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2890 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 Voltage-Dependent Load Models in Unbalanced Optimal Power Flow Using Power Cones Sander Claeys , Student Member, IEEE, Geert Deconinck , Senior Member, IEEE, and Frederik Geth, Member, IEEE Abstract—Mathematical models representing the behavior of electrical loads are an important part of any optimal power flow problem. The current state-of-the-art in unbalanced optimal power flow mostly considers wye-connected, constant power loads. However, for applications such as conservation voltage reduction, it is crucial to model how the consumption of the loads depends on the voltage of the network. This article develops a unified framework to handle a wide variety of load types: delta- or wye-connected, constant power, constant current and exponential load models. Furthermore, it proposes a novel convex relaxation for the exponential model, using power cones, that is intersected next with a well-known semi-definite relaxation of unbalanced OPF. Finally, numerical results on the LVTestCase feeder are included for both the exact, non-linear equations and the convex relaxation, which show how considering the voltage sensitivity and connection type can lead to different objective values and voltage profiles. Additional results for 128 low-voltage networks, examine the quality of the solutions obtained with the proposed relaxation. Index Terms—Unbalanced optimal power flow, load models, voltage-dependent, power cones, convex relaxation. N OMENCLATURE Sets B P C U Ui Y Yi D Di G Buses. Phases, = {a, b, c}. Conductors, = {a, b, c, n}. Units (all loads and generators). Units connected to bus i ∈ B. All wye-connected units, ⊂ U. Wye-connected units at bus i ∈ B, ⊂ U. All delta-connected units, ⊂ U. Delta-connected units at bus i ∈ B, ⊂ U. Generators, ⊂ U. Manuscript received May 6, 2020; revised August 14, 2020 and November 18, 2020; accepted December 19, 2020. Date of publication January 19, 2021; date of current version June 21, 2021. This work was supported in part by CSIRO and the Ph.D. School of InnoEnergy and in part by the Australian Renewable Energy Agency (ARENA). The work of Sander Claeys was supported in part by Research Foundation—Flanders (FWO) through a Doctoral (Ph.D.) Strategic Basic Research under Grant 1S82518N. Paper no. TSG-00688-2020. (Corresponding author: Sander Claeys.) Sander Claeys and Geert Deconinck are with the Department of Electrical Engineering, KU Leuven/EnergyVille, 3001 Leuven, Belgium (e-mail: sander.claeys@kuleuven.be; geert.deconinck@kuleuven.be). Frederik Geth is with the Commonwealth Scientific and Industrial Research Organization, Newcastle, NSW 2304, Australia (e-mail: frederik.geth@csiro.au). Color versions of one or more figures in this article are available at https://doi.org/10.1109/TSG.2021.3052576. Digital Object Identifier 10.1109/TSG.2021.3052576 H L T Tr Loads, ⊂ U. Lines. Topology, ⊂ L × B × B. Reverse topology, = {lji|lij ∈ T }. Variables Ui pn Ui Udi Iu Idu Ilij Ilji Slu Plu,p Qlu,p Su Slij Slji Sm lij Sm lji Wi [P] Wdi Ll [P] Ldu Ml Xu Vu,p Voltages at bus i ∈ B. Phase-to-neutral voltages at bus i ∈ B. Phase-to-phase voltages at bus i ∈ B. Currents going to unit u ∈ U. Phase-to-phase currents for unit u ∈ D. Sending-end series current for line l ∈ L. Receiving-end series current for line l ∈ L. = Plu + jQlu . Per-phase consumption for unit u ∈ U. Active consumption for phase p ∈ P, u ∈ U. Reactive consumption for phase p ∈ P, u ∈ U. Per-conductor consumption for unit u ∈ U. Sending-end power for line l ∈ L. Receiving-end power for line l ∈ L. Sending-end matrix power var. for line l ∈ L. Receiving-end matrix power var. for line l ∈ L. Lifted phase volt. variable for bus i ∈ B. Lifted phase-to-phase volt. var. for bus i ∈ B. Lifted phase series current var. for line l ∈ L. Lifted ph.-to-ph. current var. for unit u ∈ D. PSD-constraint block matrix for line l ∈ L. Auxilary relaxation variable for unit u ∈ D. Load voltage magnitude, u ∈ H, p ∈ P. Parameters Umin i Umax i Uid,min Uid,max Td Zl Zkr l Ykr l au,p bu,p αu,p Bus voltage magnitude lower bound, i ∈ B. Bus voltage magnitude upper bound, i ∈ B. Bus ph.-to-ph. volt. magn. lower bound, i ∈ B. Bus ph.-to-ph. volt. magn. upper bound, i ∈ B. Delta transformation matrix for line l ∈ L. Four-wire series impedance for line l ∈ L. Kron-reduced series impedance for line l ∈ L. Kron-reduced series admittance for line l ∈ L. Active power prop. constant, u ∈ H, p ∈ P. Reactive power prop. constant, u ∈ H, p ∈ P. Active power exponent, u ∈ H, p ∈ P. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ CLAEYS et al.: VOLTAGE-DEPENDENT LOAD MODELS IN UNBALANCED OPTIMAL POWER FLOW USING POWER CONES βu,p P0u,p Q0u,p 0 Vu,p min Vu,p max Vu,p g,min Pu g,max Pu g,min Qu g,max Qu Reactive power exponent, u ∈ H, p ∈ P. Nominal active power, u ∈ H, p ∈ P. Nominal reactive power, u ∈ H, p ∈ P. Nominal voltage magnitude, u ∈ H, p ∈ P. Load volt. magn. lower bound, u ∈ H, p ∈ P. Load volt. magn. upper bound, u ∈ H, p ∈ P. Generator active power lower bound, u ∈ G. Generator active power upper bound, u ∈ G. Generator reactive power lower bound, u ∈ G. Generator reactive power upper bound, u ∈ G. I. I NTRODUCTION NCREASED penetration of distributed generation may lead to voltage quality issues in low-voltage distribution networks. In order to enable further increases, active control methods have been proposed. These methods often build on optimal power flow (OPF) variants. Some methods exclusively consider reactive power originating from inverters and capacitor banks [1], [2], referred to as ‘volt-var control’. Additionally, also active power can be dispatched, typically through curtailment, which may raise fairness considerations as to whom is curtailed [3], [4]. Given the crucial role of OPF in these approaches, recent work has made open-source implementations available [5], [6]. The consumption of loads depends in varying degrees on the voltage. Conservation voltage reduction (CVR) refers to lowering the voltage magnitude to reduce consumption [7]. Through load models, this effect can be included in OPF methods to better estimate the network capacity. Several load models have been proposed, amongst which exponential load models are widely used [8]. In the exponential model, the power consumption is proportional to the applied voltage magnitude raised to some power, P ∼ V α and Q ∼ V β . Table I shows typical values for the exponents α and β by device type. The reactive power is typically more sensitive to the voltage magnitude (β > α). An alternative load model is the ZIP model, which fits the voltage-dependent power consumption as a linear combination of terms with 0, 1 and 2 as exponent, respectively. Carneiro et al. compared the ZIP model and exponential model for fitting field measurements of a distribution substation. Both worked well for the active power (α = .767), but the ZIP model lacks higher order terms to represent the reactive power well in this study (β = 4.784 for the exponential model) [9]. Optimal power flow refers to a class of mathematical optimization problems which model the steady-state physics of power grids. The power flow equations are non-linear and lead to a non-convex feasible space. Convex relaxations encode such feasible space in a larger