Decentralized Event-Triggered Frequency Control With Guaranteed L--Gain for Multi-Area Power Systems

Page 1

IEEE CONTROL SYSTEMS LETTERS, VOL. 5, NO. 2, APRIL 2021

373

Decentralized Event-Triggered Frequency Control With Guaranteed L∞-Gain for Multi-Area Power Systems Luwei Yang , Graduate Student Member, IEEE, Tao Liu , Member, IEEE, and David J. Hill , Life Fellow, IEEE

Abstract—This letter proposes a novel decentralized event-triggered control algorithm to replace the conventional periodic sampling/communication mechanism of automatic generation control (AGC) for multi-area power systems. For each control area, event-triggering rules that only rely on local measurements are designed to decide the sampling/communication instants. Further, a strictly positive dwell time is introduced to exclude Zeno behaviours. The L∞ -stability of the multi-area power system with the developed event-triggered control law and essentially bounded net load disturbances is studied, and a sufficient stability criterion is established. Finally, case studies demonstrate the effectiveness and efficiency of the proposed method. Index Terms—Power system frequency regulation, AGC, event-triggered control, L∞ -stability.

I. I NTRODUCTION UTOMATIC generation control (AGC) plays a crucial role in multi-area power systems to maintain the system frequency and scheduled net inter-area power exchanges [1]. Under the current practice of AGC, each control area only takes care of its local demand by automatically tuning the power outputs of some selected generators. To achieve this target, the power flows along the tie lines that connect with the other areas are measured locally in each control area and are transmitted to the corresponding control center via some communication channels [2]. Then, the control center updates control signals based on the latest received tie-line flows, and sends the new control signals to the local generating units.

A

Manuscript received March 18, 2020; revised May 24, 2020; accepted June 5, 2020. Date of publication June 15, 2020; date of current version June 29, 2020. This work was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region under the Theme-Based Research Scheme through Project under Grant T23701/14-N, and in part by the General Research Fund Through Project under Grant 17256516. Recommended by Senior Editor C. Prieur. (Corresponding author: Luwei Yang.) Luwei Yang and Tao Liu are with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong (e-mail: lwyang@eee.hku.hk; taoliu@eee.hku.hk). David J. Hill is with the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, and also with the School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2016, Australia (e-mail: dhill@eee.hku.hk). Digital Object Identifier 10.1109/LCSYS.2020.3002422

Conventional AGC executes information transmission between local devices and control center in a synchronous and periodic way with a typical period 2-4 s [3]. However, it is pointed out in [4] that this periodic mechanism may sacrifice the frequency control performance as the period is usually selected to handle the worst case, and is irrespective of what is happening in the system. To overcome this issue, event-triggered control (ETC) has been introduced in networked control systems and advocated as a promising alternative of periodic communication-based control [5]–[7], where information transmission is determined by a well-defined event-triggering rule (ETR) on the basis of what is happening in the system. Hence, it is a more natural way to reflect the actual communication needs of the system. Recently, some related results of using ETC for power system frequency regulation have been reported (e.g., [4], [8]–[13]). References [8]–[11] focus on the centralized ETC for AGC, where a control center on the top of all control areas is needed to collect information of the entire system and check the designed ETRs. In fact, such a systemlevel control center exists in some real-world multi-area power systems (e.g., the regional transmission organization (RTO) in the Australian NEM [14]), but does not exist in some other power networks (e.g., EU Electricity Market [15]). Moreover, even if in a grid with a system-level control centre, the centralized ETRs proposed in [8]–[11] that rely on real-time state information of all control areas cannot be evaluated by the control center of the system without communication, and thus fall short in practical applications. To cope with this issue, decentralized event-triggered control (DETC) has been proposed for AGC in [4], [12], [13], where the real-time values of the net tie-line power of the corresponding control area are used in the designed ETRs. But, in each area, only the power flows along the tie lines connecting the neighbouring control areas can be locally measured at the tie-line interconnection points. Then, the net tie-line power can be calculated at the control center by summing up all related tie-line flows received from the interconnection points. Hence, the net tie-line power of each control area cannot be monitored at any single point without communication, and thus cannot be used in the ETR design. In view of the abovementioned issues, this letter proposes a novel DETC-based AGC for multi-area power systems. The proposed decentralized ETR for each control area only relies

c 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2475-1456 See https://www.ieee.org/publications/rights/index.html for more information.

Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 07,2020 at 11:11:50 UTC from IEEE Xplore. Restrictions apply.

