A Generalized PSS Architecture for Balancing Transient and Small-Signal Response

1 A Generalized PSS Architecture for Balancing T ransient and Small-Signal Response Ryan T . Elliott, Payman Arabshahi, and Daniel S. Kirschen Abstract —For decades, power system stabilizers paired with high initial response automatic voltage regulators have served as an effectiv e means of meeting sometimes conflicting system stability requir ements. Driven primarily by increases in power electronically-coupled generation and load, the dynamics of large- scale power systems ar e rapidly changing. Electric grids are losing inertia and traditional sources of v oltage support and oscillation damping. The system load is becoming stiffer with respect to changes in voltage. In parallel, advancements in wide-area measurement technology have made it possible to implement control strategies that act on information transmitted over long distances in nearly real time. In this paper , we present a power system stabilizer architectur e that can be viewed as a generalization of the standard ∆ ω -type stabilizer . The control strategy utilizes a real-time estimate of the center-of-inertia speed derived from wide-area measurements. This approach cr eates a flexible set of trade-offs between transient and small-signal response, making synchronous generators better able to adapt to changes in system dynamics. The phenomena of interest are examined using a two-area test case and a reduced-order model of the North American W estern Interconnection. T o validate the key findings under realistic conditions, we employ a state-of-the- art co-simulation platform to combine high-fidelity power system and communication network models. The benefits of the proposed control strategy are retained even under pessimistic assumptions of communication network performance. Index T erms —automatic voltage regulator , co-simulation, lin- ear time-varying systems, phasor measurement unit, power sys- tem stabilizer , real-time control, wide-area measurement systems. I . I N T R O D U C T I O N T HE delicate balance between synchronizing and damping torque components in a synchronous machine creates a conflicting set of stability-oriented exciter performance requirements [1]–[4]. Po wer system stabilizers (PSS) have long played a critical role in satisfying these requirements; howe v er , changes in bulk system dynamics pose challenges to existing control strategies. As in v erter-coupled v ariable generation displaces synchronous machines, electric grids lose inertia and traditional sources of voltage support and oscillation damping. Correspondingly , the rapid growth of po wer electronic loads may make the system load stiffer with respect to changes in voltage [5], [6]. In parallel with these changes, wide-area measurement systems (W AMS) have transformed power system monitoring. The deployment of phasor measurement units (PMUs) has made it is possible to implement control strategies that act on information transmitted over long distances in nearly R. T . Elliott, P . Arabshahi, and D. S. Kirschen are with the Depart- ment of Electrical and Computer Engineering, University of W ashington, Seattle, W A 98195 USA (e-mail: ryanelliott@ieee.org; paymana@uw .edu, kirschen@uw .edu). (Corresponding author: Ryan T . Elliott.) real time [7]–[10]. Despite the proliferation of inv erter-coupled resources, it is projected that synchronous generation will account for a significant fraction of the capacity of large-scale power systems for decades to come [11]. As the dynamics of these systems change, it may become necessary to rethink how synchronous machines are controlled. In this paper, we deri ve a ne w PSS architecture that can be viewed as a generalization of the standard ∆ ω -type stabilizer . This control strategy stems from a time-varying linearization of the equations of motion for a synchronous machine. It utilizes a real-time estimate of the center -of-inertia speed deri ved from a set of wide-area measurements. The proposed strategy improv es the damping of both local and inter-area modes of oscillation. The ability of the stabilizer to improve damping is decoupled from its role in shaping the system response to transient disturbances. Consequently , the interaction between the PSS and automatic v oltage regulator (A VR) can be fine- tuned based on voltage requirements. This approach creates a flexible set of trade-offs between transient and small-signal response, making synchronous generators better able to adapt to changes in system dynamics. Analysis and simulation show that this strategy is tolerant of communication delay , traf fic congestion, and jitter . A. Liter atur e Review The role that PSSs play in shaping the dynamic system response to sev ere transient disturbances, such as generator trips, is explored in [12]–[15]. In [15], Dudgeon et al. show that the actions of PSSs and A VRs are dynamically interlinked. High initial response A VRs support transient stability b ut can reduce the damping of electromechanical modes of oscillation. The primary function of PSSs is to improve oscillation damping, but they can also de grade transient stability by counteracting the voltage signal sent to the exciter by the A VR. Managing these interactions through coordinated A VR and PSS design is studied in [16]–[18]. In [14], Grondin, Kamwa, et al. present a multi-band PSS compensator aimed at improving transient stability by adding damping to the lowest natural resonant frequency . The objectiv es of this compensation approach are similar to those we outline in Section II. W e present a PSS architecture that features a ne w type of multi-band compensator that le verages wide-area measurements to achiev e amplitude response attenuation. Employing remote, or global, input signals to improve the performance of power system damping controllers has inspired many research efforts including [19]–[24]. In [19], Aboul-Ela et al. propose a PSS architecture with two inputs, a local 2 signal used mainly for damping the local mode and a global signal for damping inter-area modes. For the global signal, [19] considers tie-line activ e power flo ws and speed difference signals that provide observ ability of specific inter-area modes. As stated in [14], the ideal stabilizing signal for a PSS “should be in phase with