space which is convex [11]. This has several advantages. It can certify that a solution is globally optimal, or at least provide a lower bound on the global optimum. Furthermore, infeasibility of a relaxation certifies that the original problem is also infeasible [12]. Some authors propose methods to strengthen the results when the convex relaxation is not exact. Wei et al. solve a sequence of convexified penalization problems [13]. Liu et al. propose a method to recover a feasible solution when the relaxation is inexact [14]. Convex relaxations also play an important role I 2891 TABLE I VAN C UTSEM AND VOURNAS P ROVIDE F ITTED VALUES FOR THE E XPONENTS IN THE E XPONENTIAL L OAD M ODEL , α (ACTIVE P OWER ) AND β (R EACTIVE P OWER ), BY D EVICE T YPE [10] in approaches for global optimization of the non-linear OPF problem [15]. Unbalanced (O)PF generalizes the power flow equations to include the physics of phase unbalance. This is a more appropriate modelling framework for low-voltage distribution networks, which often contain single-phase loads. To the best of the author’s knowledge, Dall’Anese et al. were the first to publish a convex relaxation of unbalanced OPF [16]. Gan and Low advanced the understanding of convex relaxations of unbalanced OPF for radial networks [17], both with theoretical results and an empirical study on common test feeders. Note that these papers contain exclusively Kron-reduced branch models and constant-power, wye-connected loads, so the test feeders were simplified appropriately. Recently, several authors have extended this work further by incorporating other network elements. Liu et al. include gangoperated on-load tap changers [18] and Bazhrafshan et al. model voltage regulators (autotransformers) [19]. Zhao et al. first considered delta-connected constant-power loads [20]. Usman et al. derive extensions for ZIP loads in unbalanced 4-wire OPF and propose a first-order approximation for the constant-current component [21]. Molzahn et al. take another approach and propose a convex relaxation of the constantcurrent component. They introduce a variable for the voltage magnitude, and link it to the lifted voltage variable through a positive semi-definite constraint. The constant-current component is then set to be proportional to voltage magnitude [22]. This article proposes a convex relaxation using power cones for exponential loads in general. Power cones have recently been used to reformulate convex optimization problems to conic form [23], and are a natural candidate for convex relaxations of exponential load models. Popular test feeders for unbalanced power flow (PF), such as IEEE13, contain a variety of load models [24]. Firstly, loads can be both wye- and delta-connected. Secondly, loads are modelled as constant power, constant impedance or constant current. To use these feeders as benchmarks, unbalanced OPF should be able to represent all of their features. This article takes an important first step in doing so, by describing a load model that includes at least all of the load types that appear in common IEEE unbalanced PF benchmarks. To the best of the author’s knowledge, this work is the first to use the mathematical objects called power cones to model the convex relaxation of general voltage-dependent loads and generators. A case study across 128 low-voltage networks examines various aspects of the relaxation: computational complexity, 2892 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 constraint violation and optimality gap with respect to the exact, non-linear load model. Implementations of the models presented in this article are publicly available as part of a community-reviewed opensource project [6]. The non-linear formulation has been validated numerically against OpenDSS, a popular tool for unbalanced power flow. The next section defines preliminaries on the network model with a strong focus on how loads and generators are connected to general four-conductor buses, in ‘wye’ and ‘delta’. Section III, the core contribution of this work, develops extensions that model the voltage-dependent behavior. Section IV summarizes the non-convex network model as a feasible set, and introduces convex relaxations in the lifted variable space. This is followed by numerical results for a test case in Section V. Fig. 1. A wye-connected unit u ∈ Y connects to all four bus conductors. The consumed power is defined as the product of the phase current and the phase-to-neutral voltage. II. P RELIMINARIES The network consists of a set of buses, B. Each bus i ∈ B has four conductors. The first three are the phase conductors P = {a, b, c}, and the fourth is the neutral conductor n. Vectors have a bold face, and matrices additionally have a dark blue color. When a vector V is indexed on C, then V[P] is the vector of elements indexed on P ⊂ C. A. Buses and Units Each bus i ∈ B has an associated bus voltage vector Ui and pn phase-to-neutral voltage Ui , T Ui [P] , (1) Ui = Ui,a Ui,b Ui,c Ui,n = Ui,n T pn Ui = Ui,a − Ui,n Ui,b − Ui,n Ui,c − Ui,n . (2) A unit u ∈ U draws an amount of current and power in each conductor, T I [P] , (3) Iu = Iu,a Iu,b Iu,c Iu,n = u Iu,n Su = Ui (Iu )* , (4) where denotes the element-wise multiplication. A unit is either connected in wye or in delta. Figures 1 and 2 illustrate both. B. Wye-Connected Units Figure 1 illustrates a wye-connected unit. The power consumption Slu is defined as ⎡ ∗⎤ Ui,a − Ui,n Iu,a pn ∗ Slu = Ui (Iu [P])* = ⎣ Ui,b − Ui,n Iu,b ⎦, (5) ∗ Ui,c − Ui,n Iu,c Note that the power drawn in phase a for example, Su,a = l = Su,a − Ui,a (Iu,a )∗ , is generally not the same as Su,a ∗ Ui,n (Iu,a ) ; only when the neutral conductor is perfectly grounded, then Ui,n = 0, and the two power definitions become the same if Ui,n = 0 : Slu = Su [P]. (6) Fig. 2. A delta-connected unit u ∈ D connects to the three phase bus terminals. The consumed power is defined as the product of the phase current and the phase-to-phase voltage. C. Delta-Connected Units Figure 2 illustrates a delta-connected unit. The power consumption Slu is now defined in the delta reference frame instead. The delta voltage Uid is the phase-to-phase voltage, a linear transformation of the phase voltages. ⎤ ⎡ ⎤ ⎡ 1 −1 0 Ui,ab 1 −1⎦Ui [P] = Td Ui [P] (7) Udi = ⎣Ui,bc ⎦ = ⎣ 0 −1 0 1 Ui,ca The delta current Iud is a vector of currents drawn from one phase and injected in the next one, as illustrated by Figure 2. The phase current Iu [P] is a linear transformation of Idu , derived by applying KCL at every internal node of the delta configuration. The delta-connected unit does not have a neutral conductor, ∀u ∈ Di : ⎡d ⎡ ⎤ d ⎤ Iu,ab − Iu,ca 1 0 −1 d d ⎦ − Iu,ab 1 0⎦ Idu (8) = ⎣−1 Iu [P] = ⎣Iu,bc d d 0 −1 1 Iu,ca − Iu,bc (Td )T Iu,n = 0 (9) For delta-connected units, Sul is defined differently, ∀u ∈ Di : * (10) Slu = Udi Idu . CLAEYS et al.: VOLTAGE-DEPENDENT LOAD MODELS IN UNBALANCED OPTIMAL POWER FLOW USING POWER CONES 2893 D. Loads and Generators G. Lifted Network Model Generators u ∈ G are modelled as wye-connected units, G ⊂ Y. The generated apparent power, −Sul , is bounded by box constraints, There are two broad categories of network models: bus injection models (BIM) and branch flow models (BFM). Unlike BIM, BFM introduces a current variable for each branch, which appears in the branch power constraints. Dall’anese et