A Comprehensive Multi-Period Optimal Power Flow Framework for Smart LV Networks
Iason-Iraklis Avramidis, Florin Capitanescu, Geert Deconinck
aa r X i v : . [ ee ss . S Y ] O c t A Comprehensive Multi-Period Optimal PowerFlow Framework for Smart LV Networks
Iason I. Avramidis,
Student Member, IEEE , Florin Capitanescu, and Geert Deconinck,
Senior Member, IEEE
Abstract —This paper presents an extensive multi-period op-timal power flow framework, with new modelling elements, forsmart LV distribution systems that rely on residential flexibilityfor combating operational issues. A detailed performance assess-ment of different setups is performed, including: ZIP flexibleloads (FLs), varying degrees of controllability of conventionalresidential devices, such as electric vehicles (EVs) or photovoltaics(PVs), by the distribution system operator (DSO) (adheringto customer-dependent restrictions) and full exploitation of thecapabilities offered by state-of-the-art inverter technologies. Acomprehensive model-dependent impact assessment is performed,including phase imbalances, neutral and ground wires and loaddependencies. The de-congestion potential of common residentialdevices is highlighted, analyzing capabilities such as active powerredistribution, reactive power support and phase balancing. Saidpotential is explored on setups where the DSO can make onlypartial adjustments on customer profiles, rather than (as iscommon) deciding on the full profiles. The extensive analysiscan be used by DSOs and researchers alike to make informeddecisions on the required levels of modelling detail, the connecteddevices and the degrees of controlability. The formulation iscomputationally efficient, scaling well to medium-size systems,and can serve as an excellent basis for building more tractableor more targeted approaches.
Index Terms —Multi-Period Optimal Power Flow, ResidentialFlexibility, Smart Distribution Systems, Unbalanced Systems N OMENCLATURE
A. Sets E Set of electric vehicles (EVs) F , Z Sets of “phases”: Z = { a,b,c } , F = Z∪{ n,g }I Set of nodes L Set of flexible loads (FLs) P Set of photovoltaics (PVs) T Set of time periods T nce Set of time periods when EV e may not charge B. Parameters c F L
Price of FL active power modification, e /kWh c DS Price of EV active discharge to grid, e /kWh c EV Price of EV active charge alteration, e /kWh c I/E
Price of import/export from/to MV level, e /kWh c P P V
Price of PV active production curtailment, e /kWh c QP V
Price of PV reactive capability utilization, e /kVar c v Penalty for technical limits violation, e /p.u. M F L
Maximum FL alteration, %
The authors acknowledge the funding from Luxembourg National ResearchFund (FNR) in the framework of gENESiS project (C18/SR/12676686). I.Avramidis and F. Capitanescu are with Luxembourg Institute of Science andTechnology (iason.avramidis,fl[email protected]); G. Deconinck is withKU Leuven Belgium ([email protected]). M P V
Maximum PV curtailment, % S genp,z,t PV apparent power generated, phase z , period t , p.u. P e,t EV original active charge, phase z , period t , p.u. R ij,fθ Resistance of branch ij , between phases f , θ , p.u. X ij,fθ Reactance of branch ij , between phases f , θ , p.u. C. Variables P I/Ez,t
Active power import/export, phase z , period t , p.u. S invp,z,t PV inverter apparent power, phase z , period t , p.u. P injp,z,t PV active grid injection, phase z , period t , p.u. Q P Vp,z,t
PV reactive power, phase z , period t , p.u. P Oce,z,t
EV active “overcharge”, phase z , period t , p.u. P Uce,z,t
EV active “undercharge”, phase z , period t , p.u. P dse,z,t EV active discharge, phase z , period t , p.u. P Di,z,t
Active load demand, bus i , phase z , period t , p.u. Q Di,z,t
Reactive load demand, bus i , phase z , period t , p.u. Q I/Ez,t
Reactive power import/export, phase z , period t , p.u. u i,f,t Voltage magnitude, bus i , phase f , period t , p.u. σ upi,z,t Overvoltage violation, bus i , phase z , period t , p.u. σ downi,z,t Undervoltage violation, bus i , phase z , period t , p.u. σ ij,z,t Thermal violation, branch ij , phase z , period t , p.u.I. I NTRODUCTION
A. Motivation
In adhering with the smart grid vision, distribution systemsare transforming into evermore active systems, characterizedby high shares of distributed energy resources (DERs), andhigh degrees of operational controllability by distributionsystem operators (DSOs) [1]. The proper management of suchdistribution systems is crucial; having been designed under the(now) archaic philosophy of fit-and-forget, they are usually ill-equipped to handle the uncoordinated, large-scale integrationof DERs [2]. Especially in LV networks, which are inherentlyunbalanced, operational issues are usually more prevalent andof elevated severity. For the highest penetration levels, thestress inflicted on such systems can result in harmful voltagespikes or dips and damaging thermal loading of the distributionequipment, all of which are difficult to effectively contain [3].To fully understand the benefits of residential flexibilityresources (FRs), detailed models of the various devices andthe distribution systems themselves are needed. Points ofinterest include the impact of the load modelling detail onthe operational profile, the behavior of the neutral and groundvoltages (for protection studies) and the interactions betweendifferently loaded phases. However, most research worksutilize simplified load and network models, or/and convexrelaxations, targeting scalability rather than accuracy. . Literature review
