Bi-Level Volt-VAR Optimization to Coordinate Smart Inverters With

Home , P (Los Angeles Railway)

arXiv:1912.03596v1 [math.OC] 8 Dec 2019 oy atleSateRsac etr etl,W 80 U 98109 WA Seattle, Center, Research Seattle [email protected]. Battelle tory, Pul University, [email protected],[email protected],liu@ State e-mail: Washington Science, Computer and ing f09 u ihreeg aigi eotddrn the during reported is ste saving OpenDSS. approximation using voltage energy and validated results feeder thoroughly higher The are average conditions. A load an pu. minimum maintains 0.96 approach voltag of bus the proposed reducing by appr the demand control power feeder’s coordinated reduce proposed objecti help the CVR that the demonstrated achieving in is framework 329-b the demonstrate a of results applicability The and feeder. feeders taxonomy test PNNL IEEE three-phase three-phase validated 123-bus a is while objective and approach (MILP) 13-bus linear proposed with Program The (NLP) Linear constraints. Program Integer quadratic Nonlinear Mixed a feasib a as and as Level-2 Le optimal model. modeled OPF is an nonlinear f 1 approximate obtain the parameters to solving control approxima by adjusted solution the linear are o Level-2, a inverters control In using smart VV flow. the inverters power optimizes bi-level smart un- three-phase Level-1 and A three-phase devices where, system. legacy proposed a and distribution is for linear power approach approximations developing electric that by flow pow balanced approach proposed optimal power VVO is An nonlinear control. formulation a inverter (OPF) present coord smart flow and Th with to devices operation devices. control is their voltage control paper legacy system’s power voltage this controls customer feeder’s of reduce controlling objective to by methods demand (VVO) optimization VAR rbtdgnrtr,trepaeotmlpwrflow. power optimal three-phase generators, tributed au ajnJha, Ranjan Rahul .Ja .Dby n ..Luaewt h colo Electrica of School the with are Liu C.C. and Dubey, A. Jha, R. .P cnie swt h aicNrhetNtoa Labora National Northwest Pacific the with is Schneider P. K. p I Φ ( Variables ne Terms Index Abstract N E E G Sets Φ N N ,j i, ij DG,i p p T ij i DG C ( = iLvlVl-A piiaint Coordinate to Optimization Volt-VAR Bi-Level ) mr netr ihVlaeCnrlDevices Control Voltage with Inverters Smart N Cnevto otg euto CR ssVolt- uses (CVR) reduction voltage —Conservation , E ) Vl-A piiain mr netr,dis- inverters, smart optimization, —Volt-VAR rjce e-hs ciepwrgnrtdby i generated power active per-phase Projected iueand nitude pnigt phase to sponding I { ietdgahfrdsrbto system distribution for graph Directed iebac)cnetn nodes connecting Line(branch) e fbace ihvlaeregulator voltage with branches of Set e fbss(oe)in (nodes) buses of Set e fdsrbto ie bace)in (branches) lines distribution of Set e fpae fbus of phases DGs of connected Set inverter smart with nodes of Set e fndswt aaio banks capacitor with nodes of Set th ij ( p pq Ga h urn ieinstance. time current the at DG = : ) | N tdn ebr IEEE, Member, Student I p ij p OMENCLATURE ∈ | ∠ δ Φ δ ij p ij p i q , screpnigpaeangle. phase corresponding is scmlxln urn corre- current line complex is ∈ p Φ ∈ i j } where, Φ G ei .Schneider, P. Kevin i where, Φ eecs.wsu.edu. i i mn A 99164 WA, lman, and nmk Dubey, Anamika { ⊆ | I ij p Ae-mail: SA | ,b c b, a, Engineer- l j G smag- is inates using e It ve. oach vel- } the es; nd ps us or er te O le e f - eirMme,IEEE Member, Senior at eetsuysosta V a eprdc the [1]. reduce States help United on the can implemented throughout CVR when feeders distribution 3.04% that all by shows consumption I study energy savings. annual energy recent attractive pilo a achieve and fact, help studies several can help on CVR voltage Based projects, [2]. the [1], decreasing custom demand to where the of reduce voltages sensitivity control the service voltage to to due of loads demand realized benefits are power savings The energy customer control. reducing voltage through by systems distribution C n-fln esrmnsad3 sn nertdVolt-Var on integrated based using 2) 3) approach, and rule-based proposed: measurements or been end-of-line have autonomous methods voltages using VVC pu) 1.05 1) service several - literature, (0.95 the In limits voltage [3]. maintaining service ANSI lower recommended still a the within while at operated range is contro Volt-VAR voltage feeder using loa The regulators techniques. banks, voltage (VVC) capacitor and as changers, such tap devices control voltage legacy z z u V s S u rdtoal,CRi copihdb otoln feeder’ controlling by accomplished is CVR Traditionally, q s s q cap,i DG,i ebr IEEE, Member, ij ij L,i p DG,i rated,p rated,p p pq tap,i p i cap,i p ij pq p g oices h nryefiinyo lcrcpower electric of efficiency energy the increase technol- a to is ogy (CVR) reduction voltage ONSERVATION ( meac matrix impedance ope he-hs meac arxfor matrix impedance line three-phase Complex z a oiincnetdt phase regulator to voltage connected position for tap variable control Binary n eciecmoet,respectively. components, reactive and V where, p tbus at s eciepwrdmn,respectively. demand, power reactive S etdt phase con- to bank nected capacitor for variable control Binary nbranch in Ga h urn ieinstance. time current the at DG ae eciepwrgnrtdb capacitor phase by to connected generated bank power reactive Rated ae e-hs paetpwrcpct for bus to capacity connected power DG apparent per-phase Rated usainbswhere, bus Substation vial e-hs eciepwrfrom power reactive per-phase Available pq ij L,i p L,i p pq i ij pq p ) = = ( = ∈ = and ,j i, .I I. hnCigLiu, Chen-Ching | r i V p Φ P ij pq orsodn ophase to corresponding i L,i p P ) p ij ij pq NTRODUCTION ij | + E ∈ pq q ( ∠ ,j i, L,i p + jx and θ + i p jq ) ij pq scmlxvlaefor voltage complex is r orsodn cieand active corresponding are jQ p L,i p orsodn to corresponding Q sa lmn ftecomplex the of element an is fbus of z ij pq ij pq ij scmlxpwrdemand power complex is r orsodn active corresponding are i scmlxpwrflow power complex is o branch for s N ∈ N ∈ p elw IEEE, Fellow, i where, fbus of DG p p ( ∈ ( fbus of i i ,j i, pq N ∈ Φ ) ) i p ∈ where, where, C ∈ i and Φ Φ i ij th er i d n s s 1 t , l 2 control (IVVC) based on real-time measurements [4]–[6]. device and smart inverters using the linear OPF model. Level-2 Several commercial VVC products are also available that solves a NLP (with linear objective and quadratic constraints) perform IVVC function mostly using heuristic [7]. Unfor- based on nonlinear OPF to further optimize the smart inverter tunately, the available products only optimize the operation parameters and result in a feasible power flow solution. Note of legacy control devices. Recently, the integration of dis- that although energy savings are reported higher for meshed tributed generations (DGs) has increased in the distribution networks [24], a majority of distribution feeders in the United grid [8]. Most DGs are equipped with smart inverters that are States are operated in radial topology. Therefore, this paper fo- capable of absorbing and supplying reactive power and thus cuses on optimizing the operation of radially operated feeders. controlling the feeder voltages locally that can help achieve The specific contributions of this paper are detailed below. additional CVR benefits [9]. Several researchers have worked • Models for Voltage-dependent Loads and Voltage Control on optimizing the reactive power dispatch from DGs and Devices: Mathematical models for voltage-dependent loads have proposed methods for smart inverter control using: 1) and grid’s voltage control devices including capacitor banks, autonomous control, 2) distributed control, and 3) centralized voltage regulators and smart inverters are proposed. For control using optimal power flow (OPF) [10]–[14]. loads, a novel CVR-based load model is proposed that The existing literature, however, presents several limitations. approximates the ZIP load model. The proposed models can First, the available literature mostly fails to coordinate the con- be easily absorbed into both levels of optimization problems trol of feeder’s legacy devices with smart inverters connected without changing the types of equations. to DG. When mathematically modeling the VVO problem • Linear and Nonlinear Models for Three-phase Power Flow: for both legacy and new devices, the optimization requires We develop valid linear and nonlinear approximations for solving an OPF problem with both discrete and continuous three-phase unbalanced power flow model. The linear ap- variables. This results in a Mixed Integer Nonlinear Program proximation is inspired by distflow equations but formulated (MINLP) that includes power flow equations for a three- for a three-phase unbalanced system. The nonlinear power phase unbalanced system. Some recent articles attempt to flow is a new formulation and obtained by approximating the solve this problem [15]–[21]. Unfortunately, these methods nonlinearities associated with the mutual coupling between do not jointly optimize the control of legacy devices and new the phases. Compared to the standard three-phase power devices, do not apply to three-phase unbalanced system, or flow formulations, the proposed model results in a reduced do not scale well even for mid-size system as the underlying number of variables and introduces only nonlinearity of the problem is a MINLP. Second, the methods based on OPF nature of quadratic equality