al. proposed a semi-definite programming (SDP) relaxation for the multiphase BIM model [16]. Gan and Low also proposed a SDP relaxation of the BFM model, specifically for radial networks. They showed that the BIM and BFM relaxations are equivalent for radial networks, and BFM is numerically more stable because it avoids ill-conditioned constraints [17]. Since the low-voltage networks examined in the case study are radial, the BFM-SDP proposed by Gan and Low is used. First, define the following lifted variables Pg,min ≤ −Plu ≤ Pg,max , Qg,min ≤ −Qlu ≤ Qg,max . u u u u (11) All other units are loads, H = U \ G. For loads, Sul is determined by the voltage magnitude it sees, following the voltage dependency described in Section III. E. Lines The buses are connected by a set of lines. A line l ∈ L connects bus i to bus j, lij ∈ T . When line-charging is ignored, each line is characterized by its impedance matrix Zl . Ohm’s law relates the sending-end and receiving-end currents to the voltage, Uj = Ui − Zl Ilij Ilij + Ilji = 0. The power flow on both ends of the line is defined as * * Slij = Ui Ilij Slji = Uj Ilji . (12) (13) When the neutral is perfectly grounded at both ends of the line, then Ui,n = Uj,n = 0 and we can apply Kron-reduction to Zl , leading to if Ui,n = Uj,n = 0 : Uj [P] = Ui [P] + Zkr l Ilij [P]. (14) Zkr l is a transformation of Zl , directly linking the voltage drop in the phase conductors to only the current in the phase conductors [25]. Rewriting (14) and substituting it in the definition of Slij [P] and Slji [P], leads to a non-linear constraint linking these variables directly to Ui and Uj , * Slij [P] = Ui [P] Ykr , (15a) l Ui [P] − Uj [P] * , (15b) Slji [P] = Uj [P] Ykr l Uj [P] − Ui [P] kr −1 where Ykr l = (Zl ) . F. Kirchoff’s Current Law Kirchoff’s current law (KCL) for the phase conductors, is (16) Iu [P] + Ilij [P] = 0, u∈Ui lij∈T ∪T r where the summation over T ∪ T r captures both the sendingend and receiving-end branch currents. Note that some authors write (16) with separate terms for the sending-end and receiving-end currents, i.e., one term for elements over T and another for T r . Since the line model is not the main focus, this work uses the more compact notation in (16). KCL can be defined in terms of power variables, by multiplying the conjugate of (16) element-wise with the bus voltage vector Ui , (17) Su [P] + Slij [P] = 0. u∈Ui lij∈T ∪T r H m H Sm lij ← Ui [P](Ilij [P]) Slji ← Ui [P](Ilji [P]) , (18a) Wi [P] ← Ui [P](Ui [P])H Wj [P] ← Uj [P](Uj [P])H , (18b) H Ll [P] ← Ilij [P] Ilij [P] , (18c) where (.)H denotes the conjugate transpose. The arrow ← indicates this is the definition of a lifted variable, not a constraint which is included in any optimization problem. Note that Sm lij is a matrix variable, of which only the diagonal elements correspond to the usual definition of line flow power, H diag(Sm lij ) = Slij . By right-multiplying (14) with (Il ) and by itself, respectively, and substituting the lifted variables (18), we obtain the BFM constraints m kr Sm lij [P] + Slji [P] = Zl Ll [P], (19a) H kr Wj [P] = Wi [P] + Zkr l Ll [P] Zl H H kr kr m [P] [P] S Z − Sm − Z (19b) lij l l lij Sm Wi [P] lij [P] H Ml = m , Ml 0, (19c) Ll [P] Slij [P] where x 0 denotes that x is positive semidefinite (PSD). The relaxation is exact when rank(Ml ) = 1, ∀l ∈ L. For this condition to be sufficient, the topology has to be radial, and the lines have to be directed away from the reference bus (this can be achieved easily by changing the orientation if needed). For more details, refer to [17]. Gan and Low only considered wye-connected loads with grounded neutrals [17]. When the neutral is grounded, the power consumed by the load between each phase and the neutral, is equal to the power drawn in each phase alone (6). In this case, the load consumption Sul appears directly in the power balance constraint (17). The same does not hold for delta-connected loads, which cannot be represented as a set of convex constraints in the lifted variable space. Instead, Zhao et al. proposed a convex relaxation for delta-connected loads [20]. Define Xu , the outer product of the phase voltage Ui [P] and the delta current Idu , ∀u ∈ Di : H Xu ← Ui [P] Idu , (20) 2894 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 and introduce a PSD constraint Wi [P] Xu (Xu )H Ldu ∀u ∈ Di : 0. (21) Using the identities H Ui [P](Iu [P]) = Ui [P] Idu Td = Xu Td , H H Udi Idu = Td Ui [P] Idu = Td Xu , H (22) (23) The main contribution of this work is allowing the load consumption Slu to depend on the magnitude of the applied voltage. Section III discusses in detail how this is modelled. For now, note that the square of the phase voltage magnitude appears in the lifted network model as the diagonal elements of Wi [P], ⎡ ⎤⎡ ⎤H Ui,a Ui,a Wi [P] ← ⎣Ui,b ⎦⎣Ui,b ⎦ Ui,c Ui,c ⎡ ∗⎤ Ui,a 2 . . . Ui,a Ui,c ⎥ ⎢ .. .. .. (25) =⎣ ⎦. . . . ∗ Ui,c Ui,a ... |Ui,c |2 pn When the neutral is grounded, Ui = Ui [P], so when modelling wye-connected loads, the square of the applied voltage magnitudes is directly available as the diagonal elements of Wi [P]. For delta-connected loads, the applied voltages are the phase-to-phase voltage magnitudes instead. Define Wid , T Wdi = Td Wi [P] Td . (26) If the relaxation is exact, the diagonal elements of Wid will contain the square of the phase-to-phase voltage magnitudes, similar to how the diagonal elements of Wi [P] contain the square of the phase voltage magnitudes. The relaxation of the exponential model in the next section, uses either the diagonal elements of Wi [P] or Wdi , depending on whether the load is wye or delta-connected. III. VOLTAGE -D EPENDENT L OAD M ODEL A multi-phase load u consists of a number of individual l in loads, each consuming an amount of apparent power Su,p l steady-state, p ∈ {a, b, c}. Su,p has a real and imaginary part, the active power Plu,p and the reactive power Qlu,p . (27) The individual loads of a multi-phase load can be connected in two different ways: in ‘wye’ or in ‘delta’. For wye-connected units, Vu,p refers to the magnitude of the phase-to-neutral voltage, whilst for delta-connected units, it refers to the phase-to-phase voltage, pn (28a) ∀u ∈ Yi : Vu,p = Ui,p , (28b) where |x| denotes the magnitude of the complex variable x. Plu,p and Qlu,p depend on a voltage magnitude Vu,p , Plu,p = au,p (Vu,p )αu,p , Qlu,p = bu,p (Vu,p )βu,p , Su [P] and Slu can be written as linear transformations of Xu , ∀u ∈ D: (24) Su [P] = diag Xu Td Slu = diag Td Xu . l = Plu,p + jQlu,p Su,p d Vu,p = Ui,p , (29) where au,p and bu,p are the proportionality constants, and αu,p and βu,p are the exponents. The proportionality constants are 0 ) and usually not specified directly, but by pairs (P0u,p , Vu,p 0 0 0 (Qu,p , Vu,p ). For example, when Vu,p is equal to Vu,p , then the consumed active power will be P0u,p . The proportionality constants are related to these by P0u,p αu,p , au,p = 0 Vu,p Q0u,p bu,p = βu,p . 