the deviation of the generator speed from the av erage speed of the entire system. ” T o approximate this ideal signal, the rotor speed is typically passed through a washout (highpass) filter , which may insufficiently attenuate steady-state changes in rotor speed and/or introduce excess phase lead into the bottom end of the control band. In contrast, we explore the implications of combining local measurements with a real-time estimate of the center-of-inertia speed. The research community is activ ely working to de velop sim- ulation techniques for studying the impact of communication networks on po wer systems. F ederated co-simulation en vir on- ments consist of two or more independent simulation platforms combined so that they exchange data and software execution commands. In [25], Hopkinson et al . present EPOCHS, a co- simulation en vironment that combined Network Simulator 2 ( ns-2 ) with PSLF and PSCAD/EMTDC. Man y subsequent research efforts followed, including [26]–[28]. In this paper, the Hierarchical Engine for Large-scale Infrastructure Co- Simulation (HELICS) is employed [28]. W e use this state-of- the-art framework to federate a communication network model dev eloped in ns-3 with a power system model de veloped in the MA TLAB-based Power System T oolbox (PST). B. P aper Or ganization The remainder of this paper is organized as follows. Sec- tion II deri ves a generalization of the standard ∆ ω -type PSS enabled by wide-area measurements. The impact of this control strategy on a two-area test system is examined in Section III. Section IV ev aluates how the main results scale to large systems using a reduced-order model of the North American W estern Interconnection. In Section IV -C , we study the effect of nonideal communication network performance using co- simulation. Section V summarizes and concludes. I I . P R O P O S E D M E T H O D The proposed PSS architecture arises from a time-varying linearization of the equations of motion for a synchronous machine. Here we briefly restate some key concepts and definitions from the theory of continuous-time linear time- varying systems. In the control strategy deriv ation, these concepts will be applied to the nonlinear form of the swing equation. A. Linear T ime-V arying Systems Let f : R n × R m → R n denote a nonlinear vector field ˙ x ( t ) = f ( x ( t ) , u ( t )) , (1) where x ( t ) ∈ R n is the system state at time t and u ( t ) ∈ R m the input. Recall that a time-varying linearization of f takes the form ∆ ˙ x ( t ) = A ( t )∆ x ( t ) + B ( t )∆ u ( t ) , (2) where ∆ x ( t ) = x ( t ) − x ( t ) and ∆ u ( t ) = u ( t ) − u ( t ) . The time-v arying trajectory about which the system is linearized is determined by x ( t ) and u ( t ) . The state-space matrices can be expressed compactly as A ( t ) = D x f ( x ( t ) , u ( t )) (3) B ( t ) = D u f ( x ( t ) , u ( t )) , (4) where the operator D x returns the Jacobian matrix of partial deri vati ves with respect to x ev aluated at time t , and D u returns the analogous matrix of partials with respect to u . In general, the state-space representation is time-varying when x ( t ) and u ( t ) define a nonequilibrium trajectory . B. Contr ol Strate gy Derivation This deriv ation applies the concepts introduced in Sec- tion II-A to the equations of motion for a synchronous machine. Stating the nonlinear swing equation in terms of the per-unit accelerating power , we have ˙ ω ( t ) = − D 2 H [ ω ( t ) − ω 0 ] + 1 2 H ω ( t ) [ P m ( t ) − P e ( t )] , (5) where ω 0 is the per-unit synchronous speed, D the damping coef ficient, and H the inertia constant [3], [4]. Linearizing (5) about a nonequilibrium trajectory yields ∆ ˙ ω ( t ) = −  D 2 H + P m ( t ) − P e ( t ) 2 H ω ( t ) 2  ∆ ω ( t ) + 1 2 H ω ( t ) [∆ P m ( t ) − ∆ P e ( t )] , (6) where ∆ ω ( t ) = ω ( t ) − ω ( t ) , ∆ P m ( t ) = P m ( t ) − P m ( t ) , and ∆ P e ( t ) = P e ( t ) − P e ( t ) . A new damping coefficient arises from analysis of (6) D ( t ) = D + P m ( t ) − P e ( t ) ω ( t ) 2 . (7) Using this coefficient, (6) can be restated as ∆ ˙ ω ( t ) = − D ( t ) 2 H ∆ ω ( t ) + 1 2 H ω ( t ) [∆ P m ( t ) − ∆ P e ( t )] . (8) Hence, as with a standard ∆ ω -type stabilizer it is possible to add damping in the L TV reference frame by creating a component of electrical torque that is in phase with the rotor speed de viations. The difference is that the speed deviations are defined such that ∆ ω ( t ) = ω ( t ) − ω ( t ) , where ω ( t ) is a function of time that tracks changes in the ov erall system operating point. The time-varying reference ω ( t ) makes it possible to almost completely wash out steady-state changes in rotor speed from the control error . C. Nonequilibrium Speed T rajectory In this paper , we will examine the implications of treating ω ( t ) as a real-time estimate of the center-of-inertia speed ω ( t ) ≈ P i ∈I H i ω i ( t ) P i ∈I H i , (9) where i is the unit index and I the set of all online con ventional generators. The right-hand side of (9) corresponds to the 3 classical center-of-inertia definition, dating back to at least [29]. A related quantity that incorporates the machine apparent power ratings is studied in [30]. This alternative approach may be more eff ectiv e than (9) in capturing the discrepancy in size between large and small machines with similar inertia constants. The question of how to compute ω ( t ) for real-time control applications is an interesting research problem in itself that is mostly outside the scope of this paper . A promising method is presented in [31]. At the time of this writing, rotor speed measurements are seldom av ailable through wide- area measurement systems; howe ver , a straightforward way of estimating (9) would be a weighted average of frequency measurements ω ( t ) = 1 f 0 X k ∈K α k f k ( t ) , (10) where k is the sensor index, and f 0 the nominal system frequency . The frequency signal reported by the k th sensor is denoted by f k ( t ) , and the associated weight by α k . The weights are nonnegati ve and sum to one, i.e., 1 T α = 1 . For simplicity , we will consider the arithmetic mean in which α k = 1 / |K| for all k , where |K| denotes the cardinality of K or simply the number of av ailable sensors. The research contrib utions presented in this paper do not depend strongly on this choice. There are