constraints. assume a constant power load model thus fail to incorporate • Scalable Model for Coordinated Control of Legacy and New the voltage dependency of customer loads that is critical for Voltage Control Devices for CVR: The optimization problem modeling CVR effects. A voltage-dependent load model that for coordinated control of both legacy and new devices for can be easily incorporated within the optimization framework a three-phase distribution feeder is a hard MINLP problem. is called for. Third, the existing literature mostly solves a To reduce complexity and enable scalability, we propose a single-phase OPF problem. Distribution systems are largely bi-level approach by decomposing the MINLP into a MILP unbalanced and require complete three-phase modeling to and a NLP. The scalability is demonstrated using IEEE deliver a reasonable result. Recently, there has been some 123-bus feeder (with 267 single-phase nodes) and 329-bus advancements in solving three-phase OPF problem, however, feeder (with 860 single-phase nodes). The proposed model due to associated nonlinearities and mutual couplings, solving solves 123-bus feeder within 4-mins and 329-bus feeder a three-phase OPF with only continuous decision variables isa (after reduction) within 9-mins. challenging problem [22], [23]. Introducing discrete variables • Validation using Multiple Test Feeders: The proposed ap- to three-phase OPF problem makes it even more challenging proach and all approximation steps are thoroughly validated to solve. The aforementioned gap in literature calls for further against OpenDSS. First, the proposed three-phase approxi- research on enabling CVR for a modern distribution system. mate power flow models are validated using IEEE 13-bus, The objective of this paper is to develop an OPF based bi- IEEE 123-bus and 329-bus PNNL taxonomy feeders. Next, level approach for VVO to achieve CVR benefits for a three- the accuracy of the proposed CVR-based load model is phase unbalanced radial distribution system by simultaneously thoroughly validated against equivalent ZIP load models. controlling both legacy devices and smart inverters. The pro- Finally, the results for OPF are validated using OpenDSS. posed approach aims at addressing the aforementioned gaps The rest of the paper is organized as follows. Section II in the literature and presents a scalable model for coordinated presents the proposed approximate models three-phase power control of grid’s all voltage control devices for CVR benefits. flow. Section III details mathematical models for distribution First, models for voltage-dependent loads and systems voltage system equipment. Section IV presents the proposed bi-level control devices are developed to model CVR benefits, and VVO approach followed by results in Section V and conclu- valid formulations are proposed that can be easily incorporated sion in Section VI. within the three-phase OPF model. Next, we develop valid linear and nonlinear approximations for the three-phase power II. THREE-PHASE UNBALANCED ELECTRIC POWER flow equations. Based on the proposed linear and nonlinear DISTRIBUTION SYSTEM power flow approximations, a bi-level framework is proposed This section introduces the mathematical formulation for for CVR. Level-1 solves MILP to obtain set points for legacy three-phase power flow based on branch-flow equations. Valid 3 approximations are proposed to reduce the original formula- a distribution system as specified by the ANSI limits for bus tion into a linear and an equivalent quadratic formulation. voltages and phase unbalance [3]. a b c A. Three-Phase Power Flow using Branch Flow Model Vi Vi Vi j∗2π/3 2 b ≃ c ≃ a = e = a (5) Let, there be directed graph G = (N , E) where N denotes Vi Vi Vi set of buses and E denotes set of lines. Each line connects Further note that ordered pair of buses (i, j) between two adjacent nodes i and pq p q qq q q j. Let, {a,b,c} denotes the three phases of the system and Φi Sij = Vi × Iij and Sij = Vi × Iij (6) denotes set of phases on bus i. For each bus i ∈ N , let, phase pq qq p Using (5) and (6), we express Sij as a function of Sij i.e. p complex voltage is given by Vi and phase p complex power p p p p p V V demand is s . Let, V V Φ and s s Φ . pq i qq i L,i i := [ i ]p∈ i L,i := [ L,i]p∈ i Sij = q × Sij , where, q is a constant (5). The off- p Vi Vi For each line, let, p phase current be Iij and define, Iij := p diagonal elements of Sij are approximated using diagonal [I ] Φ . Let, z be the phase impedance matrix. ij p∈ i ij terms which help reduce the number of power flow variables. H H H vj = vi − (Sij zij + zij Sij )+ zij lij zij (1) Note that the above conditions do not imply that a single- phase power flow will be sufficient to represent a distribution diag(S − z l )= diag(S )+ s (2) ij ij ij jk L,j system. First, an equivalent single-phase model cannot repre- : → kXj k H sent two-phase or single-phase lines and loads. Second, it is vi Sij Vi Vi imperative to solve for an unbalanced power flow even though H = (3) S lij I I ij ij ij the degree of voltage unbalance is less. vi Sij 2) Assumption 2 - Approximating Angle Difference between H : −Rank-1PSDMatrix (4) Sij lij Phase Currents: On expanding (1) and (2), nonlinearities are introduced as trigonometric functions of angle difference A three-phase power flow formulation for a radial system between the phase currents on a given three or two-phase based on branch flow relationship is given in [22] and de- line. Let, for a given line (i, j), the phase currents for phases tailed in (1)-(4). Here, (1) represents voltage drop equation, p and q be Ip = |Ip |∠δp , and Iq = |Iq |∠δq . Then the (2) corresponds to power balance equation, (3) are variable ij ij ij ij ij ij angle difference between the phase currents of a given bus i definitions for power flow quantities, and (4) is a Rank-1 i.e. δpq = ∠δp − ∠δq . We observe terms corresponding to constraint that makes the associated optimization problem non- ij ij ij sin(δpq) and cos(δpq) in power flow expressions. These terms convex. In the literature, methods are proposed to obtain a ij ij significantly increase the complexity of the OPF problem. relaxed convex problem [25], [26], however, it is difficult to In the proposed formulation, the phase angle difference obtain a feasible solution from relaxed problem for a three- between branch currents are approximated and modeled as phase system [23]. Moreover, it is difficult to extend the power a constant variable. An approximate value of δpq for branch flow model detailed in (1)-(4) for voltage-dependent loads and ij (i, j) is calculated by solving an equivalent distribution power system’s legacy control devices. A new three-phase power flow flow with system loads modeled as constant impedance loads. model for OPF problem is called for that can easily incorporate We assume δpq to be constant and equal to the one obtained system’s critical components while not significantly increasing ij by solving power flow with constant impedance load model. the inherent nonlinearity. Note that the constant impedance load model is only used to B. Approximate Three-Phase Power Flow Equations pq approximate δij . This assumption does not limit the type of In this section, we present valid linear and nonlinear power load that can be incorporated in the proposed OPF model; it flow equations by approximating (1)-(4). Fundamentally, there can easily incorporate all load types including constant power are two reasons for nonlinearity in power flow equations: and constant current loads, as detailed in Section IV. nonlinear relationship between power, voltage, and currents, 3) Power Flow Equations: The power flow equations de- and mutual coupling in a three-phase system. In the proposed fined in (1)-(4) are expanded. Using the above approximations, formulation, the nonlinearity resulting from mutual coupling we are able to redefine the power flow equations in (1)-(4) as a between the three phases is approximated. A phase-decoupled set of linear and non-linear equations shown in (7)-(11). Here, p pq pq formulation by decoupling the branch power flow and voltage (7)-(9) are linear in vi , lij , and Sij . Note that the total number equations on a per-phase basis is obtained. The resulting of variables in the proposed formulation are 15×(n−1), where three-phase power flow model characterizes the power flow n is the number of nodes; the original formulation (1)-(4) had equations using a fewer number of variables. a total of 36 × (n − 1) variables. p p 2 pq p pp pq pq pq pq pq Define: v = (V ) where, p ∈ Φi, l = (|I | × P l r δ x δ i i ij ij ij − q∈Φj ij ij cos( ij ) − ij sin( ij ) |Iq |) where, (pq) ∈ Φ , δpq = δp − δq , Spq = ij ij ij ij ij ij P pp pp pq pq pq pq P = k:j→k jk + L,j (7) Pij + jQij , where, (pq) ∈ Φij , and zi,j = rij + 2 pp pq pq pq pq pq pq pq pp qq Q − Φ l x cos(δ )+ r sin(δ ) jxij , where, (pq) ∈ Φij . Note that, (lij ) = lij × lij and ij q∈ j Pij ij ij ij ij δpq is angle difference between branch currents Ip and Iq . Qpp qp ij ij ij P = k:j→k jk + L,j (8) 1) Assumption 1 - Approximating Phase Voltages: The ◦ phase voltages are assumed to exactly 120 degree apart. p p P pq pq ∗ pq qq v = v − Φ 2Re S (z ) + Φ z l Moreover, it is assumed that the degree of unbalance in j i q∈ j ij ij q∈ j ij ij 1 1 2 1 2 2 Re zpq lq q ∠ δq q zpq ∗ (9) voltage magnitude is not large. Both conditions are valid for + q1,q2∈ΦjP,q16=q2 2 ij ij ( ijP ) ( ij ) P h i 4