0 Vu,p (30) We allow negative values for au,p (or P0u,p ), since loads at the low-voltage level often represent the aggregate of several devices, including photovoltaic generators, which can make the net consumption negative when there is a lot of irradiance. The same goes for bu,p (or Q0u,p ), to support both inductive and capacitative loads. The exponents on the other hand are always non-negative. By fixing the exponents to specific values, the equations become equivalent to those describing common load models, i.e.,: • αu,p = βu,p = 0, constant power • αu,p = βu,p = 1, constant current • αu,p = βu,p = 2, constant impedance Since a ZIP-model is nothing more than a linear combination of these three, a set of three loads can equivalently represent a ZIP-model. A. Bounds The voltage magnitude Vu,p is bounded below and above min and V max , by Vu,p u,p min max ≤ Vu,p ≤ Vu,p . 0 ≤ Vu,p (31) The voltage-dependency (29) implies bounds on Plu,p and Qlu,p once the voltage magnitude is bounded, α α l,min min u,p l,max max u,p = au,p Vu,p , Pu,p = au,p Vu,p . (32) Pu,p The same applies to bu,p and Qlu,p . B. Lifting The lifted variable space contains variables W which represent the product of ordinary voltage variables U. Introduce the 2 lifted variable Wu,p ← Vu,p , of which the definition differs for wye and delta-connected units, ∀u ∈ Yi : Wu,p = Wi,pp , (33a) ∀u ∈ Di : (33b) d Wu,p = Wi,pp , d where, e.g., Wi,pp denotes the element (p, p) of the matrix d Wi defined in II-G. We will now reformulate the equations CLAEYS et al.: VOLTAGE-DEPENDENT LOAD MODELS IN UNBALANCED OPTIMAL POWER FLOW USING POWER CONES 2/αu,p ≥ Wu,p ⇔ Plu,p /au,p 2895 (37) Therefore, we can write this as the cone inclusion constraint Plu,p /au,p , 1, Wu,p ∈ P3 2/αu,p . (38) Similarly, when 0 < αu,p < 2 applies, (34) is a concave function and defines an upper bound on Plu,p instead. α /2 Plu,p ≤ au,p Wu,p u,p (39) αu,p /2 ⇔ Wu,p ≥ Plu,p /au,p (40) Fig. 3. The black line indicates the true voltage-dependency of Plu,p . The convex relaxation consists of expanding the black line to the gray area, which is bounded above or below by the affine function l(Wu,p ). au,p and αu,p determine whether the convex relaxation is bounded above or below by the black line. The brown arrows indicate qualitatively the gradient of the unbalanced OPF problem defined in the case study; see Section V for further discussion. Qlu,p behaves identical (but substitute au,p → bu,p and αu,p → βu,p ). This leads to a slightly different cone inclusion constraint, Wu,p , 1, Plu,p /au,p ∈ P3 αu,p /2 . (41) min , Pl,min /a ) Finally, the line connecting the points (Wu,p u,p u,p max l,max and (Wu,p , Pu,p /au,p ) is an affine underestimator if 0 < αu,p < 2, and an overestimator if 2 < αu,p , l(Wu,p ) = in terms of Wu,p instead of Vu,p , for use in the lifted variable 1 space. Substituting Vu,p = Wu,p 2 into (29) leads to Plu,p = au,p Wu,p αu,p 2 , Qlu,p = bu,p Wu,p βu,p 2 . (34) For the cases where the exponents αu,p , βu,p are 0 (constant power) or 2 (constant impedance), (34) simplifies to a linear function. For the other cases, this is either a stricly convex or concave function. Figure 3 shows graphically how αu,p and the sign of au,p determine whether the function is strictly convex or concave. C. Relaxation If αu,p is not exactly 0 or 2, then (34) describes either a strictly convex or strictly concave function. Figure 3 illustrates these functions graphically as black curves. Think of these black curves as sets of feasible points (Wu,p , Plu,p ); such sets are not convex. However, the sets indicated as gray areas, are convex and include all elements of the black curves. Therefore, replacing the black curves by the gray areas, constitutes a convex relaxation. In fact, the gray areas are the ‘smallest’convex sets which contain all feasible points, i.e., they are convex hulls, for known voltage magnitude bounds. The rest of this section will describe these convex hulls mathematically, using power cones and halfplanes. Only Plu,p is discussed. Qlu,p is analogous, with the mapping Plu,p → Qlu,p , αu,p → βu,p and au,p → bu,p . Let P3 (e) denote the three-dimensional power cone with exponent 0 < e < 1, formally defined (35) P3 (e) = x ∈ R3 | x1e x21−e ≥ |x3 |, x1 , x2 ≥ 0 . Whilst Plu,p can be both negative and positive, Plu,p /au,p is always positive. What follows is valid for all values of au,p , both positive and negative. For αu,p > 2, (34) is a convex function, and defines a lower bound on Plu,p . α /2 Plu,p ≥ au,p Wu,p u,p (36) l,max − Pl,min Pl,min 1 Pu,p u,p u,p min W + − W .(42) u,p u,p max min au,p Wu,p − Wu,p au,p For the constant-current case (αu,p = 1), the power cone simplifies to a rotated second-order cone of dimension 3, RSoC(3). The constant power and impedance models are modelled by affine sets instead. In summary, the voltage-dependent unit model is αu,p = 0 Plu,p = P0u,p , l Pu,p /au,p ≥ l(Wu,p ), αu,p = 1 Wu,p /2, 1, Plu,p /au,p ∈ RSoC(3), (43a) (43b) αu,p = 2 Plu,p = au,p Wu,p , (43c) l Pu,p /au,p ≥ l(Wu,p ), 0 < αu,p < 2 Wu,p , 1, Plu,p /au,p ∈ P3 αu,p /2 , 2 < αu,p (43d) l P u,p /αu,p ≤ l(Wu,p ), Plu,p /au,p , 1, Wu,p ∈ P3 2/αu,p . (43e) Finally, the maximum gap Pmax u,p between the power cone and , can the affine constraint l, which occurs at lifted voltage Wu,p be calculated analytically max )αu,p − (V min )αu,p 2/(αu,p −2) (V 2 u,p u,p = , (44a) Wu,p max )2 − (V min )2 αu,p (Vu,p u,p αu,p 2 − l(W ). (44b) Pmax u,p = au,p (Wu,p ) u,p Note that substituting αu,p ← 1, leads to the same formula derived by Molzahn et al. for the constant-current case [22] (using the identity a2 − b2 = (a − b)(a + b) and cancelling the max − V min ) in nominator and denominator). term (Vu,p u,p The maximum gap relative to the nominal consumption, Pmax u,p /|au,p |, depends only on the exponent value and the voltage bounds. Table II shows numerical values for several exponent values and voltage bounds. The largest gap is 4.1% of the consumption at nominal voltage. The model for reactive power is obtained by simply replacing Plu,p by Qlu,p , au,p by bu,p and αu,p by βu,p . 2896 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 TABLE II T HIS TABLE S HOWS THE M AXIMUM D EVIATION I NTRODUCED BY THE R ELAXATION FOR THE ACTIVE AND R EACTIVE P OWER C ONSUMPTION , FOR VARIOUS VALUES OF THE E XPONENT AND VOLTAGE B OUNDS A. Voltage Bounds The magnitude of the phase-to-phase and phase-to-neutral voltages is bounded, to ensure proper functioning of the connected devices. When the neutral is grounded, Ui,n = 0, then the phase-to-neutral bounds become if Ui,n = 0 : D. Constant-Current Loads Constant-current loads are defined to have a power consumption proportional to the applied voltage magnitude, P, Q ∼ V. This is a special case of the exponential model, where αu,p = βu,p = 1. Molzahn et al. propose a relaxation for constant-current loads [22]. They introduce a variable Vu,p representing the voltage magnitude. Linking it to the lifted network model, would require Vu,p = Wu,p . This defines a non-convex space in (Wu,p ,Vu,p ), so instead Molzahn et al. relax this constraint 2 , which they cast as a 2x2 PSD constraint, to Wu,p ≥ Vu,p min , and add an affine constraint going through the points (Wu,p min ) and (W max , V max ). Then finally, Molzahn et al. define Vu,p u,p u,p l Pu,p = c Vu,p , for some proportionality constant c, where we ignore the Z and P component. We will now argue that this is in fact equivalent to (43b), under mild conditions. Substituting Vu,p ← 1c Plu,p in Wu,p ≥ 2 leads to c2 W l 2 Vu,p u,p ≥ (Pu,p ) . This is identical to the square of (39) with αu,p = 1 and c = au,p (squaring does not create additional solutions since Wu,p ≥ 0 and Plu,p /au,p ≥ 0). Similary, the affine constraint introduced by Molzahn