numerous implementation-related issues posed by any wide-area control scheme, such as how to handle missing or corrupted data. For examples of how these problems may be addressed, see [10], [32]. D. A Generalization of the ∆ ω -T ype PSS The control strategy implied by (8) can be generalized to encompass the standard ∆ ω -type PSS. Splitting the linear time-in v ariant (L TI) control error ∆ ω ( t ) = ω i ( t ) − ω 0 into two constituent parts and taking the linear combination yields ∆ ν ( t ) , β 1 [ ω i ( t ) − ω ( t )] + β 2 [ ω ( t ) − ω 0 ] , (11) where β 1 and β 2 are independent tuning parameters restricted to the unit interval. In Section IV, we show ho w the open-loop frequency response between the input to the exciter and the output of the PSS can be shaped by adjusting these parameters. The first term in (11) follows directly from (8), and the second makes it possible to implement a standard ∆ ω -type PSS using the same framework. As in Section II-C , ω ( t ) is a real-time estimate of the center -of-inertia speed. Fig. 1 shows the block diagram corresponding to this control strategy where v s is the output of the PSS. If necessary , more than one lead-lag compensation stage may be employed. The fr equency re gulation mode of a po wer system is a very lo w-frequency mode, typically belo w 0.1 Hz , in which the rotor speeds of all synchronous generation units participate [14], [33]. As a consequence of synchronism, the shape of this mode is such that all conv entional generators are in phase with one another . As its name implies, the frequency regulation mode is sensiti ve to load composition, turbine governor time constants, and droop gains. When β 1 > β 2 , the control tuning prioritizes the damping of inter-area and local modes of oscillation while de-emphasizing the frequency regulation mode. The conv erse Σ β 1 Σ sK T w 1 + sT w 1 + sT a 1 + sT b Σ β 2 ω i ω ω 0 v s + − − + + + Fig. 1. Generalized ∆ ω -type PSS block diagram. T ABLE I E FF EC T O F C O N T RO L P AR A M E T ER S O N P S S T U N I NG Parameter V alues T uning Description β 1 > β 2 T argets inter-area and local modes β 1 < β 2 T argets the frequency regulation mode β 1 = β 2 6 = 0 Standard ∆ ω -type PSS β 1 = β 2 = 0 No PSS control is true when β 1 < β 2 . In the special case that β 1 = β 2 , we hav e a conv entional ∆ ω -type PSS. The resulting control error in this case is exactly the same as in the standard formulation presented in [3]. T able I summarizes the effect of β 1 and β 2 on the PSS tuning. The diagram shown in Fig. 1 accurately illustrates the control strategy; howe ver , the structure can be clarified further . Expanding the second term in (11) gi ves ∆ ν ( t ) = β 1 [ ω i ( t ) − ω ( t )] + β 2 ω ( t ) − β 2 ω 0 . (12) Thus, we can construct the control error in (11) with a constant reference and a single feedback signal ∆ ν ( t ) = ν ( t ) − ν ref , where (13) ν ref = β 2 ω 0 , and (14) ν ( t ) = β 1 [ ω i ( t ) − ω ( t )] + β 2 ω ( t ) . (15) The results described in this paper are based on the strategy defined by (13)–(15) and illustrated in Fig. 1. For the sake of completeness, we present a further refinement that permits the per-unit synchronous speed ω 0 to serve as the reference. The basic idea is to divide (12) by β 2 , taking care to account for the case where β 2 = 0 . Beginning with the control error , we hav e ∆ e ω ( t ) , e ω ( t ) − ω 0 . (16) The feedback signal e ω ( t ) is then gi ven by e ω ( t ) = ( ( β 1 /β 2 ) [ ω i ( t ) − ω ( t )] + ω ( t ) , for β 2 > 0 β 1 [ ω i ( t ) − ω ( t )] + ω 0 , for β 2 = 0 . (17) This construction is similar to a conv entional ∆ ω -type stabilizer where the local speed measurement ω i ( t ) has been replaced by e ω ( t ) . Further illustrating this connection, when β 1 and β 2 are equal and nonzero the feedback signal becomes e ω ( t ) = ω i ( t ) . Fig. 2 shows how the simplified PSS block diagram fits in the context of an excitation system with an A VR. This form is equiv alent to the one outlined in (13)–(15) provided that the downstream gain K is scaled appropriately . 4 sK T w 1 + sT w 1 + sT a 1 + sT b Σ G e ( s ) V ref 1 1 + sT r E fd ∆ e ω E t v t v s + − + A VR PSS Fig. 2. Excitation system with A VR and PSS, where G e ( s ) represents the transfer function of the exciter . E. Comparison W ith Existing PSS Models This subsection compares the generalized ∆ ω -type PSS with two industry-standard stabilizer designs: PSS2C and PSS4C. As described in the IEEE recommended practice for excitation system models [34], PSS2C represents a flexible dual-input stabilizer . This model supersedes and is backward compatible with PSS2A and PSS2B. It may be used to represent two distinct implementation types: 1) stabilizers that utilize two inputs to estimate the integral of accelerating power , and 2) stabilizers that utilize rotor speed (or frequency) feedback and incorporate a signal proportional to the electrical power as a means of compensation. The generalized ∆ ω -type PSS presented here bears similarities to the second of these types. The term in (11) multiplied by β 1 also represents a local rotor speed combined with an auxiliary signal. The first ke y difference is that ω ( t ) in (11) is synthesized from from wide-area, rather than strictly local, measurements. The second is that the generalized ∆ ω PSS also provides the ability to independently adjust the amount of steady-state error included in the feedback. In contrast, PSS2C does not feature a multi-band compensation mechanism. For multi-band compensation, we turn to PSS4C which builds upon the structure originally proposed in [14]. As discussed in Section I-A , the primary difference between the generalized ∆ ω PSS and PSS4C is the way the compensation is implemented. The PSS4C structure uses parallel lead-lag compensators to delineate the frequenc y bands, whereas the generalized ∆ ω PSS uses the linear combination of steady-state and small-signal components in (11). As shown in Section IV, the latter strategy provides of a means of achieving selective attenuation with minimal impact on the phase response. For an in-depth comparison of PSS2B and PSS4B, the precursors of the models discussed here, see [35]. I I I . T W O - A R E A S Y S T E M A N A L Y S I S T o study the impact of the control strategy outlined in Section II, a combination