Works that opt for solving exact (i.e., non-relaxed) formula-tions usually make non-generic, case-specific simplifications,such as ignoring the neutral wire or assuming small loadimbalances to name a few. The seminal paper on multi-periodoptimal power flow (MP-OPF) for active distribution systems,[4], employed the single-phase (1 Φ ) network representationand the constant P/Q load model. While subsequent papershave since introduced more advanced models for both MVand LV systems, see [5], most papers prioritize the solutiontechnique rather than the model, assuming that non-generic,case-specific simplifications are always expected to hold.The MP-OPF problems between the MV and LV level sharesome conceptual similarities, such as the radial network struc-ture, the unbalanced system conditions or the non-negligibleimpact of line resistance/reactance. In MV systems, the DSOhas various resources at its disposal, such as capacitors banks,network switches (reconfiguration), distributed generators andtap changers, see [6]–[8]. However, the situation is very differ-ent in LV systems, where the DSO has far less controllabilityand equipment available. System management is achievedprimarily though electric vehicles (EVs) and photovoltaics(PVs), and rarely through energy storage (ES) systems.In terms of MP-OPF features, the authors of [9] eliminatethe neutral phase through Kron reduction and calculate anoptimal control strategy through an iterative approach (con-stant P/Q load). The technique is also used in [10], where thecurrent-based formulation is employed instead (simultaneousstudy of MV and LV network). The work [11] proposes acurrent-mismatch MP-OPF to optimize a feeder’s operation,ignoring the grounding and assuming full controlability ofresidential ES systems by the DSO (constant P/Q load).In [12], a local EV charging strategy is applied withrespect to the PV’s operation, though assuming that the EVis available at all times (constant P/Q load, 3-phase network).A multi-period approach is proposed in [13] for designingthe entire charging strategy of some EVs, subject to todynamic electricity pricing (current mismatch formulation, 3-phase network). The authors of [14] combine central and localcontrol strategies for managing distribution systems throughPV utilization (constant P/Q load, neutral consideration).Approaches based on PVs and EVs that are more special-ized have also been proposed. For example, in [15], the 3-stage, centralized, single-period PV curtailment and reactivemanagement problem is solved for 24 consecutive hours fora four-wire LV distribution system (constant P/Q load). Theauthors of [16] employ three-phase (3 Φ ) inverters for phasebalancing in pure 3 Φ networks (constant P/Q load). The usageof ES is more rarely tackled, due to their low penetrationsin LV networks and the undesirable level DSO involvementin household equipment management. In [17], the centralizedcontrol of ES in unbalanced networks is studied, thoughtheir temporal constraints are ignored (neutral consideration,constant P/Q load). The authors of [18] present a highlydetailed ES model specifically for LV systems and propose a local area control strategy involving several network agents(Kron reduction, constant P/Q load).While the constant P/Q load model is the one most com-monly employed, more intricate models have occasionallybeen explored. Explicit ZIP coefficients for commercial, res-idential and industrial loads have been proposed for con-servation of voltage reduction (CVR) studies in unbalanceddistribution systems [19], though this customer variety is onlyfound in MV systems. A common ZIP load structure has evenbeen employed in studies following model-free approaches ofdistribution system optimization, see [20] for example (phasorregulation strategy), with the work assuming the simplest formof an unbalanced system. Comparisons between different loadmodels have been performed, though these are either on apurely technical level, i.e., behavior of single, standalone load[21], or comparisons between the standard ZIP model anddifferent approximations of it [22]. Comprehensive studiesof all possible ZIP