pp 2 pp 2 p pp p (Pij ) + (Qij ) = vi lij (10) value [27]. Let, a be the turn ratio for the voltage regulator p (lpq)2 = lpplqq (11) connected to phase p of line (i, j). Then a can take values ij ij ij between 0.9 to 1.1 with each step resulting in a change of • (7) is written for all (i, j) ∈ E and represents the equation 0.00625 pu. An additional node i′ is introduced to model the for active power flow on branch (i, j) for phase pp ∈ Φij . current equations. The control for regulator is defined using pq pq p Since cos(δij )and sin(δij ) are assumed to be constant, this binary variables. Let, for u ∈{0, 1} be a binary variable pp pp pq tap,i equation is linear in Pij , Pjk , and lij . defined for each regulator step position i.e. i ∈ (1, 2, ..., 32). • (8) is written for all i, j and represents the equation p ( ) ∈ E Also define a vector bi ∈ {0.9, 0.90625, ..., 1.1}. Then Vi , for reactive power flow on branch i, j for phase pp . p p p ( ) ∈ Φij Vj , Iii′ , and Ii′j where p ∈ Φi ∩ Φj are given as follows: Similar to (7), (8) is linear in Qpp, Qpp, lpq. ij jk ij p p p p p p p • (9) represents the equation for voltage drop between the Vj = Vi′ = a Vi and Iii′ = a Ii′j (14) two nodes of branch (i, j) corresponding to phase p. The 32 32 p p pq p p p equation is also linear in vi , vj , and lij . where, a = biutap,i and utap,i =1. i=1 i=1 • (10) relates per phase complex power flow in branch (i, j) p p 2 p In order toP express (14) asP a function of vi = (Vi ) , vj = to phase voltage and phase currents. This is a non-linear p 2 pp p 2 pp p 2 (Vj ) , lii′ = (Iii′ ) , and li′j = (Ii′j ) we take square of quadratic equality constraint. 2 2 (14) and define ap = Ap and bi = Bi. Further realizing that • (11) simply relates current variables previously defined, i.e. p 2 p pq p q (utap,i) = utap,i, (14) can be reformulated as (15). lij = (|Iij | × |Iij |), pq ∈ Φij . p p p pp p pp C. Linear Three-Phase AC Power Flow Approximation vj = A × vi and lii′ = A li′j (15) The linear approximation assumes the branch power loss are B. Capacitor Banks relatively smaller as compared to the branch power flow [22]. The per-phase model for capacitor banks is developed. p The impact of power loss on active and reactive power branch The reactive power generated by capacitor bank, qcap,i, is p flow equations and on voltage drop equations is ignored. After defined as a function of binary control variable ucap,i ∈{0, 1} approximating (7)-(11), we obtain linearized AC branch flow indicating the status (On/Off) of the capacitor bank, its rated rated,p equations as shown in (12)-(13). Here (12) corresponds to per-phase reactive power qcap,i , and the square of the bus p linearized active and reactive power flow and (13) corresponds voltage at bus i for phase p, vi . to voltage drop equations. p p rated,p p pp pp p pp pp p qcap,i = ucap,iqcap,i vi (16) Pij = Pjk + pL,j and Qij = Qjk + qL,j (12) : → : → kXj k kXj k The capacitor bank model is assumed to be voltage depen- p p Re pq pq ∗ dent and provides reactive power as a function of vp when vj = vi − 2 Sij (zij ) ∀j ∈ Yi (13) i q∈Φj connected, i.e. ucap,i =1. For a three-phase capacitor bank, a X common control variable, up , is defined for each phase. The AC linearized power flow is significantly accurate in cap,i representing bus voltages. The linearized AC power flow, C. Distributed Generation with Smart Inverters although does not include the impact of power loss on voltage A per-phase model for reactive power support from smart drop, it does incorporate the impact of power flow due to load. inverter connected to DGs is developed. The DGs are modeled Since power losses are significantly small as compared to the as negative loads with a known active power generation equal branch flow due to load demand, the obtained feeder voltages to the forecasted value. The reactive power support from DG are good approximation of the actual feeder voltages [22]. depend upon the rating of the smart inverter. Let, the rated per- phase apparent power capacity for smart inverter connected to th rated,p III. DISTRIBUTION SYSTEM EQUIPMENT MODELS i DG be sDG,i and the forecasted active power generation p p This section details the models for capacitor banks, voltage be pDG,i. The available reactive power, qDG,i from the smart regulators, smart inverters and voltage-dependent customer inverter is given by (17) which is a box constraint. loads. The approximate power flow equations developed in − (srated,p)2 − (pp )2 ≤ qp ≤ (srated,p)2 − (pp )2 Section II are a function of vp V p 2. The equipment q DG,i DG,i DG,i q DG,i DG,i i = | i | (17) models are, therefore, parameterized based on respective con- p D. Voltage-Dependent Model for Customer Loads trol variables and the vi . A new CVR based load model is p The most widely acceptable load model is the ZIP model developed to represent the power demand as a function of vi . The ZIP coefficients for the load are used to obtain equivalent which is a combination of constant impedance (Z), constant CVR coefficients. Note that the equipment and load models current (I) and constant power (P)) characteristics of the load proposed in this section are specifically designed so that they [28]. The mathematical representation of the ZIP model for can be easily absorbed within the approximate power flow the load connected at phase p of bus i is given by (18)-(19). equations defined in (7)-(13) without changing their type. p 2 p p p Vi Vi A. Voltage Regulator pL,i = pi,0 kp,1 + kp,2 + kp,3 (18) " V0 V0 # A 32-step voltage regulator with a voltage regulation range p 2 p of ±10% is assumed. The series and shunt impedance of p p Vi Vi qL,i = qi,0 kq,1 + kq,2 + kq,3 (19) the voltage regulator are ignored as these have very small V0 V0 " # 5

