Data-Driven Wide-Area Control Design of Power System Using the Passivity Shortage Framework

IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 1 Data-Dri v en W ide-Area Control Design of Po wer System Using the Passi vity Shortage Frame work Y ing Xu, Member , IEEE, Zhihua Qu, F ellow , IEEE, Roland Harve y , Student Member , IEEE, and T oru Namerikaw a, Member , IEEE, Abstract —A novel wide-area control design is presented to mit- igate inter-ar ea power frequency oscillations. A large-scale power system is decomposed into a network of passivity-short subsys- tems whose nonlinear interconnections ha ve a state-dependent affine form, and by utilizing the passi vity shortage framework a two-lev el design procedure is developed. At the lower level, any generator control can be viewed as one that makes the generator passivity-short and L 2 stable, and the stability impact of the lower -level contr ol on the overall system can be characterized in terms of two parameters. While the system is nonlinear , the impact parameters can be optimized by solving a data-driven matrix inequality (DMI), and the high-level wide-area control is then designed by solving another L yapunov matrix inequality in terms of the design parameters. The proposed methodology makes the design modular , and the resulting control is adaptive with respect to operating conditions of the power system. A test system is used to illustrate the proposed design, including DMI and the wide-area control, and simulation results demonstrate effectiveness in damping out inter -area oscillations. Index T erms —wide-area control, data-driven control, matrix inequality , L yapunov stability , passivity-short systems, power systems I . I N T RO D U C T I O N I NTER-AREA oscillations observed in lar ge-scale power systems are typically recognized as lo w frequency problems on the order of 0.1-1.0 Hz. As the system expands and energy interchanges between interconnected systems increase, these low-frequenc y inter-area oscillations often become poorly damped. Recently , this problem has been e ven more chal- lenging due to the fast de velopment and high penetration of renew able resources. T o solve this problem, tremendous ef fort has been made in the past decades. In the traditional damping design, each subsystem is treated as an independent control, and each one is capable of acting on its own. For example, using power system stabilizers (PSS) is a typical local control design, forming an additional part of the generation control system. Howe ver , it is well-known that local designs may not always be effecti ve to damp out the inter-area modes of oscillations for the following reasons: • The design is usually based on the internalization of each individual subsystem under certain operation conditions, Y . Xu, Z. Qu and R. Harv ey are with Department of Electrical and Computer Engineering, University of Central Florida, Orlando 32816, USA. Emails: ying.xu@ucf.edu , qu@ucf.edu . This work is supported in part by US National Science Foundation under grants ECCS-1308928 and ECCS-1552073, by US Department of Energy’ s awards DE-EE0006340, DE-EE0007327 and DE- EE0007998, by Leidos’ contract P010161530, and by T exas Instruments’ grants. T . Namerikawa is with Keio Univ ersity , Japan. thus, the stability may not be ensured under any local design as the operation point of a power system changes. • Although an overall centralized control (e.g. AGC) has been used in po wer systems for years, a systematic design has not been reported yet to assure the o verall stability . W ith the advent of time-synchronized phasor measurement units (PMUs) and fast-speed communication technologies, the concept of wide-area measurement systems (W AMS) has attracted research interest and has become indispensable in addressing such issues as instability detection and control, se- curity assessment and enhancement in modern power systems. Progress has been reported in wide-area control of power systems, and detailed results on improving inter-area oscilla- tion damping are presented in [1]–[9]. Most of the existing literature on this topic usually focus upon one or a few of the following aspects: power system model recognition, wide-area damping controller design, and parameter tuning methodology and performance v alidation. The difficulty of processing the large amount of data captured by W AMS ov er geographically dispersed locations has been one of the major issues in the application of PMU data. In [10], inter-area dynamics of the overall system are represented by a reduced-order model based on the estimation of aggregated system angles and velocities by using a non- linear Kalman filter . blueA dynamic eigensystem realization algorithm is presented in [11] to identify the inter-area os- cillation mode, and based on that a linear quadratic Gaus- sian controller is implemented through PSS. A method using PMU measurements on specific points is presented in [12] to construct dynamic inter-area system models by aggregating the generators inside each area. This approach establishes feasibility of the wide-area control through aggregating groups of generators. Sev eral approaches based on rob ust control theories and linear matrix inequalities (LMIs) have been applied to wide- area damping control designs [13], [14]. For instance in [13], a general procedure is proposed to design a wide- area damping controller by applying an LMI approach to the problem of regional pole placements. Designs based on artificial neural networks has become more popular recently , but those methods require an appropriate set of training data [15]. A model free method is presented in [16] by applying online reinforcement learning as the data-driven wide-area oscillation-damping control. Although the aforementioned results present significant progress, there are unresolved issues that limit the application and performance of wide-area control in actual power systems. IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 