374

on the communication structure of standard AGC and local information that can be derived from related devices. Inspired by [16], a dwell time is introduced into the designed ETR to exclude Zeno behaviours. Moreover, due to the time-varying property of renewable and load changes and the fact that the L∞ -gain is widely used as performance measure for control systems with disturbances, we adopt the L∞ -gain to quantify the stability performance of the interconnected power system with the proposed DTEC-based AGC [5]. We prove that the system with DETC has a finite L∞ -gain in the presence of essentially bounded renewable and load disturbances. Notations: Denote Z≥0 , R>0 , R, Rn , Rm×n as the sets of non-negative integers, positive real numbers, all real numbers, n-dimensional real vectors and (m×n)-dimensional real matrices, respectively. Denote 1n , In , 0m×n as the n-dimensional vector with all entries equal to 1, n-dimensional identity matrix, and (m × n)-dimensional zero matrix, respectively. Let col(x1 , . . . , xk ) = (x1T , . . . , xkT )T be the column vector with xi ∈ Rni , i = 1, . . . , k; and diag(a1 , . . . , ak ), diag(A1 , . . . , Ak ) be the diagonal and block diagonal matrices with the diagonal entries being ai ∈ R and Ai ∈ Rmi ×ni , i = 1, . . . , k, respectively. Let | · | be the absolute value of scalars a ∈ R, and · be the Euclidean norm of vectors x ∈ Rn and the induced 2-norm of matrices A ∈ Rm×n . Given a symmetric matrix A ∈ Rn×n , we denote λmin (A) (λmax (A)) as its minimum (maximum) eigenvalue; and use A 0 (A 0) to represent that A is positive definite (semi-definite). For a function f (t) : [0, +∞) → Rn , let f L∞ = esssupt∈[0,+∞) f (t) be its L∞ -norm, then it is essentially bounded, denoted as f (t) ∈ L∞ [0, +∞), if its L∞ -norm is finite, i.e., f L∞ < +∞.

II. M ODEL D ESCRIPTION We consider a power transmission network with n control areas, where each area under standard AGC is equipped with a control center that determines control signals for local generators. For simplicity, we treat all generators in each area as an equivalent generating unit [2]. Hence, each control area can be represented by the equivalent generating unit and its own control center. We assume the power grid fulfils the following assumptions: (i) voltage magnitudes are fixed for all areas in the system; (ii) the frequency of a control area is a common factor and can be measured at any interior point in the control area; (iii) reactive power flows do not affect the frequency; (iv) the tie lines that connect neighbouring control areas are lossless and characterized by their synchronizing torque coefficients Tij = Tji > 0. In fact, these assumptions are widely used in control and analysis for transmission networks (e.g., [1], [17]). Assumption (i) can be guaranteed by fast voltage control approaches. Assumption (ii) is due to the fact that the frequency in the same control area is almost the same everywhere. Assumption (iii) is due to the approximate decoupling of active and reactive power control partly caused by the small resistance of transmission lines that is usually ignored in transmission network analysis and control. Assumption (iv) is also due to the ignorable resistance of tie lines. For details of the justification for Assumptions (i)-(iv), please refer to [2].

IEEE CONTROL SYSTEMS LETTERS, VOL. 5, NO. 2, APRIL 2021

Fig. 1. The diagram of the ith control area under the DETC-based AGC.

We further assume the underlying graph that describes the physical interconnections between control areas is connected and the power network initially operates at an equilibrium point at t = t0 . Without loss of generality, let t0 = 0. We use the linearized model at the initial operating point to describe the system dynamics. For each control area i ∈ N with N = {1, . . . , n}, let ωi , pmi , pvi , pci be the increments of the frequency, mechanical power, governor valve position, and load reference set-point of the equivalent generating unit with respect to their initial nominal values, respectively; pdi be the change in the local net demand which represents the disturbance from both local renewable generation and load consumption; and Mi , Di , Ri , Tmi , Tvi be the moment of inertia, load-damping constant, droop-control gain, governor time constant and turbine time constant, respectively. Denote Ni as the index set of the areas that are directly connected with the ith control area, and pij , j ∈ Ni as the increment of the power flow in the direction i → j from its initial value. In practice, the power flows pij , j ∈ Ni , are measured locally at the tie-line connection points in the ith control area, i ∈ N , and then are transmitted to the control center via communication. The control signal pci is updated at the control center based on the latest received tie-line flows and the frequency measured in the control center, and then is sent to the local generating unit (see Fig. 1). For the ith control area, define ij the increasing time sequence {sk } and {tki } with j ∈ Ni and k ∈ Z≥0 as the instants when the sampled values of pij are sent to the control center and the instants when the updated control decision pci is sent to the generating unit, respectively. Then, the dynamics of the equivalent generating unit in the ith control area can be described as follows [2] Mi ω̇i = −Di ωi + pmi − pdi − pij j∈Ni

Tmi ṗmi = −pmi + pvi

Tvi ṗvi = −R−1 i ωi − pvi + p̄ci ṗij = Tij (ωi − ωj ), j ∈ Ni

(1)

i where p̄ci = p̄ci (t) = pci (tki ), ∀t ∈ (tki , tk+1 ] is the latest control input that the generating unit receives. The control center of the ith control area updates the AGC signal as ṗci = −Ki (βi ωi + p̄ij ) (2) j∈Ni

ij

ij

ij

where p̄ij = p̄ij (t) = pij (sk ), ∀t ∈ (sk , sk+1 ] is the most recently updated tie-line flow that the control center receives; and βi , Ki are respectively the frequency-bias coefficient and integral control gain. We assume system (1) with (2) is well

Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 07,2020 at 11:11:50 UTC from IEEE Xplore. Restrictions apply.