p where, kp,1 + kp,2 + kp,3 =1, kq,1 + kq,2 + kq,3 =1, pi,0 and A. Level 1 - MILP Formulation for Coordinated Control p qi,0 are per-phase load consumption at nominal voltage, V0. The objective of this stage is to minimize the power The ZIP load model represented in (18)-(19) are a function p p p 2 consumption for the feeder by controlling voltage regulators, of both Vi and vi = (Vi ) . Including (18) and (19) to OPF capacitor banks, and smart inverters while ensuring that the formulation will make (7),(8),(13), and (14), earlier linear in p voltage limits are satisfied. The control of legacy devices vi , nonlinear. Here we develop an equivalent load model for introduces integer variables into the optimization problem. voltage-dependent loads using the definition of CVR factor. A linear three-phase AC power flow is used and resulting Next, an equivalence between ZIP parameters and proposed problem is a MILP formulation as detailed in (26)-(37). The CVR factors is obtained. objective is to minimize the sum of three-phase active power CVR factor is defined as the ratio of percentage reduction flowing out of the substation bus at time t (26). Here, s ∈ N in active or reactive power to the percentage reduction in denotes the substation bus. Since, the distribution feeder is bus voltage. Let CVR factor for active and reactive power radial, the substation power equals net feeder power demand. reduction be CV R , and CV R , respectively defined in (25). p q Variables: p p p p p p pp pp pq dpL,i V0 dqL,i V0 utap,i(t), ucap,i(t), qDG,i(t), vi (t), Pij (t), Qij (t), Sij (t) CV Rp = p p and CV Rq = p p (20) p dV q dV p i,0 i i,0 i Minimize: P (t) (26) X sj p p p p p p p∈Φs,j:s→j where, pL,i = pi,0 + dpi and qL,i = qi,0 + dqi . Furthermore, p p 2 p p p p vi = (Vi ) . Therefore, dvi =2Vi dVi . Assuming Vi ≈ V0 Subject to: p p 2 and dvi = vi − (V0) , we obtain: pp pp p p Pij (t)= Pjk (t)+ pL,j (t) − pDG,i(t) ∀i ∈N (27) p p X p k:j→k p p i,0 vi p = p 0 + CV Rp − 1 (21) L,i i, 2 V 2 Qpp(t)= Qpp(t)+ qp (t) − qp (t) − qp ∀i ∈N 0 ij X jk L,j DG,i C,i p p k:j→k q 0 v p p i, i (28) qL,i = qi,0 + CV Rq 2 − 1 (22) 2 V0 p p pq pq ∗ v (t)= v (t) − 2Re S (t)(z ) ∀j ∈ Yi (29) j i X ij ij Note that the CVR based load model detailed in (21) and (22) q∈Φj p is linear in v , thus can be easily included in approximate p i p p pi,0(t) p p (t)= p (t)+ CV Rp(t) (v (t) − 1)∀i ∈NL power flow equations (7)-(13). The CVR factors, CV Rp and L,i i,0 2 i CV Rq are estimated from the ZIP coefficients of the load. (30) p On differentiating the ZIP model detailed in (18) and (19) p p qi,0(t) p q (t)= q (t)+ CV Rq(t) (v (t) − 1)∀i ∈NL and assuming V0 =1 p.u., we obtain: L,i i,0 2 i p (31) dpL,i p p p p p v (t)= A (t)v (t)∀(i, j) ∈ET (32) p = pi,0 (2kp,1Vi + kp,2) (23) j i i dVi 32 32 p p p p dq A (t)= Biu (t), u (t) = 1∀(i, j) ∈ET (33) L,i p p i X tap,i X tap,i = q (2k 1V + k 2) (24) =1 =1 dV p i,0 q, i q, i i i qp (t)= up (t)qrated,pvp(t) ∀(i) ∈N (34) p C,i cap,i cap,i i C Using (20), (23), (24) and assuming V ≈ V0, we obtain i p rated,p 2 p 2 (25). Using (25), the CVR factors for customer loads can be qDG,i(t) ≤ q(sDG,i ) − (pDG,i) (t) ∀(i) ∈NDG (35) obtained from the ZIP coefficients. p rated,p 2 p 2 qDG,i(t) ≥−q(sDG,i ) − (pDG,i)(t) ∀(i) ∈NDG (36) 2 2 CV Rp =2kp,1 + kp,2 and CV Rq =2kq,1 + kq,2 (25) p (Vmin) ≤ vi (t) ≤ (Vmax) ∀i ∈N (37)

IV. PROPOSED BI-LEVEL VOLT-VAR OPTIMIZATION • Constraints (27)-(29) are linear AC power flow constraints. • Constraints (30)-(31) define CVR based load model. The primary function of VVO is to use voltage control • Constraints (32)-(33) define regulator control equations. to 1) reduce energy consumption, 2) reduce system losses, • Constraint (34) defines equations for capacitor control. and 3) regulate feeder voltages. The problem of coordinating • Constraints (35)-(36) define control equations for reactive the control of system’s legacy devices and smart inverters power dispatch at time t from smart inverters. results in a MINLP problem. To reduce complexity and ensure • Constraints (37) defines operating limits for feeder voltages. scalability, a bi-level approach is proposed. 1) Level 1: Develops a 15-min schedule for legacy devices B. Level 2 - NLP Problem for Smart Inverter Control and smart inverter reactive power demand set-points with the objective of minimizing the active power consumption Level-1 uses a linear three-phase power flow model that for the feeder based on a MILP formulation. approximates the losses. The solutions although feasible for 2) Level 2: Develops revised 15-min schedule for smart linear power flow formulation, may violate the critical oper- inverter controls using a NLP formulation to achieve ating constraints of the feeder. The objective of this stage is feasible three-phase power flow solutions. Level-2 uses to adjust the set-points of smart inverter control variables in the nonlinear power flow formulation proposed in Section order to obtain an optimal and feasible three-phase nonlinear power flow solution. The discrete control variables, up t , II.C and obtains revised set points for smart inverter tap,i( ) up t , are assumed to be fixed as obtained in Level-1. The control that ensure feasible power flow solutions. cap,i( ) 6

32 29 250 33 30 251 51 111 110 112 113 114 28 50 151 300 optimal control set points for reactive power dispatch from 31 49 109 107 2548 47 46 26 108 451 smart inverters are obtained by solving the NLP problem (with 27 45 64 106 104 23 44 43 65 103 450 105 102 63 100 linear objective and quadratic constraints) defined in (38)-(50). 24 42 41 21 66 101 40 99 71 p p pp pp pq pq 22 62 197 98 Variables: q (t), v (t), P (t), Q (t), S (t), l (t) 38 39 70 DG,i i ij ij ij ij 36 97 69 19 135 35 20 18 68 75 160 67 p 37 60 74 Minimize: Psj (t) (38) 73 X 14 58 57 Φ : 11 59 72 85 p∈ s,j s→j 61 610 79 9 7778 2 10 52 5453 56 Subject to: 152 55 76 7 8 13 80 94 X X X 84 pp pq pq pq pq pq 149 1 34 96 88 150 12 17 92 90 81 Pij (t) − lij (t) rij cos(δij (t)) − xij sin(δij (t)) X 83 Φ 15 95 87 86 82 q∈ j 3 93 91 89 X 5 6 16 pp p p 4 = P (t)+ p (t) − p (t) ∀i ∈N (39) X jk L,j DG,i QRV11 . QRV11 . :]:H1 Q` k:j→k Fig. 1. IEEE 123-bus distribution test feeder. Qpp(t) − lpq (t) xpq cos(δpq(t)) + rpq sin(δpq (t)) ij X ij ij ij ij ij q∈Φj using OpenDSS. Next, the proposed voltage-dependent load = Qpp(t) − qp (t) − qp (t) ∀i ∈N (40) X jk DG,i C,i models are validated against equivalent ZIP load models. k:j→k Finally, we demonstrate the proposed VVO approach using vp(t)= vp(t) − 2Re Spq (t)(zpq)∗ + zpqlqq (t) j i X ij ij X ij ij the aforementioned three test feeders. All simulations are done Φ Φ q∈ j q∈ j on MATLAB platform. Level-1 problem, modeled as MILP, is 1 1 2 1 2 2 + 2Re zpq lq q (t) ∠(δq q (t)) (zpq )∗ X ij ij ij ij solved using CPLEX 12.7 and Level-2 problem, modeled as 1 2 Φ 1= 2 q ,q ∈ j ,q 6 q NLP, is solved using fmincon function in MATLAB optimiza- (41) tion toolbox. A computer with core i7 3.41 GHz processor with pp 2 pp 2 p pp ∀ ∈E (Pij (t)) +(Qij (t)) = vi (t)lij (t) (i, j) (42) 16 GB of RAM has been used for the simulations. The results pq 2 pp qq (lij (t)) = lij (t)lij (t) ∀(i, j) ∈E (43) obtained from MATLAB are validated against OpenDSS. p p p pi,0(t) p IEEE-13 bus is a small highly loaded unbalanced distribu- p (t)= p (t)+ CV Rp(t) (v (t) − 1)∀i ∈NL L,i i,0 2 i tion feeder operating at 4.16 kV making it a good candidate (44) p for testing VVO applications. This test feeder includes a three- p p qi,0(t) p q (t)= q (t)+ CV Rq (t) (v (t) − 1)∀i ∈NL phase and a single-phase capacitor bank and a voltage regu- L,i i,0 2 i (45) lator at the substation. A PV with smart inverter of 575 kVA 32 rated capacity is installed at node 671. IEEE-123 bus feeder p p p p p v (t)= A (t)v (t),A (t)= Biu (t)∀(i, j) ∈E also presents unbalanced loading conditions and several single- j i i i X tap,i T i=1 phase lines and loads with voltage drop problems making it a (46) good candidate for demonstration of VVO application. There p p rated,p p qC,i(t)= ucap,i(t)qcap,i vi (t) ∀(i) ∈NC (47) are four voltage regulators and four capacitor banks deployed p rated,p 2 p 2 along the feeder as shown in Fig. 1. The feeder is modified q (t) ≤ (s ) − (p ) (t) ∀(i) ∈NDG (48) DG,i q DG,i DG,i to include three DGs of capacity 345 kVA, 345 kVA, and p rated,p 2 p 2 qDG,i(t) ≥−q(sDG,i ) − (pDG,i)(t) ∀(i) ∈NDG (49) 690 kVA at nodes 35, 52, and 97 respectively (see Fig. 1). 2 p 2 The 329-bus feeder is used to demonstrate the scalability of (Vmin) ≤ v (t) ≤ (Vmax) ∀i ∈N (50) i the proposed approach. Notice that 329-bus feeder includes • Constraints (39)-(43) are approximate nonlinear AC power 329 physical nodes and a total of 860 single-phase nodes. flow equations defined at time t. Compared to the state-of-art, this is a significantly large test • Constraints (44)-(45) define CVR based load model. system to demonstrate the coordinated control of all voltage • Constraints (46) define regulator control equations. Note that p control devices. The feeder includes one voltage regulator, one utap,i is known from Level-1 solution. 600 kVAr three-phase capacitor bank, three 100 kVAr single- • Constraint (47) defines control equations for capacitor banks p phase capacitor banks, and three DGs of capacity 23kVA, at time t. Note that ucap,i is known from Level-1 solution. 57.5kVA and 115kVA (see Fig. 2). • Constraints (48)-(49) define control equations for reactive Customer loads are assumed to have a CVR factor of 0.6 power dispatch at time t from smart inverters. for active power and 3 for reactive power [2]. Note that the • Constraints (50) defines operating limits for feeder voltages. CVR values are arbitrary and can be easily adjusted based on the parameters for ZIP model of the load, if available, as V. RESULTS AND DISCUSSION detailed in Section III D. To demonstrate the applicability of The proposed VVO approach is validated using following the proposed approach for different load mix, additional cases three test feeders: IEEE 13-bus, IEEE 123-bus [29], and are simulated using a combination of residential and small and PNNL 329-bus taxonomy feeder [30]. First, the proposed large commercial loads. The daily load and generation profiles linear and nonlinear approximate power flow formulations are are simulated in 15-min interval and are based on example validated against the actual power flow solutions obtained profiles provided in OpenDSS (see Fig. 3). 7