et al. is equivalent to l(Wu,p ) after substituting out Vu,p . In fact, Molzahn et al. introduce a voltage magnitude variable V per bus, and not per load. Therefore, if multiple loads are present at a bus, the same voltage magnitude approximation error is shared by all loads, whilst in our formulation, this can be chosen independently for each load. For multiple constant-current loads at a single bus, the equivalence does not hold anymore. In practice however, this rarely occurs, as loads are often aggregated per bus anyways. In summary, our model is complementary to that of Molzahn et al., as it is practically equivalent for αu,p = βu,p = 1, whilst extending it to values for αu,p , βu,p other than 0, 1 and 2. IV. F ORMULATIONS An optimization problem has variables, constraints and an objective function. The variables and constraints define a feasible set: the set of all values of those variables for which all constraints are satisfied. In this section, we will summarize explicitly all variables and constraints that together define the feasible set of the optimization problem. Later, in the case study, we add an objective function to obtain a well-defined optimization problem. First, we introduce the main network constraints of interest: bounds on the phase-to-phase and phase-to-neutral voltage magnitude. Thereafter, we summarize the exact, non-convex formulation, followed by a convex relaxation of the entire model. Umin ≤ |Ui [P]| ≤ Umax . i i (45) The phase-to-phase voltages are linear transformations of the phase voltages, and are bounded by (46) Uid,min ≤ Td Ui [P] ≤ Uid,max . B. Non-Convex Formulation SU This exact, non-convex formulation of the problem has only power (S) and voltage variables (U). When a wye-connected unit is connected to a bus with a perfectly grounded neutral, then Ui,n = 0, and the current variables disappear from (5), if Ui,n = 0 : Slu = Su [P]. (47) For delta-connected units however, expressing Su [P] as a function of Ui [P] and Slu is more involved, ∀u ∈ Di : ⎤ ⎡ l l Su,a Su,c U − ⎢ i,a Ui,a −Ui,b Ui,c −Ui,a ⎥ ⎢ l ⎥ l ⎢ ⎥ Su,b Su,a ⎥ (48) U Su [P] = ⎢ − ⎢ i,b Ui,b −Ui,c Ui,a −Ui,b ⎥. ⎢ ⎥ l l ⎣ ⎦ Su,b Su,c Ui,c Ui,c −U − Ui,b −U i,a i,c Combining this with the non-convex branch model and nodal power balance equations, we obtain the exact, nonconvex formulation of the problem. feasible set SU variables Su [P], Slu ∀u ∈ U Ui [P] ∀i ∈ B Slij [P], Slji [P] ∀lij ∈ T subject to (29) ∀u ∈ H, (11) (6) ∀u ∈ Yi , (48) (17), (28) (45), (46) ∀u ∈ G ∀u ∈ Di ∀i ∈ B, (15) ∀lij ∈ T . C. Convex Relaxation SWL This formulation combines the BFM-SDP model by Gan and Low [17], the convex relaxation of delta-connected loads by Zhao et al. [20], and the proposed convex relaxation of exponential loads Section III-C. The BFM-SDP model contains power flow variables (S), lifted voltage variables (W) and lifted current variables for the lines (L); therefore, this formulation is referred to as SWL. Most of the constraints are defined in the previous sections; only the voltage bounds remain. CLAEYS et al.: VOLTAGE-DEPENDENT LOAD MODELS IN UNBALANCED OPTIMAL POWER FLOW USING POWER CONES Fig. 4. The LVTestCase feeder supplies power to 55 households, which are modelled as single-phase loads. Two generators are added to the feeder, V and G. The diagonal elements of Wid contain the square of the phase-to-phase voltage magnitude, Uid , (46) becomes T ≤Uid,max Uid,max . Uid,min Uid,min ≤diag Td Wi [P] Td (49) Similarly, the phase-to-neutral bounds (45) become Umin Umin ≤ diag(Wi [P]) ≤ Umax Umax . i i i i (50) In summary, feasible set SWL variables Su [P], Slu ∀u ∈ U Wi [P] ∀i ∈ B m Sm lij [P], Slji [P], Ll [P] ∀lij ∈ L Xu , Ldu ∀u ∈ D subject to (43) ∀u ∈ H, (11) ∀u ∈ G (6) ∀u ∈ Yi , (24), (21) ∀u ∈ Di (17), (33) (49), (50) ∀i ∈ B, (19) ∀lij ∈ T . V. C ASE S TUDY A. Exponent Sensitivity Study The case study in this section minimizes the generation cost to supply a set of loads in the IEEE benchmark feeder LVTestCase (at time step 1000). Two things are studied: the error made by not considering the voltage-dependency of the loads, and the gap introduced by the proposed relaxation. The SU formulation we developed, can also be used to solve unbalanced PF cases. By doing so, we validated it against OpenDSS for constant power loads, and matched it very closely (the error on the bus voltage magnitude was less than 1E-7 in p.u.) [6]. The results presented here, were obtained with Ipopt as a solver for the SU formulation, and with Mosek v9.1 for the SWL formulation. To the best of the author’s knowledge, Mosek (and only from v9 onwards) is 2897 Fig. 5. α and β are set to the same value, and are varied from 0 to 3. The optimality gap is nearly zero when α, β ≥ 2. The objective is normalized with respect to the constant power objective value (α = β = 0), f = $142. the only solver which accepts semi-definite and power cone constraints simultaneously. In order to match the scope of the model presented in this article, we slightly modify LVTestCase; we remove the substation transformer and update the nominal voltage level to 230V LN, which is more common in continental Europe. For the details, please refer to the full OpenDSS specification [26]. The original network is very lightly loaded, so we multiply each load by a factor of 3, increasing the total loading per phase to [53.2, 48.4, 42.5] kW and [17.5, 15.9, 14.0] kvar. LVTestCase is an IEEE benchmark feeder for unbalanced PF. Therefore, we have to augment it with OPF specific data. Firstly, we impose bounds on the phase-to-neutral voltage pn magnitude, 207V ≤ Ui √ ≤ 253V, d and on the √ phase-to-phase voltage magnitude, 207 3V ≤ Ui ≤ 253 3V (0.9 and 1.1 in per-unit). Next, Figure 4 shows where two generators are added to the network. A first generator G generates at a cost cG = 1.0$/kW in the middle of the network. It can inject in each phase at most a third of two times the total loading. A second generator, V, is added at the start of the network, with a cost which is 20% larger, cV = 1.2$/kW. It represents the exchange with the medium-voltage network; the voltage at the bus it is connected to, is fixed to a balanced phasor of 1 p.u., and the exchange is unbounded, acting as a slack bus generator in this system. The objective is minimizing the total generation cost to supply the loads, (51) cu 1T Pgu . f = u∈G Note that it is not required that the exponents for active power (α) and for reactive power (β) are the same; this is only done to simplify the figures. First, all loads are wye-connected as specified in the original LVTestCase. Figure 5 shows the objective for the SU and SWL formulation, for different values of the voltage sensitivity exponent α and β. If the voltage sensitivity is present but not included in the model, then the setpoint obtained for generator G will be suboptimal; this is illustrated by the ‘constant power decision’ case. Figure 6 illustrates what is happening when the voltage sensitivity increases. These results are local optima obtained for 2898 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 Fig. 8. The behavior is similar to wye-connected loads, besides that ignoring the voltage sensitivity leads to a bigger increase in the generation cost, (α = β = 0). The objective is normalized with respect to f = $142. Fig. 6. This figure illustrates the distribution of the phase-to-neutral