of time- and frequency-domain analysis was employed. A custom dynamic model based on the block diagram shown in Fig. 1 was implemented in the MA TLAB-based Power System T oolbox (PST) [36]. This application facilitates not only time-domain simulation of nonlinear systems b ut also linearization and modal analysis. D1 D2 G1 G2 G3 G4 10 20 3 101 13 120 110 Fig. 3. Oneline diagram of the two-area test system. T wo test systems were studied: a small model based on the Klein-Rogers-Kundur (KRK) two-area system [37], and a reduced-order model of the W estern Interconnection. This section summarizes the results of analyzing the two-area test system. It comprises 13 buses, 14 branches, and 4 synchronous generators. A oneline diagram of the system is shown in Fig. 3. In both models, the acti ve component of the system load is modeled as constant current and the reactiv e component as constant impedance. T o permit study of transient disturbances, several modifica- tions were made to the original KRK system. The synchronous machines in the standard case are representati ve of aggregate groups of generators concentrated in each area. Each unit has the same capacity and inertia. Hence, tripping any one generation unit of fline would be equi v alent to losing 25 % of the rotating inertia online in the system. T o facilitate the study of realistically-sized generator trips, the capacity was redistributed such that each area possessed one machine representativ e of a collection of generators and the other a large individual plant. Generators G 1 and G 3 were scaled such that they each represented 5 % of the overall system capacity . The remainder was equally split between G 2 and G 4 . Every unit in the system was then outfitted with the generalized ∆ ω PSS described in Section II. A. Sensitivity of System P oles to the PSS T uning P ar ameters Here we examine the effects of sweeping the PSS tuning parameters β 1 and β 2 on the poles of the system. The modal analysis was performed by linearizing the system dynamics and then solving for the eigen values and eigen vectors of the system matrix. The main result is that the oscillatory modes effecti vely split into two groups, one that is sensiti ve to changes in β 1 and the other β 2 . Let us begin by examining the effect of the tuning parameters on the inter-area and local modes. Consider the inter-area mode indicated by the blue x located at 0.76 Hz in Fig. 4. The shape of this mode observ ed through the machine speeds is shown in Fig. 5(a). Recall that mode shape is defined by the elements of the right eigen v ector corresponding to the states of interest [38]. As demonstrated by Fig. 5(a), this mode is characterized by generators G1 and G2 oscillating against G3 and G4. The two-area system is tuned such that this inter-area mode is unstable without supplemental damping control. The plots in Fig. 4 show the sensitivity of the system poles to the PSS tuning parameters. T o generate these plots, either β 1 or β 2 was swept o ver an interv al while the other was held at zero. The tuning parameters for all of the PSS units were swept in unison, and the gain was uniformly held fixed at K = 25 . Sweeping the tuning parameters for all units together facilitates 5 − 6 − 4 − 2 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 local inter-area freq. reg. control control Real axis Imaginary axis (Hz) Set A Set B Set C (a) β 1 parameter sweep. − 6 − 4 − 2 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 local inter-area freq. reg. control control Real axis Imaginary axis (Hz) Set A Set B Set C (b) β 2 parameter sweep. Fig. 4. Sensitivity of the system oscillatory modes to the PSS tuning parameters. The modes in A are sensitive to changes in β 1 , those in B to β 2 , and those in C to both. 60 90 120 150 180 210 240 270 300 330 0 30 0 . 5 1 ω 1 ω 2 ω 3 ω 4 (a) 0.76 Hz inter-area mode. 60 90 120 150 180 210 240 270 300 330 0 30 0 . 5 1 ω 1 ω 2 ω 3 ω 4 (b) 0.09 Hz freq. regulation mode. Fig. 5. Normalized mode shape plots for the two-area system. study of the effect of PSSs on the frequency regulation mode. In Fig. 4(a), β 1 was swept ov er the interval [0 , 1] while β 2 was held at zero. As β 1 increases, the inter-area mode moves to the left and decreases slightly in frequency . The local modes, indicated by the blue x’ s in the upper right quadrant of Fig. 4(a), mov e to the left and increase slightly in frequency . A well- controlled exciter mode marked by the blue x in the upper left quadrant mov es up and to the right but remains comfortably in the left half of the complex plane. For all intents and purposes, the frequency regulation mode is unaffected by changes in β 1 . Hence, β 1 dictates the extent to which the PSS damps inter-area and local modes of oscillation. The parameter β 2 primarily influences the frequenc y regula- tion mode. This mode is indicated by the red triangle located at 0.09 Hz in Fig. 4. The shape of the frequency regulation mode observed through the machine speeds is sho wn in Fig. 5(b). All of the machine speeds are in phase and have nearly identical magnitudes. In Fig. 4(b), the parameter β 2 was swept over the interval [0 , 1] while β 1 was held at zero. As β 2 increases, the frequency regulation mode moves to the left. The higher- frequency exciter mode marked with a diamond moves upward. This control mode exhibits some sensitivity to both β 1 and β 2 . In contrast, the inter -area and local modes are relativ ely unaf fected by changes in β 2 . The dependence of the frequency regulation mode on β 2 indicates that PSSs, in aggre gate, play an important role in shaping the system response to transient disturbances. T o demonstrate this phenomenon, and the effects of the PSS tuning parameters more broadly , we present a collection of time-domain simulations. 0 5 10 15 20 − 1 . 0 0 . 0 1 . 0 Relativ e speed ( 10 − 3 pu) β 1 = 0 . 33 β 1 = 0 . 67 β 1 = 1 . 00 0 5 10 15 20 0 . 99 1 . 00 1 . 01 Time (s) G4 T er m. voltag e (pu) β 1 = 0 . 33 β 1 = 0 . 67 β 1 = 1 . 