QRV11 . :]:H1 Q` QRV11 .

Fig. 2. PNNL 329-bus taxonomy distribution test feeder.

PV Generation Profile TABLE I 1.2 Base Load Demand Profile at Nominal Voltage COMPARISON OF APPROXIMATE LINEAR AND NONLINEAR POWER FLOW 1 FORMULATIONS AGAINST OPENDSS SOLUTIONS 0.8 Largest Error in Linear Power Flow wrt. OpenDSS Solutions

0.6 Test Feeder % Loading Pflow(%) Qflow(%) V (pu.) 0.4 IEEE 13 Bus 75% 5.1287 4.938 0.0075 IEEE 13 Bus 100% 7.227 6.442 0.0096 0.2 IEEE 123 Bus 75% 5.248 9.502 0.0054

Power Generation (pu) 0 IEEE 123 Bus 100% 5.328 11.313 0.0074 2 4 6 8 10 12 14 16 18 20 22 24 PNNL 329 Bus 75% 1.16 6.9 0.001 Time (Hours) PNNL 329 Bus 100% 1.55 9.51 0.002 Fig. 3. Load demand and PV generation in 15-min interval. Largest Error in Nonlinear Power Flow wrt. OpenDSS Solutions A. Verification of Approximate Power Flow Formulations Test Feeder % Loading Pflow(%) Qflow(%) V (pu.) This section validates the proposed approximate power IEEE 13 Bus 75% 0.2414 1.668 0.0015 flow models. The results obtained from the proposed linear IEEE 13 Bus 100% 0.297 2.034 0.0025 and nonlinear power flow models are compared with the IEEE 123 Bus 75% 0.505 2.58 0.0014 IEEE 123 Bus 100% 0.606 3.88 0.0016 power flow solution obtained using OpenDSS (see Table I). PNNL 329 Bus 75% 0.3 2.2 0.0001 The largest errors in active and reactive power flow and bus PNNL 329 Bus 100% 0.6 3.4 0.0002 voltages are reported for the three test feeders in Table I for different loading conditions. Note that the three-phase linear power model is sufficiently accurate in modeling power flow equations for an unbalanced system. Since, the losses are 329-bus feeder is relativelyTABLE more II balanced and, therefore, ignored in flow equations (equation (12)), the linear model MAXIMUM ERRORINAPPROXIMATINGPHASEANGLEDIFFERENCES incurs higher error in flow quantities (Pflow, and Qflow). incurs less error in voltages as compared to the rest of the Test Feeder % Load error in δpq error in θpq However, since the voltage drop due to flow quantities is two feeders. ij i included in linear model (equation (13)), the bus voltages are IEEE 13 Bus 75% 1.8 1.8 IEEE 13 Bus 100% 2.1 2.2 well approximated. Another key observation is an increase in IEEE 123 Bus 75% 0.8 0.9 error in Qflow vs. Pflow for 123-bus and 329-bus feeders. The IEEE 123 Bus 100% 1.13 1.3 approximation errors in flow quantities using the linearized PNNL 329 Bus 75% 0.5 0.55 model depend upon relative values of line resistance and PNNL 329 Bus 100% 0.9 1.05 reactance. The line reactance is higher than the line resistance The proposed nonlinear power flow formulation is based for these two feeders leading to more reactive power losses on two approximations: 1) difference between phase angles pq ◦ and hence a higher error in approximating Qflow quantities of node voltage (θi ) is close to 120 , 2) difference between pq using linearized model. The nonlinear power flow model phase angles of branch currents (δij ) is close to those obtained includes losses in the formulation and, therefore, results in using a constant impedance load model. These approximations lesser error in both flow quantities and bus voltages. The are validated in Table II. The table reports largest deviation maximum error in bus voltages during peak load using linear between actual quantities obtained using OpenDSS vs. the and nonlinear models are: 0.0096 pu and 0.0025 pu for the approximated ones used in this paper. As it can be seen, the 13-bus, 0.0074 pu and 0.0016 pu for the 123-bus, and 0.002 largest error is less than 2◦ for both voltage and current phase and 0.0002 pu for 329-bus systems, respectively. Note that angle difference. 8

TABLE III TABLE IV COMPARISON OF APPROXIMATE NONLINEAR POWER FLOW ZIP COEFFICIENTSFORDIFFERENTCLASSOFLOADS FORMULATIONS AGAINST OPENDSS SOLUTIONS Load Zp Ip Pp Zq Iq Qq CV Rp CV Rq Error in Nonlinear Power Flow wrt. OpenDSS for unbalanced case Class