YANG et al.: DECENTRALIZED EVENT-TRIGGERED FREQUENCY CONTROL WITH GUARANTEED L∞ -GAIN FOR MULTI-AREA POWER SYSTEMS

initialized at t = 0, i.e., pij (t), pci (t), ∀i ∈ N , j ∈ Ni are sampled and shared between the generating unit and control center in the ith control area at t = 0, which is commonly used in the field of ETC (e.g., [5], [6], [9]–[11], [16], [21]). To guarantee this, time synchronization between different control areas is needed in this letter. These assumptions have conservativeness in practice, and how to relax them deserves attention. Further, for simplicity, we assume no time delays during the signal sampling and transmission, and no data losses in the communication channels. III. DETC A LGORITHM In this section, we propose a DETC algorithm for system (1) with (2) to decide the communication instants, i.e., the instants ij {sk }, j ∈ Ni , {tki } with k ∈ Z≥0 , for each control area i ∈ N . To be specific, we propose the following ETRs ij

ij

sk+1 = inf{t ≥ sk + T | |epij (t)| > σ1 |ωi (t)|} i tk+1

= inf{t ≥

tki

+ T | |eci (t)| > σ2 |ωi (t)|}

(3a) (3b)

where epij (t) = p̄ij (t) − pij (t), eci (t) = p̄ci (t) − pci (t) are the errors between the latest updated values and current values of the tie-line flow pij as well as control decision pci , respectively; and σ1 , σ2 , T ∈ R>0 are three constants to be selected. In particular, T is the dwell time introduced to exclude Zeno behaviours, i.e., a strictly positive minimum inter-event time interval is guaranteed and is no less than T . According to (3a), each control area i, continuously monitors the tie-line flow pij (t), j ∈ Ni , and frequency ωi (t) at ij the tie-line interconnection point. For time t > sk , the next ij ij event happens at t = sk+1 only if both t ≥ sk + T and |epij (t)| > σ1 |ωi (t)| are satisfied. Then, p̄ij (t) will be updated ij by the latest sampled tie-line power flow pij (sk+1 ), and hence epij (t) will be reset to zero. The update of p̄ci (t) is similar as that of p̄ij (t) but according to ETR (3b). Remark 1: In general, centralized ETC may achieve better control performance than decentralized one. Nevertheless, as discussed in the Introduction, the centralized ETC for AGC usually depends on real-time state information of all control areas (e.g., [8]–[11]), which however cannot be obtained at the system-level control center without communication. In fact, for a multi-area power network, AGC usually requires each control area to be controlled separately but help each other through tie-line power flows during a transient period. From this point of view, ETRs (3) only rely on locally available information in each area, and thus the proposed DETC-based AGC can be implemented in a fully decentralized way. Moreover, unlike [4], [12], [13], ETR (3a) can be evaluated right at the tie-line interconnection point, and thereby avoids the requirements on the net tie-line power exchange of the whole control area. IV. L∞ -S TABILITY A NALYSIS In this section, we study the L∞ -stability of system (1) with (2) under the designed event-triggered control law (3) for the cases pdi ∈ L∞ [0, +∞), ∀i ∈ N , i.e., the net load changes are assumed to be essentially bounded. Let ptiei be the deviation of the net tie-line power of control area i from its nominal value. From ptiei = j∈Ni pij , we have

375

ṗtiei = j∈Ni Tij (ωi − ωj ) and i∈N ptiei = 0, as pij = −pji , ∀i ∈ N , j ∈ Ni . Moreover, let x = col(ω, pm , pv , ptie , pc ) with ω = col(ω1 , . . . , ωn ), pm = col(pm1 , . . . , pmn ), pv = col(pv1 , . . . , pvn ), ptie = col(ptie1 , . . . , ptien ), and pc = col(pc1 , . . . , pcn ); pd = col(pd1 , . . . , pdn ); and e = col(ep , ec ) with ep = col(ep1 , . . . , epn ) and ec = col(ec1 , . . . , ecn ). Here, epi = col(epi1 , . . . , epin ) ∈ Rn with epij = 0, ∀i ∈ N , j ∈ N \ Ni . Then, system (1) with (2) as well as (3) can be rewritten as ẋ = Ax + Be + Fpd

(4)

where matrix F = [ − M −1 , 0]T ∈ R5n×n , and A ∈ R5n×5n , 2 B ∈ R5n×(n +n) are defined by ⎡ ⎤ −M −1 D M −1 0 − M −1 0 ⎢ 0 0 ⎥ 0 − Tm−1 Tm−1 ⎢ ⎥ −1 −1 −1 ⎢ A = ⎢ −Tv R 0 − Tv 0 Tv−1 ⎥ ⎥ ⎣ L 0 0 0 0 ⎦ −Kβ 0 0 −K 0

