POWER system operational reliability is monitored and

1 Generalized Injection Shift Factors Yu Christine Chen, Meber, IEEE, Sairaj V. Dhople, Meber, IEEE, Alejandro D. Doínguez-García, Meber, IEEE, and Peter W. Sauer, Life Fellow, IEEE Abstract Generalized injection shift factors ISFs are the sensitivities of active-power line flows to active-power bus injections. They are coputed without designating an artifactual slack bus in the power network; therefore, conceptually, they reflect sensitivities of power flows ore accurately than conventional ISFs, the values of which depend on the choice of the slack bus. This paper derives analytical closed-for expressions for generalized ISFs fro a perturbative analysis of the AC circuit equations. In addition, they are coputed fro a syste of linear equations that arise fro high-frequency synchronized easureents collected fro phasor easureent units. As an application, generalized ISFs are used to predict active-power line flows during the transient period following a contingency by leveraging inertial and governor power flows over appropriate tie horizons. Index Ters Distribution factor, injection shift factor, onitoring, phasor easureent units, sensitivity. I. INTRODUCTION POWER syste operational reliability is onitored and aintained with a suite of online static and dynaic security-assessent tools that allow operators to assess whether or not the syste is capable of withstanding a wide variety of disturbances, such as sudden loss of a generator or load. Dynaic security assessent tools indicate the ability of the syste to withstand transients induced by disturbances prior to reaching steady-state operation at a new operating point [1]. On the other hand, static security assessent tools exaine the syste in steady-state operation. A coon security assessent tool is real-tie N 1 contingency analysis, in which operators deterine whether or not the syste would eet operational reliability requireents in case of an outage in any one particular asset [2]. With an accurate odel of the syste, operators can perfor the N 1 security analysis by repeatedly solving the nonlinear power flow equations. However, for a large power syste with any possible contingencies, this process is coputationally expensive, and therefore could take prohibitively long periods The work of S. V. Dhople was supported in part by the National Science Foundation under CAREER award ECCS-1453921, and in part by a Minnesota s Discovery, Research and Innovation Econoy grant. The work of A. D. Doínguez-García and P. W. Sauer was supported in part by the U.S. Departent of Energy under the Consortiu for Electric Reliability Technology Solutions, and in part by the Power Systes Engineering Research Center. Y. C. Chen is with the Departent of Electrical and Coputer Engineering at the University of British Colubia, Vancouver, BC, V6T 1Z4, Canada. E-ail: chen@ece.ubc.ca. S. V. Dhople is with the Departent of Electrical and Coputer Engineering at the University of Minnesota, Minneapolis, MN, 55455, USA. E-ail: sdhople@umn.edu. A. D. Doínguez-García and P. W. Sauer are with the Departent of Electrical and Coputer Engineering at the University of Illinois at Urbana-Chapaign, Urbana, IL, 61801, USA. E-ails: {aledan, psauer}@illinois.edu. of tie. An alternative is to use an estiate of the current operating point together with linear sensitivity distribution factors DFs in order to predict values of different variables following the occurrence of a contingency [3]. For exaple, the socalled power transfer distribution factor PTDF approxiates the changes in active-power line flows due to an exchange of active power between two buses. In general, such line flow DFs can all be derived fro the injection shift factor ISF, which approxiates the change in active-power flow across a transission line due to a change in active-power generation or load at a particular bus. In this paper, we develop the notion of a generalized ISF. Coputed via a perturbative analysis of the AC circuit equations around a particular operating point, the generalized ISF is agnostic to the location and even existence of a slack bus. Indeed, the concept of the generalized ISF represents an iportant step in reoving the dependence on slack bus location in steady-state power syste studies. Additionally, leveraging results fro previous work in [4], we estiate generalized ISFs by using real-tie easureents without relying on an offline odel of the syste. Such easureent-based ethods have been shown to be robust to undetected syste-topology and operating-point changes [4]. The proposed generalized ISFs are utilized based on the propagation of disturbances over tie-scales for which inertial and governor power flows are valid [5] to predict active-power line flows during the transient period following a disturbance. Conventional ISFs are coputed by linearizing the power flow equations around an operating point with a designated slack-bus location; then, they can be used to predict postcontingency steady-state active-power line flows [6]. The generalized ISFs presented in this work offer three copelling advantages over conventional ISFs. First, conventional ISF values differ depending on the choice of slack bus [7]. This is because conventional ISFs are derived fro the power flow equations forulated by designating a slack bus, with the corresponding generator assued to copensate for any power ibalances in the syste. Second, the coputation of conventional ISFs relies on an accurate odel of the syste, which ay not be available due to poor records or erroneous teleetry fro reotely onitored circuit breakers [8]. On the other hand, generalized ISFs can be coputed fro highfrequency synchronized easureents collected fro phasor easureent units PMUs. Finally, traditionally, ISFs and DFs in general have been used exclusively to verify that operational reliability requireents are et see, e.g., [6] [10]. Based on the propagation of disturbances over tiescales for which inertial and governor power flows are valid, we estiate active-power line flow liit violations not only at the post-contingency steady-state operating point, but also