2 The most critical issue to be addressed is an overall stability analysis of interconnected dynamic systems. T o this end, a nov el systematic control design is proposed in this paper , and it uses the framework of passivity shortage, outlined in [17], applied to po wer systems. It should be noted that intercon- nection of passive systems was studied in [18], [19] among others but, in power systems, devices such as synchronous generators are not passiv e. Hence, passi vity-short systems and their properties must be applied. It is worth recalling that passivity-short systems and their properties are in vestigated in [20], and generator dynamics are alw ays passi vity-short [21]. In this paper , this energy-based approach is applied to design a two-lev el control by taking advantage of wide-area measurement data. Using the passivity-shortage framework, stability of the ov erall system is inv estigated as the interconnection of blue- subsystems or groups of coherent generators, their individual generator controls, and their wide-area control. By nature, each of the subsystems is passivity-short and L 2 stable. W ith W AMS data, reduced-order load flow equations can be identified, and the impact of passi vity-short bluesubsystems of coherent generators and their interconnections can be quanti- fied by two parameters using data-driv en matrix inequalities. As such, their impacts can be minimized by the design of individual controls for the subsystems. And, the high-level control can then be synthesized to ensure the overall system stability and hence to ef fectively damp out potential inter-area oscillations. The proposed data-dri ven control design allows the controls to adapt themselves to both the power system operating conditions and their transient behaviors, resulting in superior performance. The remainder of this paper is organized as follo ws. In sec- tion II, blue(aggre gated) models of synchronous and renew able generators are cast into the passi vity-short framew ork, and a two-le vel control design problem is formulated. In section III, a four-step design process is presented for both local and wide- area controls, data driv en optimization is done, and rigorous analysis of the ov erall system is performed. In section IV , communication topology , the control design algorithm, and its robustness are discussed. In section V , simulation results are presented to illustrate the effecti veness of the proposed design, followed by concluding remarks in section VI. I I . P RO B L E M F O R M U L A T I O N Consider the class of interconnected dynamic systems de- scribed by the following heterogeneous nonlinear model: for the i th subsystem, ˙ x i = A i ( x i ) x i + B i ( x i ) v i + X j ∈N i H ij ( y i , y j )( y j − y i ) , y i = C i ( x i ) x i , (1) where N = { 1 , · · · , n } is the index set, i ∈ N , N i ⊂ N denotes the neighboring set of system i , x i ∈ < n g i is the state vector of the i th subsystem, y i ∈ < l is its output vector , v i ∈ < m is its control input, matrices A i ( x i ) , B i ( x i ) , C i ( x i ) are state-dependent and of proper dimensions, and coupling matrices H ij ( y i , y j ) are output-dependent and of proper dimensions. It is shown in appendix A that dynamics of a po wer system of n generators, bluewhich could be either con ven- tional or renew ables or their aggregated models in certain geographical regions, are described by a set of nonlinear differential-algebraic equations. Those equations can be recast into system equations in the form of (1) and with respect to their equilibrium. In the appendix, vectors x i , y i , and v i as well as matrices A i , B i , and C i are detailed so the proposed control design becomes directly applicable. Specifically , the state/output in (A.5) and the matrices in (A.7) should be used for con ventional generators, while in verter -based ener gy sources have the state/output and their corresponding matrices defined by (A.8) and (A.9), respectiv ely . For the i th system in (1), the proposed wide-area control is of form v i = − K i x i + u i , (2) where the first term − K i x i is the self-feedback control 1 , and u i is the W AMS-enabled control as u i = − k c i X j ∈N c i S c ij ( t )( y j − y i ) , (3) in which k c i > 0 is a control gain, and S c ij ( t ) denotes the pos- sibly time-v arying communication matrix for W AMS: giv en communication neighboring set N c i ( t ) of the i th subsystem, S c ij ( t ) = ( 1 if j ∈ N c i 0 otherwise . Under control (2), system (1) can be re written as ˙ x i = A i ( x i ) x i + B i ( x i ) u i + X j ∈N i H ij ( y i , y j )( y j − y i ) , y i = C i ( x i ) x i , (4) where A i ( x i ) = A i ( x i ) − B i ( x i ) K i . Its nominal controlled dynamics (excluding the interconnection) are e xpressed as ˙ x i = A i ( x i ) x i + B i ( x i ) u i , y i = C i ( x i ) x i . (5) Design of control (2) for all the subsystems in volves choices of feedback gain matrices K i , communication neighboring sets N c i , and cooperati ve control gain matrix K c = diag { k c i } . Feedback gain matrices and the cooperative control gain matrix will be synthesized in section III, and choices of communication topology will be discussed in section IV -A. The proposed design employs two novel tools. One is the analytical framework of passi vity-short dynamic systems, as summarized in the following definition. The concept of passivity-short systems is used because most physical systems, including the swing equation in power systems, are not pas- siv e. Definition 2.1: Nominal subsystem (5), or its input-output pair ( u i , y i ) , is said to be passivity-short with respect to storage function V i if inequality ˙ V i ≤ u T i y i +  ii 2 k u i k 2 − ρ i 2 k y i k 2 , (6) 1 If needed, a nonlinear self-feedback control of form − K i ( x i ) x i could be designed. IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 3 holds for some  ii > 0 . The subsystem is said to be L 2 stable if inequality (6) holds for some ρ i > 0 and a positiv e definite V i . The subsystem is said to be passive if inequality (6) holds for some  ii = 0 (and ρ i = 0 ). The second tool employed is an efficient computational algorithm in volving data-driven matrix inequality (DMI). T o illustrate the idea, consider nominal subsystem (5) and suppose that the storage function V i in Definition 2.1 is chosen to be quadratic as V i = 1 2 x T i P i x i , (7) where P i is a positiv e definite matrix. Then, we know from L yapunov’ s direct method that the nominal subsystem is passivity-short provided that " A T i P i + P i A i + ρ i C T i C i P i B i − C T i B T i P i − C i −  ii I # < 0 . By Schur complement lemma [22], the abo ve matrix inequality is equi v alent to M i ( x i ) 4 = A T i P i + P i A i + ρ i C T i C i + 1  ii k P i B i − C T i k 2 < 0 . (8) Matrix inequality (8) is state-dependent but can efficiently be solved real-time for K i (within matrix A i ),  ii and ρ i . This design based on data-dri ven matrix inequality (8) not only applies to individual subsystems b ut also to the interconnected system as a whole, and it also makes it possible to modularly synthesize a multi-level control for the resulting system, as shown by the four-step design process outlined in the subse- quent section. I I I . M O D U L A R C O N T R O L D E S I G N In this section, a modular control design is presented for the interconnected system consisting of (2), (3), and (4). The first step is to determine H ij using real-time measurement data. The second step is to design indi vidual feedback control gain K i so that individual nominal system (5) passi vity-short. The third step is to quantify the impact of interconnections among the subsystems on stability of the overall system. The fourth step is to mak e the ov erall system synchronize by appropriately synthesizing the wide-area control (3). When applied to po wer systems, steps two to four quantitati vely prescribe the impacts of generator controls, load flow equations, and wide-area control on the overall power system stability and performance, respectiv ely . Design steps two and three are expressed and solved as a real-time optimization problem so the overall system performance is ensured and enhanced. A. Data-Driven Calculation of H ij blueAssuming that the (aggregated) power system has N b buses, we know that power flow equations are described by P i − P L i = N b X j =1 V i V j ( g ij cos θ ij + b ij sin θ ij ) , Q i − Q L i = N b X j =1 V i V j ( g ij sin θ ij − b ij cos θ ij ) , where V i is the nodal voltage, θ i is the phase angle, P i is the power injection, P L i is the load, all at the i th bus; { g ij , b ij } are the real- and imaginary-part of the ( i, j ) th element in the power network admittance matrix, and θ ij = θ i − θ j . Due to the expansi ve nature of po wer transmission/distribution networks, it is impossible to monitor all the bus voltages, load variations, and topology changes (i.e., parameter variations of g ij and b ij ) within the o verall system. W ith the de velopment of W AMS, phase angles ( δ i ) and power injections ( P g i , Q g i ) blueat all geographical regions (i.e. major groups of power generation units) are monitored real- time. Accordingly , a reduced-order set of power equations can equiv alently be established at the power generation lev el as follows: P g i = X j ∈N E i E j ( G ij cos δ ij + B ij sin δ ij ) , (9a) Q g i = X j ∈N E i E j ( G ij sin δ ij − B ij cos δ ij ) , (9b) where E i is the inner bus voltage behind transient reactance of the i th generator , { P g i , Q g i } are the activ e and reactive power injections by the i th generator , G ij and B ij are the real and imaginary parts of the ( i, j ) th entry in the reduced- order network admittance matrix, and δ ij = δ i − δ j is the angle difference between the i th and j th generators. blueBy collecting recent time series measurement of E i , δ ij , P g i and Q g i , parameters G ij and B ij can be estimated by applying such standard techniques as the least square method [23] to equi v alent network equations in (9). Although real-time estimation is possible, system topology may change, and estimation error may not be neglectable. Hence, the proposed control design only requires the ranges of parameters G ij and B ij rather than not their accurate estimates, as shown below and in section IV .C. It follo ws from (A.7) or (A.9) in appendix A that H ij ( y i , y j ) =   0 0 h ij ( y i 1 ,y j 1 ) M i 0 0 0   , where y i 1 = δ i − δ ∗ i , δ ∗ i is the equilibrium of δ i , and h ij = E i E j G ij (cos δ ij − cos δ ∗ ij ) + B ij (sin δ ij − sin δ ∗ ij ) δ ij − δ ∗ ij , (10) and that po wer system dynamics hav e the follo wing property . Pr operty 3.1: Matrices A i , B i , C i , and H ij may be in general nonlinear but they are uniformly bounded in the whole state space. Further discussion will be provided in section IV -C to illustrate this property as well as robustness of the proposed control design. It should be noted that equi v alent network equations (9a) and (9b) are linear in system parameters. Hence, using the data locally a vailable from W AMS, the linear equations in (9) can be solved distributiv ely to estimate the system parame- ters G ij and B ij . For instance, a distributed algorithm was proposed in [24] to solve linear equations and determine the IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 4 system topology matrix using only local information, and this approach has successfully been applied in [25] to determine a reduced-order dynamic model of power systems. In short, distributed and ef ficient algorithms can be developed to moni- tor both the system parameters and the v alues of h ij ( y i 1 , y j 1 ) online. blueOnce again, the proposed design needs only the knowledge of upper and lower bounds on h ij ( y i 1 , y j 1 ) in order to ensure rob ustness. B. Individual Contr ol Design Consider nominal subsystem (5) and its corresponding storage function (7). It follows from (6) that