Test Feeder Vunbal(%) Pflow(%) Qflow(%) V (pu.) Residential 0.96 -1.17 1.21 6.28 -10.16 4.88 0.75 2.4 IEEE 123 Bus 3.2 0.61 3.88 0.002 Small Com- 0.77 -0.84 1.07 8.09 -13.65 6.56 0.7 2.53 mercial IEEE 123 Bus 5.7 0.68 3.94 0.004 Large Com- 0.4 -0.41 1.01 4.43 -7.99 4.56 0.39 0.87 PNNL 329 Bus 2.8 0.52 3.74 0.0007 The proposed nonlinear power flow is verified at heavily mercial PNNL 329 Bus 4.5 0.95 4.51 0.0012 unbalanced loading for IEEE 123 node and 329 node system. The unbalanced in the system is created by increasing the load for one of the phase. The voltage unbalance defined in (51) according to IEEE definition, is used to quantify the effect of load unbalance created in the system. demand, both active and reactive, due to change in bus voltage are similar for both CVR-based load model and equivalent ZIP max.deviation V = ∗ 100 (51) load model. unbalance V | avg| 104 120 The IEEE-123 node system has the inherent apparent power 115 102 110 unbalance of 23.2%, which creates a maximum voltage un- 105 100 balance of 3.2%. Further, to produce more unbalance in the 100 Residential Residential cvr cvr Residential 95 Residential system the apparent power of phase A is increased which ZIP ZIP SmallCommercial SmallCommercial cvr 90 cvr 98 SmallCommercial SmallCommercial results in apparent power unbalance of 39.6%. Due to in- ZIP 85 ZIP LargeCommercial Reactive Power (kVAr) cvr LargeCommercial Active Power Demand (kW) 80 cvr creased power unbalance the maximum voltage unbalance in LargeCommercial ZIP LargeCommercial 96 ZIP the system is 5.7%. The maximum error in Pflow ,Qflow and 0.95 1 1.05 0.95 1 1.05 Node Voltage (pu) voltage is shown in Table III for IEEE-123 node system for Node Voltage (pu) Fig. 4. Comparison of proposed load model with ZIP model: (a) Active power both the test case. It is known that the 329 bus system is a demand, (b) Reactive power demand. balanced system. Hence, to generate unbalance in the system the apparent power of phase B is increased which originates to C. CVR using proposed VVO approach apparent power unbalanced of 40.72% and 57.39%. The effect The proposed bi-level VVO approach is validated using of unbalanced power results in voltage unbalanced of 2.8% and IEEE test feeders. The optimal control set points are obtained 4.5 % respectively. The maximum error in Pflow ,Qflow and for both legacy and smart inverter control devices for the entire voltage with respect OpenDSS power flow results is shown in day. The results demonstrate that the proposed formulation Table III. It is required to mention that the nonlinear power ensures that feeder operates closer to minimum voltage range flow is solved at flat start. According to ANSI C84.1 electric while not violating the voltage limit constraints and therefore, system can have the maximum voltage unbalance of 3%. is effective in achieving CVR objectives. Hence, Table III uphold the proposed power flow can be used 1) IEEE 13-bus test system: The control variables for this for the heavily unbalanced system. feeder include a 32-step three-phase voltage regulator, a three- phase capacitor bank (Cap1), a single-phase capacitor bank B. Validation of Proposed CVR-based Load Model on Phase C (Cap2), and one three-phase DG with smart The proposed CVR-based voltage dependent load model inverter control. The model is simulated in 15-min interval derived in equations (21)-(22) is validated against equivalent for 1 day. The results obtained for the day during minimum ZIP load models detailed in equations (18)-(19). When ac- and maximum loading conditions are shown in Table IV. As curately modeled, the CVR-based load model should require it can be seen that the feeder is unbalanced with a largest the same power demand as the equivalent ZIP load model for difference of around 0.24 MW during peak load condition. the acceptable range of operating voltages (0.95pu-1.05pu). Based on the table, with the increase in load, regulator tap Therefore, to validate the load models, the active and reactive position changes from -13 at minimum load condition to 14 power consumption for CVR-based load models are compared at maximum load condition. The three-phase capacitor bank against the power consumption for ZIP load model for varying is OFF for both load conditions, while single-phase capacitor node voltages. ZIP models for residential, small commercial, in ON during peak load condition. For each phase, DG is and large commercial loads are used for validation. The ZIP supplying reactive power in order to maintain the feeder coefficients for the different class of loads are obtained from voltages within the ANSI limits except for phase C during [28] and converted to CVR-based load model using equation maximum load condition. This is because the single-phase (25) (also see Table IV). capacitor is ON and supplies the required reactive power. It The simulation case is detailed here. For each load class, should also be noted that the absorbed power supplied by DG the base active pi,0 and qi,0 reactive power are taken as 100 increases with the loading. The average feeder voltage seen at kW and 100 kVAr, respectively. The voltage at the load point both maximum and minimum loading conditions are close to is varied from 0.95 to 1.05 pu. The active and reactive power 0.96 pu. The proposed VVO approach is, therefore, successful demand for the two load models are shown in Fig. 4. It can be in maintaining feeder voltages close to minimum voltage limit, observed that for different load classes, the variation in power thus help extract the CVR benefits. 9

2.5 100 TABLE V without CVR control VOLT-VAR OPTIMIZATION RESULTS FOR IEEE 13-BUS FEEDER with proposed VVO approach for CVR Reduction in demand due to CVR (CV Rp = 0.6 AND CV Rq = 3) 2 80

IEEE-13 Minimum Load Maximum Load 1.5 60 OPF solution from MATLAB

Phase A B C A B C 1 40 Regulator Tap -13 -13 -13 14 14 14

Cap1 Status OFF OFF OFF OFF OFF OFF 0.5 20 power demand (MW)

Cap2 Status — — OFF — — ON Total three-phase substation p DG1 qDG(MVAR) -0.04 -0.13 -0.03 -0.30 -0.45 0.12 0 0 Optimal substation power flow and voltages using MATLAB 0 2 4 6 8 10 12 14 16 18 20 22 24 Reduction in Active Power Demnad (kW) Load (MW) 0.143 0.132 0.091 0.866 0.622 0.685 Time (hours) Fig. 5. IEEE-13 Bus CVR benefits observed using the proposed approach Min. Voltage (pu) 0.955 0.956 0.955 0.95 0.955 0.95 (CV Rp = 0.6 and CV Rq = 3). Max. Voltage (pu) 0.97 0.97 0.97 1.03 1.03 1.03 TABLE VII Avg. Voltage (pu) 0.958 0.956 0.958 0.972 0.971 0.972 VOLT-VAR OPTIMIZATION RESULTS FOR IEEE 123-NODE FEEDER Validation of substation power flow and voltages using OpenDSS (CV Rp = 0.6 AND CV Rq = 3) Load (MW) 0.144 0.135 0.095 0.87 0.625 0.69 Min. Voltage (pu) 0.955 0.955 0.955 0.95 0.953 0.95 IEEE-123 Minimum Load Maximum Load Max. Voltage (pu) 0.97 0.97 0.97 1.03 1.03 1.03 Phase A B C A B C Avg. Voltage (pu) 0.958 0.956 0.958 0.971 0.97 0.971 OPF solution from MATLAB Reg1 Tap -13 -13 -13 -8 -8 -8 The results obtained from the proposed approach are val- Reg2 Tap 0 — — -2 — — idated using OpenDSS. The optimal controls obtained from Reg3 Tap 1 — 1 7 — 2 MATLAB for both maximum and minimum load conditions Reg4 Tap 0 0 0 0 0 0 are implemented on 13-bus test feeder. The test feeder, with Cap1 Status OFF OFF OFF ON ON ON Cap2 Status OFF — — ON — — given statuses of voltage control devices, is solved using Cap3 Status — OFF — — OFF — OpenDSS and substation power demand and minimum, max- Cap4 Status — — OFF — — OFF DG1 qp (MVAR) -0.03 0.045 0.012 -0.028 0.03 0.04 imum, and average node voltages are recorded (see Table DGp DG2 qDG(MVAR) 0.04 -0.03 0.03 -0.025 0.039 -0.01 V). It is observed that the system parameters obtained from p DG3 qDG(MVAR) -0.08 -0.02 -0.08 -0.09 0.045 -0.09 MATLAB and OpenDSS closely match. This is expected given Optimal substation power flow and voltages using MATLAB the accuracy of the proposed nonlinear power flow model. Load (MW) 0.20 0.13 0.18 0.99 0.78 1.02 The optimal power consumption as recorded at the sub- Min. Voltage (pu) 0.955 0.955 0.955 0.951 0.953 0.951 Max. Voltage (pu) 0.965 0.965 0.965 0.995 0.995 0.995 station transformer after implementing the proposed VVO Avg. Voltage (pu) 0.957 0.957 0.958 0.963 0.965 0.966 strategy for the day is shown in Fig. 5. The total power Validation of substation power flow and voltages using OpenDSS demand is compared with the case when VVO is not enabled. Load (MW) 0.205 0.134 0.183 1.00 0.79 1.024 For this case, the capacitors and voltage regulators work in Min. Voltage (pu) 0.954 0.954 0.954 0.95 0.95 0.95 Max. Voltage (pu) 0.965 0.965 0.965 0.995 0.995 0.995 autonomous control mode while DG is operating at unity Avg. Voltage (pu) 0.956 0.956 0.956 0.96 0.961 0.963 power factor. Except for peak demand duration, the proposed approach results in a reduction in net power demand. The are obtained within 9 sec for the 13-bus system. The largest largest reductions are seen at low loading condition. time taken for solving Level-2 problem is 14 sec. To further validate the proposed approach, we include 2) IEEE 123-bus test system: Similarly, the proposed bi- additional test results with realistic load models for residential, level VVO approach is implemented using IEEE 123-node commercial and large commercial loads. The ZIP coefficients system for 1 day at 15-min interval. The results obtained from details in Table IV are used to obtain CVR factors for each VVO for 123-node system are shown Table VII. The feeder is case with different load mix. The total load demand for unbalanced with Phase B load being less than Phase A and C. minimum and maximum load conditions are reported in Table The voltage regulator, Reg1, located at substation transformer VI. It can be observed that the reduction in active power (see Fig. 1), is at −13 tap for minimum load and −8 tap at demand is lower for load mix with large commercial load as maximum load. The voltage regulator, Reg4 is always at tap it shows less sensitivity to voltage. TABLE VI 0. Voltage regulators 2 and 3 are single and two-phase devices CVR FOR IEEE 13-BUS FEEDER respectively and operate as optimization program instructs. Cap1 is a three-phase device and is OFF during minimum Load Minimum Load Maximum Load load and ON at maximum load condition and supplies required Composition CVR No CVR CVR No CVR 100% R 0.463 0.481 2.096 2.233 reactive power to maintain the voltage profile. Cap2, Cap3 and 70% R, 30% SC 0.465 0.480 2.098 2.234 Cap4 are single phase devices and are ON/OFF depending 50% R, 30% SC, 0.468 0.475 2.103 2.237 upon the load demand. The DGs are located at three-phase 20% LC nodes (see Fig. 1). Compared to minimum load condition, 1R-Residential, SC-Small Commercial, LC-Large Commercial the reactive power demand or generation for DG1 and DG3 does not change significantly, except DG3. In contrast with minimum loading, DG3 is absorbing reactive power in Phase B Computational Complexity On an average on a dual core during maximum load condition. Since Reg3 does not change i7 3.41 GHz processor with 16 GB of RAM, the Level-1 the tap position, Phase B of DG3 adjusts the set points to solutions are obtained in less than 5 sec and Level-2 solutions account for the increase in load. Similarly, since there is 10

