Robust Power Flow and Three-Phase Power Flow Analyses

Transcription

1 Robust Power Flow and Three-Phase Power Flow Analyses 1 Amritanshu Pandey 1, Graduate Student Member, IEEE, Marko Jereminov 1, Graduate Student Member, IEEE, Martin R. Wagner 1, Graduate Student Member, IEEE, David M. Bromberg 1, Member, IEEE, Gabriela Hug 2, Senior Member, IEEE and Larry Peggi 1, Fellow, IEEE Abstract Robust simulation is essential for reliable oeration and lanning of transmission and distribution ower grids. At resent, disarate methods exist for steady-state analysis of the transmission (ower flow) and distribution ower grid (threehase ower flow). Due to the non-linear nature of the roblem, it is difficult for alternating current (AC) ower flow and threehase ower flow analyses to ensure convergence to the correct hysical solution, articularly from arbitrary initial conditions, or when evaluating a change (e.g. contingency) in the grid. In this aer, we describe our equivalent circuit formulation aroach with current and voltage variables that models both the ositive sequence network of the transmission grid and three-hase network of the distribution grid without loss of generality. The roosed circuit models and formalism enable the extension and alication of circuit simulation techniques to solve for the steadystate solution with excellent robustness of convergence. Examles for ositive sequence transmission and three-hase distribution systems, including actual 75k+ nodes Eastern Interconnection transmission test cases and 8k+ nodes taxonomy distribution test cases, are solved from arbitrary initial guesses to demonstrate the efficacy of our aroach. Index Terms circuit simulation methods, continuation methods, convergence roblems, equivalent circuit aroach, ower flow, robust convergence, steady-state analysis, three-hase ower flow, Tx steing method A I. INTRODUCTION n interconnected electric grid is a network of synchronized ower roviders and consumers that are connected via transmission and distribution lines and oerated by one of multile entities. Reliable and secure oeration of this electric grid is of utmost imortance for maintaining a country s economy and well-being of its citizens. To oerate the grid reliably and securely under all conditions, as well as adequately lan for the future, it is essential that one can robustly analyze the grid off-line and in real-time. At resent, numerous analysis methods exist for oeration and lanning of the grid. These can be broadly categorized into one of the following: i) steady-state analysis in the frequency domain (ower flow, three hase ower flow, and harmonic analyses), ii) transient and steady-state analysis in time domain, iii) analysis for otimal disatch of resources, and iv) other market disatch-based analyses. Among these analyses, fundamental frequency based steady-state analysis (ower flow and three-hase ower flow) is essential for the day-to-day oeration as well as future lanning of the grid. Furthermore, the solution to the steady-state analysis serves as the initial state for transient analysis as well as the otimal ower flow roblem. Due to its critical imortance, research has roduced significant advances toward imroving the convergence of these solution methods [5]-[12]. At resent, steady-state simulation is divided into two domains, high-voltage transmission systems and sub-station level voltage distribution systems. Disarate methods exist for analyzing these two (transmission and distribution) systems. The steady-state solution for the high voltage transmission system is obtained via ositive sequence AC ower flow analysis (often referred to as ower flow analysis), whereas the steady-state oerating oint for the distribution system is obtained via three-hase AC ower flow analysis. The industry standard for solving the ositive sequence AC ower flow roblem is the PQV formulation [1], wherein nonlinear ower mismatch equations are solved for bus voltage magnitudes and angles that further define the steady-state oerating oint of the system. In contrast, the backward-forward swee method [2] and the current injection method (CIM) [3] are rimary used for obtaining the steady-state solution of the three-hase ower flow roblem for the distribution grid. In their existing forms, the solution methods for ower flow and three-hase ower can suffer from lack of convergence robustness [5], [10]. The PQV based formulation for the ositive sequence ower flow roblem is known to diverge or converge to non-hysical solutions for l-conditioned [2] and large scale (>50k buses) systems [20], where a non-hysical solution corresonds to a system that contains low voltages or demonstrates angular instabity. For distribution system roblems, the backward-forward swee method that was roosed to solve radial and weakly meshed distribution systems with high R/X ratio [2] has difficulties converging for heavy meshed systems with more than a single source of generation [12]. The CIM method based on Dommel s work in 1970 [4], like the equivalent circuit aroach roosed in this aer, reresents the currents and voltages in terms of rectangular coordinates, but is challenged by incororation of multile PV buses in the system [13]-[14]. In general, of the known challenges associated with convergence for existing ower flow and three-hase ower flow solution methods, the This work was suorted in art by the Defense Advanced Research Projects Agency (DARPA) under award no. FA for the RADICS rogram. 1 Authors are with the Electrical and Comuter Engineering Deartment, Carnegie Mellon University, Pittsburgh, PA USA, (e-ma: {amritan, mjeremin, mwagner1, dbromber, eggi}@andrew.cmu.edu). 2 Author is with the Power System Laboratory, ETH Zurich, (e-ma: hug@eeh.ee.ethz.ch). Digital Object Identifier /TPWRS IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information.