T 0 0 0 0 − CK B= 0 0 Tv−1 0 0 with M = diag(M1 , . . . , Mn ), D = diag(D1 , . . . , Dn ), Tm = diag(Tm1 , . . . , Tmn ), Tv = diag(Tv1 , . . . , Tvn ), R = diag(R1 , . . . , Rn ), K = diag(K1 , . . . , Kn ), β = 2 diag(β1 , . . . , βn ), C = diag(1n , . . . , 1n ) ∈ Rn ×n . Matrix L = [lij ] ∈ Rn×n is defined by lij = −Tij , if i = j, j ∈ Ni ; lij = 0, if i = j, j ∈ N \ Ni ; and lii = j∈Ni Tij . Based on system (4), we provide the definition of L∞ stability from the disturbance pd to state x [18]. Definition 1: The closed-loop system (4) under DETC (3) is said to be L∞ -stable from the disturbance pd to state x with an L∞ -gain no greater than θ ∈ R>0 if and only if x L∞ ≤ θ pd L∞ , ∀pd ∈ L∞ [0, +∞).

(5)

From Assumption (iv), we have Tij > 0, ∀i ∈ N , j ∈ Ni , which together with the definition of L leads to L 0. In fact, L is a Laplacian matrix with only one zero eigenvalue for the connected multi-area power network [16]. Applying elementary operations on A gives an equivalent upper triangular block matrix à whose block diagonal entries are Tv−1 , Tm−1 , M −1 , −K and L. Since Tv , Tm , M, K are all diagonal matrices with nonzero diagonal entries, we have rank(A) = rank(Ã) = 5n − 1 ([19, Th. 2.5]), and hence A has only one zero eigenvalue. Further, since the Laplacian matrix L is symmetric, there exists an orthogonal matrix = [φ1 , φ2 , . . . , φn ] ∈ Rn×n , i.e., T = In , such that T L = diag(λ1 , λ2 , . . . , λn ) with 0 = λ1 < λ2 ≤ . . . ≤ λn , where φi is the eigenvector of L corresponding to the eigenvalue λi . In particular, we choose φ1 = √1n 1n for λ1 . Define the orthogonal matrix H as H = diag(I3n , , In ) ∈ R5n×5n . Then, H T AH and A are similar matrices and have the same eigenvalues. From φ1T L = 0, we have that all entries in the (3n + 1)th row of H T AH are zero, which corresponds to the zero eigenvalue of matrix H T AH. We introduce a new transformation HT AH ∈ R(5n−1)×(5n−1) with H = diag(I3n , , In ) ∈ R5n×(5n−1) and = [φ2 , . . . , φn ] ∈ Rn×(n−1) to remove the (3n+1)th row and column of H T AH. Then, the remaining matrix HT AH with no zero eigenvalue has the same nonzero eigenvalues as H T AH.

Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 07,2020 at 11:11:50 UTC from IEEE Xplore. Restrictions apply.

376

IEEE CONTROL SYSTEMS LETTERS, VOL. 5, NO. 2, APRIL 2021

Further, matrix has the following properties T = In−1 , and T = In − 1n 1n 1Tn [16] . Motivated by the above observations, to analyse the L∞ stability of the system (4) under DETC (3), we define y = HT x. Moreover, we have Hy = HHT x = diag(I3n , T , In )x = x (6) where we use the fact i∈N ptiei = 0. Therefore, system (4) can be rewritten in the new coordinate as ẏ = Āy + B̄e + F̄pd (7) with Ā = HT AH, B̄ = HT B, F̄ = HT F. From (6), we have x = Hy ≤ H y = y (8) where the last equality is due to the definition of H. Inequality (8) leads to x L∞ ≤ y L∞ . Therefore, we have the following lemma. Lemma 1: If system (7) with (3) is L∞ -stable from pd to y, then system (4) with (3) is L∞ -stable from pd to x. Based on the input-to-state stability (ISS) theory for linear systems [20], a necessary condition for the L∞ -stability of system (7) with respect to the exogenous disturbance pd is that matrix Ā is Hurwitz. Hence, we assume the control gains β, K in (2) are selected such that Ā is Hurwitz. Let no = maxi∈N {ni } with ni being the total number of neighbouring control areas that are directly connected with control area i, and To = maxi∈N ,j∈Ni {Tij } be the maximum synchronizing torque coefficient among all tie lines. We now present the main results of this letter. Theorem 1: Consider system (4) under DETC (3). Suppose there exist constants δ, γ , μ ∈ R>0 and matrix P 0 such that the following linear matrix inequalities (LMIs) hold ⎤ ⎡ ϒ11 ϒ12 ϒ13 ⎣ ϒ T ϒ22 ϒ23 ⎦ 0 (9a) 12 T ϒT ϒ ϒ13 33 23 −P Ā − Ā P 0 T

(9b)

where ϒ11 = Q − δI5n−1 − 4no To2 Uω − ĀT Uc Ā, ϒ12 = −P B̄ −ĀT Uc B̄, ϒ13 = −P F̄, ϒ22 = γ I(n2 +n) − B̄ T Uc B̄, ϒ23 = 0(n2 +n)×n , and ϒ33 = μIn with Q = −P Ā − ĀT P, Uω = diag(In , 0) ∈ R(5n−1)×(5n−1) , and Uc = diag(0, In ) ∈ R(5n−1)×(5n−1) . Then, for any given parameters σ1 , σ2 , T ∈ R>0 in (3) satisfying