3 300 without CVR control TABLE IX with proposed approach for CVR VOLT-VAR OPTIMIZATION RESULTS FOR 329-NODE FEEDER 2.5 Resuction in demand due to CVR 250 200 2 IEEE-329 Minimum Load Maximum Load 150 Phase A B C A B C 1.5 100 OPF solution from MATLAB 1 Reg1 Tap -6 -6 -6 1 1 1 50 Cap1 Status OFF OFF OFF ON ON ON 0.5 0 Cap2 Status OFF — — OFF — — Substation Power Demand (MW) 0 -50 Cap3 Status — OFF — — OFF — 0 2 4 6 8 10 12 14 16 18 20 22 24 Cap4 Status — — OFF — — OFF Reduction in Active Power Demand (kW) Time (Hours) DG1 qp (MVAR) 0.02 0.02 0.02 0.01 0.01 0.01 Fig. 6. IEEE-123 CVR Benefits Observed using the Proposed Approach DGp DG2 qDG(MVAR) -0.06 -0.06 -0.06 -0.03 -0.03 -0.03 (CV Rp = 0.6 and CV Rq = 3). p DG3 qDG(MVAR) -0.11 -0.11 -0.08 0.055 0.035 0.022 Optimal substation power flow and voltages using MATLAB no other VVC device between Reg1 and DG2, there is a Load (MW) 0.444 0.459 0.434 2.86 2.97 2.775 drastic change in optimal DG behaviour between the two load Min. Voltage (pu) 0.958 0.958 0.958 0.955 0.955 0.955 conditions. The feeder voltage characteristics are also shown in Max. Voltage (pu) 0.962 0.962 0.962 1.0063 1.0063 1.0063 Table VII. On an average the feeder operates close to minimum Avg. Voltage (pu) 0.959 0.959 0.959 0.974 0.972 0.976 Validation of substation power flow and voltages using OpenDSS voltage limit, i.e. 0.96 pu, for both load conditions. Load (MW) 0.445 0.462 0.438 2.87 2.98 2.79 The bi-level VVO approach is validated against OpenDSS. Min. Voltage (pu) 0.958 0.958 0.958 0.954 0.953 0.954 The optimal status of capacitor banks switch, voltage regula- Max. Voltage (pu) 0.962 0.962 0.962 1.0063 1.0063 1.0063 tor tap, and reactive power reference to the DGs, obtained Avg. Voltage (pu) 0.959 0.959 0.959 0.971 0.97 0.973 from MATLAB, are implemented on OpenDSS model for the 123-bus system. The substation power demand and feeder the 15-min control interval. Note that 123-bus test feeder voltage characteristics obtained using MATLAB are validated represents a practical mid-size primary distribution circuit. The against OpenDSS (see Table VIII). The system parameters test feeder has 123 buses and a total of 267 single-phase nodes. obtained from MATLAB closely match to those obtained from It should be noted that the Level-1 formulation scales well OpenDSS thus validating the VVO model. for larger feeders. This is because Level-1 solves an MILP that TABLE VIII is relatively easier to solve even for a large set of constraints. CVR FOR IEEE 123-BUS FEEDER The NLP problem in Level-2, however, is more difficult to scale for a large distribution system. In such cases, network Load Minimum Load Maximum Load reduction techniques are needed to represent the system with Composition CVR No CVR CVR No CVR fewer equations [31]. In the following section, we demonstrate 100% R 0.588 0.777 2.726 2.842 70% R, 30% SC 0.588 0.776 2.727 2.846 the scalability of the proposed approach using a 329-bus three- 50% R, 30% SC, 0.589 0.748 2.728 2.859 phase distribution feeder with the help of a simple network 20% LC reduction technique. 2R-Residential, SC-Small Commercial, LC-Large Commercial 3) 329-bus PNNL Taxonomy Feeder: The selected PNNL Finally, the CVR benefits obtained using the proposed taxonomy feeder includes 329 buses, where, the number of approach are reported. The total three-phase substation load nodes for phases A, B and C are 288, 298 and 274, respectively demand is compared to the case when VVO control is not (total 860 single-phase nodes) (see Fig. 2). The proposed enabled as shown in Fig. 6. On an average a reduction bi-level approach is implemented on 329-bus system. It is of around 150 kW is reported in net feeder active power observed that Level-1 problem (MILP) takes on an average demand. As expected the largest savings are reported during 20-sec. to solve, however, Level-2 problem (NLP) takes on an the minimum load condition. average 20-mins. Note that Level-2, for 329-bus system, solves The proposed approach is further validated using ZIP load for 4233 variables. In order to scale the Level-2 problem and to models for residential, commercial and large commercial obtain a solution within 15-min interval, the 329-bus system loads. The ZIP coefficients detailed in Table IV are used to is reduced using a simple network reduction technique. To obtain CVR factors for each case with different load mix. The reduce the network, we used the property of radial distribution total feeder load demand for the minimum and maximum load feeders; the nodes that do not include branches, loads, or condition are reported in Table VIII. voltage control devices are combined using the equations Computational Complexity: On an average on a dual core for the series system for the corresponding branches. Using i7 3.41 GHz processor with 16 GB of RAM, the Level-1 this method, the 329-bus system is reduced to a 184-bus solutions are obtained in less than 5 sec for 123-bus system. system where, the number of nodes in phase A , B and C For 123-bus, Level-1 solves 800 MILP equations with 1160 are 163, 171 and 156, respectively. After network reduction, variables. On an average it takes 2-min to solve Level- the total number of variables for the Level-2 problem are 2 problem for 123-bus system. The largest time taken for reduced to 2415. Since network reduction is exact both models solving Level-2 problem for 123-bus system is 4 min. The result in same power flow quantities. The Level-2 problem is Level-2 problem for 123-bus system solves 795 linear and implemented using 184-bus reduced network. The maximum 528 nonlinear equations (quadratic equalities) and with 1263 computation time required to solve the Level-2 problem for variables. The Level-1 and Level-2 solution times are within the reduced network model is 9 mins. 11

flow approximations are reasonably accurate, 2) the proposed 10 with proposed approach for CVR 500 without CVR control approach successfully coordinates the operation of legacy and reduction in demand due to CVR 8 400 new devices for CVR benefits, and 3) both Level-1 and Level-2 solutions are computationally efficient for a realtime operation. 6 300