voltage magnitude, across all buses and phases. As the load becomes more sensitive to the voltage, the optimal dispatch will lower the resulting voltage profile, to prevent increasing the load too much. These results are local optima obtained for the SU formulation. Fig. 7. When α ≥ 2, the relaxation only introduces additional, non-physical solutions which consume more active power. This will increase the objective of an OPF problem that minimizes the generation cost, and therefore the gap p vanishes for α ≥ 2. the SU formulation. When the loads are constant power, then the cheaper generator G will supply as much load as it can, constrained by the bounds on the voltage magnitude. However, once the active power exponent α is greater than zero, this comes at a tradeoff: utilizing generator G to the fullest, leads to high voltage magnitudes and increased load consumption. Therefore, as α is increased, eventually the optimal dispatch redispatches the generators to yield a voltage profile with lower voltage magnitudes. In order to examine the gap due to the relaxation of the voltage-dependency, define αu,p p l 2 , (52) = Pu,p − au,p Wu,p u∈H p∈P a measure for the distance of the solution to the ‘black line’ in Figure 3, which we will refer to as the ‘active power gap’ (and similarly q for the reactive power gap). When the relaxation is tight, this is zero. Figure 7 shows how the gap evolves for different values of α/β. As indicated by Figure 3, when 0 ≤ α ≤ 2, the relaxation allows additional solutions which Fig. 9. The optimal voltage profile for delta-connected loads, looks quite different. The optimal dispatch results in a voltage profile closer to 1 p.u. than for wye-connections. consume less power. When the objective is to minimize generation cost, this is exploited to reduce the effective load. When α ≥ 2, the relaxation only introduces additional solutions where the load consumes higher amounts of active power; this explains why p vanishes. Note that this is an empirical observation, and most likely cases exist where this does not hold. For example, a case which is infeasible before the relaxation due to overvoltages, could be made feasible by increasing the active power consumed by the loads. When the loads are connected in delta instead, the results change. Figure 8 shows the same comparison of the obtained objective values, which follow a similar trajectory as the wyeconnected loads. Figure 9 shows the corresponding voltage profile. Unlike the wye-connected loads, the optimal dispatch lowers the voltage profile more for the same voltage sensitivity exponent α/β. B. Large Scale Study The previous case study focused on gaining a qualitative understanding of the proposed optimization problem, and how the solution changes with the exponent values α, β. Next, the CLAEYS et al.: VOLTAGE-DEPENDENT LOAD MODELS IN UNBALANCED OPTIMAL POWER FLOW USING POWER CONES same optimization model is applied to a large set of different networks. Fig. 5 shows that the optimality gap was relatively large for, e.g., α, β = 1.0, whilst for α, β ≥ 2.0 it was small. Therefore, two scenarios are considered: α, β = 1.0 and α, β = 3.0. Rigoni et al. published a data set of 128 synthetic, lowvoltage networks. These are based on real feeders in the UK, and are supplied with several time series for, e.g., loads and photovoltaic generators in both summer and winter [27]. This case study applies the summer load profiles with a power factor of 0.95, at time step 144 (11:55-12:00). Furthermore, the networks were slightly modified in two ways. The original networks contain a single 11/0.416 kV transformer between the reference bus and the rest of the feeder; this transformer was removed, and the reference bus voltage was set to 0.230 kV LN instead. Secondly, the original networks have a very high ratio of lines to loads; this is sometimes caused by tracing the networks from GIS data. Note that these lines have zero admittance to ground. When they are connected in series without any other components between them, they are merged into a single line for this case study. The same phase-to-neutral voltage bounds are applied as pn before, 207V ≤ Ui ≤ 253V. The generator V is added at the reference bus at the head of the feeder. The location of generator G is systematically determined by the following procedure. First, solve the power flow for the network when only generator V is active. Next, calculate the maximum phaseto-neutral voltage magnitude at each load bus and sort them. Finally, place generator G at the load bus in the middle of this list (or just before for an even number of load buses). This systematically yields a placement similar to the previous case study. For the exact placement of the generators, refer to [28]. The generator cost remains the same. The generator bounds are unconstrained, in order to ensure a fair comparison. If these bounds are active at the solution, then this would hide differences in the dispatch between the SWL and SU formulation. This yields a library of 128 instances of the optimization problem, where the number of loads varies from 2 up to 302. The number of buses is proportional to the number of loads, with on average 2.04 buses for each load. Figure 10 shows that the solve time increases linearly in the number of buses; the SWL formulation takes on average 3.3 times longer to solve than the exact SU formulation. These results were obtained with constant power loads. Adding power cones to model exponential loads (α = β = 3.0) in the SWL formulation, did not impact the solve time in a clear way. On average, it took 4% longer, but varied from 57% faster up to 177% slower. Note that 5/128 instances did not converge due to numerical issues for αu,p = βu,p = 3.0, and are left out of the results. As discussed in Section II-G, for the BFM-SDP relaxation to be exact, Ml should be of rank 1 for all lines l ∈ L. A PSD matrix of rank 1, has a single non-zero, positive eigenvalue. Due to numerical errors, even when the relaxation is exact, Ml will have small non-zero eigenvalues. Practically, we evaluate whether the matrix has rank 1 by requiring that λ1 /λ2 , the ratio of the highest to the second highest eigenvalue, is sufficiently high. Figure 11 shows that the lowest eigenvalue ratio for all 2899 Fig. 10. The solve time increases approximately linearly in the number of buses. Solving the SWL formulation takes on average 2.6 times longer than SU. These instances were solved with constant power loads, α = β = 2.0; adding the power cones to model α = β = 3.0, only increased the SWL solve time by 4% on average. Fig. 11. For the BFM-SDP relaxation to be exact, the ratio λ1 /λ2 should be sufficiently high for all lines. This figure shows the distribution of the lowest ratio across all lines, for all 128 instances. lines and all networks, is at least 1E5. Past experience with Mosek showed that ratios above 1E4 were sufficient to yield practically identical results as the SU formulation. However, the relaxation as a whole is not exact, because the power cone relaxation is not tight. Figure 12 shows the active power gap. In order to compare it across network instances of different size and total loading, the gap has been normalized with respect to the total apparent power at nominal voltage, ⎞2 ⎛ ⎞2 !