structures as they related to optimizingdistribution system behavior are lacking from the literature.In terms of solution techniques, convex relaxations are oftenemployed for multi-phase networks. The second-order coneprogramming (SOCP) and semi-definite programming (SDP)relaxations are popular choices that have spawned differentvariations (based on types of connection and imbalance),though the neutral wire and ground are rarely included [23],[24]. The branch flow formulation is also commonly em-ployed in radial systems [25]. While extendable to unbalancedsystems, it usually includes power flow linearizations or as-sumptions of very small imbalances [26], [27]. However, mostrelaxations hold very rarely for realistic power systems [28].Approximations for specialized versions of the unbalancedMP-OPF have also been proposed. The authors of [29] em-ploy the SDP relaxation for distribution systems with neutralcables and fixed ZIP loads. The work [30] proposes linearand quadratic simplifications of an MI optimization problemthat addresses different operating conditions between MVand LV systems (exponential ZIP load). Aside from thesimplifications, these formulations are not entirely practicalin representing the behavior of realistic distribution networks.In recapitulating, all proposed approaches (simplified or not)do produce promising results. However, if the solved problemis a relaxed one, the original system is largely simplified andthe results are rarely feasible/reliable. On the other hand, inexact approaches, the neutral and ground cables are usuallyignored, while the effects of the mutual couplings and theload types are not always properly represented. C. Contributions & Paper structure
To focus on scalability, most works sacrifice some accu-racy, leaving the behavior of modern (vast flexibility optionsarray) and realistic LV systems in MP settings insufficientlyaddressed. This work develops a comprehensive, versatile andeasily reproducible MP-OPF tool for unbalanced distributionsystems and realistic device models.This work draws inspiration from and significantly extendsseveral past works. The current-based formulation is a combi-ation of [9], [10], [31], extended to include the interactionswith the neutral and ground wires and the rotation of eachphase to a common reference plane. The load modelling isadapted from [5], also accounting for partial load flexibility.The PV modelling is adapted from [5], [9], also including lim-ited curtailment capability and a realistic production capabilitycurve (PCC). The modelling of EV flexibility is original. Forthe phase balancing through 3 Φ inverters, this work extends[16], in accounting for the re-scheduling of the user-drivencharging profile. The proposed approach for remuneratingcustomer participation is also a novel addition.On top of the novel modelling elements, the more concep-tual contributions of the paper can be summarized as follows: • It provides an extensive analysis of a comprehensive setof available modelling decisions (many often disregarded)for the optimal management of LV distribution systems. • It construct a generic MP-OPF model that can providevaluable information on how to unlock the full flexibilitypotential of common LV networks. The framework iseasily reproducible, adaptable to each researcher’s needsand ideal as a basis for more sophisticated developments. • As the topic is underaddressed, the paper proposes a firstcrude DSO/customer collaboration framework throughwhich the DSO can utilize residential devices to achievebetter system management. The importance of the frame-work is paramount, given the DSO’s traditionally minimalinvolvement in managing LV systems.The great advantage of the developed tool is its adaptabilityto the specific needs of each problem (load/network/devicemodel). It can be used under a vast array of optimizationsetups, including active power redistribution, reactive powermanagement schemes or phase balancing. This offers greatinsight to DSOs, which can explore the behavior of oftenneglected system parts in a reliable manner, and unlock thenetwork’s full potential for residential flexibility utilization.Informed decisions can be made on which modelling elementsare required and which can be reliably ignored.The remainder of this paper is structured as follows. Theproblem formulation and the main assumptions are presentedin Section II. The case study is extensively analyzed in SectionIII. Conclusions are drawn in Section IV.II. P
ROBLEM F ORMULATION
A. Problem assumptions