00 Fig. 6. T ime-domain simulations of generator G3 being tripped offline for various values of β 1 . The top subplot shows the relative speed between G2 and G4. B. T ime-Domain Simulations The two-area system was simulated in PST for a variety of PSS tunings. As in the frequency-domain analysis, all PSS units were tuned alike and used the same gain. The contingency of interest in this set of simulations is a trip of generator G3. This ev ent was selected because it initiates a transient disturbance that excites not only the inter-area and local modes but also the frequency regulation mode. In the first set of simulations, β 1 was varied over the set { 0 . 33 , 0 . 67 , 1 } while β 2 was held fixed at 0 . 33 . In the second set of simulations, β 1 was held fixed at 0 . 33 while β 2 was v aried over the set { 0 , 0 . 33 , 0 . 67 } . For all simulations, the ov erall PSS gain was set to K = 18 . The case where β 1 = β 2 = 0 . 33 corresponds to a standard ∆ ω stabilizer with a gain of K = 6 . This set of simulations assumes ideal communication in the construction of the time-varying reference ω ( t ) . Section IV -C addresses the effect of nonideal communication network performance. Fig. 6 shows the key results for the case where β 1 is varied. The upper subplot shows the difference in speed between generators G2 and G4, i.e., ω 2 ( t ) − ω 4 ( t ) . The oscillatory content in this signal is dominated by the 0.76 Hz inter-area mode. As β 1 increases, the damping of this mode also increases. The lower subplot shows the terminal voltage of generator G4. As β 1 is varied, the large-signal trajectory of the terminal voltage and its post-disturbance v alue are unchanged. This reflects the fact that varying β 1 only alters the small-signal characteristics of the field current. Fig. 7 shows the key results for the case where β 2 is varied. The upper subplot shows the system frequency response, which readily shows the behavior of the frequency regulation mode. The results sho w that β 2 plays a ke y role in determining the depth of the frequency nadir . The frequency nadir improves significantly as β 2 increases from 0 to 0.33 , and modestly as it goes from 0.33 to 0.67 . Effecti vely , β 2 determines the lev el of overshoot in the system step response. The lower subplot shows the terminal voltage of generator G4. As β 2 is increased, the terminal voltage following the generator trip 6 0 5 10 15 20 59 . 8 59 . 9 60 . 0 Sys tem frequency (Hz) β 2 = 0 . 00 β 2 = 0 . 33 β 2 = 0 . 67 0 5 10 15 20 0 . 99 1 . 00 1 . 01 Time (s) G4 T er m. voltag e (pu) β 2 = 0 . 00 β 2 = 0 . 33 β 2 = 0 . 67 Fig. 7. T ime-domain simulations of generator G3 being tripped offline for various values of β 2 . becomes incrementally more depressed. This can be attributed to the fact that β 2 controls the extent to which steady-state changes in rotor speed are included in the PSS control error . The process that causes β 2 to affect the frequency nadir is indirect. Increasing β 2 amplifies the steady-state component of the control error in (11). This depresses the field current supplied by the e xciter and causes the voltage induced in the stator to dip. The electrical load decreases in response to this v oltage dip with the amount of relief depending on the sensitivity of the load with respect to v oltage. This tends to reduce the time-varying mismatch in mechanical and electrical torque, which improv es the frequency nadir . This effect depends on the load composition, and the amount of improvement in the nadir decreases as the fraction of constant power load increases. Thus, there is a trade-off between improving the frequency nadir and degrading the voltage response. As explained in [15], the tendency of the PSS to counteract the voltage signal sent to the exciter by the A VR can reduce synchronizing torque and degrade transient stability . The control strategy presented in this paper makes it possible to fine-tune the interaction between the PSS and A VR without affecting the damping of inter -area and local modes, and vice versa. I V . L A R G E - S C A L E T E S T S Y S T E M A NA L Y S I S For the two-area system discussed in Section III, the inter-area and local modes were influenced by β 1 , and the frequency regulation mode by β 2 . This section addresses whether this property is preserved for large-scale systems. W e consider a reduced-order model of the W estern Interconnection named the miniWECC , in reference to the W estern Electric Coordinating Council (WECC). It comprises 122 buses, 171 ac branches, 2 HVDC lines, and 34 synchronous generators. This system spans the entirety of the interconnection including British Columbia and Alberta. Its modal properties have been extensi vely validated against real system data [9], [39]. The aim of this analysis is to illustrate the fundamental behavior of various aspects of the proposed architecture in a controlled setting. Prior to implementation, high-fidelity simulation studies − 3 − 2 − 1 0 0 . 0 0 . 5 1 . 0 1 . 5 Real axis Imaginary axis (Hz) Set A Set B Set C (a) β 1 parameter sweep. − 3 − 2 − 1 0 0 . 0 0 . 5 1 . 0 1 . 5 Real axis Imaginary axis (Hz) Set A Set B Set C (b) β 2 parameter sweep. Fig. 8. Sensitivity of the miniWECC oscillatory modes to the PSS tuning parameters. The modes in A are sensitive to changes in β 1 , those in B to β 2 , and those in C to both. that account for variation in PSS structure and the dynamics of in verter -coupled generation would be required. A. Sensitivity of System P oles to the PSS T uning P ar ameters T o examine the sensitivity of the oscillatory modes to the PSS tuning parameters, the method described in Section III-A was applied to the miniWECC. Every generation unit in the system was outfitted with a generalized ∆ ω PSS with the gain set to K = 25 . In practice, WECC polic y dictates that “a PSS shall be installed on ev ery synchronous generator that is larger than 30 MV A , or is part of a complex that has an aggregate capacity larger than 75 MV A , and is equipped with a suitable excitation system” [40]. Fig. 8 shows the movement of the system poles in response to changes in the tuning parameters. In each subplot, either β 1 or β 2 was swept over an interval while the other was held at zero. The main result matches the one observed for the two-area system. The inter-area and local modes are influenced by β 1 , and the frequency regulation mode by β 2 . For the miniWECC, there is one well-controlled exciter mode near 0.28 Hz that exhibits sensitivity to both parameters. This mode is marked in Fig. 8 with a diamond. B. Open-Loop F r equency Response Analysis The frequency-domain analysis presented