⎛ ! ! P0u,p ⎠ + ⎝ Q0u,p ⎠ . St = "⎝ u,p (53) u,p As observed in the first case study, the active power gap goes to zero when αu,p = 3.0. Even then, the relaxation is not tight, because the reactive power gap remains non-zero as shown in Figure 13. In the remainder of this section, we will examine the ‘quality’ of the inexact solution obtained by solving the SWL formulation. Solving the optimization problem, yields a set point for generator G. When the SWL formulation is inexact, applying this set point will cause different voltage levels than indicated by the Wi variables in the SWL solution. The actual network 2900 Fig. 12. The active power gap p is several order of magnitude smaller when αu,p is 3.0 instead of 1.0; for αu,p = 3.0, it is practically zero. Fig. 13. The reactive power gap q is not influenced much by the voltage sensitivity exponent βu,p . state is determined by solving a power flow problem, where the power injection by generator G is fixed to the obtained value. Figure 14 shows that the error on the voltage magnitude, is always smaller than 1E-3 pu, and typically smaller than 1E-4. From this, it follows that when using the SWL formulation in a control scheme to generate set points for generator G, the voltage constraint violation is at most 1E-3 across all 128 networks (and in 90% of the instances, smaller than 1E-4). Similarly, the total network loss predicted by the inexact SWL solution was at most 1% different, and less than 0.1% in 93% of the network instances. Finally, it is of interest how different the inexact SWL solution is from the best known solution to the SU formulation. Note that solutions obtained with the SU formulation, are local optima which are not guaranteed to be the global optimum, and will always have an objective value which is larger than or equal to the SWL solution for the same minimization problem. Figure 15 shows the ‘optimality gap’, which is the difference between the SWL and SU objective values, relative to the SU value. For the case where αu,p = βu,p = 3.0, the gap is several orders of magnitude smaller. Figure 16 shows how much the active power set point for SWL deviates relatively IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 Fig. 14. The voltage magnitude predicted by the inexact SWL solutions, differs at most by 1E-3 pu from the actual voltages obtained when applying the SWL generator set points, and typically by less than 1E-4 pu. Fig. 15. The optimality gap is the difference between the objective of the SU and SWL formulation, relative to the SU objective value. The gap is several orders of magnitude lower for αu,p = βu,p = 3.0. from the set point for SU. The deviation is very small when αu,p = 3.0, which is consistent with earlier results; the active power gap for the power cones is nearly zero for those cases, and the inexact SWL solution dispatches better active power set points. For the absolute generator set points, refer to [28]. Figure 17 shows that the same does not hold for the reactive power; whether βu,p is smaller or larger than 2, has a limited effect on the reactive power gap. This is to be expected, as increasing/decreasing the reactive power consumption does not affect the generation cost in an obvious way. Consuming more reactive power might bring down the voltage magnitude and the associated voltage-dependent active consumption, but it might also increase the network losses. This differs from the active power consumption, where an increase will clearly increase the generation cost. C. Multiple Distributed Generators The previous case studies are likely to differ from a realistic scenario because they have identical exponents for active and reactive power and they only contain a single distributed generator (DG). This helped simplify the analysis and discussion CLAEYS et al.: VOLTAGE-DEPENDENT LOAD MODELS IN UNBALANCED OPTIMAL POWER FLOW USING POWER CONES 2901 TABLE III T HE P ROBLEM IS S OLVED FOR T WO PAIRS OF N ON -I DENTICAL α, β AND FOR VARIOUS N UMBERS OF DG S . D EVIATIONS ARE R EPORTED FOR THE DG S ET P OINTS AND VOLTAGE M AGNITUDE E RROR . N OTE T HAT ‘S UM ’ D ENOTES THE S UM OF THE A BSOLUTE VALUES , AND ‘M AX ’ THE M AXIMUM Fig. 16. The deviation of the active power set point for generator G (between SWL and SU), is determined as the sum of the absolute deviations per phase, relative to the total amount of active power. The deviations are several orders of magnitude smaller for αu,p = βu,p = 3.0. results. For the set point deviation, the maximum decreases as the number of DGs increases, which can be expected as the size of each individual DG decreases as well. The sum of the set point error increases with the number of DGs, but remains small relative to the delivered load (i.e., at most 2.28 kW cummulative error for 144 kW load). The error on the voltage magnitude is calculated as in the previous case study, and remains small, with 0.0012 pu as the highest deviation. VI. C ONCLUSION Fig. 17. The deviation of the reactive power set point for generator G, is affected less than the active power by whether αu,p , βu,p are larger or smaller than 2. of the results. This final case study shows that αu,p = βu,p and increasing the number of DGs, lead to similar results. The case study is constructed by adding a variable number of DGs to the LVTestCase feeder. This feeder has a total of 55 loads. If, e.g., 10 DGs are added, they are assigned to the 10 last buses (ordered by name) which have a load attached to them. The total available dispatchable generation is 432 kW and 141 kvar, about three times the total consumption. This level ensures that even for 30 DGs, at least 20% of them have inactive generation bounds at the solution, allowing for a fair comparison of the obtained SU and SWL set points. The available dispatchable generation is divided equally across the DGs and phases; note that therefore the bounds are the same, but not necessarily the set points. Table III summarizes the results of this case study with multiple DGs. The exponent values correspond to loads consisting of air conditioners and battery charge (Table I). The difference between the DG set points obtained with the SU DG and SWL formulation, PDG g,p +jQg,p , is significantly smaller when α is 2.59 instead of 0.5, consistent with the previous This article presents a general load model flexible enough to represent both wye- and delta-connected, constant power, constant impedance, constant current and ‘exponential’ load models. This includes all load types present in common unbalanced power flow test feeders [26]. Additionally, convex relaxations were derived using power cones for the voltagedependent load model. As illustrated by the case study, these models can be used to consider conservation voltage reduction in optimization models for distribution networks when minimizing the generation cost. Though the relaxation is not tight in all conditions, it yields high-quality solutions which can certify the near-optimality of solutions to the non-linear problem. Molzahn et al. observed that a suitable relaxation of constant-current loads with a negative proportionality coefficient often leads to a tight solution, unlike ones with a positive coefficient [22]. This article extends this observation to general voltage-dependent units, with a qualitative argument and numerical results. When the voltage-dependency of the active power consumed by loads is quadratic or rises even faster as a function of the voltage, the proposed relaxation is significantly tighter. In this case, the relaxation only allows the active power consumption to increase, which is rarely more optimal when the objective is to minimize generation cost. This was confirmed by a case study covering 128 multiphase, low-voltage networks. The voltage-dependent model using power cones can be used in balanced OPF problems as well. Future work will focus on extending unbalanced OPF further to incorporate also explicit four-wire models, which are not Kron-reduced. 