the nominal subsystem is passi vity-short and L 2 stable if K i is designed to ensure data-driv en matrix inequality (8). There are many choices of self-feedback control gain K i (or gain function K i ( x i ) ). As shown in appendix A, traditional generators and in verter -based energy resources hav e the relativ e degrees of two or higher , and hence the inter-area po wer dispatch control problem always in v olves passi vity-short subsystems. Recall that property 3.1 holds for power systems. Accordingly , individual subsystems can always be made passivity-short, as the same conclusion is drawn in [26] for any linear stabilizable systems. From the perspecti ve of minimizing the impact of subsystem dynamics on the overall system, an ef fectiv e self-feedback control can be designed by minimizing  ii and maximizing ρ i . That is, the follo wing real-time optimization problem with constraints of the data-dri ven matrix inequality (DMI) can be solved: min K i , ii ,ρ i [ α ii  ii − (1 − α ii ) ρ i ] s.t. P i > 0 , M i ( x i ) ≤ 0 ,  ii , ρ i ≥ 0 , (11) where α ii ∈ (0 , 1) is a design parameter . At any instant of time t , x i ( t ) becomes known as well as matrix A i ( x i ) , and hence K i can be designed adaptiv ely by using av ailable (or previously determined) L yapunov function P i > 0 . C. Quantifying P assivity-Shortage Impact of Inter connections The following lemma, whose proof is included in appendix B, provides a useful property for subsystem (4). In particular , the quadratic terms  ij k y j k 2 quantify the impact of nonlinear interconnections on subsystem (4) in a way parallel to that of  ii k u i k 2 . Lemma 3.2: Subsystem (4) has the property that ˙ V i ≤ u T i y i +  ii 2 k u i k 2 − ρ i 2 k y i k 2 + 1 2 X j ∈N i  ij k y j k 2 , (12) provided that M 0 i ( x i , y j ) ≤ 0 , where M 0 i 4 = M i − X j ∈N i  P i H ij C i + C T i H T ij P i − 1  ij P i H ij H T ij P i  . (13) While u i can be designed to damp out inter-area oscilla- tions, transient impacts of those oscillations must be mini- mized. Accordingly , the following optimization problem can be solved by using the DMI in (13): min  ij X j ∈N i α ij  ij s.t. P i > 0 , M 0 i ( x i , y j ) ≤ 0 ,  ij , α ij ≥ 0 , X j ∈N i α ij = 1 . (14) It is worth noting that the optimization problems in (11) and (14) can be combined into one as: min K i , ii , ij ,ρ i   X j ∈N i ∪{ i } α ij  ij −   1 − X j ∈N i ∪{ i } α ij   ρ i   s.t. P i > 0 , M 0 i ( x i , y j ) ≤ 0 ,  ii ,  ij , ρ i ≥ 0 , α ii + X j ∈N i α ij < 1 . (15) The abo ve DMI-based optimization problem can be solved by using any of the standard techniques a vailable to solve either LMIs or bilinear matrix inequality (BMI) optimizations [27]. It is apparent that, if H ij = 0 for any j ,  ij ≡ 0 is the corresponding solution. D. Communication-Enabled W ide-Area Contr ol Design It follo ws from (3) that the network le vel cooperati ve control can be written in the vector form u = − K c Ly , (16) where K c = diag { k c i } , S c = { S c ij } , D = diag { S c 1 } , L = ( D − S c ) is the Laplacian of communication network of wide-area control. Design of the network level control depends on properties of individual subsystems, specifically , bluetheir impact coef ficients and L 2 parameters are quantified by {  ii , · · · ,  ij , · · · } and ρ i , respectiv ely . F or con venience of expression, let us denote W = diag {  ii } , Γ = diag { γ i } , and Φ = diag { φ i } , where γ i are entries of the first left eigen vector of L (that is, γ T L = 0 ) and φ i = γ i ρ i − X j =1: n ; i ∈N j γ j  j i . Stability of the ov erall system can be achiev ed by the choice of gain matrix K c , as sho wn by the follo wing theorem. Theor em 3.3: Under inequality (12), system (1) exponen- tially con verges to the desired output consensus under coop- erativ e control (16) provided that gain k c i ≈ k c is chosen as follows: i) If Φ ≥ 0 , then 0 ≤ k c < λ 0 (Γ L + L Γ) λ max ( L T W L ) . (17) IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 5 ii) If some φ i are negati ve but both inequalities P i φ i > 0 and λ 2 b + 4 λ a min( φ i ) ≥ 0 holds, then k c ∈ λ b − p λ 2 b + 4 λ a min( φ i ) 2 λ a , λ b + p λ 2 b + 4 λ a min( φ i ) 2 λ a ! , (18) where λ a = λ max ( L T W L ) , λ b = λ 0 (Γ L T + L Γ) , and λ 0 ( · ) and λ max ( · ) denote the dominant eigen v alue (the smallest non- zero) and the largest eigen value, respecti vely . Proof . It follows from Perron-Frobenius theorem [28] that, as long as Laplacian L (or equiv alently S c ) is strongly connected, its left eigen vector γ = vec { γ i } is positive. Given storage functions (7) for interconnected subsystems (4), choose the ov erall storage function as V = n X i =1 γ i k c i V i . It follo ws from (12) that ˙ V = n X i =1 γ i k c i ˙ V i ≤ n X i =1 γ i k c i y T i u i + 1 2 n X i =1 γ i k c i  ii k u i k 2 − 1 2 n X i =1 γ i k c i ρ i k y i k 2 + 1 2 n X i =1 X j ∈N i γ i k c i  ij k y j k 2 = − 1 2 y T Qy , (19) where Q = Γ L T + L Γ − L T K c W L + Ψ , (20) and Ψ = diag { ψ i } 4 = diag    γ i ρ i k c i − X j =1: n ; i ∈N j γ j k c j  j i    . (21) Hence, the overall system is exponentially stable if matrix Q is positiv e definite, and it has both L yapunov stability and an output consensus if matrix Q is both positive semi-definite and of rank ( n − 1) . Should k c i = k c , equation (20) reduces to Q = − k c L T W L + Γ L T + L Γ + Φ k c . (22) It follo ws from [28] that, if L is strongly connected, (Γ L + L Γ) is positiv e semi-definite and of rank ( n − 1) , and so is L T K c W L . Therefore, (Γ L + L Γ − k c L T W L ) is positi ve semi- definite and of rank ( n − 1) for all small values of k c satisfying (17). Hence, stability can be concluded for the case that Φ ≥ 0 . In the ev ent that Φ 6≥ 0 , some of φ i must be negati ve, and stability will be established in two steps. First, note that both (Γ L T + L Γ) and L T W L and Φ are positiv e semi-definite and of rank ( n − 1) , in particular , 1 T [ − k c L T W L + (Γ L T + L Γ)] 1 = 0 if and only if x = c 1 , where 1 is the vector of 1 s. It follo ws that 1 T Q 1 = 1 k c 1 T Φ 1 > 0 . Second, it follows from (22) that, for x / ∈ { c 1 } with c ∈ < and c 6 = 0 , x T Qx ≥ − k c λ max ( L T W L ) k x k 2 + λ 0 (Γ L T + L Γ) k x k 2 + min( φ i ) k c k x k 2 , (23) which is positive for all k c satisfying inequality (18). This concludes the proof. 