2 2 two that are most detrimental are convergence to low-voltage or unaccetable solutions [15] and divergence [5]. The objective and contribution of the aroaches described in this aer is to rovide robust ower flow and three-hase ower flow convergence. Secifically, a generalized aroach for ower flow and three-hase ower flow analyses that can ensure convergence to correct hysical solution indeendent of the choice of initial conditions. The factors that are the most fundamental toward making these roblems challenging are the use of non-hysical reresentations for modeling the ower grid comonents, and in the case of the PQV formulation, the use of inherently nonlinear ower mismatch equations to formulate the roblem. The non-hysical reresentations of the system equiment may not cature the true behavior of the model in the entire range of system oeration. For examle, an aroximated macro-model of a generator that is reresented via ositive sequence or threehase PV node can result in convergence to a low-voltage solution or divergence due to its quadratic voltage characteristics. Simarly, the inherent non-linearities in the PQV formulation almost always cause divergence for large (>50k) and l-conditioned test cases [20] when solved from an arbitrary set of initial conditions. This lack of a hysics-based formulation, along with the methods that can constrain the nonhysics based models in their hysical sace, is what renders the existing ower flow and three-hase ower flow roblem and solution aroaches to be non-robust. To develo a robust solver for the steady-state solution of the ower grid, it is imerative that the solver can efficiently and effectively navigate through the aforementioned challenges whe converging to a solution that is both meaningful and correct. Intuitively and hysically, both the transmission and distribution electric grids corresond to an electric circuit. Our aroach toward solving the ower flow and three-hase ower flow roblems is to utize circuit modeling and formalism to develo new algorithms that wl robustly solve them. Toward this goal, we roose a two-ronged aroach. First, the use of an equivalent circuit formulation in terms of the true state variables of currents and voltages [16]-[18] to model both the transmission and distribution ower grid (Sect. III.). Secondly, the adatation and alication of circuit simulation methods [19]-[22] to ensure robust convergence to correct hysical solutions (Sect. IV.) for ower flow and three-hase ower flow roblems. To demonstrate the interaction between the two, Sect. V of this aer discusses the general algorithm for solving the ower flow and three-hase ower flow roblems. Several examles are shown which demonstrate the efficacy of our aroach. II. BACKGROUND A ower grid in its simlest form can be reresented by a set of N buses, where the sets of generators G and load demands L are subsets of N, which are further connected by a set of line elements, T X and set of transformers xfmrs. Furthermore, there is a set of slack buses (one for each island in the system) reresented by ξ. In addition to these, the ower grid may contain other elements, such as shunts, flexible alternating current transmission system (FACTS), etc. The objective of steady-state analysis of the ower grid is to model the fundamental frequency comonent of the ower grid and solve for the comlex voltages at its buses. The high voltage transmission network of the grid generally oerates under balanced conditions, and therefore, the steady-state solution of the transmission network is obtained via ositive sequence ower flow model and analysis. In contrast, the distribution network of the ower grid can oerate under unbalanced conditions, and therefore we must aly three-hase ower flow model and analysis to find the steady-state solution of the distribution grid. In the following sub-sections, we discuss the current formulations used for steady-state analysis of transmission and distribution networks and highlight their limitations. A. PQV based Formulation for Positive Sequence Power Flow Problem The PQV based ower flow formulation is the industry standard for solving for the steady-state solution of the high voltage transmission network. In this formulation, a set of 2(N ξ) G ower mismatch equations are solved for unknown comlex voltage magnitudes and angles of the system using the Newton Rahson (NR) method. The set of ower mismatch equations are defined as follows: P G i P L i = V i V l (G Y cos + B Y sin ) IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information. N l=1 N Q G i Q L i = V i V l (G Y sin B Y cos ) l=1 where, P G i + jq G i and P L i + jq L i are the comlex generation and comlex load at the node i and G Y + jb Y is the comlex admittance between the nodes i and l. In order to solve for unknown comlex voltages V i i in the system, the real and reactive ower mismatch equations given by (1)-(2) are solved for the set of (N ξ G ) buses in the system, whereas only real mismatch equations (1) are solved for the set of buses with generators G connected to it. Imortantly, this PQV formulation is inherently non-linear, since the set of network constraints result in non-linear ower mismatch equations indeendent of hysics of the models used. For examle, in the PQV formulation, a linear network consisting of linear models for the slack bus, the transmission lines and the loads would corresond to a non-linear set of ower mismatch equations, a feature that could result in convergence difficulties for systems even trivial in size. B. Current Injection Method for Three-Phase Power Flow Problem Unt recently, the backward forward swee method was the most commonly used method for the steady-state analysis of the radial and weakly meshed distribution systems [2]. The method was referred over the PQV formulation due to the radial nature of the distribution grid and high R/X ratios of the distribution lines, both of which are known to cause convergence difficulties for the NR method [2] with PQV formulation. However, the backward forward swee method itself is rone to convergence difficulties for systems that are highly meshed or have multile sources [12]. The current injection method (CIM) for the three-hase ower flow roblem [3] was roosed to address challenges (1) (2)