2 during the post-disturbance transient period. This paper builds on preliinary work reported in [11], and enhances it in several directions. First, fro a circuittheoretic vantage point, we establish an analytical derivation for generalized ISFs that does not depend on the location of the slack bus. The process uncovers and foralizes intuitive connections between power-flow and current-flow sensitivity factors. Furtherore, by revisiting assuptions underlying the definition of widely accepted power syste linear sensitivity factors, we extend the utility of power-flow sensitivities to large-scale power-syste dynaic operational reliability onitoring. Finally, to deonstrate these aspects, we show how generalized ISFs can predict post-contingency transient line flows via case studies involving the siplified Western Electricity Coordinating Council WECC and the New England test systes. The reainder of the paper is organized as follows. In Section II, we establish atheatical notation, introduce voltage phase-angle and agnitude sensitivities, define the conventional ISFs, and describe the conventional odel-based ethod to obtain the. In Section III, we define the generalized ISFs and provide odel- and easureent-based ethods to obtain the. Generalized ISFs, as we show in Section IV, can be anipulated to recover conventional ISFs, and to predict transient active-power line flows after a disturbance. In Section V, we illustrate applications of generalized ISFs via case studies involving the WECC 3-achine 9-bus and the New England 10-achine 39-bus test systes. Finally, concluding rearks are provided in Section VI. II. PRELIMINARIES This section first introduces the notation and power syste odel used in the reainder of the paper. Next, we derive sensitivities of bus-voltage phase-angles and agnitudes with respect to active-power bus injections, which are used to copute conventional ISFs. Finally, we provide an exaple to otivate the need for generalized ISFs. A. Notation and Power Syste Model Matrix transpose will be denoted by T, coplex conjugate by, coplex-conjugate transposition by H, real and iaginary parts of a coplex nuber by Re{ } and I{ }, respectively, and j := 1. A diagonal atrix fored with entries of the vector x is denoted by diagx; and diagx/y fors a diagonal atrix with the ith entry given by x i /y i, where x i and y i are the ith entries of vectors x and y, respectively. The spaces of N 1 real-valued and coplex-valued vectors are denoted by R N and C N, respectively. Given a vector function fx = [f 1 x,..., f M x] T : R N R M, x fx returns the M N Jacobian atrix [ x f 1 x T,..., x f M x T ] T, where, for each i = 1,..., M, the gradient x f i x = [ f i x/ x 1,..., f i x/ x N ]. The N N identity atrix is denoted by I N. The M N atrices with all zeros and ones are denoted by 0 M N and 1 M N, respectively; e i denotes a colun vector of all zeros except with the ith entry equal to 1; and e ij denotes a colun vector of all zeros except with the ith and jth entries equal to 1 and 1, respectively. For a vector x = [x 1,..., x N ] T, x i [ π, π] i = 1,..., N, cosx := [cosx 1,..., cosx N ] T and sinx := [sinx 1,..., sinx N ] T. Finally, Π l := [e 1,..., e l 1, e l+1,..., e N ] R N N 1. Consider a power syste with N buses, and let L denote the set of transission lines and N the set of buses in the syste. Denote the bus adittance atrix by Y C N N. Further, let V = [V 1,..., V N ] T = θ, where V i = V i θ i C represents the voltage phasor at bus i, and let I = [I 1,..., I N ] T, where I i C denotes the current injected into bus i through a path not included in Y. Then, Kirchhoff s current law for the buses in the power syste can be copactly represented in atrix-vector for as follows: I = Y V. 1 In 1, Y C N N is the adittance atrix defined as y +,k L y k, if = n, [Y ] n := y n, if, n L, 0, otherwise, where y = g + jb = y +,k N y sh k denotes the total shunt adittance connected to bus with N representing the set of neighbours of bus and y C any passive shunt eleents connected to bus. Denote the vector of coplex-power bus injections by S = [S 1,..., S N ] T = P + jq, with P = [P 1,..., P N ] T and Q = [Q 1,..., Q N ] T. [By convention, P i and Q i are positive for generators and negative for loads.] Then, coplex-power bus injections can be copactly written as 2 S = diag V I. 3 The bus-voltage angles are collected in the vector θ = [θ 1,..., θ N ] T, θ i [ π, π]. The following auxiliary busvoltage-angle and power related variables will be useful: θ := θ θ 1 N 1, θ := Π T θ, P := Π T P, Q := Π T Q, 4 where θ is obtained by setting the syste angle reference as θ the voltage angle at bus. In 4, θ, P, and Q denote the N 1-diensional vectors that result fro reoving the th entry of the N-diensional vectors θ, P, and Q, respectively. For exaple, suppose N = 3, and θ = [θ 1, θ 2, θ 3 ] T. For = 2, it follows that θ 2 = [θ 1 θ 2, 0, θ 3 θ 2 ] T, and θ 2 = [θ 1 θ 2, θ 3 θ 2 ] T. B. Bus-voltage Phase-angle and Magnitude Sensitivities Let the angle reference be established by the voltage phase angle at bus. Then, leveraging the definitions in 4, the power-flow equations can be written copactly as follows: f θ,, P, Q = 0 2N 1 1, 5 where the superscript indicates that f is forulated by setting the angle reference to bus. In 5, the dependence on network paraeters, such as line series and shunt ipedances, is iplicit in the forulation of the function f.