4 200 REFERENCES

Total three-phase 2 100 [1] K. P. Schneider, J. C. Fuller, F. K. Tuffner, and R. Singh, “Evaluation of conservation voltage reduction (CVR) on a national level,” tech. rep.,

substation Power Demand (MW) 0 0 PNNL, Richland, WA (US), 2010. 2 4 6 8 10 12 14 16 18 20 22 24 Reduction in Active power Demand (kW) [2] K. Forsten, “Green circuits: Distribution efficiency case studies.,” tech. Time (hours) rep., EPRI,Palo Alto, CA: 2011. 1023518., 2011. Fig. 7. 329-bus CVR Benefits Observed using the Proposed Approach [3] “ANSI C84.1, American National Standard For Electric Power Systems (CV R = 0.6 and CV R = 3). p q and EquipmentVoltage Ratings (60 Hertz),” 2011. The CVR results obtained for maximum and minimum [4] M. E. Baran and M.-Y. Hsu, “Volt/var control at distribution substations,” IEEE Transactions on Power Systems, vol. 14, no. 1, pp. 312–318, 1999. load conditions are shown in Table IX. Note that the Level- [5] T. T. Hashim, A. Mohamed, and H. Shareef, “A review on voltage control 1 problem is implemented using full 329-bus feeder and the methods for active distribution networks,” Electrical review, 2012. Level-2 problem is implemented using reduced 184-bus feeder. [6] H. V. Padullaparti, Q. Nguyen, and S. Santoso, “Advances in volt-var control approaches in utility distribution systems,” in 2016 IEEE Power As the load is closely balanced, the behavior of each phase and Energy Society General Meeting (PESGM), pp. 1–5, July 2016. is almost similar. The voltage regulator at the substation is [7] B. Green, “Grid Strategy 2011: Conservation Voltage Reduction and at -6 tap for the minimum load and at 1 tap position for the Volt VAR Optimization in the Smart Grid,” tech. rep., EPRI,Palo Alto, CA: 2011. 1024482., 2010. maximum load condition. At minimum load, the three-phase [8] G. L. Barbose, “U.S. Renewables Portfolio Standards: 2017 Annual as well as single-phase capacitor banks are OFF. However, at Status Report,” 07/2017 2017. the maximum load condition, the three-phase capacitor is ON [9] A. Bokhari, A. Raza, M. Diaz-Aguil, F. de Len, D. Czarkowski, R. E. Uosef, and D. Wang, “Combined effect of CVR and DG penetration and single-phase capacitor banks are OFF. The reactive power in the voltage profile of low-voltage secondary distribution networks,” support from DG1 is same for all phases for both maximum IEEE Transactions on Power Delivery, vol. 31, pp. 286–293, Feb 2016. and minimum load conditions. The minimum voltage for all [10] M. Farivar, R. Neal, C. Clarke, and S. Low, “Optimal inverter var control in distribution systems with high pv penetration,” in 2012 IEEE Power the phases is at 0.958 pu at minimum load condition and and Energy Society General Meeting, pp. 1–7, July 2012. at 0.955 pu at maximum load. The average voltage along [11] H. Zhu and H. J. Liu, “Fast local voltage control under limited reactive the feeder is 0.959 and 0.972 at minimum and maximum power: Optimality and stability analysis,” IEEE Transactions on Power load conditions, respectively. The substation power demand Systems, vol. 31, pp. 3794–3803, Sept 2016. [12] V. Kekatos, L. Zhang, G. B. Giannakis, and R. Baldick, “Fast localized and feeder voltage characteristics obtained using MATLAB voltage regulation in single-phase distribution grids,” in 2015 IEEE are validated against OpenDSS (see Table IX). The system International Conference on Smart Grid Communications (SmartGrid- parameters obtained from MATLAB closely match to those Comm), pp. 725–730, Nov 2015. [13] E. DallAnese, S. V. Dhople, B. B. Johnson, and G. B. Giannakis, obtained from OpenDSS, validating the proposed VVO model. “Decentralized optimal dispatch of photovoltaic inverters in residential The CVR benefits obtained using the proposed approach for distribution systems,” IEEE Transactions on Energy Conversion, vol. 29, 24-hour duration are reported in Fig. 7. The total three-phase pp. 957–967, Dec 2014. [14] X. Su, M. A. S. Masoum, and P. J. Wolfs, “Optimal pv inverter reactive substation load demand is compared to the case when VVO power control and real power curtailment to improve performance of control is not enabled. On an average a reduction of around unbalanced four-wire lv distribution networks,” IEEE Transactions on 200kW is reported in the net feeder active power demand. Sustainable Energy, vol. 5, pp. 967–977, July 2014. [15] H. Ahmadi, J. R. Mart, and H. W. Dommel, “A framework for volt-var optimization in distribution systems,” IEEE Transactions on Smart Grid, VI. CONCLUSION vol. 6, pp. 1473–1483, May 2015. [16] T. V. Dao, S. Chaitusaney, and H. T. N. Nguyen, “Linear least-squares This paper presents a VVO approach for CVR by coor- method for conservation voltage reduction in distribution systems with dinating the operation of distribution system’s legacy volt- photovoltaic inverters,” IEEE Transactions on Smart Grid, vol. 8, age control devices and smart inverters. A bi-level VVO pp. 1252–1263, May 2017. [17] Z. Wang, J. Wang, B. Chen, M. M. Begovic, and Y. He, “Mpc- framework based on mathematical optimization techniques is based voltage/var optimization for distribution circuits with distributed proposed to efficiently handle the discrete and continuous generators and exponential load models,” IEEE Transactions on Smart control variables. The proposed approach solves OPF for a Grid, vol. 5, pp. 2412–2420, Sept 2014. [18] D. Ranamuka, A. P. Agalgaonkar, and K. M. Muttaqi, “Online voltage three-phase unbalanced electric power distribution system. The control in distribution systems with multiple voltage regulating devices,” Level-1 solves a MILP problem to obtain control setpoints IEEE Transactions on Sustainable Energy, vol. 5, pp. 617–628, 2014. for both legacy devices and smart inverters using linear [19] M. S. Hossan, B. Chowdhury, M. Arora, and C. Lim, “Effective cvr planning with smart dgs using minlp,” in 2017 North American Power approximation for three-phase OPF. Next, Level-2 freezes Symposium (NAPS), pp. 1–6, Sept 2017. the control for legacy devices and solves a NLP problem [20] F. Ding, A. Nguyen, S. Walinga, A. Nagarajan, M. Baggu, (with linear objective and quadratic constraints) to obtain a S. Chakraborty, M. McCarty, and F. Bell, “Application of autonomous smart inverter volt-var function for voltage reduction energy savings feasible and optimal solution by adjusting the setpoints for and power quality in electric distribution systems,” in 2017 IEEE Power DG control using an approximate nonlinear OPF model. The Energy Society Innovative Smart Grid Technologies Conference (ISGT), approach is thoroughly validated using three test feeders, pp. 1–5, April 2017. [21] M. Liu, C. A. Canizares, and W. Huang, “Reactive power and voltage IEEE 13-bus, IEEE 123-bus, and PNNL 329-bus taxonomy control in distribution systems with limited switching operations,” IEEE feeders. The results demonstrate that: 1) the proposed power Transactions on Power Systems, vol. 24, no. 2, pp. 889–899, 2009. 12

[22] L. Gan and S. H. Low, “Convex relaxations and linear approximation for optimal power flow in multiphase radial networks,” in 2014 Power Systems Computation Conference, pp. 1–9, Aug 2014. [23] W. Wang and N. Yu, “Chordal conversion based convex iteration algorithm for three-phase optimal power flow problems,” IEEE Transactions on Power Systems, vol. 33, pp. 1603–1613, March 2018. [24] J. Wang, A. Raza, T. Hong, A. C. Sullberg, F. de León, and Q. Huang, “Analysis of energy savings of cvr including refrigeration loads in distribution systems,” IEEE Transactions on Power Delivery, vol. 33, no. 1, pp. 158–168, 2018. [25] M. Farivar and S. H. Low, “Branch flow model: Relaxations and convexification: Part I,” IEEE Transactions on Power Systems, vol. 28, pp. 2554–2564, Aug 2013. [26] S. H. Low, “Convex relaxation of optimal power flow Part II: Exactness,” IEEE Transactions on Control of Network Systems, vol. 1, pp. 177–189, June 2014. [27] W. H. Kersting, Distibution System Modeling and Analysis. thrid edition, CRC, 2012. [28] A. Bokhari, A. Alkan, R. Dogan, M. Diaz-Aguil, F. de Len, D. Czarkowski, Z. Zabar, L. Birenbaum, A. Noel, and R. E. Uosef, “Experimental determination of the ZIP coefficients for modern residential, commercial, and industrial loads,” IEEE Transactions on Power Delivery, vol. 29, pp. 1372–1381, June 2014. [29] K. P. Schneider, B. A. Mather, B. C. Pal, C. W. Ten, G. J. Shirek, H. Zhu, J. C. Fuller, J. L. R. Pereira, L. F. Ochoa, L. R. de Araujo, R. C. Dugan, S. Matthias, S. Paudyal, T. E. McDermott, and W. Kersting, “Analytic considerations and design basis for the ieee distribution test feeders,” IEEE Transactions on Power Systems, vol. 33, pp. 3181–3188, May 2018. [30] Y. C. S. T. D. C. KP Schneider, DW Enge and R. Pratt, “Modern grid initiative distribution taxonomy,” November 2008. [31] A. Nagarajan, A. Nelson, K. Prabakar, A. Hoke, M. Asano, R. Ueda, and S. Nepal, Network Reduction Algorithm for Developing Distribution Feeders for Real-Time Simulators: Preprint. Jun 2017.