For the sake of clarity, we lay out the main problem assump-tions. This is day-ahead (planning), multi-period (24-hourhorizon, hourly resolution), centralized control optimizationproblem, where the DSO has partial controllability of theavailable flexibility resources (FRs). The work is containedto the deterministic setting, assuming the DSO uses a most-likely-to-occur forecast scenario for its planning (real-time de-viations are addressed on-the-spot, though this is out of scope).We assume the existence of a digital platform through whichcustomers inform the DSO of their ideal controllable deviceschedules, initially designed through a rule-based approach or by a sophisticated software (e.g., energy management system).Each customer may or may not receive new set-points for theircontrollable devices. The expected difference (not the real-time deviations) between the original customer profile and thedesigned, post-request profile, is the basis for the customer’sremuneration. All the necessary software is pre-installed. Thegeneric formulation is applicable for most setups.This work covers radial, LV distribution systems. Giventhe proposed framework’s generic structure, it can be usedto model 4-wire, multi-grounded systems (found in NorthAmerica, Europe), 3-wire, grounded or ungrounded systems(found in Europe, UK), single-wire with earth return (foundin Australia), and cases of highly specialized grounding (highresistance/reactance, Petersen coils) [32]. More intricate con-figurations, e.g., 5-wire systems, are out of scope.This work fits within the generic MP-OPF formulationoriginally developed for 1 Φ networks representations in [4].The proposed formulation employs the rectangular coordinatesformulation for voltages and currents (1)-(2): ( v i,z,t ) = ( v rei,z,t ) + ( v imi,z,t ) (1) ( i i,z,t ) = ( i rei,z,t ) + ( i imi,z,t ) (2) B. Objective function
The DSO plans an hourly cost-optimal control strategy forthe various FRs, to maintain acceptable conditions across itssystem. The objective (3) is composed of the following costs:import/export (4), PV utilization (5)-(6), FL alteration (7), EVprofile re-design (8) and limit violation (9): min( C I/E + C P P V + C QP V + C F L + C EV + C V ) (3) C I/E = c I/E · X t,z (cid:0) P Iz,t + P Ez,t (cid:1) (4) C P P V = c P P V · X t,z,p pf p,z,t · ( S genp,z,t − S invp,z,t ) (5) C QP V = c QP V · X t,z,p S invp,z,t · q − pf p,z,t (6) C F L = c F L · X t,z,l ( P Od, l,z,t + P Od, l,z,t + P Ud, l,z,t + P Ud, l,z,t ) (7) C EV = X t,z,e c EV · P Oce,z,t + P Uce,z,t ξ EV · c DS · P dse,z,t ! (8) C V = c V · X t,z,i,j : i = j ( σ upi,z,t + σ downi,z,t + σ ij,z,t ) (9)Active power imported/exported from/to the MV level in-curs a proportional cost for the DSO (4). PV active powercurtailment is highly priced, while the cost for utilizing reac-tive capabilities is tied to the active power that is not injecteddue to producing reactive power instead (5)-(6). Increasingor decreasing the customer-forecasted load demand has aproportionally associated cost (7). Deviations from an EV’scustomer-desired profile, or discharging, are proportionallypriced (8). Technical limits violations carry a high penalty (9).ach cost is chosen based on the desired activation priorityorder (PO). For example, the less desirable PV curtailmenthas higher associated costs than load profile alteration. Theapplied costs (Table I) simply reflect a certain PO. However,the framework is applicable with any (more realistic) costs. C. Device modelling
For all subsequent equations, for any device a (load, PV,EV), the symbol u a,z,t refers to the voltage difference thatconcerns device a , originally connected to phase z , at timeperiod t : for 4-wire systems, the difference is between phaseand neutral (Wye) or phase (delta); for 3-wire systems, it isbetween phase and ground (Wye) or phase (delta).
1) Customer loads:
The active and reactive demand of aload is based on fixed impedance, current and consumptionelements, i.e., the ZIP model (10)-(11), represented by theirrespective consumption percentages (12). A power factor ( pf )of 0.95 is assumed for all loads. However, the user may defineany pf they deem appropriate. The nominal active and reactivepowers, P l,z,t , Q l,z,t , can thus be interrelated through (13),where Ω represents the type of pf (1 for lagging, -1 forleading). The load currents are calculated based on (14)-(15).The following hold ∀ l ∈ L , ∀ z ∈ Z , ∀ t ∈ T : P Dl,z,t = P l,z,t h a ZP ( u l,z,t ) + a IP ( u l,z,t ) + a Pp,t i (10) Q Dl,z,t = Q l,z,t h a ZQ ( u l,z,t ) + a IQ ( u l,z,t ) + a Qp,t i (11) a Zq,t + a Iq,t + a Pq,t = a Zp,t + a Ip,t + a Pp,t = 1 (12) Q l,z,t = P l,z,t | sin (arccos ( pf )) |· pf − · Ω (13) i D,rel,z,t = (cid:0) P Dl,z,t u rel,z,t + Q Dl,z,t u iml,z,t (cid:1) · u − l,z,t (14) i D,iml,z,t = (cid:0) P Dl,z,t u iml,z,t − Q Dl,z,t u rel,z,t (cid:1) · u − l,z,t (15)A customer load may be flexible (FL) in some ways, inwhich case its nominal power can be generalized, as shownin (16). Highly accurate models split the FL into 4 parts:fixed consumption, P fxl,z,t , fully flexible consumption, P fcl,z,t ,energy (E) shiftable consumption, P esl,z,t and time (T) shiftableconsumption P tsl,z,t . The first two parts are most commonlyconsidered in the vast majority of cases. The fully flexibleconsumption is complemented by the variables P Odl,z,t , P