3 Denote the solution to 5 by θ, V, P, Q, and assue f is continuously differentiable with respect to θ,, P, and Q at θ, V, P, Q. Let θ = θ + θ, = V +, P = P + P, and Q = Q + Q. Then, by assuing that θ,, P, and Q are sufficiently sall, we can approxiate 5 as f θ,, P, Q f θ, V, P, Q + J [ θ ] + D P + E Q, 6 where J = [ θ,] f θ,, P, Q θ, V, P D = P f θ,, P, Q θ, V, P E = Q f θ,, P, Q θ,q, V, P,Q.,Q The subscript for the quantities J, D, and E indicates that they are obtained by setting the angle reference to bus. Since θ, V, P, Q is a solution to 5, then f θ, V, P, Q = 0 2N 1 1. Also, for all application studies in the paper, we assue power flow balance is satisfied, which requires that f θ,, P, Q = 0 2N 1 1. Furtherore, θ,, P, and Q are assued to be sall. With these in ind, it follows fro 6 that 0 J [ θ 7 ] + D P + E Q. 8 In addition, since J is the Jacobian of the power flow equations evaluated at a noinal non-stressed operating point, we assue it is invertible around θ, V, P, Q, so we can rearrange ters in 8 to obtain [ ] θ J 1 D P + E Q = J 1 D Π T P + E Q, 9 where the second equality in 9 follows fro 4. C. Conventional Injection Shift Factors Denote, by Ψ l,n,i, the ISF of line, n L assue positive real power flow fro bus to n easured at bus with respect to bus i, which is the linear approxiation of the sensitivity of the active power flow in line, n with respect to the active-power injection at bus i, with the slack bus location specified as bus l. In the conventional power flow forulation, the slack bus sets the angle reference, and the generator connected to it absorbs any power ibalance in the syste. With this assuption in place, siilar to 5, the power flow equations can be written copactly as f l θl,, P l, Q = 0 2N 1 1. 10 Following an analogous developent as in 5 9, where all instances of are substituted with l, and neglecting Q by setting Q = 0 N 1, 9 becoes [ θl ] J 1 l D l P l. 11 Denote the active-power injection at bus i and flow in line, n by P i and P,n, respectively. Further, assue all quantities in the syste reain constant except a sall change in P i, denoted by P i. Denote the resulting change in P,n by P,n,i. Then, define Ψ l,n,i := P,n P i P,n,i P i. 12 Conventional ISFs can be coputed with a power flow odel of the syste obtained offline. We begin by expressing the current through line, n L as I,n = y,n V V n + y sh,n V, 13 where y,n C\{0} is the adittance of the line connecting buses and n, and y,n sh C \ {0} is the shunt adittance at bus. Denote, by S,n = P,n +jq,n, the coplex power flowing across line, n, which is expressed as S,n = V I,n. 14 Substituting 13 into 14, and isolating the real part, we obtain P,n = Re{S,n } =: h,n θ l,. 15 Under the sae sall θ l and assuption used to derive 11, we obtain an expression for sall variations P,n due to θ l and, as follows: [ ] θl P,n [w 1, w 2 ], 16 where θl w 1 = θlh,n, θl w 2 = h,n θl,,, V θl., V Substituting 11 into 16, we get that P,n Ψ l,n P l, Ψ l,n R 1 N 1, where the conventional ISFs in 12 are recovered as the entries of Ψ l,n = [w 1, w 2 ]J 1 l D l. 17 D. Motivation for Generalized Injection Shift Factors The derivation of the sensitivity factor in 17 assues steady-state operation and a predefined slack bus, i.e., bus l. However, a power syste exhibits dynaic behaviour, and the notion of a single slack bus that absorbs all power ibalances in the syste is only valid for static power-flow analysis. As a result, conventional ISFs do not provide accurate results when syste dynaic behaviour is considered, a shortcoing that we exeplify via the following exaple. Exaple 1 3-Bus Syste: To illustrate the otivation for deriving generalized ISFs, consider a siple 3-bus syste connected in a ring foration. Transission lines are odelled with luped paraeters, where y 12 = 1.3652 j11.6041 p.u., y12 sh = j0.088 p.u., y 23 = 0.7598 j6.1168 p.u., y23 sh = j0.153 p.u., y 13 = 1.1677 j10.7426 p.u., y13 sh = j0.079 p.u. In this syste, generators are connected to buses 1 and 2, injecting P 1 = 1.5973 p.u. and P 2 = 0.7910 p.u. respectively.

4 A load is connected to bus 3, with active-power injection P 3 = 2.35 p.u. and reactive-power injection Q 3 = 0.5 p.u.. The voltage agnitude at buses 1 and 2 are regulated at V 1 = 1.04 p.u. and V 2 = 1.025 p.u., respectively. Suppose bus 1 is set as the slack bus, i.e., l = 1. Let Ψ 1,n = [Ψ1,n,1, Ψ1,n,2, Ψ1,n,3]. Coputed using 17, the conventional ISFs for this syste are Ψ 1 1,2 = [ 0 0.7529 0.2792 ], Ψ 1 2,3 = [ 0 0.2478 0.2792 ], Ψ 1 1,3 = [ 0 0.2466 0.7415 ]. By definition, the ISFs with respect to the slack bus are 0. On the other hand, if l = 2, then conventional ISFs are Ψ 2 1,2 = [ 0.7368 0 0.2479 ], Ψ 2 2,3 = [ 0.1301 0 0.4128 ], Ψ 2 1,3 = [ 0.4254 0 0.5037 ]. The above deonstrates that vastly different nuerical ISF values can result depending on the designated slack bus location. The concept of generalized ISFs, which do not depend on an arbitrary slack bus, iproves upon this. Moreover, Section IV deonstrates that the proposed generalized ISFs can be used to recover conventional ISFs as well as to predict active-power line flows during the transient period between two steady-state operating points. III. MODEL- AND MEASUREMENT-BASED ESTIMATION OF GENERALIZED ISFS Consider the power syste odel described in Section II-A. Beginning with a valid power flow solution, instead of designating an explicit slack bus l as in the derivation of conventional ISFs see Section II-C, define the generalized ISF of line, n with respect to bus i as Γ,n,i := P,n P i P,n,i P i, i N, 18 where P i denotes sall variation in P i and P,n,i represents the change in active power flow in