3 associated with the PQV formulation and the backwardforward swee method. In the CIM formulation, the non-linear current mismatch equations for the system buses are solved via NR method for each individual hase with comlex rectangular real and imaginary voltages (V Ri + jv Ii ) as the unknown variables. The current mismatch equations for the three-hase ower flow roblem are defined as follows [3]: ΔI Ri = (Pis ) V Ri + (Q s i ) V Ii (V Ri ) 2 + (V Ii ) 2 (3) N (G t V t Ri B t V t Ii ) l=1 tε set ΔI Ii = (P s i ) V Ii (Q s i ) V Ri (V Ri ) 2 + (V Ii ) 2 N (G t V t Ii B t V t Ri ) l=1 tε set where ΔI Ri + jδi Ii is the net current mismatch in hase at node i and (P s i ) + j(q s i ) is the secified comlex ower injection at node i. The set set includes hases a, b and c. Although, the CIM method is known to imrove the convergence roerties for heavy and weakly meshed threehase radial distribution systems with high R/X ratio, the method is known to diverge for test-cases with high enetration of PV buses [13]. Traditionally, the number of PV buses in the distribution system were limited to a small number; however, with the advent of large scale installation of distributed generation (DGs) and voltage control devices in the distribution system this is no longer true. Therefore, it is essential that a robust three-hase ower flow formulation can robustly handle high enetration of PV buses and other voltage control devices in the system. III. EQUIVALENT CIRCUIT FORMULATION We extend the equivalent circuit aroach in [16]-[20] for steady-state analysis of the transmission and distribution ower grid to tackle the challenges exhibited by the existing formulations. This aroach for generalized modeling of the ower system in steady-state (i.e. ower flow and three-hase ower flow) reresents both the transmission and distribution ower grid elements in terms of equivalent circuit elements without loss of generality. It was shown that each of the ower system comonents can be directly maed to an equivalent circuit model based on the underlying relationshi between current and voltage state variables. Imortantly, this formulation can reresent any hysics based model or measurement based semi-emirical models as a sub-circuit, as shown in [24], [25] and [26], and these models can be combined hierarchically with other circuit abstractions to bud larger aggregated models. In the following section, we discuss generic equivalent circuit reresentations of ower system comonents for both the ositive sequence ower flow roblem and the three-hase ower flow roblem. Note that throughout the aer, the symbol suerscrit in the mathematical exressions of the equivalent circuit models reresents a hase from the set set of three hases a, b and c for the three-hase (4) roblem and reresents the ositive sequence () comonent for the ower flow roblem. A. PV Bus or the Generator Model In the equivalent circuit aroach, the generator (PV) bus model is modeled via a comlex current source [19] and has the same behavior as of the PV node in ower flow and three-hase ower flow roblems. To enable the alication of NR, this comlex current source is slit into real and imaginary current sources ( and I IG, resectively). This is necessary due to the non-analyticity of comlex conjugate functions [16]. The resulting equations for the PV model in the ower flow and three-hase ower roblem are: I RG = P G V RG + Q G V IG (5) ) 2 + (V IG ) 2 (V RG = P G V IG Q G V RG (6) ) 2 + (V IG ) 2 (V RG Additional constraints that allow the generators to control the voltage magnitude either at its own node or any other remote node in the system are modeled by a control circuits, as shown in the following subsection. In the case of ower flow roblem, a single control circuit is needed whereas for the three-hase ower flow roblem, three such control circuits are needed for each PV bus in the system. The reactive ower Q G of the generator acts as the additional unknown variable for the additional constraint that is introduced due to voltage control. In case of three-hase ower flow, three such additional variables and constraints are introduced. As an examle, the equivalent circuit for the ositive sequence model for a PV bus used in ower flow is shown in Fig. 1 for the k + 1 th iteration of NR. It is constructed by linearizing the set of equations (5)-(6) for the ositive sequence arameters and then reresenting the resulting equations using fundamental circuit elements (detaed rocedure for this rovided in [16]). To construct the PV bus equivalent circuit for three-hase ower flow roblem, three such circuits are first constructed and then are connected in grounded-wye configuration. + 1 V RG _ 1 R = V RG Real Circuit V IG V V I RG RG RG V V IG IG Figure 1: Equivalent Circuit Model for PV generator model. B. Voltage Regulation of a Bus Numerous ower grid elements such as generators, FACTS devices, transformers, shunts etc., can control a voltage magnitude at a given node in the system. Moreover, they can control the voltage magnitude (V set ) at either their own node O or a remote node W in the system. In equivalent circuit formulation, we reresent the control of the voltage magnitude by a control circuit (Fig. 2), which is governed by the following exression: Q G Q Q G G F W (V set ) 2 (V RW + 1 V IG Imag. Circuit V RG 1 V IG R _ I I = V V RG RG ) 2 (V IW ) 2 = 0 (7) 3 V V IG IG Q G Q Q G G IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information.

4 + _ IEEE TRANSACTIONS ON POWER SYSTEMS 4 The circuit in Fig. 2 is derived from the linearized version of (7). For the ower flow roblem, it is stamed (i.e. values are added to the Jacobian in a modular way) for each node W in the system whose voltage is being controlled such that there exists at least one single ath between the node W and the equiment s node O that is controlling it. Simarly, for threehase ower flow three of these circuits are stamed for each node W. The additional unknown variables for these additional constraints are deendent on the ower system device that is controlling the voltage magnitude. For examle, the additional unknown variable for a generator is its reactive ower Q, whereas in the case of transformers, it is the transformer ta tr, and for FACTS devices it is the firing angle φ. The revious section already described how the additional unknown variable Q for PV buses is integrated into the resective equivalent circuits for generators. Figure 2: Voltage magnitude constraint control equivalent circuit. C. ZIP Load Model In this section, we derive the ositive sequence and threehase model for the ZIP load. The ZIP load models the aggregated load in the system as a mix of constant imedance (Z + jz Q ), constant current (I + ji Q ), and constant ower (S + js Q ) load behavior, which can be mathematically reresented as follows: (P i ZI ) = Z ( V i ) 2 + I ( V i ) + S (8) (Q i ZI ) = Z Q ( V i ) 2 + I Q ( V i ) + S Q (9) In the equivalent circuit aroach, the equations for the ZIP load model can be re-written as: where: V hist set ZI (I Ri ) = Y V Ri Y Q V Ii + S V Ri + S Q V Ii (V Ri ) 2 + (V Ii ) 2 (10) + ( I 2 + I 2 Q ). cos ( i I f ) ZI (I Ii ) = Y V Ii + Y Q V Ri + S V Ii S Q V Ri (V Ri ) 2 + (V Ii ) 2 (11) + ( I 2 + I 2 Q ). sin ( i I f ) I f = tan 1 ( I Q I ) (12) i = tan 1 ( V Ii V Ri (13) 1 Y P + jy Q = Z P + jz Q (14) For the load model given in (10) through (14), the constant imedance art of the load is linear, whereas the constant current and constant ower art of the aggregated load is nonlinear. Once, (10)-(11) are linearized, they are used to construct the equivalent circuit models for both the ower flow and three-hase ower flow roblem. The constructed threehase model of the ZIP load model can either be connected in wye or delta formation. As an examle, ZIP load model connected in wye and delta formation is shown in Fig. 3. Figure 3: Real circuit for a) wye connected ZIP Load Model (on left) b) delta (D) connected ZIP load model (on right). It is imortant to note that both the ZIP and PQ load models result in non-linear network constraints for both the PQV and CIM formulations. In the PQV formulation the non-linearities in the network constraints are due to the use of ower mismatch equations whereas in the CIM, the non-linearities are due to PQ and ZIP model equations. These added non-linearities are one of the rimary causes of divergence and convergence to low voltage solutions. To address this roblem, we have roosed an accurate and yet linear BIG load model [25]-[27]. D. BIG Linear Load Model The BIG aggregated load model was roosed based on the circuit theoretic aroach in [25]-[27] and aims to create a linear load model that can cature the true measure and sensitivity of the aggregated load in the system. The model is comrised of a suscetance (B), indeendent current source (I), and conductance (G). The comlex governing equation of the generalized load current for the BIG load model is given by: (I R BIG ) + j(i I BIG ) = ( R BIG ) + j( I BIG ) (15) + ((V R BIG ) + j(v I BIG ) ) ((G BIG ) + j(b BIG ) ) where ( BIG R ) + j( BIG I ) reresents the base value for the modeled aggregated load and the corresonding comlex admittance ((G BIG ) + j(b BIG ) ) catures the voltage sensitivities. For instance, a negative conductance in conjunction with comlex current (( BIG R ) + j( BIG I ) ) mimics the inverse current/voltage sensitivity relationshi, simar to constant ower (PQ) load behavior and ositive conductance in conjunction with comlex current source wl reresent the ositively correlated current/voltage sensitivity relationshi, simar to the imedance load behavior. Both the ositive and negative imedances cature the change in load with voltage with resect to the ortion of the load that is modeled by the current source. Fig. 4 shows the ositive sequence () BIG load model. Simar to the ZIP load model, the three-hase BIG load model can be constructed by connecting the equivalent circuits of individual hases in wye or delta formation. BIG I R Figure 4: Equivalent circuit of a BIG load model IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information. + BIG V R Real Circuit BIG G B + BIG BIG R BIG I I + BIG V I _ Imag. Circuit BIG G B BIG BIG I

5 5 IV. CIRCUIT SIMULATION METHODS Decades of research in circuit simulation have demonstrated that circuit simulation methods can be alied for determining the DC state of highly non-linear circuits using NR. These techniques have been shown to make NR robust and ractical for large-scale circuit roblems [21]-[22] consisting of blions of nodes. Most notable is the abity to guarantee convergence to the correct hysical solution (i.e. global convergence) and the caabity of finding multile oerating oints [28]. We have reviously roosed analogous techniques for ensuring convergence to the correct hysical solution for the ositive sequence ower flow roblem [19]-[20]. In this section, we extend these methods to be used with ositive-sequence ower flow and three-hase ower flow roblems alike. A. General Methods 1) Variable Limiting The solution sace of the system node voltages in a ower flow and three-hase ower flow roblem are well defined. Whe solving these roblems, a large NR ste may ste out of this solution sace and result in either divergence or convergence to a non-hysical solution. It is, therefore, imortant to limit the NR ste before an invalid ste out of the solution sace is made. In [19], we roosed the variable limiting method to achieve the ostulated goal for ower flow roblem. In this technique, the state variables that are most sensitive to initial guesses are damed when the NR algorithm takes a large ste out of the re-defined solution sace. Note, however, that not all the system variables are damed for the variable limiting technique, as is done for traditional damed NR. Circuit simulation research has shown that daming the most sensitive variables rovides suerior convergence comared to damed NR in general [21]. To aly variable limiting in our rototye simulator for the ower flow and three-hase ower flow roblem, the mathematical exressions for the PV nodes in the system are modified as follows: 1 = ς I CG (V RG k 1 V RG I CG V RG + ς I CG V (V IG 1 V IG ) IG V IG V RG k ) + I k CG + I CG Q (Q G 1 Q G ) G (16) where, 0 ς 1. The magnitude of ς is dynamically varied through heuristics such that convergence to the correct hysical solution is achieved in the most efficient manner. The heuristics deend on the largest delta voltage ( V RG, V IG ) ste during subsequent NR iterations. If during subsequent NR iterations, a large ste ( V RG, V IG ) is encountered, then the factor ς is decreased. The factor ς is scaled back u if consecutive NR stes result in monotonically decreasing absolute values for the largest error. 2) Voltage Limiting An equally simle, yet effective, technique is to limit the absolute value of the delta ste that the real and imaginary voltage vectors can make during each NR iteration. This is analogous to the voltage limiting technique used for diodes in circuit simulation, wherein the maximum allowable voltage ste during NR is limited to twice the thermal voltage of the diode [22]. Simarly, for the ower flow and three-hase ower flow analyses, a hard limit is enforced on the normalized real and imaginary voltages in the system. The mathematical imlementation of voltage limiting in our formulation is as follows: (V C ) k 1 = min Vmin C Vmax C ) k + δ S min( (V C ) k, V max C )) C (17) V c, if x > Vmax c min min V C Vmax V min min c, if x < V c C x, otherwise (18) and δ S = sign ( (V C ) k ) and C {R, I} reresents the laceholder for real and imaginary arts. Analogously, other system variables such as the reactive ower Q G of the PV buses, can be limited by limiting the calculated currents I C + (I C ) k at NR ste k + 1 and then 1 finding the new Q G from inverse function (f 1 ) of limited (I C + (I ). C ) k B. Homotoy Methods Limiting methods may fa to ensure convergence for certain l-conditioned and large test systems when solved from an arbitrary set of initial guesses. To ensure convergence for these network models to the correct hysical solutions indeendent of the choice of initial conditions, we roose the use of homotoy methods. Homotoy methods in ast have been used to study the voltage collase of a given network or to determine maximum avaable transfer caabity [8]-[9]. They have also been researched for locating all solutions to a ower flow roblem [11], [30]. However, their usage for enabling convergence for hard to solve ositive sequence and threehase ower flow roblems has been limited at best. Of the roosed methods for better convergence [5], [23], most have suffered from convergence to low voltage solutions or divergence. On the other hand, some of them have been develoed for formulations that are not standard for both ositive sequence as well as three-hase ower flow [6]. Furthermore, none of the reviously roosed homotoy methods are known to scale u to test systems that are of the scale of the Euroean or the US grids and in general are not extendable to the three-hase ower flow roblem. In the homotoy aroach, the original roblem is relaced with a set of sub-roblems that are sequentially solved. The set of sub-roblems exhibit certain roerties, namely, the first sub-roblem has a trivial solution and each incrementally subsequent roblem has a solution very close to the solution of the rior sub-roblem. Mathematically this can be described via the following exression: where λ [0, 1]. H(x, λ) = (1 λ)f(x) + λg(x) (19) The method begins by relacing the original roblem F(x) = 0 with H(x, λ) = 0. The equation set G(x) is a reresentation of the system that has a trivial solution. The homotoy factor λ has the value of 1 for the first sub-roblem and therefore the initial solution is equal to trivial solution of G(x). For the final sub-roblem that corresonds to the original roblem, the homotoy factor λ has the value of zero. To generate sequential sub-roblems, the homotoy factor is dynamically decreased in small stes unt it has reached the value of zero IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. 