Udl,z,t ,i.e., “overdemand” and “underdemand”, respectively. “Overde-mand” means higher consumption than originally forecastedTABLE I: Characteristics of available FRs and DSO options
FR or DSO option Service Type PO Cost ( e / kW )IE Energy Import U 2 1Energy Export D 2 1PV Active curtailment D 5 10Reactive management U/D 1 0.5FL Consumption profile U/D 3 1.5alterationEV Charging profile U/D 4 4.5alterationSlacks Limit violations U/D 6 30 U: Upward, D: Downward and vice versa for “underdemand”. The two variables arelimited by a maximum alteration, M F L , and designed to notsimultaneously be positive (sub-optimal, see [4]).While they are less common, the last two parts are alsoadded for the sake of model comprehensiveness. The thirdpart of the FL is energy-shiftable, i.e., in-day alterations areallowed (similarly to the second part), but their net sum mustbe zero (fixed daily energy consumption), (18). The final partof the FL is time-shiftable, i.e., it can be moved throughtime, though unaltered. This is modelled through the use ofbinary variables ( δ l,t ), see (19) which dictates that the fixedpart, P SLl,z,t , must be active exactly for A time periods (akinto cycle time). The resulting MINLP problem is generallyintractable and requires the use of approximations or heuristicsto effectively manage, see [33]. P l,z,t → P fxl,z,t + P fcl,z,t + P esl,z,t + P tsl,z,t (16)Common: ( P fcl,z,t → P fcl,z,t + P Od, l,z,t + P Ud, l,z,t ≤ ( P Od, l,z,t , P Ud, l,z,t ) ≤ M F L · P cfl,z,t (17)E-shiftable: P enl,z,t → P enl,z,t + P Od, l,z,t + P Ud, l,z,t ≤ ( P Od, l,z,t , P Ud, l,z,t ) ≤ M F L · P enl,z,t P t ( P Od, l,z,t − P Ud, l,z,t ) = 0 (18)T-shiftable: (cid:26) P tsl,z,t → P tsl,z,t + P SLl,z,t · δ l,t P t ( δ l,t ) = A (19)
2) Photovoltaics:
The apparent power generated by theapplied solar irradiation, S genp,z,t , can be curtailed up to acertain percentage, M P V (20), resulting into the apparentpower at the inverter level, S invp,z,t . The PV’s power factor(pf) is partially flexible, allowing for the utilization of a PV’sreactive power (21)-(22). Said flexibility is limited by thePV’s traditional PCC, as dictated by (23). The injected PVcurrents are calculated through (24)-(25). When the PV is notconnected to a particular phase, all corresponding variables areequal to zero. The following hold ∀ p ∈ P , ∀ z ∈ Z , ∀ t ∈ T : S genp,z,t (1 − M P V ) ≤ S invp,z,t ≤ S genp,z,t (20) pf p,z,t = P injp,z,t · ( S invp,z,t ) − (21) pf minp,z,t ≤ pf p,z,t ≤ pf maxp,z,t (22) ( Q P Vp,z,t ) + ( P injp,z,t ) = ( S invp,z,t ) (23) P injp,z,t = u rep,z,t i P V,rep,z,t + u imp,z,t i P V,imp,z,t (24) Q P Vp,z,t = u ip,z,t i P V,rep,z,t − u rep,z,t i P V,imp,z,t (25)The above hold for PV inverters. However, using 3 Φ inverters instead allows for redistributing a PV’s output amongthe three phases [16]. For simplicity, we ignore the PCC (unity pf ) when phase balancing is possible; nonetheless, per-phaseformulations of (21)-(23) can be defined. The total injectionamong the three phases must be equal to the original njection (26). There is a limit to the redistribution of activepower, based on the rate of each 1 Φ inverter, P P V,ratep , (27).An individual inverter may consume active power, hence thepossible negative values for Φ PV injection. This can be fullyexploited at night, when PV production is zero [16]. X z P injp,z,t = X z S invp,z,t · pf p,z,t (26) − P P V,ratep / ≤ P injp,z,t ≤ P P V,ratep / (27)
3) Electric vehicles:
An EV can “overcharge” or “under-charge” with respect to its originally forecasted profile (similarlogic as in FLs), or discharge if the V2G capability is available.Since V2G is a much more sought-after service, its cost ismuch higher than that of profile rescheduling. This designensures that the “undercharge” capability (i.e., charging modu-lation) will be first fully utilized before before discharging canactually be activated (see also [4]). The EV can only interactwith the grid when the resident is home (28). The EV must“conclude” the optimization horizon under the same statusthat its owner originally intended (29)-(30), where n c , n d are the charging and discharging efficiencies. Constraint (32)dictates the technical limits of the EV’s charging/dischargingbehaviour. The binary parameter ξ EV represents whether theEV can only charge ( ξ EV = 0 ) or if it has the V2G capability( ξ EV = 1 ), in which case it also dabbles as a traditionalenergy storage system. Constraint (33) ensures that the EVdoes not charge/discharge above/below certain energy limits, E cape , E mine . The EV currents are calculated by (34)-(35). EVsare assumed to operate with a fixed pf . Advanced EVs mayeven be capable of employing 4-quadrant control [34], thoughsuch instances are rare; (21)-(23) could be easily adapted andemployed if need be. When the EV is not connected to aparticular phase, all corresponding variables are equal to zero.The following hold ∀ e ∈ E , ∀ z ∈ Z , ∀ t ∈ T : P Oce,z,t = P Uce,z,t = P dse,z,t = 0 ∀ t ∈ T nce (28) P demande,z,t = P Ce,z,t + P Oce,z,t − P Uce,z,t − ξ EV · P dse,z,t (29) ( P Oce,z,t , P