1 1 ρδ , T ≤ arctan( ) (10) max{σ1 , σ2 } ≤ (no + 1)γ α ρ √ where ρ ∈ (0, 1) and α = γ + ε with some ε ∈ R>0 , system (4) under DETC (3) is L∞ -stable for any pd ∈ L∞ [0, +∞); and no Zeno behaviours occur in the system. Proof: The existence of a strictly positive lower bound on the inter-event time interval of each communication channel is ij ij ensured by the positive dwell time T > 0, i.e., sk+1 − sk ≥ T , i i tk+1 − tk ≥ T , ∀i ∈ N , j ∈ Ni , k ∈ Z≥0 . Consequently, no Zeno behaviours will occur in the system with DETC (3). According to Lemma 1, we consider L∞ -stability of (7) instead of (4) in the sequel. For system (7), consider the following function V(ξ ) = U(y) + Wpij (epij , τpij ) i∈N j∈Ni + Wci (eci , τci ) (11) i∈N

where ξ = col(y, e, τp , τc ), τp = col(τp11 , . . . , τp1n , . . . , ij τpn1 , . . . , τpnn ), τc = col(τc1 , . . . , τcn ) with τpij = t − sk , i τci = t − tk being respectively the time elapsed since ij the last communication instants sk and tki . Functions U(y), Wpij (epij , τpij ) and Wci (eci , τci ) are defined as U(y) = yT Py, Wpij (epij , τpij ) = αϕ(τpij )e2pij , and Wci (eci , τci ) = αϕ(τci )e2ci , ∀i, j ∈ N , where ϕ(·) : [0, +∞) → [0, ρ −1 ] is a monotone non-increasing function defined as −tan(ατ − arctan( ρ1 )) τ ∈ [0, τ ∗ ] ϕ(τ ) = (12) 0 τ ∈ (τ ∗ , +∞) with τ ∗ = α −1 arctan(ρ −1 ) ∈ R>0 . Denote Nc− (t), Nc (t), Nc+ (t) as the index sets of the control areas satisfying 0 ≤ τci < τ ∗ , τci = τ ∗ , and τci > τ ∗ at any time t ∈ [0, +∞), respectively. Similarly, for the ith control area, denote Np−i (t), Npi (t), Np+i (t) as the index sets of its neighbouring control areas j ∈ Ni that satisfy 0 ≤ τpij < τ ∗ , τpij = τ ∗ , and τpij > τ ∗ at time t, respectively. Thus, we have Nc− (t) ∪ Nc (t) ∪ Nc+ (t) = N and Np−i (t) ∪ Npi (t) ∪ Np+i (t) = Ni , ∀t ∈ [0, +∞). In what follows, we denote these sets by dropping the time argument t. Then, inspired by [21], the proof of Theorem 1 will be conducted with respect to V(ξ ) in two steps. Step 1: We shall show that V̇ ≤ −κV + μ pd 2 with some κ ∈ R>0 for any time t ∈ [0, +∞) when no communication is triggered and Np1 ∪ · · · ∪ Npn ∪ Nc = ∅. According to ETR (3), epij , eci , ∀i ∈ N , j ∈ Ni , are piecewise continuous, i.e., they are continuous between any two consecutive communication instants. Then, due to the differentiability of ϕ(τ ) for any τ = τ ∗ , the functions Wpij , Wci are differentiable, and thus V is differentiable. Taking the derivative of U(y) in V along system (7) gives U̇ = −yT Qy + 2yT P B̄e + 2yT P F̄pd ≤ −δ y 2 + γ e 2 + μ pd 2 − 4no To2 Uω y 2 − Uc Āy + Uc B̄e 2

(18)

where the inequality is derived from (9a). Further, from (8) and x = Hy, we have i∈N Hi y 2 = i∈N xi 2 ≤ y , which together with (18) yields (−δ Hi y 2 + μp2di + γ e2ci − (Uci Āy + Uci B̄e)2 U̇ ≤ i∈N + (γ e2pij − (Tij Uωi y − Tij Uωj y)2 )) (19) j∈Ni

where xi = col(ωi , pmi , pvi , ptiei , pci ) is the vector derived by collecting entries of x from the ith control area; and Hi , Uωi , Uci ∈ R1×(5n−1) are the submatrices of H, Uω , Uc obtained by refining the ith row of H, Uω and Uc , respectively. To deduce (19), we use the following facts (Tij Uωi y − Tij Uωj y)2 i∈N j∈Ni ≤ 2To2 ((Uωi y)2 + (Uωj y)2 ) i∈N j∈Ni = 2To2 2ni (Uωi y)2 ≤ 4no To2 Uω y 2 . (20) i∈N

Taking the time derivative of Wpij gives Ẇpij = 0, ∀i ∈ N , j ∈ Np+i

(21a)

Ẇpij = −2αϕ(τpij )epij (Tij Uωi y − Tij Uωj y) − α 2 (ϕ 2 (τpij ) + 1)e2pij , ∀i ∈ N , j ∈ Np−i (21b)

Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 07,2020 at 11:11:50 UTC from IEEE Xplore. Restrictions apply.