line, n easured at bus resulting fro P i. Also, collect Γ,n,i s in the row vector Γ,n R 1 N. Note that Γ,n captures sensitivities with respect to active-power injections at all buses, which is in contrast to Ψ l,n in 17. The reainder of this section details two ethods for obtaining the generalized ISFs defined in 18 by: i coputing the fro the AC circuit equations, and ii estiating the using active-power injection and flow easureents obtained in real tie. The derivation of the generalized ISF relies on a few preliinaries; these include: i a derivation of sensitivities of line-current flows to current injections, ii a atrix-based representation of active-power line flows, and iii a perturbative analysis of active-power line flows that is agnostic to explicitly designating a slack bus. A. Current Injection Sensitivities The derivation of current injection sensitivities begins by rewriting the current through line, n fro 13 as I,n = y,n e T n + y,n sh et V, 19 where V C N is the vector of bus voltage phasors. If the bus adittance atrix, Y, is invertible, 1 then fro 1, the bus voltages can be expressed as V = Y 1 I. Subsequently, 19 can be written as I,n = a T,n I, where a T,n = y,n e T n + y,n sh et Y 1 C 1 N. 20 The entries of a,n correspond to the current injection sensitivity factors of line, n with respect to the current injection at bus i. Note that the current injection sensitivity factors only depend on network paraeters, and not on the operating point. B. Active-power Injection Sensitivities Our derivation of the generalized injection shift factors is grounded on coputing the sensitivity of a atrix-related expression for the active-power line flows and recognizing their relationship to line currents. To this end, instead of the approach in 14 17, we cobine 20 together with 14 to obtain S,n = V a H,n I = e T V a H,n I. 21 Eliinating I fro 21 using 3 yields S,n = e T V a H,n diag V 1 S. 22 Using the phasor for of the voltages, 22 becoes S,n = V θ a H,n diag 1N 1 S. 23 θ Since line power flows are often expressed as functions of angle differences, it is useful to rewrite 23 as S,n = V a H,n diag 1N 1 1 θ S θ = V a H,n diag 1N 1 θ S, 24 where θ = θ θ 1 N 1. To focus on active-power line flows, it is useful to decopose the current injection sensitivity factors as a,n = α,n + jβ,n, 25 which enables us to express the real part of 24 as P,n = e T u T P + v T Q where cos θ u := diag α,n diag =: p,n θ,, P, Q, 26 sin θ β,n, 27 1 The inclusion of the shunt adittance ter in 19 intrinsically guarantees invertibility of Y by inducing diagonal doinance [12].

5 sin θ v := diag α,n + diag cos θ β,n. 28 Note that p,n in 26 is a function of active- and reactivepower injections at all buses. With these preliinaries in place, next we state the ain result of this paper. Consider the active-power flow in line, n. Suppose the originating bus sets the angle reference and the solution to 5 is denoted by θ, V, P, Q. Then, sall variations in the active-power flow through line, n are related to sall variations in the active-power bus injections by way of P,n Γ,n P, where Γ,n = [r 1, r 2 ]J 1 D Π T + s 1, 29 with r 1 R 1 N 1, r 2 R 1 N, s 1 R 1 N given by r 1 = e T u T Π diag Q v T Π diag P, 30 r 2 = P T u + Q T v P e T e T u T diag + v T diag Q, 31 s 1 = e T u T, 32 and u and v as defined in 27 and 28. The quantities u, v, r 1, r 2, and s 1 are all evaluated at the noinal power flow solution θ, V, P, Q. The derivation of 29 is provided in Appendix A. The expressions in 29 and 30-32 yield sensitivity factors of P,n with respect to entries of the active-power injection vector, P, for a special case. Assue a flat voltage profile, i.e., V i = 1 p.u. and θ i 0, for all i N. With these assuptions, 29 reduces to Γ,n = α T,n. 33 In other words, the power-flow sensitivities reduce to the line current-flow sensitivities, which akes intuitive sense. C. Measureent-based Estiation of Generalized ISFs In Section III-B, we derived closed-for expressions for the generalized ISF using an up-to-date odel of the syste. On the other hand, PMUs, which provide synchronized voltage, current, and frequency easureents as any as 60 ties per second [13], enable us to estiate generalized ISFs in real tie. For case studies in Section V, a sapling rate of 30 Hz is used. The easureent-based ethod relies only on inherent fluctuations in easureents of load and generation and is thus adaptive to operating point and topology changes [4]. This section tailors the easureent-based ISF estiation approach proposed in [4] and further refined in [14] to obtain generalized ISFs in real-tie, using active-power bus injection and line flow easureents. Let P i t and P i t + t denote the active-power injection at bus i at ties t and t + t, respectively t > 0 can be interpreted as the PMU sapling period. Define P i t = P i t + t P i t and denote the change in active power flow in line, n resulting fro P i t by P,n,i t. The generalized ISF, Γ,n,i, can be coputed based on the approxiation in 18. However, 18 requires P,n,i t, which is not readily available fro PMU easureents. Instead, assue that the net variation in active power through line, n, denoted by P,n t, is available fro PMU easureents. Further, decopose this net variation as the su of active power variations in line, n due to activepower injection variations at each bus i: P,n t = N P,n,i t. 34 i=1 Substitution of 18 into each ter in 34 yields P,n t N P i tγ,n,i. 35 i=1 Suppose M +1 sets of synchronized active-power bus injection and line flow easureents are available. Let P i [k] = P i k + 1 t P i k t, P,n [k] = P,n k + 1 t P,n k t, k = 1,..., M; and define P,n = [ P,n [1] P,n [M] ] T, P i = [ P i [1] P i [M] ] T. For ease of notation, let P represent the M N atrix [ P 1,..., P i,..., P N ], where N denotes the nuber of buses in the syste; then, it follows that P,n = P Γ T,n. 36 If M N, then 36 is an overdeterined syste. Via weighted least-squares WLS estiation, the solution for Γ,n can be obtained as [4]: Γ T,n = P T W P 1 P T W P,n, 37 where W R M M is a weighting atrix. Often, W is designed to place ore iportance on recent easureents and less on earlier ones. Note that, if W = I M, i.e., all easureents within the estiation tie window are weighted equally, then 37 reduces to the solution of a least-squares errors LSE estiation proble. The forulation in 37 assues that all buses are equipped with PMUs. To relax this requireent, consider the intuition that ost line flows are significantly affected by only a sall set of electrically nearby buses this intuition is verified in [14]. With this in ind, 37 can be odified so as to reduce the nuber of required easureent locations in the syste. This observation, in cobination with the fact that, in a practical large-scale power syste, only ajor transission corridors are onitored, would greatly reduce the nuber of PMUs required in the easureent-based ISF estiation ethod. IV. APPLICATIONS OF GENERALIZED ISFS While the derivation of generalized ISFs does not depend on the location of a slack bus or any power allocation schees, in this section, we show that the generalized ISFs can be used in conjunction with power allocation schees to

6 yield eaningful line-flow predictions. Most generally, power allocation schees are based on the idea of a distributed slack bus, where any syste power ibalance is absorbed by several buses [15], [16]. For exaple, in the context of contingency analysis, ultiple generators ay respond to a loss in generation or increase in load. Based on this fact, we utilize generalized ISFs to predict the active-power line flows during the transient period following a loss of generation or increase in load. Moreover, conventional ISFs can be recovered as a special case by assuing that a single generator at a designated slack bus absorbs all power ibalances caused by the contingency. A. Obtaining Participation Factor-based ISFs Suppose we are interested in the sensitivity of the activepower flow in line, n with respect to a particular injection at bus i, denoted as P i, and this injection is balanced by soe linear cobination of injections at other buses, i.e., P j = γ j P i, j i γ j = 1. 38 The participation factors γ j s can be chosen based on insights gleaned fro econoic dispatch, governor control, or synchronous generator inertia [5]. For exaple, generator inertiabased participation factors are obtained by defining γ j = H j j H, 39 j where H j denotes the inertia of the synchronous generator j in the syste [17, Ch. 5]. Siilarly, governor participation factors are obtained by defining γ j = 1/R j j 1/R, 40 j where 1/R j represents the steady-state governor gain for generator j [17, Ch. 4]. The participation factors described in 39 and 40 describe realizations of the distributed slack bus based on power-frequency characteristics of generators in the syste [5]. Denote by Ω,n,i the sensitivity of P,n with respect to P i with consideration for generator participation factors. Substitution of 38 into 35 results in P,n Ω,n,i P i, where Ω,n,i = Γ,n,i Γ,n,j γ j. 41 j i Siilar procedure for easureent-based generalized ISFs yields P,n Ω,n,i P i, where Ω,n,i = Γ,n,i Γ,n,j γ j. 42 j i The participation factor-based ISFs in 41 and 42 can be used to predict the transient active-power line flows following a disturbance. Since the inertial response is faster than that of the governor, we expect the inertia-based ISFs obtained using 41 and 42 together with 39 to be valid for a short tie after the occurrence of the disturbance. Following this, we expect that the governor-based ISFs obtained using 41 and 42 together with 40 to be valid until participation factors arising fro econoic dispatch becoe relevant. P 1,2 [p.u.] P 2,3 [p.u.] P 1,3 [p.u.] 0.08 0.06 0.04 0.88 0.86 1.6 1.55 0 5 10 15 Tie [s] Fig. 1: Line flows in 3-bus syste due to a 0.1 p.u. increase in active power deand at bus 3. B. Obtaining Conventional ISFs The conventional ISF is siply the sensitivity of the active power flow across line, n with respect to an active-power injection at bus i, which is balanced entirely by the generator at the designated slack bus l, i.e., in the context of 38, γ l = 1 and all other γ j = 0, j i, l. Thus, as a special case of 41, the conventional ISF can be recovered as Ω l,n,i = Γ,n,i Γ,n,l, 43 and Ω l,n,i can be analogously recovered fro 42. C. Illustrative 3-Bus Syste Exaple In Exaple 1, we otivated the need to develop generalized ISFs that do not rely on a slack bus. Here, we revisit the 3- bus syste fro Exaple 1 and illustrate concepts described in Sections III and IV. First, the current injection sensitivity factors for this syste are coputed using 20. For each line, n L, a,n = α,n + jβ,n, where α 1,2 = [ 0.5178 0.2329 0.2485 ] T, α 2,3 = [ 0.2443 0.4934 0.0289 ] T, 44 α 1,3 = [ 0.4822 0.2329 0.2485 ] T. For brevity, we refrain fro reporting values for β,n. Next, using 29, generalized ISFs Γ,n are coputed for each line, n L in this syste, to yield Γ 1,2 = [ 0.5178 0.2353 0.2493 ], Γ 2,3 = [ 0.2457 0.4934 0.0283 ], 45 Γ 1,3 = [ 0.4822 0.2353 0.2493 ]. Note that, indeed, the generalized ISFs obtained directly above are very siilar to α,n s reported in 44. This siilarity verifies that, indeed, generalized ISFs can be interpreted as the power analogue of current injection sensitivity factors, as pointed out in 33. Furtherore, like the current injection shift factors, the derivation of generalized ISFs does not depend on a predefined power allocation schee. To illustrate the flexibility of generalized ISFs, we first verify the validity