in Sections III and IV -A focused on a system-wide perspecti ve. Here we provide a unit-specific analysis of the open-loop frequency response for a single generator . Outfitting a single unit with a PSS yields the state-space representation ˙ x ( t ) = Ax ( t ) + B p u ( t ) (18) y ν ( t ) = C ν x ( t ) , (19) where B p describes how the system states are affected by changes in the PSS control input. The closed-loop control action determined by (13) can be implemented with the input u ( t ) = − K y ν ( t ) = − K C ν x ( t ) (20) u ( t ) = − K  0 γ 1 γ 2 . . . − β 1         b x ( t ) f 1 ( t ) f 2 ( t ) . . . ω i ( t )        , (21) 7 ν ref Σ G c ( s ) G p ( s ) F ( s ) ν ∆ ν v s + { f k } k ∈K − Fig. 9. Feedback loop for a single generation unit outfitted with a generalized ∆ ω PSS, where G c ( s ) represents the controller, G p ( s ) the plant, and F ( s ) the feedback process. where K is a scalar gain. The output matrix C ν combines the states to form the PSS feedback signal ν ( t ) . Note the presence of the extra negati ve sign to conform to the negati ve feedback con vention. The state vector x in (21) is organized with the unused states b x on top, followed by the frequency measurements and the local rotor speed. For the k th sensor γ k = α k ( β 1 − β 2 ) /f 0 , where α k stems from the linear com- bination in (10), and f 0 is the nominal system frequency . In this analysis, the frequencies were computed by applying a deriv ati ve-filter cascade to the bus voltage angles as described in [10]. Hence, the unity-gain open-loop transfer function between a change in the PSS reference ν ref and the feedback signal ν is H ( s ) = C ν ( sI − A ) − 1 B p . (22) Fig. 9 shows a high-level block diagram of the feedback loop for a single generation unit outfitted with a generalized ∆ ω PSS. Here G c ( s ) represents the PSS, G p ( s ) the plant, and F ( s ) the feedback process. The exciter dynamics are included in G p ( s ) , and the input to the plant represents a change in the exciter voltage reference V ref . By commutativity , it holds that H ( s ) = G c ( s ) G p ( s ) F ( s ) = G p ( s ) F ( s ) G c ( s ) . (23) Hence, the loop transfer function between ∆ ν and ν is the same as the transfer function between a change in the exciter voltage reference V ref and the output of the PSS v s . Using this function, we can ev aluate the ef fect of the PSS tuning parameters on the open-loop frequency response. For this analysis, only the unit being studied was outfitted with a PSS. Fig. 10 shows the effect of β 1 on the open-loop frequency response for generator G2, a hydroelectric unit in eastern British Columbia, where β 2 = 1 for all traces. The peak in the amplitude response near 0.04 Hz corresponds to the frequency regulation mode. As the plot sho ws, β 1 has no effect on the gain of the system at this frequency . This corroborates the system- wide modal analysis done in Sections III and IV -A at the unit le vel. As expected, β 1 does change the amplitude response for the inter-area and local modes of oscillation. Unlike traditional compensation methods, this approach does not degrade the phase response in the attenuation region. As β 1 is varied, the phase response at the frequencies of the dominant amplitude peaks ( 0.37 Hz , 0.62 Hz , and 1.0 Hz ) is essentially unchanged. The observed transition in phase through 0° at the resonant frequencies is ideal for damping control. Fig. 11 shows the effect of β 1 on the overall PSS compen- sation. As in [41], the uncompensated open-loop frequency response, including the washout filter dynamics, is pro vided for comparison. The o verall compensation comprises both the lead-lag compensator and the tuning determined by β 1 , β 2 . 10 − 2 10 − 1 10 0 10 1 − 60 − 40 − 20 Magnitude (dB) β 1 = 1 . 0 β 1 = 0 . 5 β 1 = 0 . 2 10 − 2 10 − 1 10 0 10 1 − 180 − 90 0 90 180 Frequency (Hz) Phase (deg) β 1 = 1 . 0 β 1 = 0 . 5 β 1 = 0 . 2 Fig. 10. The effect of β 1 on the open-loop frequency response between the input to the exciter and the output of the generalized PSS for generator G2. 10 − 2 10 − 1 10 0 10 1 − 60 − 40 − 20 0 20 Uncomp . Magnitude (dB) β 1 = 1 . 0 β 1 = 0 . 5 β 1 = 0 . 2 10 − 2 10 − 1 10 0 10 1 − 180 − 90 0 90 180 Uncomp . Frequency (Hz) Phase (deg) β 1 = 1 . 0 β 1 = 0 . 5 β 1 = 0 . 2 Fig. 11. The effect of β 1 on the ov erall PSS compensation for generator G2 with the washout filter included in the uncompensated frequency response. When β 1 = β 2 = 1 , the tuning stage has a gain of unity and imparts no phase shift. Hence, all of the compensation stems from the lead-lag compensator . This is expected because the case where β 1 = β 2 = 1 yields a standard ∆ ω stabilizer as shown in T able I. Fig. 12 shows the effect of β 2 on the open-loop frequency response where β 1 = 1 for all traces. As β 2 is varied, the amplitude response at the frequencies corresponding to the local and inter-area modes is effecti vely unchanged. In contrast, the gain at the frequency regulation mode is reduced by roughly 14 dB as β 2 goes from 1 to 0.2 . F or β 2 = 0 . 2 , the phase response at the frequency regulation mode leads the case where β 2 = 1 by roughly 35° . This suggests that if a β 2 v alue below some nominal threshold is required for a particular application, it may be necessary to retune the lead-lag compensator and/or washout filter to ensure satisfactory low-frequency performance. The effect of β 2 on the overall PSS compensation is shown in Fig. 13. 8 10 − 2 10 − 1 10 0 10 1 − 60 − 40 − 20 Magnitude (dB) β 2 = 1 . 0 β 2 = 0 . 5 β 2 = 0 . 2 10 − 2 10 − 1 10 0 10 1 − 180 − 90 0 90 180 Frequency (Hz) Phase (deg) β 2 = 1 . 0 β 2 = 0 . 5 β 2 = 0 . 2 Fig. 12. The effect of β 2 on the open-loop frequency response between the input to the exciter and the output of the generalized PSS for generator G2. 10 − 2 10 − 1 10 0 10 1 − 60 − 40 − 20 0 20 Uncomp . Magnitude (dB) β 2 = 1 . 0 β 2 = 0 . 5 β 2 = 0 . 2 10 − 2 10 − 1 10 0 10 1 − 180 − 90 0 90 180 Uncomp . Frequency (Hz) Phase (deg) β 2 = 1 . 0 β 2 = 0 . 5 β 2 = 0 . 