YANG et al.: DECENTRALIZED EVENT-TRIGGERED FREQUENCY CONTROL WITH GUARANTEED L∞ -GAIN FOR MULTI-AREA POWER SYSTEMS

where equation (21b) is derived by noting that ėpij = −ṗij = −Tij (ωi − ωj ) = −Tij (Uωi y −Uωj y), τ̇pij = 1, ∀i ∈ N , j ∈ Np−i , and ϕ̇(Ď„ ) = âˆ’Îą(Ď• 2 (Ď„ ) + 1), âˆ€Ď„ ∈ (0, Ď„ ∗ ) from (12). Similar to (21), ėci = −ṗci = −Uci ẏ = −Uci (AĚ„y + Uci BĚ„e) and τ̇ci = 1, ∀i ∈ Nc− lead to Ẇci = 0, ∀i ∈ Nc+ Ẇci = −2ιϕ(Ď„ci )eci (Uci AĚ„y + Uci BĚ„e) − Îą 2 (Ď• 2 (Ď„ci ) + 1)e2ci , ∀i ∈ Nc− .

(22a) (22b)

From (19), (21) and (22), the time derivative of V(Ξ ) along system (7) satisfies inequality (23), as shown at the bottom of the page. According to ETR (3a), the ith control area, i ∈ N , transmits the sampled values of the tie-line flow pij , j ∈ Ni , only when Ď„pij ≼ T and |epij | > Ďƒ1 |ωi | are both satisfied. As no communication occurs in the case of Step 1, we get |epij | ≤ Ďƒ1 |ωi |, âˆ€Ď„pij ≼ T . Similarly, we have |eci | ≤ Ďƒ2 |ωi |, âˆ€Ď„ci ≼ T from ETR (3b). Combining (10) with |ωi | ≤ xi = Hi y , ∀i ∈ N gives Ď Î´ Hi y 2 , ∀i ∈ N , j ∈ Np+i ni + 1 Ď Î´ Hi y 2 , ∀i ∈ Nc+ . Îł e2ci ≤ ni + 1

γ e2pij ≤

(24a) (24b)

i∈N

j∈Npi

i∈Nc−

Îľe2ci

(25)

√ which results from Îą = Îł + Îľ and a2 + b2 ≼ −2ab, ∀a, b ∈ R. Moreover, from Ď•(Ď„ ) ∈ (0, Ď âˆ’1 ), âˆ€Ď„ ∈ (0, Ď„ ∗ ) and yT Py ≤ Îťmax (P) y 2 = Îťmax (P) HT Hy 2 ≤ Îťmax (P) HT 2 Hy 2 with HT H = I(5n−1) , we have Îą 2 V ≤ Îťmax (P) HT 2 Hy 2 + e − i∈N j∈Npi Ď pij Îą 2 e . + (26) i∈Nc− Ď ci As a consequence, we have V̇ ≤ âˆ’ÎşV + Îź pd 2

(27)

Ď Îľ with Îş = min{ Îť (1âˆ’Ď )δ T 2 , Îą }. max (P ) H Step 2: We shall show that V is non-increasing at any time t when at least one communication channel is activated or Np1 âˆŞ ¡ ¡ ¡ âˆŞ Npn âˆŞ Nc = ∅. For the case that no event is triggered at time t, we have V is continuous and thus non-increasing at time t due to the continuity of Ď•(Ď„ ) at Ď„ = Ď„ ∗ . For the cases that there are events triggered at time t, let NĚƒc (t) be the set of all control

i∈N \Nc

V(Ξ + )

where V(Ξ ), are the values of function V at the communication instant t and t+ , respectively. Therefore, V is non-increasing at any time t when at least one communication channel is activated or Np1 âˆŞ ¡ ¡ ¡ âˆŞ Npn âˆŞ Nc = ∅. According to the comparison principle in [20], it follows from (27) and (28) that t V(Ξ ) ≤ eâˆ’Îşt V(Ξ(0)) + Îź eâˆ’Îş(t−s) pd 2 ds 0

Îź (1 − eâˆ’Îşt ) pd 2L∞ (29) Îş for all t ∈ [0, +∞), where Ξ(0) is the initial condition at t = 0. Since the system is assumed to be initially operate at a nominal point and well initialized with pĚ„ij (0) = pij (0) = pĚ„ci (0) = pci (0) = 0, ∀i ∈ N , j ∈ Ni , we have Ξ(0) = 0 and epij = eci = 0. Thus, from (29), yT Py ≼ Îťmin (P) y 2 , x ≤ y and φ(Ď„ ) ≼ 0, âˆ€Ď„ ≼ 0, we have x 2 ≤ y 2 ≤

Îź 1 V(Ξ ) ≤ pd 2L∞ Îťmin (P) κΝmin (P)

(30)

for all t ∈ [0, +∞), which leads to

x L∞ ≤ θ pd L∞

(31)

with θ = κΝminÎź(P ) . Therefore, system (4) with ETR (3) is L∞ -stable with an L∞ -gain less than or equal to θ . Remark 2: In model (1) with (2), each control area is treated as an equivalent generating unit. In practice, each control area often has more than one generators. For such a system, the network-reduced model can be used to include the dynamics of the governor and turbine of each generator (see [1] for more details). This will lead to higher-order system matrices AĚ„, BĚ„, FĚ„ in (4). But, the obtained results in this letter can be easily extended to such a system. We will show this by using a threearea system that has three generators in each control area in the case study.