7 P 5,7 [p.u.] P 7,8 [p.u.] 0-0.5-1 0.95 0.9 0.85 0.8 P 26,28 [p.u.] P 10,11 [p.u.] -1.2-1.4-1.6 3.5 3 2.5 2 P 3,9 [p.u.] 0.8 0.6 0.4 0 5 10 15 20 25 Tie [s] Fig. 2: Line flows in WECC 3-achine 9-bus syste due to loss of load at bus 6. of 43, with bus 1 designated as the slack bus. By using the Γ,n s in 45, we obtain Ω 1 1,2 = [ 0 0.7531 0.2685 ], Ω 1 2,3 = [ 0 0.2477 0.2741 ], 46 Ω 1 1,3 = [ 0 0.2469 0.7315 ]. Indeed, the Ω 1,n s obtained here are very siilar to corresponding Ψ 1,n s obtained in Exaple 1. Finally, we predict transient active-power line flows using participation factor-based ISFs as given in 41, for a 0.1 p.u. increase in the load at bus 3. To obtain the actual effect due to the 0.1 p.u. active load change disturbance described above, the dynaic siulation tool Power Syste Toolbox PST [18] is used to acquire active-power line flows at pre- and post-disturbance operating points. The load at bus 3 is odelled as a constant-power sink. The synchronous generators at buses 1 and 2 are odelled with the subtransient achine dynaic odel equipped with a DC exciter and a turbine governor see, e.g., [1]. In these odels, the rotational inertia are set to H 1 = 8 s and H 2 = 3.01 s for generators connected to buses 1 and 2, respectively. The governor droop for both generators are set to 1/R 1 = 1/R 2 = 25 p.u. Based on these paraeters, the inertia-based participation factors for the two synchronous generators in the syste are γ 1 = 8/11.01 and γ 2 = 3.01/11.01. Using these factors in concert with 41, we obtain the change in active power flow through the three lines as P 1,2 = 0.0066 p.u., P 2,3 = 0.0339 p.u., and P 1,3 = 0.0661 p.u. Siilarly, the governor-based participation factors are γ 1 = γ 2 = 1/2, and the corresponding changes in power flow are P 1,2 = 0.0106 p.u., P 2,3 = 0.0394 p.u., and P 1,3 = 0.0606 p.u. Siulation results are plotted as the solid trace in Fig. 1. Superiposed onto the actual active-power line flows, we also plot the line flows predicted by the inertia-based and governor-based ISFs in dash and dash-dot traces, respectively. Indeed, we observe that inertia-based ISFs provide a good approxiation to the line flows iediately after the load increase, while governor-based ISFs provide a good estiate over longer tie horizons. P 6,7 [p.u.] P 1,2 [p.u.] 4 3 2 0-1 -2-3 0 5 10 15 20 25 30 35 Tie [s] Fig. 3: Line flows in New England syste due to loss of load at bus 8. V. CASE STUDIES In this section, we illustrate soe applications of generalized ISFs with odels of the WECC 3-achine 9-bus and the New England 39-bus systes. We copute both conventional and generalized ISFs using the odel-based ethod described in Section III-B. Loads are odelled as constant-power sinks and fluctuations in active-power deand are siulated with the rando tie-series data: P i [k] = P o,i [k] + σν[k], 47 where P o,i [k] is the noinal active-power load at bus i and σν[k] is a pseudorando nuber drawn fro a noral distribution with 0-ean and standard deviation σ = 0.03 p.u. We assue a PMU sapling rate of 30 Hz. A. WECC 3-Machine 9-Bus Syste Model Using the pre-contingency base case, we create synthetic power-injection profiles with 47, for i = 5, 6, 8, i.e., buses that have loads, which are inherently variable due to the rando nature of electricity deand fro end users. Using PST, we conduct tie-doain dynaic siulations, fro which we extract M = 74 sets of easureents of activepower injections at all buses and flows across all lines. Fro these siulated easureents, we estiate generalized ISFs using 37 with W = I M, and further copute inertiaand governor-based ISFs via 42. These ISFs are then used to predict active-power line flows in the post-disturbance transient period before a new steady-state is reached. In this case study, we consider a loss-of-load contingency at bus 6, i.e., P 6 = 1.63 p.u., at tie t = 1 s. We use the sae generalized ISFs that were estiated under the precontingency pseudo-steady-state operating point. In Fig. 2, line flows predicted by generalized ISFs are superiposed over the dynaic response. Even with the severe loss of load contingency, the generalized ISFs accurately capture postcontingency line flows. As a easure of error, we copute the

8 P 22,23 [p.u.] 4.5 4 3.5 3 2.5 Model-based Model-based Meas.-based Meas.-based obtained at the pre-disturbance steady-state operating point, we show, through nuerical exaples and case studies, that they can be easily anipulated to predict active-power line flows during the transient period following a disturbance. As future work, we will explore dynaic odel reduction techniques to obtain tie-dependent expressions for participation factors so as to ake accurate line flow predictions throughout transients. 0 5 10 15 20 25 30 35 Tie [s] Fig. 4: Line 22, 23 flow in New England syste due to loss of load at bus 8, with an undetected outage in line 16, 21. Flows predicted by odel-based ISFs are delineated by, while those predicted by easureent-based ISFs are delineated by. absolute value of the difference between the predicted and actual active-power line flows at t = 25 s. The average prediction error is 0.004 p.u., with a axiu error of 0.0061 p.u. B. New England 39-Bus Syste Model This odel contains 39 buses and 10 synchronous generators odelled with the subtransient odel [1]. All generator odels include a governor/turbine and nine include voltage regulators/exciters. To validate the values of generalized ISFs obtained via both the odel- and easureent-based ethods, we introduce a loss of load contingency at t = 1 s, at bus 8, and copare actual post-contingency lines flows with generalized ISF predictions. 