Uce,z,t , P dse,z,t ) ≥ (30) X z,t " n c · ( P Oce,z,t − P Uce,z,t ) − ξ EV · P dse,z,t n d ≡ M = 0 (31) − P ratee · ξ EV ≤ P demande,z,t ≤ P ratee (32) E mine ≤ M + t X z,t =1 P Ce,z,t ′ · n c ≤ E cape − ξ EV · E e (33) P demande,z,t = u ree,z,t i EV,ree,z,t + u ime,z,t i EV,ime,z,t (34) Q EVe,z,t = u ime,z,t i EV,ree,z,t − u ree,z,t i EV,ime,z,t (35)EVs may also be equipped with inverters [16]. Con-straint (32) is modified, due to the limit on the redistribution ofactive power, based on the rate of each inverter, P EV,ratee ,(36). While power can be injected in a phase, the EV type isessentially the same (net charger); pure net discharge (V2G) is not possible, as this would involve a different (and farmore expensive) kind of EV. In addition, the variables andparameters P Ce,t , P
Oce,t , P
Uce,t are now only considered for theEV as a whole (no z index). (29) is thus modified as (37): − P EV,ratee / ≤ P demande,z,t ≤ P EV,ratee / (36) X z P demande,z,t = P Ce,t + P Oce,t + P Uce,t (37)
4) Stand-alone battery:
The EV model is easily adaptableto a stand-alone battery model. This is achievable through thefollowing modifications, where P De,z,t is the original, user-driven discharging profile, and P Odse,z,t , P
Udse,z,t are the “over-discharge” and “under-discharge” variables, respectively:(29) : P dse,z,t → P De,z,t + P Odse,z,t − P Udse,z,t (38)(30) → (30) & ( P Uc ≤ P Ce,z,t ) &( P Uds ≤ P De,z,t ) (39)(31) : P dse,z,t → P Odse,z,t − P Udse,z,t (40)(33) : P Ce,z,t · n c → P Ce,z,t · n c − P De,z,t /n d (41) D. Low voltage network technical constraints
The distribution line model depicted in Fig. 1 is adopted.Each “phase” f ∈ F is characterized by its self-impedance, Z ff and its mutual coupling with other “phases” θ ∈ F : f = θ , Z fθ . A grounding impedance, Z gr , may also bepresent. The neutral and ground currents are calculated usingcurrent dividers. As is common for LV networks, devices areassumed to be wye-connected, though the adaptation to deltaconnections is also presented. For neighboring nodes, currentsare assumed equal at origin and destination (small line shunts[35]).For the LV network, we have the current injection bal-ance (42)-(43), where (42) concerns wye-connected devicesand (43) details the necessary modification to address deltaconnections (the superscript demand refers to any considereddevice, e.g., PV, E, FL, and z, z ∗ , z ∗∗ represent differentphases). The modification is necessary, since employing theconventional Y → ∆ transformations would only lead to anapproximated MP-OPF solution. We also have the matching ofthe injections of the three phases with the neutral phase (44),Fig. 1: 3 Φ , four-wire, multi-grounded distribution line [35]here binary parameter φ ∆ represents the existence of a deltaconnection, the constraints that guarantee the current balanceat common ground points (45), the voltage drop across lines(46)-(47), the voltage and branch currents technical limits (48)-(49). The above hold both for the real and imaginary parts.Note that all slack variables are positive (50). No constraintsare enforced for the neutral and ground “phases” since theyare completely dependent on phases a, b, c. The balancesbetween the currents at the neutral-ground connection areenforced by the current dividers; no additional constraintsare necessary to capture this behavior. The following hold ∀ i, j ∈ I : i = j, ∀ z ∈ Z , ∀ f ∈ F , ∀ t ∈ T : Y : − i P V,re/imi,z,t + i D,re/imi,z,t + i EV,re/imi,z,t = X i re/imij,z,t (42) ∆ : i device,re/imi,z,t → i device,re/imi,z − z ∗ ,t + i device,re/imi,z − z ∗∗ ,t (43) X z i reij,z,t = i rei,n,t · φ ∆ & X z i imij,z,t = i imi,n,t · φ ∆ (44) X i reij,g,t = i rei,g,t & X i imij,g,t = i imi,g,t (45) u rei,f,t − u rej,f,t = X θ ∈F (cid:0) R l,fθ i rel,θ − X l,fθ i iml,θ (cid:1) (46) u imi,f,t − u imj,f,t = X θ ∈F (cid:0) R l,fθ i iml,θ + X l,fθ i rel,θ (cid:1) (47) V min − σ downi,z,t ≤ u i,z,t ≤ V max + σ upi,z,t (48) − I max − σ ij,z,t ≤ i ij,z,t ≤ I max + σ ij,z,t (49) σ upi,z,t , σ downi,z,t , σ ij,z,t ≥ (50) E. Voltage shifts