2 Fig. 13. The effect of β 2 on the ov erall PSS compensation for generator G2 with the washout filter included in the uncompensated frequency response. C. Co-Simulation of P ower and Communication Systems All of the analysis presented in Sections III through IV -B was performed under the assumption of ideal communication. In this section, we analyze the effect of communication delay in the frequency domain and verify the findings in the time domain, as in [42]. The mathematical modeling de veloped here represents the real-time exchange of synchronized phasor measurement data ov er a network. As described in the IEEE standard gov erning data transfer in PMU networks [43], communication delays in W AMS are typically in the range of 20 – 50 ms ; ho wev er , the combined delay must also account for the ef fect of transducers, processing, concentrators, and multiplexing [44]– [46]. In [44], the delay attrib uted to these factors is estimated at 75 ms , which yields an approximate range of 95 – 125 ms for the combined delay . This range is reflectiv e of systems that utilize fiber-optic communication. It aligns closely with the experimental results reported in [10], 69 – 113 ms , but may vary depending on the communication method employed, e.g., 10 − 2 10 − 1 10 0 10 1 − 60 − 40 − 20 Magnitude (dB) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s 10 − 2 10 − 1 10 0 10 1 − 180 − 90 0 90 180 Frequency (Hz) Phase (deg) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s Fig. 14. The effect of the combined delay τ on the open-loop frequency response for generator G2 with β 1 = 1 and β 2 = 0 . 5 . wired vs. wireless. Here we ev aluate scenarios with delays that are 5 to 10 times greater than the high end of this range. Modifying the state-space output matrix in (21) to account for delays as in [47], we ha ve b C ν ( s ) =  0 γ 1 e − sτ 1 γ 2 e − sτ 2 . . . − β 1  (24) b H ( s ) = b C ν ( s ) [ sI − A ] − 1 B p , (25) where τ k is the delay of the k th sensor . Thus, the output matrix changes as a function of frequency . The open-loop transfer function with delay is given by (25). Fig. 14 sho ws the results of using (25) to e v aluate the effect of delay on the open-loop frequency response for generator G2 with β 1 = 1 and β 2 = 0 . 5 . For simplicity , τ k = τ for all k . The entries of (24) correspond to the case where the local signal is not delayed, and the local and remote measurements are not time-aligned upon arri val. As a result, b H ( s ) 6 = H ( s ) e − sτ . In the extreme case where τ = 1 . 25 s shown in Fig. 14, the gain and phase are altered slightly in the neighborhood of the frequency regulation mode; howe v er , the control performance and stability margins are essentially unchanged. T o study the impact of nonideal communication performance in the time domain, we used a co-simulation framew ork called HELICS [28]. A communication network model for the miniWECC was developed in ns-3 [48]. It features PMU endpoints that communicate with the controllers via the User Datagram Protocol (UDP). This model includes transmission delay , congesting traffic, and packet-based error emulation. Each generation unit in the PST model was outfitted with a generalized ∆ ω PSS where β 1 = 1 , β 2 = 0 . 5 , and K = 9 . Fig. 15 sho ws time-domain simulations of generator G26, a large nuclear plant in Arizona, being tripped offline for various expected delays τ . The results are in close agreement with the frequency-domain analysis shown in Fig. 14. Thus, for this example, the benefits of the control strategy are retained e ven u nder pessimistic assumptions of communication network performance. In the miniWECC examples discussed herein, the center-of- inertia speed estimate ω ( t ) was synthesized using 30 sensors 9 0 5 10 15 20 25 30 35 40 59 . 8 59 . 9 60 . 0 Sys tem frequency (Hz) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s 0 5 10 15 20 25 30 35 40 0 . 99 1 . 00 1 . 01 1 . 02 1 . 03 Time (s) G2 T er m. voltag e (pu) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s Fig. 15. Simulations of generator G26 being tripped offline for various average combined delays where β 1 = 1 and β 2 = 0 . 5 . geographically distributed throughout the system. The effect of delay on the open-loop frequency response is dependent on the number and placement of the frequency (or speed) sensors. These factors determine how well ω ( t ) tracks the target defined in (9). By extension, they also influence its spectral content. When ω ( t ) approximately tracks the true center-of-inertia speed, its amplitude spectrum is dominated by very low-frequency content, generally ≤ 0 . 1 Hz . If too few sensors are used to synthesize this estimate, and/or those sensors are not adequately distributed, the amplitude spectrum of ω ( t ) may include significant higher-frequency content, in and above the range of the electromechanical modes. If this occurs, the delay may impart lar ger deviations in the phase response abov e the frequency regulation mode than shown in Fig. 14. For similar reasons, the coefficients of the linear combination in (10) also af fect the relationship between the combined delay and the frequency response. The other main factor influencing this relationship is the tuning determined by β 1 , β 2 . Analysis indicates that tunings where β 1 < β 2 may be more susceptible to the ef fects of delay than those where β 1 ≥ β 2 . T o explore this behavior , we will analyze the entries of the output matrix b C ν ( s ) . Let b γ k = γ k /β 1 . The matrix b C ν ( s ) may then be expressed as b C ν ( s ) = β 1  0 b γ 1 e − sτ 1 b γ 2 e − sτ 2 . . . − 1  , (26) where b γ k = α k f 0  1 − β 2 β 1  . (27) Recall from (10) that the weights α k are nonnegativ e and sum to one. For all real ω τ , it holds that | e − j ωτ | = 1 . Thus, the relationship between the magnitudes of the entries of b C ν ( s ) corresponding to the delayed and non-delayed system states is primarily determined by the ratio β 2 /β 1 . T able II shows a breakdown of the possible cases. The term inside the brackets of (26) corresponding to the local rotor speed alw ays has a magnitude of one. The magnitudes of the remaining entries may either be zero, bounded, or unbounded. T ABLE II E FF EC T O F C O N T RO L P AR A M E T ER S O N b γ k C O EFFI C I E N T S Parameter Ratio Coefficient Range β 2 /β 1 < 1 β 2 /β 1 = 1 β 2 /β 1 > 1 0 < b γ k ≤ α k /f 0 0 ≤ b γ k ≤ 0 −∞ ≤ b γ k < 0 When β 2 /β 1 = 1 the controller is immune to delay because b γ k = 0 for all k . This aligns with expectations because the case where β 1 = β 2 6 = 0 corresponds to a standard ∆ ω stabilizer , as shown in T able I. When β 2 /β 1 < 1 , the magnitude of b γ k has an upper bound of α k /f 0 . This corresponds to the case where β 1 > β 2 , and the PSS prioritizes the damping of local and inter- area modes. When β 2 /β 1 > 1 , b γ k is unbounded below . Thus, the magnitude of b γ k may grow arbitrarily large as β 1 → 0 . This does not imply that b C ν ( s ) may have infinite values; rather , that the steady-state component of the control error (11) may be much larger than the small-signal component. This corresponds to the case where β 1 < β 2 , and the PSS prioritizes shaping the system response to transient disturbances. It is observed that the sensitivity of the open-loop frequency response to delay increases as the ratio β 2 /β 1 increases. W e hypothesize that the driving factor in this relationship is that as β 2 /β 1 grows, so too do the magnitudes of the entries of b C ν ( s ) corresponding to the delayed system states in relation to the non-delayed state(s). That is, when β 1  β 2 , it follows that | b γ k | > 1 for some k . This suggests that if β 1 < β 2 , the ratio β 2 /β 1 should be kept small. T o illustrate this behavior , suppose that β 1 = 0 . 5 , β 2 = 1 for generator G2, where β 2 /β 1 = 2 . Fig. 16 shows the effect of delay on the open-loop frequency response for G2 in this case. As the delay increases, a key transfer function zero changes position in the complex plane. Fig. 16 indicates that this zero is pushed across the j ω -axis between 0.1 – 0.2 Hz and into the right half of the complex plane as the delay increases. Right-half-plane zeros, especially in the neighborhood of the electromechanical modes, may erode stability margins and are generally undesirable [21]. In this case, the system remains stable when τ = 1 . 25 s because the gain at the critical frequencies is very lo w . Fig. 17 shows the system response in the time domain follo wing a trip of G26. Each generation unit in the PST model was outfitted with a generalized ∆ ω PSS where β 1 = 0 . 5 , β 2 = 1 , and K = 9 . As the state trajectories show , this tuning places more emphasis on shaping the transient response than on damping local and inter-area modes. Such parameter combinations should be used with caution. Careful stability analysis must be carried out to ensure that it is safe to employ a particular tuning giv en the performance characteristics of the measurement, communication, and control equipment. V . C O N C L U S I O N This paper presented a generalization of the standard ∆ ω - type stabilizer . It works by incorporating local information with a real-time estimate of the center-of-inertia speed. The ability of the stabilizer to impro ve the damping of electromechanical 10 10 − 2 10 − 1 10 0 10 1 − 60 − 40 − 20 Magnitude (dB) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s 10 − 2 10 − 1 10 0 10 1 − 180 − 90 0 90 180 Frequency (Hz) Phase (deg) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s Fig. 16. The effect of the combined delay τ on the open-loop frequency response for generator G2 with β 1 = 0 . 5 and β 2 = 1 . 0 5 10 15 20 25 30 35 40 59 . 8 59 . 9 60 . 0 Sys tem frequency (Hz) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s 0 5 10 15 20 25 30 35 40 0 . 99 1 . 00 1 . 01 1 . 02 1 . 03 Time (s) G2 T er m. voltag e (pu) τ = 0 s τ = 0 . 625 s τ = 1 . 250 s Fig. 17. Simulations of generator G26 being tripped offline for various average combined delays where β 1 = 0 . 5 and β 2 = 1 . modes is decoupled from its role in shaping the system response to transient disturbances. Hence, the interaction between the PSS and A VR can be fine-tuned based on v oltage requirements. Future w ork will e xplore a v ariation of the proposed architecture that permits integral of accelerating power feedback for mitigating torsional oscillations. 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Ryan Elliott receiv ed the M.S.E.E. degree in 2012 from the University of W ashington, Seattle, W A, USA, where he is currently a Ph.D. candidate in the Department of Electrical and Computer Engineering. His research focuses on renew able energy integration, wide-area measurement systems, and power system operation and control. From 2012 to 2015, he was with the Electric Power Systems Research Department at Sandia National Laboratories. While at Sandia, he served on the WECC Renew able Energy Modeling T ask Force, leading the dev elopment of the WECC model validation guideline for central- station PV plants. In 2017, he earned an R&D 100 A ward for his contributions to a real-time damping control system using PMU feedback. Payman Arabshahi receiv ed the Ph.D. degree in 1994 from the University of W ashington, Seattle, W A, USA, where he is currently an Associate Professor of Electrical and Computer Engineering and a principal research scientist with the Applied Physics Laboratory . His research focuses on wireless communications and networking, sensor networks, signal processing, data mining and search, and biologically inspired systems. From 1994 to 1996, he served on the faculty of the Electrical and Computer Engineering Department, Univ ersity of Alabama, Huntsville, AL, USA. From 1997 to 2006, he was on the senior technical staff of NASA ’s Jet Propulsion Laboratory in the Communications Architectures and Research Section. While at JPL he also served as affiliate graduate faculty at the Department of Electrical Engineering, California Institute of T echnology , Pasadena, CA, USA, where he taught the three-course graduate sequence on digital communications. Daniel Kirschen receiv ed the Electro-Mechanical Engineering degree from the Free University of Brussels, Belgium and the Ph.D. degree from the University of Wisconsin, Madison, WI, USA. He is currently the Donald W . and Ruth Mary Close Professor of Electrical and Computer Engineering at the University of W ashington, Seattle, W A, USA. His research focuses on the integration of renew able energy sources in the grid, power system economics, and power system resilience. Prior to joining the Univ ersity of W ashington, he taught for 16 years at the Univ ersity of Manchester , U.K. Before becoming an academic, he worked for Control Data and Siemens on the development of application software for utility control centers. He has co-authored two books on power system economics and reliability standards for electricity networks.