1 Base Case: We copute generalized ISFs via 29 using the base-case syste circuit odel. The average prediction error is 0.0252 p.u., with a axiu error of 0.0558 p.u. In Fig 3, we plot actual and predicted post-contingency line flows for a subset of lines. 2 Modified Network: Suppose a line outage has occurred for line 16, 21, unbeknownst to the syste operator. In this case, the odel-based generalized ISFs fro above, which are coputed fro the base-case network topology, would be inaccurate in predicting line flows for the loss-of-load contingency. The average prediction error is 0.0802 p.u., with a axiu error of 0.8619 p.u., corresponding to line 22, 23 flow, as shown in Fig. 4. Such discrepancies can lead to incorrect or suboptial operating decisions. With this case study, we highlight the advantages of the easureent-based ethod in Section III-C, which results in generalized ISF estiates that are adaptive to the current syste operating point. Indeed, using the easureent-based ethod, we obtain prediction errors coparable to those reported above in Section V-B1. VI. CONCLUDING REMARKS In this paper, we introduce the concept of generalized ISF, which can be coputed fro a sensitivity analysis of the AC circuit equations or estiated using real-tie easureents obtained fro the syste without relying on a odel of the syste obtained offline. Even though the generalized ISFs are APPENDIX A. Derivation of the Result in 29 The result in 29 is derived as follows. Around the noinal power flow solution, θ, V, P, Q, consider sall perturbations θ,, P, and Q. We then get the following expression for sall variations in the active-power line flows P,n : P,n r 1 θ + r 2 + s 1 P + s 2 Q, 48 where, with reference to p,n in 26, r 1 = θp,n, r 2 = p,n, s 1 = P p,n, s 2 = Q p,n. Substitution of 9 into 48 yields P,n [r 1, r 2 ]J 1 D P + s 1 P [r 1, r 2 ]J 1 E Q + s 2 Q = [r 1, r 2 ]J 1 D Π T + s 1 P + [r 1, r 2 ]J 1 E + s 2 Q = Γ,n P + Λ,n Q, 49 where the second equality in 49 follows fro the fact that θ = θ θ 1 N 1, as defined in 4. Next, expressions for r 1 and r 2 in 30 and 31, respectively, are derived. The derivation begins with the expression for the active-power line flows in 26, which can be rewritten as P,n = V Π T u T Π T P + Π T v T Π T Q + e T α,n P + e T β,n Q = V u T Π P + v T Π Q + e T α,n P + e T β,n Q =: p,n θ,, P, Q. 50 The above expresses the line, n active-power flow in this anner to recover a forulation that is explicitly a function of θ, with respect to which we can differentiate and recover r 1. In particular, with 50 defined above, r 1 = p θ,n θ,, P, Q 51 = V θ u T Π diag P v T Π diag Q. + θ Differentiation of u and v in 27 and 28, respectively, with respect to θ results in θ u T Π = v T Π, θ v T Π = u T Π. 52

9 Substitution of 52 into 51 returns r 1 as given by 30. Next, r 2 can be obtained by differentiating the expression in 26 with respect to to yield r 2 = p,n θ,, P, Q = V u T diagp + v T diagq + [ e u T P + v T Q ] T. 53 Fro the expressions for u and v in 27 and 28, respectively, we get u T P diagp = u T diag, v T Q diagq = v T diag. 54 Substitution of 54 into 53 returns r 2 as given by 31. Finally, s 1 and s 2 are obtained directly fro differentiating 26 with respect to P and Q as s 1 = P p,n θ,, P, Q = V u T, 55 s 2 = Q p,n θ,, P, Q = V v T. 56 The sensitivities above are all evaluated at the noinal solution of the power flow equations, denoted by θ, V, P, Q. Neglecting Q by setting Q = 0 N 1 in 49 copletes the derivation of 29. REFERENCES [1] P. Kundur, Power Syste Stability and Control. New York, NY: McGraw-Hill, Inc., 1994. [2] U.S.-Canada Power Syste Outage Task Force. 2004, Apr. Final report on the august 14th blackout in the united states and canada: causes and recoendations. [Online]. Available: https://reports.energy.gov/blackoutfinal-web.pdf [3] J. Peschon, D. S. Piercy, W. F. Tinney, and O. J. Tveit, Sensitivity in power systes, IEEE Transactions on Power Apparatus and Systes, vol. PAS-87, no. 8, pp. 1687 1696, Aug 1968. [4] Y. C. Chen, A. D. Doínguez-García, and P. W. Sauer, Measureentbased estiation of linear sensitivity distribution factors and applications, IEEE Transactions on Power Systes, vol. 29, no. 3, pp. 1372 1382, May 2014. [5] M. Lotfalian, R. Schlueter, D. Idizior, P. Rusche, S. Tedeschi, L. Shu, and A. Yazdankhah, Inertial, governor, and AGC/econoic dispatch load flow siulations of loss of generation contingencies, IEEE Transactions on Power Apparatus and Systes, vol. PAS-104, no. 11, pp. 3020 3028, Nov 1985. [6] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control. New York: Wiley, 1996. [7] P. W. Sauer, On the forulation of power distribution factors for linear load flow ethods, IEEE Transactions on Power App. Syst., vol. PAS- 100, pp. 764 779, Feb 1981. [8] FERC and NERC. 2012, Apr. Arizona-southern california outages on septeber 8, 2011: Causes and recoendations. [Online]. Available: http://www.ferc.gov/legal/staff-reports/04-27-2012-ferc-nerc-report.pdf [9] P. W. Sauer, K. E. Reinhard, and T. J. Overbye, Extended factors for linear contingency analysis, in Proceedings of the 34th Annual Hawaii International Conference on Syste Sciences, Jan 2001, pp. 697 703. [10] T. Güler, G. Gross, and M. Liu, Generalized line outage distribution factors, IEEE Transactions on Power Systes, vol. 22, no. 2, pp. 879 881, May 2007. [11] Y. C. Chen, A. D. Doínguez-García, and P. W. Sauer, Generalized injection shift factors