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6 6 In this aer, we discuss two homotoy methods that are secifically develoed for the ower flow and three-hase ower flow analyses: 1) Tx Steing We roosed the Tx Steing method in [20] secifically for the ower flow roblem. In this section, the method is further extended for the three-hase ower flow roblem. a) General Aroach In Tx steing method, the series elements in the system (transmission lines, transformers etc.) are first virtually shorted to solve the initial roblem that has a trivial solution. Secifically, a large conductance ( G ) and a large suscetance ( B ) are added in arallel to each transmission line and transformer model in the system. In case of three-hase ower flow, a large self-imedance ( Y ) is added in arallel to each hase of the transmission line and transformer model. Furthermore, the shunts in the system, are oen-circuited by modifying the original shunt conductance and suscetance values. Imortantly, the solution to this initial roblem results in high system voltages (magnitudes), as they are essentially driven by the slack bus comlex voltages and the PV bus voltage magnitudes due to the low voltage dros in the lines and transformers (as exected with virtually shorted systems). Simarly, the solution for the bus voltage angles wl lie within an -small radius around the slack bus angle. Subsequently, like other continuation methods, the formulated system roblem is then gradually relaxed to reresent the original system by taking small increment stes of the homotoy factor (λ) unt convergence to the solution of the original roblem is achieved. Mathematically, the line and transformer imedances during homotoy for the ower flow is exressed by: {T X, xfmrs} G + jb = (G + jb )(1 + λγ) (20) and for the three-hase roblem: Y aa Y ba Y ca [ Y ab Y bb Y cb Y ac Y bc Y cc Y aa ] = [ (1 + γλ) Y ba Y ca Y bb Y ab (1 + γλ) Y cb Y cc Y ac Y bc ] (21) (1 + γλ) where, G, B, and Y are the original system imedances and G, B, and Y are the system imedances used whe iterating from the trivial roblem to the original roblem. The arameter γ is used as a scaling factor for the conductances and suscetances. If the homotoy factor (λ) takes the value one, the system has a trivial solution and if its takes the value zero, the original system is reresented. Along with ensuring convergence for a roblem, Tx steing avoids the undesirable low voltage solutions for the ositive sequence ower flow and three-hase ower flow roblem since the initial roblem results in a solution with high system voltages, and each subsequent ste of the homotoy aroach continues and deviates ever so slightly from this initial solution, thereby guaranteeing convergence to the high voltage solution for the original roblem. b) Handling of Transformer Phase Shifters and Tas To virtually short a ower system, we must also account for transformer tas tr and hase shifting angles Θ. In a virtually shorted condition, all the nodes in the system must have comlex voltages that are near the slack bus or PV bus comlex voltages, which can be intuitively defined by a small eson norm ball around these voltages. Therefore, to achieve the following form, we must modify the transformer tas and hase shifter angles such that at λ = 1, their turns ratios and hase shift angles corresond to a magnitude of 1 u and 0, resectively. Subsequently, the homotoy factor λ is varied such that the original roblem is solved with original transformer ta and hase shifter settings. This can be mathematically exressed as follows: i xfmrs tr i = tr i + λ(1 tr i ) (22) i xfmrs Θ i = Θ i λθ i (23) c) Handling of Voltage Control for Remote Buses To achieve a trivial solution during the first ste of Tx steing it is essential that we also handle remote voltage control aroriately. Remote voltage control refers to a device on node O in the system controlling the voltage of another node W in the system. This behavior is highly non-linear and if not handled correctly can result in divergence or convergence to low voltage solution. Existing commercial tools for ower flow and three-hase ower flow analyses have difficulties dealing with this roblem and suffer from lack of robust convergence when modeling remote voltage control in general. With Tx steing we can handle this roblem efficiently and effectively. We first incororate a virtually short ath between the controlling node (O) and the controlled node (W) at λ = 1, such that the device at the controlling node can easy suly the current needed for node W to control its voltage. Then following the homotoy rogression, we gradually relax the system such that additional line connecting the controlling node (O) and controlled node (W) is oen at λ = 0. d) Imlementation of Tx Steing in Equivalent Circuit Formulation Unlike traditional imlementations of homotoy methods, in equivalent circuit formulation we do not directly modify the non-linear set of mathematical equations, but instead embed a homotoy factor in each of the equivalent circuit models for the ower grid comonents. In doing so we allow for incororation of any ower system equiment into the Tx steing aroach within the equivalent circuit formulation framework, without loss of generality. Furthermore, we ensure, that the hysics of the system is reserved whe modifying it for the homotoy method. Fig. 5 demonstrates how the homotoy factor is embedded into the equivalent circuit of the transformer. Figure 5: Homotoy factor embedded in transformer equivalent circuit. 2) Dynamic Power Steing Another homotoy technique that can ensure robust convergence for systems that have a low ercentage of constant voltage nodes in the system is the dynamic ower steing method. Existing distribution systems tend to belong to these tyes of systems and therefore, dynamic ower steing can be alied to robustly obtain the steady-solution of the distribution grid by solving the three-hase ower flow roblem. This method has been reviously described for the ositive-sequence IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information.