At the feeder head, voltages are defined as { ∠ ◦ , ∠ − ◦ , ∠ ◦ } → { j, − . − . j, − . . j } forphases a, b, c, respectively. While constraint (48) is perfectlyvalid, a computational inefficiency stems from the fact that thephases are rotated with respect to each other; the rectangularmodelling of voltages causes constraint (48) to be non-convex.Assuming that the voltage angles (per phase) diverge onlyslightly from their reference value, a convex reformulation isemployed, similarly to [9]. This allows overcoming one ofthe several non-convexities, while making the evaluation ofthe results far easier and more intuitive. The three phases areFig. 2: Voltage constraint reformulation (adapted from [9]) rotated by ROT = −{ ∠ ◦ , ∠ − ◦ , ∠ ◦ } so that theylie close to the reference axis ◦ , and the same feasible spaceis defined for each [9]. A visual representation is presented inFig. 2. Constraint (48) is thus re-defined as: (cid:26) ROT ( u i,z,t ) ≤ V max + σ upi,z,t Re {ROT ( u i,z,t ) } ≥ V min − σ downi,z,t } (51)A crucial point not addressed in [9] (as the paper assumeda perfectly grounded system and consequently used the Kronreduction to remove the neutral cable) is that the phase rotationaffects the interactions between phases a, b, c and the neutralwire (44). These constraints must be updated to account forthe voltage rotation performed in (51). As such, the calculatedcurrent for each phase is counter-rotated by ROT . If we definethe total current injections at node bus i , phase z , period t ,(52), then (44) are updated as (53)-(54): − X i reij,z,t = y rei,z,t & − X i imij,z,t = y imi,z,t (52) y rei,a,t − (0 . y rei,b,t + 0 . y imi,b,t ) − (0 . y rei,c,t − . y imi,c,t ) = − i rei,n,t (53) y imi,a,t − (0 . y imi,b,t − . y rei,b,t ) − (0 . y imi,c,t + 0 . y rei,c,t ) = − i imi,n,t (54) F. Remarks
The problem at hand is an MP-OPF. The conventionalsetup (1 Φ inverters, no voltage shifting) is comprised of therectangular coordinates modelling and the various costs (1)-(9), the modelling of FLs (10)-(19), PVs (20)-(25), EVs (28)-(35), and the system technical constraints (42)-(50). Proposednovel aspects of this MP-OPF pertain to goal (1) are: ZIP loadflexibility, realistic PV power management, EV original profilealteration and neutral/ground currents interactions.Further novelty allows adopting an advanced MP-OPF for-mulations. PVs are additionally complimented by the specificconstraints of their 3 Φ inverters (26)-(27). Same goes for EVs,where the original total demand constraints (32) are replacedby the updated (36)-(37). Voltage constraint (48) is replacedby the set of shifted constraints (51). The original relationsbetween the currents of phases a, b, c and the neutral current(44) are converted to reflect the voltage shift (52).The physical problem modelled is an NLP, regardless ofthe modelling choices concerning the network and the res-idential devices. The nonlinearity stems from the ZIP loadmodels, the PCC of PVs and the relations between power andvoltage/current. The complex formulation, including all thepeculiarities of MP-OPF in multi-phase LV networks, is notdirectly amenable to exact convex formulations (particularlySDP/SOCP). The authors prefer adopting an NLP formulationand solver, in order to obtain a feasible and at least localoptimal solution, as compared to the (high) risk of obtainingphysically meaningless solution from a relaxed problem [28].II. C ASE S TUDIES AND F RAMEWORK E XHIBITION
A. Simulation environment
The proposed formulation is applied on the 18-node, modi-fied CIGRE LV benchmark distribution network [36], depictedin Fig. 3. The original network dataset does not include dataregarding the neutral wire and the ground; artificial values areused, based on observations regarding the relationship betweenphases a, b, c and the missing values, as derived from [35].When grounding is included, a value of R g = 1Ω (adjusted inp.u.) is considered for all customer nodes, though other kindsof grounding are also possible (e.g., grounding impedance orPetersen coil). All other nodes are assumed to be ungrounded.Base power and (3 Φ ) voltage are chosen as kW and V ,respectively. The Z/I/P percentages are randomly assigned.The connected distributed devices throughout the networkare FLs, EVs and PVs (connection points shown in Fig. 3).The characteristics of each device are available in Table II.Their actual 24-hour profiles are available in [3]. While theframework can accommodate any residential device, in orderto better examine the impact of different flexibility setups,devices with high power rates are purposefully selected. Allsimulations are performed on a PC of 2.7-GHz and 8-GBRAM, using the general purpose NLP solver IPOPT [37],with default settings, through GAMS [38]. The frameworkwas previously evaluated (via power flow comparisons) ona number of MV and LV networks. The errors (in p.u.) oncomplex voltages varied between 10 − and 10 − . B. Modelling and operational scenarios
Apart from proposing a comprehensive MP-OPF frameworkfor realistic distribution systems, this work also provides infor-Fig. 3: CIGRE LV feeder (28% customer-to-node ratio)TABLE II: Device characteristics