4 I V . I M P L E M E N TA T I O N A N D I T S R O B U S T N E S S The proposed design yields a two-lev el control implemen- tation: a lower -le vel control in volving feedback gain matrix K i for passivity shortage and L 2 stability of the subsystems, and a higher -level control enabled by W AMS. The control architecture is shown in Fig. 1, and the details of communi- cation topology design, control algorithm implementation and robustness are illustrated in the subsequent subsections. Fig. 1. Ov erall architecture of the proposed control A. Communication T opology Design Implementation of wide-area cooperati ve control (3) or (16) in volves the choice of communication matrix S c , a mathemat- ical abstraction of W AMS. The simplest choice of S c is S c = 11 T , or S c ij ≡ 1 , which is all to all communication. W ide-area control u i in (3) is in the form of consensus law [28], and it has the property that, should subsystems i and j are coher ent (a concept established in power systems in [29] or , equi valently , in the sense that ( y i − y j ) ≈ 0 ), the corresponding control contribution is approximately zero. In other words, W AMS-enabled control (3) aims specifically at damping out inter-area oscillations, and communication neighboring set N c i should be chosen to exchange information among incoherent groups of subsystems (i.e., subsystems that are geographically apart). That is, the sparsest communication matrix is as follo w: giv en µ coherent groups of generators and for any 1 ≤ l 1 6 = l 2 ≤ µ , S c ij = 0 for all i ∈ C l 1 and j ∈ C l 2 except for one pair ( i ∗ , j ∗ ) such that S c i ∗ j ∗ = 0 with i ∗ ∈ C l 1 and j ∗ ∈ C l 2 , where C l is the index set of the l th coherent group, and ∪ µ l =1 C j = N . Any topology more dense than the sparsest would work for the proposed wide-area control. Under the topology of all-to- all communication, system parameters G ij and B ij can be solved at all the sites. Under all other possible topologies, system parameters can be estimated either by using distributed algorithm explained in section III-A or by the dispatch control center (which collects all critical information of the overall system). IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 6 B. Implementation of Data-Driven Contr ol The data used in the proposed design contains: Step 1: Local model and data at individual systems: matrices A i , B i , and C i ; local state x i and output y i ; and the outcome of indi vidual feedback gain design is P i , K i ,  ii , and ρ i ; Step 2: W ide-area data-driven interconnection model and its impact: outputs y = vec { y i } ; estimates of system parameters G ij , B ij , and values H ij ; determination of impact measures  ij through optimization; Step 3: W ide-area cooperative control: determination of k c by applying theorem 3.3. Should matrices A i , B i , and C i be constant, step 1 needs to be done only once; otherwise, step 1 needs to be dynamically computed if the matrices are state dependent. In general, step 2 needs to be dynamically calculated as wide-area data comes in. Steps 1 and 2 can be integrated into one as illustrated by (15). Step 3 needs to be repeated if  ij change noticeably . It follo ws from (10) that the changes in matrices H ij are due to their elements h ij ( y i 1 , y j 1 ) . As will be discussed in section IV -C, the change of h ij ov er time is often small and relativ ely slow . As a result, wide-area data-driv en computation can be held of f until the cumulativ e change of h ij has e xceeded certain threshold c T i . Specifically , let us define the following measure: h i ( t ) 4 = X j ∈N i | h ij ( y i 1 ( t ) , y j 1 ( t )) | , (24) then step 2 is ignored for the i th generator over time interval [ t − δ t, t ] if | h i ( t ) − h i ( t − δ t ) | < c T i . (25) In summary , the proposed data-driv en control is imple- mented using algorithm 1: Algorithm 1 Computational algorithm of data-dri ven control 1: At time t i , update H ij ( t i ) and check condition (25): if (25) holds, exit ; else, continue. 2: Initialization: K i ,  ii ,  ij , ρ i , and P i . 3: Update H ij ( t i ) by wide-area data (also update A i , B i , and C i if they are state-dependent). 4: Perform DMI optimization pr ocess at time t i until it con verges: 5: Update M 0 i according to (13); 6: Solve bilinear problem (15). 7: Update W , Φ , and choose k c according to theorem 3.3. 8: goto t i +1 . C. blueT ime Delay and Robustness Analysis It is straightforward to sho w using (10) that, while h ij ( y i 1 , y j 1 ) are nonlinear, their values are uniformly bounded from abov e and below . Indeed, upon determining system parameters G ij and B ij , the ranges of h ij ( y i 1 , y j 1 ) can easily be found for a wide operational range of δ i and its equilibrium δ ∗ i . Fig. 2 is an illustration of the range of h ij , which is drawn -1.5 -1 -0.5 0 0.5 1 1.5 ij - * ij 0 0.5 1 1.5 2 2.5 h ij t - t Fig. 2. Range of h ij values with respect to ( δ ij − δ ∗ ij ) based on the parameters from appendix C. The de viation of h ij as shown by the middle part of the curve is quite small when the system has a low-frequenc y oscillation (i.e. ( y i 1 − y j 1 ) fluctuates around zero). It follows from Property 3.1 that the optimization problem (14) remains to be solv able only with the ranges of H ij rather than their precise values. As such, a solution to the optimization problem (15) could be found without the exact knowledge of y j . blueIn case that y j was av ailable at ( t − δ t ) rather than current time instance t because of communication delay or interruption, lets define the follo wing error of h ij to analyze rob ustness: ∆ h ij = h ij ( t ) − h ij ( t − δ t ) , and hence that of H ij can be written accordingly as ∆ H ij 4 = H ij ( t ) − H ij ( t − δ t ) = ∆ h ij h ij ( t − δ t ) H ij ( t − δ t ) . Therefore, by substituting ( H ij ( t − δ t ) + ∆ H ij ) into (15), we hav e the following DMI matrix at t M 0 ( t ) = M 0 ( t − δ t ) +  ∆ h ij h ij ( t − δ t )  2 N ij ( t − δ t ) + ∆ h ij ( t − δ t ) h ij ( t − δ t ) N 0 ij ( t − δ t ) , where N ij ( t − δ t ) = 1  ij h ij ( t − δ t ) P i H ij ( t − δ t ) H T ij ( t − δ t ) P i , and N 0 ij ( t − δ t ) = 1  ij h ij ( t − δ t )  2 P i H ij ( t − δ t ) H T ij ( t − δ t ) P i −  ij ( P i H ij ( t − δ t ) C i + C T i H T ij ( t − δ t ) P i )  . Giv en M 0 ( t − δ t ) ≤ 0 , it can be observed that M 0 ( t ) ≤ 0 holds if k ∆ h ij k 2 k N ij ( t − δ t ) k + k ∆ h ij kk N 0 ij ( t − δ t ) k ≤ λ min ( M 0 ( t − δ t )) , IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 7 G2 G3 G1 3 1 AREA2 AREA3 AREA1 2 𝚫𝐏 8 6 7 9 4 5 Fig. 3. A three-area power system: IEEE 9-bus system which is equi valent to k ∆ h ij k ≤ 1 2 k N ij ( t − δ t ) k  −k N 0 ij ( t − δ t ) k + q k N 0 ij ( t − δ t ) k 2 + 4 k N ij ( t − δ t ) k λ min ( M 0 ( t − δ t )) i (26) where λ min ( · ) denotes the smallest eigenv alue. blueThe above bound can be integrated into c T i in (25), the condition for algorithm 1. Based on the fact that deviation ∆ h ij is usually small for low-frequenc y oscillations, robustness of algorithm 1 is assured, and hence each indi vidual subsystem is guaranteed to be passivity-short and L 2 stable in the presence of delay δ t . Using the passivity shortage frame work, the ov erall system sta- bility under consensus-based cooperative control (16) through delayed communication network remains to be stabilizing al- beit its performance is degraded graciously as delay increases. Due to space limitation, further analysis and analytical proof are omitted here but the readers are referred to [28], [30] for the detailed analysis on stability of interconnecting passi vity- short systems with significant communication delays. V . S I M U L A T I O N R E S U L T S blueIn this section, performance of the proposed control is illustrated using a three-area test system modeled by the standard IEEE 9-bus system shown in Fig. 3. Each of the areas is represented by one aggreg ated generator . The detailed parameters of the test system and its aggregated generator models are listed in appendix C. blueAll the simulation results are obtained using the fol- lowing setting: a short-circuit fault happens on bus 8 at t = 2 . 0 s and is remov ed at t = 2 . 1 s , and a load change of ∆ P load = 1 . p.u. occurs simultaneously at t = 2 . 1 s in area 2. In this setting, there are a large disturbance and a consequent change of operating condition. bluePerformance of the following three wide-area controls are compared under the same simulation setting: I) the pro- posed DMI control, II) the LMI-designed control [31], and III) a traditional control of typical gain choice (constant gains). F or case III, the local gain matrix is chosen to be a droop gain, and the wide area control is AGC whose gain matrix is an integral control, that is, in (2) and (A.3), K i = [0 , k i , 0 , 0 , 0] , K I i =  0 , k I i , 0 , 0 , 0  . The specific gain values in case III are set to be k i = 30 and k I i = 0 . 3 , as suggested in [32], for all three aggreg ated generators. The control gains in case II are solved using the LMI method [31] based on the operating condition. The control gains in case I are updated using the proposed DMI optimization procedure, and network-lev el control gain k c is solved subsequently . blue The system responses under the three control strategies are illustrated by both the frequency and a line power (line 8- 2), and the results are shown in Fig. 4 and Fig. 5, respectively . The results show that system frequency increases as a result of the load decrease in area 2, after sustained oscillations. The oscillation frequency is about 0.4Hz. Fig. 4 also shows that system frequency gradually goes back to nominal frequency in all the cases. 0 50 100 150 200 60 60.2 60.4 Case I 0 50 100 150 200 60 60.2 60.4 Frequency (Hz) Case II 0 50 100 150 200 Time (s) 60 60.2 60.4 Case III Fig. 4. Trajectories of ω 3 : the proposed control (case I), LMI control (case II), and traditional control (case III) 0 50 100 150 200 0 2 4 Case I 0 50 100 150 200 0 2 4 Line power (p.u.) Case II 0 50 100 150 200 Time (s) 0 2 4 Case III Fig. 5. Trajectories of po wer on line 8-2: the proposed control (case I), LMI control (case II), and traditional control (case III) blueThe representative simulation results show that the pro- posed DMI design method is more effecti ve in damping out the oscillations caused by the disturbances. Case III has the least IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 8 Fig. 6. Adaptation over time of control gains in the proposed DMI design: k i , k I i and k c . damping because the typical A GC is not ef fectiv e in oscillation suppression. The performance in case II is some what better as the control gains are optimized but only with respect to one loading condition. Changes of the operating condition are not considered in either cases II or III. In case I, the overall system stability is monitored using matrix inequalities, the control gains are adapti vely updated, and hence the best performance is achieved. Fig. 6 shows the time ev olution of the adaptive control gains in the proposed data-driv en control. From both analytical design and the simulations, we learn that k i in (15) is far more effecti ve than k I i (as k I i has to be small). blueT o illustrate performance of the proposed DMI control with delayed communication, the simulation is repeated with delay δ t = 200 ms (the same value used in [33]). The results in Fig. 7 show that the proposed DMI control remains quite effecti ve and performance degradation is in tune with the delay . 