7 7 ower flow roblem in [19] and is analogous to the source steing and gmin steing aroaches in standard circuit simulation solvers. In the ower steing method, the system loads and generation are scaled back by a factor of β unt the convergence is achieved. If these loads and generations are scaled down all the way to zero, then the constraints for the PQ buses in the system result in linear network constraints. Simarly, current source non-linearities of the PV buses that are due to the constant real ower are also eliminated. Therefore, by alying the ower steing factor, the nonlinearities in the system are greatly eased and convergence is easy achieved. Uon convergence, the factor is gradually scaled back u to unity to solve the original roblem. In this method, as in all continuation methods, the solution from the rior ste is used as the initial condition for the next ste. The mathematical reresentation of dynamic ower steing for the three-hase ower flow and ositive sequence ower flow roblem is as follows: G PV: P G = βp G (24) L PQ: P L = βp L and Q L = βq L (25) where, PQ are all load nodes and PV are all generator nodes. V. POWER FLOW AND THREE-PHASE POWER FLOW ALGORITHM No change in λ Re-stam homotoy Y H Change in λ NO Bud os-seq models Udate: λ, ς and δ Inut Test Case Stam Linear Y L Initialize X int, λ, ς & δ Re-stam Y NL Solve for X int 1 ( R) Aly Limiting NO Inut data 3-Phase? Find X int 1 Inner Loo Comlete? Is shunt and xfmr control ossible? Algorithm 1: Simulation algorithm for Positive Sequence and Three-Phase Power Flow Solver Algorithm 1 shows the recie for the solving the ositivesequence as well as three-hase ower flow roblem in equivalent circuit aroach with the use of circuit simulation methods. In this framework, the solver starts with buding the system models based on the inut fe sulied. Linear models (Y L, J L ) are then stamed in the Jacobian matrix. Inut state variables and other continuation arameters (x 0, δ, ζ, λ) are then initialized. Non-linear models are then stamed (Y NL, J NL ) and No Sto YES YES Bud three-hase models New Adjustment YES NR is alied with limiting methods to calculate the next iterate for voltages and generator reactive owers (X 1 ). Continuation and limiting arameters are then dynamically udated and homotoy models (Y H, J H ) are stamed or restamed if required to ensure convergence. Uon convergence of inner loo generator limits, switched shunts and transformer tas are adjusted and inner loo is reeated unt final solution is achieved. VI. RESULTS Examle cases were simulated in our rototye solver SUGAR (Simulation with Unified Grid Analyses and Renewables) to demonstrate that the equivalent circuit aroach along with circuit simulation techniques facitates a robust framework for ositive sequence ower flow and the three-hase ower flow analyses. The first set of results comare the solution of contingency analyses for two hard to solve cases with and without the use of circuit simulation methods to demonstrate the efficacy for these methods. All the further results comare the results of SUGAR (with circuit simulation methods) with other industry tools. The examle cases for ositive sequence ower flow analyses include known l-conditioned test cases and large network models that reresent different oerating and loading conditions for the eastern interconnection network of the US grid. For the threehase ower flow analysis, examle cases include a set of standard distribution taxonomy cases [29], high density urban test cases [31], and a meshed transmission grid test case that was modified from a ositive sequence to a three-hase network model. The results that follow demonstrate that the roosed framework along with the use of circuit simulation methods can ensure convergence to a correct hysical solution for all the ower flow and three-hase ower flow cases, indeendent of the choice of the initial guess and thus overcomes the challenges faced by existing formulations. A. Circuit Simulation Methods The urose of following set of results is to demonstrate the robustness of the solver that is enabled due to the use of circuit simulation methods. To show this, contingencies were simulated on two (2) hard to solve test-cases that reresent a real network for the subset of the US ower grid. The base case for both simulations is first solved via Tx-steing method and then used as an initial condition for the set of contingencies. The contingencies in the contingency set reresent the loss of largest 10% of online generators and highest caacity lines and transformers droed one at a time. TABLE 1: COMPARISON OF SUGAR WITH AND WITHOUT CIRCUIT SIMULATION TECHNIQUES SUGAR w/o Circuit SUGAR with Circuit IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information. Case Id # Bus # Total Cases Simulation Methods Diverge Converge /Infeasible Simulation Methods Diverge Converge /Infeasible Case Case The results in the Table 1 confirm that the circuit simulation methods when alied to equivalent circuit formulation can significantly increase the robustness of the ower flow solver.