δ Hi y 2 δ Hi y 2 δ Hi y 2 − Îł e2pij ) − − Îł e2ci ) − − Îł e2pij + Îą 2 e2pij +( −( i∈N i∈Nc i∈N j∈Npi ni + 1 ni + 1 ni + 1 δ Hi y 2 − Îł e2ci + (Tij Uωi y − Tij Uωj y)2 + 2ιϕ(Ď„pij )epij (Tij Uωi y − Tij Uωj y) + Îą 2 Ď• 2 (Ď„pij )e2pij ) − −( i∈Nc ni + 1 + Îą 2 e2ci + (Uci AĚ„y + Uci BĚ„e)2 + 2ιϕ(Ď„ci )eci (Uci AĚ„y + Uci BĚ„e) + Îą 2 Ď• 2 (Ď„ci )e2ci ) + Îźp2di (23)

V̇ ≤ −

areas fulfilling ETR (3b) at time t. Further, for the ith control area, let NĚƒpi (t) be the set of all its neighbouring control areas that satisfy ETR (3a) at time t. According to (3a), the values of epij , ∀i ∈ N , j ∈ NĚƒpi , and eci , ∀i ∈ NĚƒc will be reset to zero at t+ = lims↓t s. Thus, by the continuity of Ď•(Ď„ ) at Ď„ = Ď„ ∗ , continuity of y, epij , ∀i ∈ N , j ∈ Ni \ NĚƒpi , eci , ∀i ∈ N \ NĚƒc at time t, and Ď•(Ď„ ∗ ) = 0, we have ιϕ(Ď„pij )e2pij V(Ξ + ) = U(y) + i∈N j∈Ni \(NĚƒpi âˆŞNpi ) + ιϕ(Ď„ci )e2ci (28a) i∈N \(NĚƒc âˆŞNc ) ≤ U(y) + ιϕ(Ď„pij )e2pij i∈N j∈Ni \Npi + ιϕ(Ď„ci )e2ci = V(Ξ ) (28b)

≤ eâˆ’Îşt V(Ξ(0)) +

Substituting (24) into (23) gives

V̇ ≤ −(1 − Ď )δ Hy 2 + Îźp2di i∈N 2 − − Îľepij −

377

j∈Np+i

(

i∈N

Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 07,2020 at 11:11:50 UTC from IEEE Xplore. Restrictions apply.

378

IEEE CONTROL SYSTEMS LETTERS, VOL. 5, NO. 2, APRIL 2021

variations and load changes. Case studies have demonstrated the effectiveness of the proposed method.

R EFERENCES

Fig. 2. State responses for the DETC algorithm (black lines) and periodic communication mechanism with a typical period 2 s (red lines).

V. C ASE S TUDY We test the proposed DETC method on a three-area power network that has three generating units in each area (see [1] for details and parameters of the test system). Here, we only consider the disturbance from renewables by using the real photovoltaic (PV) generation data presented from [23]; and assume the load demand of each area keeps unchanged during the simulation period as they usually change relatively slowly within a short time period. The net load disturbances are given in Fig. 2. We set σ1 = σ2 = 0.2 and T = 0.4 s such that condition (10) is satisfied. The control performance of the DETC-based AGC is compared with conventional AGC with a typically periodic communication period 2 s. Fig. 2 gives the responses of the frequency and net tie-line power flow of each control area of the two methods, which suggests that the proposed control method has better control performance. The average communication period of the DETC-based AGC is 2.09 s that is greater than 2 s, the period of the periodic communication. This indicates the potential of our event-triggered frequency control approach in mitigating the communication burden of AGC, and the rigorous analysis will be studied in the future. Further, we adopt t the integral squared error (ISE) index defined by J = 0 ( ω 2 + ptie 2 )dt to quantify the control performance [22]. During the simulation period, we have J = 5.414 × 10−4 for DETC and J = 6.998 × 10−4 for the periodic mechanism, which implies the developed eventtriggered control law is consistent with a 22.6% performance improvement [6]. Furthermore, we have x L∞ = 0.1508 and pd L∞ = 0.0879 during the simulation period, which satisfies the results obtained in Theorem 1 with θ = 10. VI. C ONCLUSIONS This letter has developed a fully decentralized eventtriggered communication algorithm for AGC in multi-area power systems. It has been shown that the closed-loop system with the proposed algorithm is L∞ -stable against essentially bounded net demand disturbances caused by renewable