2902 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 4, JULY 2021 ACKNOWLEDGMENT The views expressed herein are not necessarily the views of the Australian government, and the Australian government does not accept responsibility for any information or advice contained herein. R EFERENCES [1] M. H. K. Tushar and C. Assi, “Volt-VAR control through joint optimization of capacitor bank switching, renewable energy, and home appliances,” IEEE Trans. Smart Grid, vol. 9, no. 5, pp. 4077–4086, Sep. 2018. [2] J. Barr and R. Majumder, “Integration of distributed generation in the Volt/VAR management system for active distribution networks,” IEEE Trans. Smart Grid, vol. 6, no. 2, pp. 576–586, Mar. 2015. [3] P. Lusis, L. L. H. Andrew, S. Chakraborty, A. Liebman, and G. Tack, “Reducing the unfairness of coordinated inverter dispatch in PV-Rich distribution networks,” in Proc. IEEE Milan PowerTech, Jun. 2019, pp. 1–6. [4] M. Z. Liu et al., “On the fairness of PV curtailment schemes in residential distribution networks,” IEEE Trans. Smart Grid, vol. 11, no. 5, pp. 4502–4512, Sep. 2020. [5] V. Rigoni and A. Keane, “Open-DSOPF: An open-source optimal power flow formulation integrated with OpenDSS,” in Proc. IEEE PES Gen. Meeting, Apr. 2020, pp. 1–5. [6] D. M. Fobes, S. Claeys, F. Geth, and C. Coffrin, “PowerModelsDistribution.jl: An open-source framework for exploring distribution power flow formulations,” Electr. Power Syst. Res., vol. 189, Dec. 2020, Art. no. 106664. [7] H. Ahmadi, J. R. Martí, and H. W. Dommel, “A framework for VoltVAR optimization in distribution systems,” IEEE Trans. Smart Grid, vol. 6, no. 3, pp. 1473–1483, May 2015. [8] Z. Wang, J. Wang, B. Chen, M. M. Begovic, and Y. He, “MPC-based voltage/var optimization for distribution circuits with distributed generators and exponential load models,” IEEE Trans. Smart Grid, vol. 5, no. 5, pp. 2412–2420, Sep. 2014. [9] A. S. Carneiro, L. F. Araujo, J. L. R. Pereira, P. A. Garcia, I. D. Melo, and M. B. Amaral, “Static load modeling based on field measurements,” in Proc. IEEE Manchester PowerTech, Jun. 2017, pp. 1–5. [10] T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. Dordrecht, The Netherlands: Kluwer, 1998. [11] D. K. Molzahn and I. A. Hiskens, “A survey of relaxations and approximations of the power flow equations,” Found. Trends Electr. Energy Syst., vol. 4, nos. 1–2, pp. 1–221, 2019. [12] S. H. Low, “Convex relaxation of optimal power flow—Part I: Formulations and equivalence,” IEEE Trans. Control Netw. Syst., vol. 1, no. 1, pp. 15–27, Mar. 2014. [13] W. Wei, J. Wang, N. Li, and S. Mei, “Optimal power flow of radial networks and its variations: A sequential convex optimization approach,” IEEE Trans. Smart Grid, vol. 8, no. 6, pp. 2974–2987, Nov. 2017. [14] Y. Liu, J. Li, and L. Wu, “ACOPF for three-phase four-conductor distribution systems: Semidefinite programming based relaxation with variable reduction and feasible solution recovery,” IET Gener. Transm. Distrib., vol. 13, no. 2, pp. 266–276, 2019. [15] S. Gopinath et al., “Proving global optimality of ACOPF solutions,” Electr. Power Syst. Res., vol. 189, Dec. 2020, Art. no. 106688. [16] E. Dall’Anese, G. B. Giannakis, and B. F. Wollenberg, “Optimization of unbalanced power distribution networks via semidefinite relaxation,” in Proc. North Amer. Power Symp., Sep. 2012, pp. 1–6. [17] L. Gan and S. H. Low, “Convex relaxations and linear approximation for optimal power flow in multiphase radial networks,” in Proc. Power Syst. Comput. Conference, Aug. 2014, pp. 1–9. [18] Y. Liu, J. Li, L. Wu, and T. Ortmeyer, “Chordal relaxation based ACOPF for unbalanced distribution systems with DERs and voltage regulation devices,” IEEE Trans. Power Syst., vol. 33, no. 1, pp. 970–984, Jan. 2018. [19] M. Bazrafshan, N. Gatsis, and H. Zhu, “Optimal power flow with stepvoltage regulators in multi-phase distribution networks,” IEEE Trans. Power Syst., vol. 34, no. 6, pp. 4228–4239, Nov. 2019. [20] C. Zhao, E. Dall’Anese, and S. H. Low, “Optimal power flow in multiphase radial networks with delta connections: Preprint,” Nat. Renew. Energy Lab. (NREL), Golden, CO, USA, Rep. NREL/CP-5D0067852, 2017. [21] M. Usman, A. Cervi, M. Coppo, F. Bignucolo, and R. Turri, “Bus injection relaxation based OPF in multi-phase neutral equipped distribution networks embedding wye- and delta-connected loads and generators,” Int. J. Electr. Power Energy Syst., vol. 114, Jan. 2020, Art. no. 105394. [22] D. K. Molzahn, B. C. Lesieutre, and C. L. DeMarco, “Approximate representation of ZIP loads in a semidefinite relaxation of the OPF problem,” IEEE Trans. Power Syst., vol. 29, no. 4, pp. 1864–1865, Jul. 2014. [23] L. T. K. Hien, “Differential properties of Euclidean projection onto power cone,” Math. Methods Oper. Res., vol. 82, no. 3, pp. 265–284, Dec. 2015. [24] W. H. Kersting, “Radial distribution test feeders,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 975–985, Aug. 1991. [25] W. H. Kersting, Distribution System Modeling and Analysis. Boca Raton, FL, USA: CRC Press, 2002. [26] K. P. Schneider et al., “Analytic considerations and design basis for the IEEE distribution test feeders,” IEEE Trans. Power Syst., vol. 33, no. 3, pp. 3181–3188, May 2018. [27] V. Rigoni, L. F. Ochoa, G. Chicco, A. Navarro-Espinosa, and T. Gozel, “Representative residential LV feeders: A case study for the North West of England,” IEEE Trans. Power Syst., vol. 31, no. 1, pp. 348–360, Jan. 2016. [28] S. Claeys. (Nov. 2020). Set Points Obtained for Generator G in Large-scale Study. [Online]. Available: https://doi.org/10.5281/zenodo.4275959 Sander Claeys (Student Member, IEEE) received the dual M.Sc. degrees from KTH, Stockholm, and UPC, Barcelona. He is currently pursuing the Ph.D. degree in engineering science with the University of Leuven, Belgium. His research focuses on the modeling and optimization of electricity distribution networks, in the context of smart grids. Geert Deconinck (Senior Member, IEEE) received the Ph.D. degree in engineering science from KU Leuven, Belgium, in 1996. He is currently a Full Professor with KU Leuven, where he heads the Research Group on Electrical Energy Systems and Applications and with EnergyVille Research Center. His research focuses on distributed coordination and control, specifically in the context of smart electric-distribution networks. Frederik Geth (Member, IEEE) received the B.Sc., M.Sc., and Ph.D. degrees from the University of Leuven, Belgium, in 2007, 2009, and 2014, respectively. He is a Research Scientist working with the CSIRO, Newcastle, Australia, in the energy systems program. His current research focus is applications of optimization models in distribution network operations, including unbalanced state estimation and optimal control of battery storage systems.