8 B. Positive Sequence Power Flow Results 1) Ill-Conditioned and Large Test cases A convergence swee was run on the l-conditioned bus PEGASE test case using the SUGAR solver and a standard commercial tool and their results were comared. Fig. 6 shows that SUGAR was able to robustly converge to the correct hysical solution indeendent of the choice of the initial conditions, whereas the standard tool was highly sensitive to the choice of the initial guess and could converge to the correct hysical solution only from a few samles for the initial guess. Figure 6: Convergence swee comarison for node PEGASE testcase between SUGAR and Standard tool. Red indicates divergence and green indicates convergence A simar convergence swee was erformed for larger test cases (> 75k+ nodes) that reresent different loading and oerating scenarios for eastern interconnection of the US grid. Simulations were erformed on three different test cases for 15 different initial conditions each. Results are shown in Table 2. The set of initial conditions for all buses were identical and were uniformly samled from: Case Name V ang [ 40, 40], V mag [0.9, 1.1]. (26) TABLE 2: CONVERGENCE PERFORMANCE FOR LARGE EASTERN INTERCONNECTION TEST CASES Standard Tool SUGAR # Nodes # # # Converge Diverge Converge Case Case Case # Diverge For the larger eastern interconnection test cases, the runtime er iteration is less than 0.4 seconds and is comarable to other simulation tools out in the market. The total comutation time in general is deendent on the choice of initial conditions. A sufficiently close initial condition may result in convergence within 7 iterations whereas a totally random set of initial guesses may take u to 100 iterations with Tx steing method. 2) Contingency Analysis In the next set of results, we erformed a set of contingency analyses with SUGAR and a standard commercial tool for two test cases that reresent different network configuration of the eastern interconnection of the US grid. The initial guess for solving the contingency cases was chosen to be the oerating oint rior to the contingency. The set of contingencies in the exeriment includes loss of generation (L G ) and loss of branches (L B ). The results are summarized in Table 3 and highlight the need for continuation methods to solve such roblems robustly. TABLE 3: COMPARISON OF CONTINGENCIES OF LARGE TEST CASES Case # Nodes Contingency* Standard Tool SUGAR Case L G 2L G + 2L B Diverge Diverge Converge Converge Case L G Diverge Converge 2L G + 2L B Diverge Converge *The number in front of L G and L B reresents the equiment outage count. (For e.g. 2L G reresents that two generators were lost during this contingency). C. Three-Phase Power Flow Results 1) Taxonomical Test Cases Table 4 documents the results obtained from SUGAR threehase solver for standard taxonomical cases and three large meshed test cases. The standard taxonomical cases include both balanced and unbalanced three-hase test cases. The first two meshed test cases are 342-Node Low Voltage Network Test Systems [31] that reresent high density urban meshed low voltage networks. The third meshed test system is a high voltage 9241 node PEGASE transmission system that was extended to a balanced three-hase model. All these cases were simulated in SUGAR three-hase solver to validate its accuracy by comaring the obtained results against a standard distribution ower flow tool GridLAB-D. Slight differences (less than 1e -2 ) in the results were observed for cases between SUGAR and GridLAB-D and are due to default values used for unsecified arameters (e.g. neutral conductor resistance) in GridLAB-D. TABLE 4: SUGAR THREE-PHASE RESULTS FOR TAXONOMICAL CASES Cases #Nodes Deviation from GridLAB-D Iter. Max. ΔV Count mag Max. ΔV ang [u] [ ] R E E-03 R E E-02 R E E-02 R E E-03 R E E-03 NetworkModel E E-03 NetworkModel E E-03 case9241egase * NA # NA # * 9241 bus PEGASE transmission test case was extended to three-hase model #The following case did not run in GridLAB-D 2) Ill-Conditioned Test Cases To solve certain hard to solve l-conditioned three-hase test cases, we made use of homotoy methods. To demonstrate one such examle, we extended the standard 145 bus transmission system model into a balanced three-hase network model. Figure 7: Convergence of 145 bus test case for three-hase ower flow with (bottom) and without (to) ower steing. For the ower steing case, the green dotted line reresents the change in continuation factor λ Fig. 7 lots the convergence results for this test case with and without the use of dynamic ower steing. Without the use of IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information.

9 9 dynamic ower steing, the test system did not converge within maximum number of allowable iterations; however, with the use of dynamic ower steing, the system robustly converged to correct hysical solution. VII. CONCLUSIONS In this aer, we have demonstrated that the equivalent circuit aroach with the use of novel circuit simulation methods can robustly solve for the steady-state solution of the transmission and distribution grid without loss of generality. This roosed formulation and the analogous circuit simulation methods can be generically alied to both the ositive sequence ower flow roblem and the three-hase ower flow roblem. Imortantly, our aroach toward steady-state analyses of transmission and distribution grid ensures robust convergence to correct hysical solutions, and in doing so enables robust contingency analyses, statistical analyses, and security constrained otimal ower flow analyses. Furthermore, the roosed generic framework for transmission and distribution grid analyses can be extended for joint simulation of transmission and distribution circuits without loss of generality. VIII. REFERENCES [1] W. F. Tinney and C. E. Hart, Power flow solutions by Newton s method, IEEE Transactions on PAS, Vol. 86, No. 11, , Nov [2] D. Shiromohammadi, H. W. Hong, A. Semlyen, G. X. Luo, A comensation-based ower flow method for weakly meshed distribution and transmission networks, IEEE Transactions on Power Systems, Vol. 3, No. 2, May [3] P. A. N. Garcia, J. L. R. Pereria, S. Carneiro Jr., M. P. Vinagre, F. V. Gomes, Imrovements in the Reresentation of PV Buses on Three-Phase Distribution Power Flow, IEEE Transactions on Power Systems, Vol. 19, No. 2, Ar [4] H.W. Dommel, W.F. Tinney, and W.L. Powell, Further develoments in Newton's method for ower system alications, IEEE Winter Power Meeting, Conference Paer No. 70 CP 161-PWR New York, January [5] W. Murray, T. T. De Rubira, and A. Wigington, Imroving the robustness of Newton-based ower flow methods to coe with oor initial conditions, North American Power Symosium (NAPS), [6] H. D. Chiang and T. Wang, "Novel Homotoy Theory for Nonlinear Networks and Systems and Its Alications to Electrical Grids," in IEEE Transactions on Control of Network Systems. [7] Power Flow Convergence and Reactive Power Planning in the Creation of Large Synthetic Grids DOI /TPWRS , IEEE Transactions on Power Systems. [8] C. Liu, C. Chang, J. A. Jiang and G. H. Yeh, "Toward a CPFLOW-based algorithm to comute all the tye-1 load-flow solutions in electric ower systems," in IEEE Transactions on Circuits and Systems I: Regular Paers, vol. 52, no. 3, , March [9] V. Ajjarau, C. Christy, The Continuation Power Flow: The Tool for Steady-State Voltage Stabity Analysis, IEEE Transactions on Power Systems, Vol. 7, No. 1, February [10] A. G. Exosito, E. R. 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Peggi, Robust Convergence of Power Flow using Tx Steing Method with Equivalent Circuit Formulation, Power Systems Comutation Conference, [21] L. Plage (Peggi), R. Rohrer, C. Visweswariah, Electronic Circuit & System Simulation Methods, McGraw-Hl, Inc., New York, NY, USA, [22] L. Nagel, SPICE2: A Comuter Program to Simulate Semiconductor Circuits, EECS Deartment, University of California Berkeley, [23] S. Cvijic, P. Feldmann, M. Ilic, Alications of homotoy for solving AC ower flow and AC otimal ower flow, IEEE PES General Meeting, San Diego, July [24] A. Pandey, M. Jereminov, X. Li, G. Hug, L. Peggi, Unified Power System Analyses and Models using Equivalent Circuit Formulation, IEEE PES Innovative Smart Grid Technologies, Minneaolis, USA, [25] M. Jereminov, A. Pandey, H. A. Song, B. Hooi, C. Faloutsos, L. Peggi Linear load model for robust ower system analysis, IEEE PES Innovative Smart Grid Technologies, Torino Italy, Setember [26] A. Pandey, M. Jereminov, X. Li, G. Hug, L. Peggi, Aggregated Load and Generation Equivalent Circuit Models with Semi-Emirical Data Fitting, IEEE Green Energy and Systems Conference (IGESC 2016) November [27] B. Hooi, H. A. Song, A. Pandey, M. Jereminov, L. Peggi, and C. Faloutsos StreamCast: Fast and Online Mining of Power Grid Time Sequences, 2018 SIAM International Conference on Data Mining. [28] L. B. Goldgeisser and M. M. Green, Using Continuation Methods to Imrove Convergence of Circuits with High Imedance Nodes, in IEEE International Symosium on Circuits and Systems, 2000, Geneva Switzerland IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information.