[1] H. Bevrani, Robust Power System Frequency Control. New York, NY, USA: Springer, 2009. [2] P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1994. [3] J. Nanda, A. Mangla, and S. Suri, “Some new findings on automatic generation control of an interconnected hydrothermal system with conventional controllers,” IEEE Trans. Energy Convers., vol. 21, no. 1, pp. 187–194, Mar. 2006. [4] L. Yang, T. Liu, and D. J. Hill, “Decentralized periodic event-triggered frequency regulation for multi-area power systems,” in Proc. Power Syst. Comput. Conf., Dublin, Ireland, 2018, pp. 1–7. [5] M. C. F. Donkers and W. P. M. H. Heemels, “Output-based eventtriggered control with guaranteed L∞ -gain and improved and decentralized event-triggering,” IEEE Trans. Autom. Control, vol. 57, no. 6, pp. 1362–1376, Jun. 2012. [6] D. J. Antunes and M. H. Balaghi I., “Consistent event-triggered control for discrete-time linear systems with partial state information,” IEEE Control Syst. Lett., vol. 4, no. 1, pp. 181–186, Jan. 2020. [7] B. A. Khashooei, D. J. Antunes, and W. P. M. H. Heemels, “Outputbased event-triggered control with performance guarantees,” IEEE Trans. Autom. Control, vol. 62, no. 7, pp. 3646–3652, Jul. 2017. [8] S. Wen, X. Yu, Z. Zeng, and J. Wang, “Event-triggering load frequency control for multiarea power systems with communication delays,” IEEE Trans. Ind. Electron., vol. 63, no. 2, pp. 1308–1317, Feb. 2016. [9] C. Peng, J. Li, and M. Fei, “Resilient event-triggering H∞ load frequency control for multi-area power systems with energy-limited DoS attacks,” IEEE Trans. Power Syst., vol. 32, no. 5, pp. 4110–4118, Sep. 2017. [10] P. Dahiya, P. Mukhija, and A. Saxena, “Design of sampled data and event-triggered load frequency controller for isolated hybrid power system,” Int. J. Elect. Power Energy Syst., vol. 100, pp. 331–349, Sep. 2018. [11] J. Liu, Y. Gu, L. Zha, Y. Liu, and J. Cao, “Event-triggered H∞ load frequency control for multiarea power systems under hybrid cyber attacks,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 49, no. 8, pp. 1665–1678, Aug. 2019. [12] Y. Yuan, P. Zhang, Z. Wang, and Y. Chen, “Noncooperative event-triggered control strategy design with Round-Robin protocol: Applications to load frequency control of circuit systems,” IEEE Trans. Ind. Electron., vol. 67, no. 3, pp. 2155–2166, Mar. 2020. [13] Y. Cui, M. Fei, and D. Du, “Convergence rate controllable eventtriggered robust load frequency control for multi-area power systems,” in Proc. UKACC 11th Int. Conf. Control, Belfast, U.K., 2016, pp. 1–11. [14] R. P. O’Neill, R. Baldick, U. Helman, M. H. Rothkopf, and W. Stewart, “Dispatchable transmission in RTO markets,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 171–179, Feb. 2005. [15] T. Slot and H. Dijk, Options for Future European Electricity System Operation, Eur. Commission, Brussels, Belgium, 2015. [Online]. Available: https://ec.europa.eu/energy/sites/ener/files/documents/153071 [16] T. Liu, M. Cao, C. De Persis, and J. M. Hendrickx, “Distributed event-triggered control for asymptotic synchronization of dynamical networks,” Automatica, no. 86, pp. 199–204, Dec. 2017. [17] T. Liu, D. J. Hill, and C. Zhang, “Non-disruptive load-side control for frequency regulation in power systems,” IEEE Trans. Smart Grid, vol. 7, no. 4, pp. 2142–2153, Jul. 2016. [18] A. J. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control. London, U.K.: Springer, 2000. [19] N. Buaphim, K. Onsaard, P. So-ngoen, and T. Rungratgasame, “Some reviews on ranks of upper triangular block matrices over a skew field,” Int. Math. Forum, vol. 13, no. 7, pp. 323–335, 2018. [20] E. D. Sontag, “Input to state stability: Basic concepts and results,” in Nonlinear and Optimal Control Theory. Heidelberg, Germany: Springer, 2006, pp. 163–220. [21] M. Abdelrahim, R. Postoyan, J. Daafouz, and D. Nešić, “Stabilization of nonlinear systems using event-triggered output feedback controllers,” IEEE Trans. Autom. Control, vol. 61, no. 9, pp. 2682–2687, Sep. 2016. [22] R. J. Abraham, D. Das, and A. Patra, “Effect of TCPS on oscillations in tie-power and area frequencies in an interconnected hydrothermal power system,” IET Gener. Transm. Distrib., vol. 1, no. 4, pp. 632–639, Jul. 2007. [23] (2020). Live Photovoltaic Data. [Online]. Available: https://pvoutput.org

Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 07,2020 at 11:11:50 UTC from IEEE Xplore. Restrictions apply.

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.