0 5 10 15 20 60 60.2 60.4 Frequency (Hz) w/o delay with delay 0 5 10 15 20 Time (s) 0 2 4 Line power (p.u.) w/o delay with delay Fig. 7. Damping performance of the proposed DMI control: W ithout or with communication delay . V I . C O N C L U S I O N A N D D I S C U S S I O N A nov el systematic approach for damping out inter-area low-frequenc y oscillations of power systems is presented. By taking advantage of the fact that all synchronous generators are passivity-short and L 2 stable, the passivity shortage frame- work is utilized to modularly design a two-le vel control. The stability of the overall system is inv estigated as the intercon- nection of subsystems, local control, and wide-area control. First, using W AMS data, a DMI algorithm is formulated to minimize the impact of individual control of subsystems and their interconnections. Then, a high-lev el control algorithm is designed and analyzed to ensure the overall system stability . Simulation on a standard test system pro ves the efficacy of the proposed method. Implementation of a passivity-short design results in a closed-form representation in which overall system stability can be guaranteed. blueThe proposed data-dri ven control is rob ust with respect to network parameter changes and communication delays or interruptions. Its implementation only requires computation of matrix inequalities, hence the proposed control is scalable to large-scale systems. Nonetheless, it is natural to model and control groups of coherent generators together . Future research will be pursued to address such technical issues as joint design of communication, on-line model reduction, and wide-area control. Such a comprehensive solution is what is needed for W AMS and EMS. R E F E R E N C E S [1] X. Zhang, C. Lu, X. Xie, and Z. Y . Dong, “Stability analysis and con- troller design of a wide-area time-delay system based on the expectation model method, ” IEEE T ransactions on Smart Grid , vol. 7, no. 1, pp. 520–529, 2016. [2] Y . Shen, W . Y ao, J. W en, and H. 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Salazar, “ A measurement-based framew ork for dynamic equiv alencing of large power systems using wide-area phasor measurements, ” IEEE Tr ansactions on Smart Grid , vol. 2, no. 1, pp. 68–81, 2011. [40] G. Scarciotti, “Low computational complexity model reduction of power systems with preservation of physical characteristics, ” IEEE T ransac- tions on P ower Systems , v ol. 32, no. 1, pp. 743–752, Jan 2017. A P P E N D I X A S TA T E - D E P E N D E N T A FFI N E M O D E L I N G O F P O W E R S Y S T E M S Consider a power system that consists of n generators. If the i th generator is con ventional, its dynamic equations are described by the following swing equations: ˙ δ i = ω i , M i ˙ ω i = P m i − P g i − D i ω i . (A.1) where M i is the inertia, P m i is the prime po wer , P g i is the electrical power , D i is the damping constant, δ i is the rotor angle, and ω i is the frequency deriv ation (aw ay from the synchronous frequency ω 0 ). Should the generator hav e a prime mov er, the mechanical power P m i is the output of a second-order turbine/gov ernor model [34]: τ i 1 ˙ P m i = Y g i − P m i , τ i 2 ˙ Y g i = U i − Y g i , (A.2) where Y g i denotes the prime mov er input (e.g. the gate position of the turbine), τ i 1 and τ i 2 are time constants, and U i is the control input. T raditionally , control U i consists of a local frequenc y con- trol (LFC) and an automatic generator control (A GC). As shown in Fig. 8 and Fig. 9, LFC is a droop control that is a spe- cial case of the proposed feedback control v i = − K i x i + u i , and A GC is an integral control of the local feedback. Hence, control U i is in general of the form U i = v i + α i + P ref g i , ˙ α i = − K I i x i , (A.3) where x i is the local state to be defined, α i is the integral control v ariable, P ref g i is the desired set point, K I i is an inte gral gain row vector (of small values), v i = − K i x i + u i , and u i is the higher -level control to be designed in the form of (3). IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 10 Fig. 8. Block diagram of droop control Fig. 9. Block diagram of automatic generator control (A GC) It follo ws from (9), (A.1), (A.2), and (A.3) that the equilib- rium of the overall system with x ∗ i = 0 is P ∗ m i = P ∗ g i (A.4a) Y ∗ g i = P ∗ g i (A.4b) P ∗ g i = P ref g i + α ∗ i (A.4c) P ∗ g i = X j ∈N E i E j ( G ij cos δ ∗ ij + B ij sin δ ∗ ij ) (A.4d) Q ∗ g i = X j ∈N E i E j ( G ij sin δ ∗ ij − B ij cos δ ∗ ij ) . (A.4e) In the abov e equations, δ ∗ ij = δ ∗ i − δ ∗ j represents the final angle differences whose values are determined by the A GC signals [32] applied at each of the generators. A GC is a fundamental function of the power system control center , which adjusts the outputs of all major plants to compensate the frequency and load changes. For a generator described by model (A.1) and (A.2), let us choose its state and output vectors as x i =       x i 1 x i 2 x i 3 x i 4 x i 5       =       δ i − δ ∗ i ω i P m i − P ∗ g i Y g i − P ∗ g i α i − α ∗ i       , y i =  y i 1 y i 2  =  x i 1 x i 2  . (A.5) It follo ws from (9) and (A.5) that P g i = P ∗ g i + X j ∈N h ij ( y i 1 , y j 1 ) × ( y i 1 − y j 1 ) , (A.6) where h ij ( y i 1 , y j 1 ) is defined by (10). It is straightforward to sho w that, under the definition of x i in (A.5), dynamic equations (A.1) and (A.2) together with control (A.3) and load flo w equations (9) are mapped into (1) with A i =        0 1 0 0 0 0 − D i M i 1 M i 0 0 0 0 − 1 τ i 1 1 τ i 1 0 0 0 0 − 1 τ i 2 1 τ i 2 − K I i        , B i =       0 0 0 1 τ i 2 0       , C i =  1 0 0 0 0 0 1 0 0 0  , H ij ( y i , y j ) =       0 0 h ij M i 0 0 0 0 0 0 0       . (A.7) For inv erter -based renew able generation or distributed en- ergy resources, there are at least two options to deriv e their dynamic equations and apply the proposed design. One is to use their nativ e dynamics to deriv e their equations, as was done in [35]. The other option is to introduce a layer of generator