10 [29] Modern Grid Initiative: Distribution Taxonomy Final Reort, Pacific Northwest National Laboratory, November 15 th, [30] K. Iba, H. Suzuli, M. Egawa, and T. Watanabe, Calculation of the critical loading with nose curve using homotoy continuation method, IEEE Trans. Power Syst., vol. 6, no. 2, , May [31] K. Schneider, P. Phanivong, J. Lacroix, IEEE 342-Node Low Voltage Network Test System, IEEE PES General Meeting Conference & Exosition, July IX. BIOGRAPHIES Amritanshu Pandey was born in Jabalur, India. He received his M. Sc. Degree in electrical engineering from Carnegie Mellon University in Pittsburgh, PA in He is resently ursuing Ph.D. degree at Carnegie Mellon University. Prior to joining as a doctoral student at Carnegie Mellon University, he worked as an electrical engineer at MPR Associates Inc. from 2012 to He has reviously interned at Pearl Street Technologies, ISO New-England and GE Global Research. His research interests include modeling and simulation, otimization and control of ower systems. Lawrence Peggi is the Tanoto rofessor of electrical and comuter engineering at Carnegie Mellon University, and has reviously held ositions at Westinghouse Research and Develoment and the University of Texas at Austin. He received his Ph.D. in Electrical and Comuter Engineering from Carnegie Mellon University in He has consulted for various semiconductor and EDA comanies, and was cofounder of Fabbrix Inc., Extreme DA, and Pearl Street Technologies. His research interests include various asects of digital and analog integrated circuit design and design methodologies, and simulation and modeling of electric ower systems. He has received various awards, including Westinghouse cororation s highest engineering achievement award, the 2010 IEEE Circuits and Systems Society Mac Van Valkenburg Award, and the 2015 Semiconductor Industry Association (SIA) University Researcher Award. He is a co-author of "Electronic Circuit and System Simulation Methods," McGraw-Hl, 1995 and "IC Interconnect Analysis," Kluwer, He has ublished over 300 conference and journal aers and holds 39 U.S. atents. He is a fellow of IEEE. 10 Marko Jereminov was born in Belgrade, Serbia. He received his B.Sc. Degree in Electrical Engineering from South Carolina State University, South Carolina, USA in 2016, and is currently ursuing Ph.D. degree in Electrical and Comuter Engineering at Carnegie Mellon University, Pittsburgh, PA. He reviously interned at Pearl Street Technologies, Pittsburgh, PA. His research interests include otimization, simulation and modeling of ower systems. Martin R. Wagner received his B.Sc. degree in Electrical Engineering (2011) and M.Sc. degree in Microelectronics (2014) from the Vienna University of Technology in Vienna, Austria. His master thesis was created in collaboration with the Austrian Institute of Technology in Vienna, Austria. He is currently ursuing a PhD degree in Electrical Engineering at Carnegie Mellon University in Pittsburgh, PA, USA. His research interests include robabistic methods alied to modeling and simulation of ower systems. David M. Bromberg was born in Brooklyn, New York. He received the BS, MS, and Ph.D. degrees, all from Carnegie Mellon University in Pittsburgh, PA, in 2010, 2012, and 2014, resectively. After his graduate studies he joined Aurora Solar in Palo Alto, CA as a senior scientist, where he develoed cloud simulation software for solar energy roduction modeling. In 2017 he co-founded Pearl Street Technologies, a ower systems software comany, and currently serves as its Chief Executive Officer. His research interests include robust, scalable algorithms for the simulation of electric ower systems. He is a member of IEEE. Gabriela Hug (S 05, M 08, SM 14) was born in Baden, Switzerland. She received the M.Sc. degree in electrical engineering and the Ph.D. degree from the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, in 2004 and 2008, resectively. After her Ph.D. degree, she was with the Secial Studies Grou of Hydro One in Toronto, Canada, and from 2009 to 2015, she was an Assistant Professor with Carnegie Mellon University, Pittsburgh, USA. She is currently an Associate Professor with the Power Systems Laboratory, ETH Zurich. Her research interest includes control and otimization of electric ower systems IEEE. Personal use is ermitted, but reublication/redistribution requires IEEE ermission. See htt:// standards/ublications/rights/index.html for more information.