and application to estiation of power flow transients, in North Aerican Power Syposiu NAPS, Sep 2014, pp. 1 5. [12] R. A. Horn and C. R. Johnson, Matrix Analysis. New York, NY: Cabridge University Press, 2013. [13] US DOE & FERC. 2006, Feb Steps to establish a real-tie transission onitoring syste for transission owners and operators within the eastern and western interconnections. [Online]. Available: http://energy.gov/sites/prod/files/oeprod/docuentsandmedia/ final 1839.pdf [14] Y. C. Chen, A. D. Doínguez-García, and P. W. Sauer, A sparse representation approach to online estiation of power syste distribution factors, IEEE Transactions on Power Systes, vol. 30, no. 4, pp. 1727 1738, July 2015. [15] M. Okaura, Y. O-ura, S. Hayashi, K. Ueura, and F. Ishiguro, A new power flow odel and solution ethod - including load and generator characteristics and effects of syste control devices, IEEE Transactions on Power Apparatus and Systes, vol. 94, no. 3, pp. 1042 1050, May 1975. [16] M. S. Ćalović and V. C. Strezoski, Calculation of steady-state load flows incorporating syste control effects and consuer self-regulation characteristics, International Journal of Electrical Power & Energy Systes, vol. 3, no. 2, pp. 65 74, 1981. [17] P. W. Sauer and M. A. Pai, Power Syste Dynaics and Stability. Upper Saddle River, NJ: Prentice-Hall, Inc., 1998. [18] J. H. Chow and K. W. Cheung, A toolbox for power syste dynaics and control engineering education and research, IEEE Transactions on Power Systes, vol. 7, no. 4, pp. 1559 1564, Nov 1992. Yu Christine Chen S 10 M 15 received the B.A.Sc. degree in engineering science fro the University of Toronto, Toronto, ON, Canada, in 2009, and the M.S. and Ph.D. degrees in electrical engineering fro the University of Illinois at Urbana-Chapaign, Urbana, IL, USA, in 2011 and 2014, respectively. She is currently an Assistant Professor with the Departent of Electrical and Coputer Engineering, The University of British Colubia, Vancouver, BC, Canada, where she is affiliated with the Electric Power and Energy Systes Group. Her research interests include power syste analysis, onitoring, and control. Sairaj V. Dhople S 09 M 13 received the B.S., M.S., and Ph.D. degrees in electrical engineering, in 2007, 2009, and 2012, respectively, fro the University of Illinois, Urbana-Chapaign. He is currently an Assistant Professor in the Departent of Electrical and Coputer Engineering at the University of Minnesota Minneapolis, where he is affiliated with the Power and Energy Systes research group. His research interests include odeling, analysis, and control of power electronics and power systes with a focus on renewable integration. Dr. Dhople received the National Science Foundation CAREER Award in 2015. He currently serves as an Associate Editor for the IEEE Transactions on Energy Conversion. Alejandro D. Doínguez-García S 02 M 07 received the degree of Electrical Engineer fro the University of Oviedo, Spain, in 2001, and the Ph.D. degree in electrical engineering and coputer science fro the Massachusetts Institute of Technology, Cabridge, MA, USA, in 2007. He is currently an Associate Professor with the Electrical and Coputer Engineering Departent, University of Illinois at Urbana-Chapaign, Urbana, IL, USA, where he is affiliated with the Power and Energy Systes area; he also has been a Grainger Associate since August 2011. He is also an Associate Research Professor with the Coordinated Science Laboratory and in the Inforation Trust Institute, both at the University of Illinois at Urbana- Chapaign. His research interests are in the areas of syste reliability theory and control, and their applications to electric power systes, power electronics, and ebedded electronic systes for safety-critical/fault-tolerant aircraft, aerospace, and autootive applications. Dr. Doínguez-García received the National Science Foundation CAREER Award in 2010, and the Young Engineer Award fro the IEEE Power and Energy Society in 2012. In 2014, he was invited by the National Acadey of Engineering to attend the U.S. Frontiers of Engineering Syposiu, and selected by the University of Illinois at Urbana-Chapaign Provost to receive a Distinguished Prootion Award. He is an editor of the IEEE TRANS- ACTIONS ON POWER SYSTEMS and the IEEE POWER ENGINEERING LETTERS.

10 Peter W. Sauer S 73 M 77 SM 82 F 93 LF 12 received the B.S. degree fro the University of Missouri at Rolla, Rolla, MO, USA, in 1969, and the M.S. and Ph.D. degrees fro Purdue University, West Lafayette, IN, USA, in 1974 and 1977, respectively, all in electrical engineering. Fro 1969 to 1973, he was an Electrical Engineer on a design assistance tea for the Tactical Air Coand at Langley Air Force Base, VA, USA, working on design and construction of airfield lighting and electrical distribution systes. He has been on the faculty at the University of Illinois at Urbana-Chapaign, Urbana, IL, USA, since 1977, where he teaches courses and directs research on power systes and electric achines. Fro August 1991 to August 1992, he served as the Progra Director for Power Systes in the Electrical and Counication Systes Division, National Science Foundation, Washington, DC, USA. He is a cofounder of the Power Systes Engineering Research Center PSERC and has served as the Illinois site director fro 1996 to the present. He is a cofounder of PowerWorld Corporation and served as Chairan of the Board of Directors fro 1996 to 2001. He is currently the Grainger Chair Professor of Electrical Engineering at Illinois. Dr. Sauer is a registered Professional Engineer in Virginia and Illinois and a eber of the U.S. National Acadey of Engineering.