emulation control [36], [37] or equiv alently the so-called virtual synchronous generator [38]. In the latter case, in verter - based ener gy sources have the same dynamic performance as synchronous machines, which is adopted in this paper for simplicity of technical presentation. Accordingly , for inv erter- based generators, dynamic equations in (A.1) are used, and the corresponding state and matrices are gi ven as follows: x i =   x i 1 x i 2 x i 3   =   δ i − δ ∗ i ω i α i − α ∗ i   , y i =  y i 1 y i 2  =  x i 1 x i 2  , (A.8) and A i =   0 1 0 0 − D i M i 1 M i − K I i   , B i =   0 1 M i 0   , C i =  1 0 0 0 1 0  , H ij =   0 0 h ij M i 0 0 0   . (A.9) It is worth noting that generator dynamics may be subject to such nonlinearity as saturation. Correspondingly , those nonlinearities can be introduced and matrix representations (A.7) and (A.9) become state dependent. blueIt should be noted that large-scale po wer systems often consist of distinct geographical regions and their coherent groups of physical generators. Rather than designing wide-area controls for all physical generators, it is both advantageous and customary to de velop an aggregated model for each of the areas. Indeed, the aggregate model can also be expressed in the form of (A.7) or (A.9), and they can be determined using one of model reduction algorithms in [25], [39], [40]). Hence, in the paper , model (1) is used to represent either an individual generator or a group of coherent generators. IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 11 A P P E N D I X B P RO O F O F L E M M A 3 . 2 It follows from subsystem (4) and storage function (7) that ˙ V i = u T i y i +  ii 2 k u i k 2 − ρ i 2 k y i k 2 + X j ∈N i  ij 2 k y j k 2 +  x T i y T j u T i  M i ( x i , y j )   x i y j u i   . Inequality (12) can be concluded by using the abov e equa- tion together with DMI (B.1). Matrix M i in (B.1) has the special structure that it has four sub-blocks as M i = " M i, 11 M i, 12 M T i, 12 M i, 22 # and the lo wer right block M i, 22 is diagonal and negati ve definite. It follows from Schur complement lemma [22] that M i ≤ 0 is equi valent to M i, 22 < 0 and M 0 i 4 = M i, 11 − M i, 12 M − 1 i, 22 M T i, 12 < 0 . It is straightforward to sho w algebraically that M 0 i is gi ven by (13). clearly , the dimension of matrix M 0 i is much lo wer than that of matrix M i . M i ( x i , y j ) =              A T i ( x i ) P i + P i A i ( x i ) + ρ i C T i C i − X j ∈N i ( P i H ij C i + C T i H T ij P i ) · · · P i H ij ( y i , y j ) · · · P i B i − C T i . . . . . . 0 . . . 0 H T ij ( y i , y j ) P i 0 −  ij I 0 0 . . . . . . 0 . . . 0 B T i P i − C i 0 0 0 −  ii I              ≤ 0 . (B.1) A P P E N D I X C T E S T S Y S T E M D AT A The network parameters are shown in T able I. The pa- rameters of aggregated generators are shown in T able II. All parameters are per unit values. T ABLE I: Network parameters of the IEEE test system Bus 1 Bus 2 R X B 1 4 0 0.0576 0 4 5 0.017 0.092 0.158 5 6 0.039 0.17 0.358 3 6 0 0.0586 0 6 7 0.0119 0.1008 0.209 7 8 0.0085 0.072 0.149 8 2 0.00 0.0625 0 8 9 0.032 0.161 0.306 9 4 0.01 0.085 0.176 T ABLE II: Generator parameters of the IEEE test system Gen. No. M i D i X 0 d τ i 1 τ i 2 1 470 0.1 0.0014 0.03 0.01 2 130 0.1 0.0023 0.03 0.01 3 62 0.1 0.0029 0.03 0.01 Y ing Xu is a postdoctoral researcher at the Depart- ment of Electrical and Computer Engineering, Uni- versity of Central Florida (UCF), USA. He received the B.Eng, M.Eng and PH.D. degrees from Harbin Institute of T echnology , China, in 2003, 2005 and 2009 respectively . From 2009-2017, he has been a Senior Engineer in North China Grid Dispatching and Control Center . His main research interests and experiences include power system analysis, system modeling and control, big-data implementation in power systems, cooperativ e control and distrib uted optimization for networked systems. Zhihua Qu (M90-SM93-F09) receiv ed the Ph.D. degree in Electrical Engineering from the Geor gia Institute of T echnology , Atlanta, in June 1990. Since then, he has been with the Univ ersity of Central Florida (UCF), Orlando. Currently , he is the SAIC Endowed Professor in College of Engineering and Computer Science, a Pegasus Professor and the Chair of Electrical and Computer Engineering, and the Director of FEEDER Center (one of DoE- funded national centers on distributed technologies and smart grid). His areas of expertise are nonlinear systems and control, resilient and cooperativ e control, with applications to energy and po wer systems. IEEE TRANSA CTIONS ON POWER SYSTEMS: COMPLETED ON JUNE 1, 2021 12 Roland Harvey is currently pursuing his Ph.D. degree in the Department of Electrical and Computer Engineering, University of Central Florida (UCF), USA. He earned his B.S. from the Department of Physics and Engineering Physics from Tulane Uni- versity . His research interests include system model- ing and control, cooperativ e control and distributed optimization for networked cyber-ph ysical systems. T oru Namerikawa (M94) recei ved the B.E., M.E., and Ph.D. degrees in electrical and computer en- gineering from Kanaza wa Uni versity , Kanazawa, Japan, in 1991, 1993, and 1997, respectively . He was as an Assistant Professor with Kanazawa University , from 1994 to 2002; with the Nagaoka University of T echnology , Nigata, Japan, from 2002 to 2005; and again with Kanazawa Uni versity , from 2006 to 2009. In 2009, he was with K eio Uni versity , Y okohama, Japan, where he is currently a Professor with the Department of System Design Engineering. He held visiting positions with the Swiss Federal Institute of T echnology , Zurich, Switzerland, in 1998; the University of California at Santa Barbara, Santa Barbara, CA, USA, in 2001; the Uni versity of Stuttgart, Stuttgart, Germany , in 2008; and Lund Univ ersity , Lund, Sweden, in 2010. His current research interests include robust control, and distributed and cooperative control and their application to power network systems.