LMI-based 516 synthesis method in Lemma 2, as shown in Fig. 13.

517
The simulation results from Fig. 13 indicate that compared 518 to other control approaches, the proposed method represents 519 superior control performance to remarkably reduce the over-520 shoots of frequency fluctuations in all three controlled areas. 521 A detailed comparison of performance indices is represented 522 in Table 3. Additionally, the proposed method surpasses the 523 method introduced in [53], whereby the frequency deviations 524 in all areas are significantly reduced. Note also that the 525 proposed strategy demonstrates strong robustness to random 526 disturbances, and outperforms other approaches in alleviating 527 the impact of random exogenous disturbances on controlled 528 output f i . This is because the proposed SMC algorithm 529 has been constructed based on the assumption of minimizing 530 the impact of external disturbances on the controlled output. 531 VOLUME 10, 2022 consequently, further improvements in the power system fre-579 quency performance are required. Motivated by the impor-580 tance of recent challenges, our future works will be devoted 581 to considering memory-based and higher-order sliding mode 582 controllers in interconnected power systems with communi-583 cation delays to address the LFC problem.