Node Device P r ( kW ) E cap ( kW h ) Phase a ZP/Q a IP/Q a PP/Q
11 EV 8 24 a – – –11 FL 1.5 – a 0.2/0.1 0.2/0.1 0.6/0.815 EV 8 24 b – – –15 FL 2 – b 0.6/0.3 0.1/0.2 0.3/0.515 PV 4 – a – – –16 FL 2.5 – c 0.05/0.3 0.15/0.1 0.8/0.616 PV 4 – b – – –17 EV 10 30 c – – –17 FL 1.5 – a 0.3/0.5 0.4/0.1 0.3/0.417 PV 5 – c – – –18 FL 2 – b 0.05/0.01 0.25/0.8 0.7/0.1 mation on the impact of each modelling aspect on the solution.The goal is to understand which are necessary and whichcan be reasonably ignored, depending on the needs of theapplication. Three major, multi-scenario cases are examined:1) Case 1: Unbalanced network model (3 Φ , perfectlygrounded neutral), PVs/EVs ( ξ EV = 0 ). Various loadmodels, L x , are examined; they are presented in TableIII. PVs can be curtailed up to 20% ( M P V = 0 . ), FLsmodified up to 10% ( M F L = 0 . ).2) Case 2: ZIP model for customer loads, PVs/EVs( ξ EV = 0 ). Various network models, N x , are examined;they are presented in Table IV. PVs can be curtailed upto 20%, FLs modified up to 10%.3) Case 3: Unbalanced network model (3 Φ plus neutral)and ZIP model for customer loads. Two scenarios ofinverter types are examined: 1 Φ inverters vs 3 Φ inverterswith balancing capabilities ( ξ EV = 0 ). Various scenariosof FR controllability are examined; the different scenar-ios S x are presented in Table V.Focusing on smart distribution feeders with high penetrationof potential FRs, the main purpose of this work is reflectedin Cases 1 and 2. That purpose is to provide a comprehensiveunderstanding to DSOs with regards to how their systemrealistically behaves, how the accuracy of the solution isaffected with cumulative simplifications/approximations, andhow practical each level of modelling detail actually is. Theinclusion of Case 3 is done in order to demonstrate to DSOshow realistic distribution systems would behave if the fullpotential of the various interconnected FRs were unlocked.TABLE III: Versions of load modelling Version L L L L L L L Z-component
X X X – X – –I-component X X – X – X –P-component X – X X – – X VD Q & LIN Q & LIN Q LIN Q LIN None
Q: Quadratic, LIN: Linear, VD: Voltage dependencyTABLE IV: Versions of network modelling
Version Grounding Neutral Mutual UnbalancedCoupling N (3 Φ , 4-wire) X X X X N (3 Φ , 4-wire) – X X X N (3 Φ , 3-wire) – – X X N (3 Φ , 3-wire) – – – X N (1 Φ ) – – – – TABLE V: FR controllability scenarios
Scenario S S S S S S S S M PV pf min M FL X X T nc – – – – – – t − t t − t ABLE VI: Results per load model: 500 Monte Carlo simulations per model with varying ( ± L L L L L L L Solution time (s) 4.59 ± ± ± ± ± ± ± e ) 340.8 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± TABLE VII: Results per network model: 500 Monte Carlo simulation per model with varying ( ± N N N N N Solution time (s) 17.45 ± ± ± ± ± e ) 397.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± C. Examining the effects of modelling choices1) Case 1:
Results for various levels of load modellingaccuracy (problem of 37,653 variables, 17,001 constraints) arepresented in Table VI. It becomes immediately apparent thatthe (often neglected) load modelling choices affect the solutionin a non-negligible way (though solution times and total vio-lations are comparable). When the Z-component is dominant,the network’s voltage profile tends to rise. Contrariwise, whenthe P-component is dominant, the voltage profile tends to drop.The I-component appears to have the smallest impact on thevoltage profile. Similar observations are much more prevalentin nodes that are further away from the substation.A higher average demand does not necessarily correspond toa reduced voltage profile. When there are binding interactionsbetween the load model and the voltage profile, it is not alwaysstraightforward to estimate how the two will be affected.Formulations that include the P-component (higher averagedemand) such as ZP, IP, P, tend to show pictures of higherstress for the system, requiring more reactionary measures inresponse (i.e., higher operational costs). Formulations that canproduce a more “malleable” demand such as ZI, Z, I, have theopposite effect (lower costs on average). The smallest, almostnegligible difference in the objective is observed under theZP load model; the I-aspect of loads could perhaps safelybe dropped from simplified formulations. Nevertheless, lessconservative load models (assuming there is sufficient datato support them) can bring about operational benefits (moreaccurate coordination of flexibility options). Do note that whilethe differences (not necessarily errors) in objective values arenot very high (on average about 3.3%), the cumulative impactof the load modelling decisions are certainly non-negligible.
2) Case 2:
Results for various levels of network modellingaccuracy are presented in Table VII (assuming N is the idealobjective). Simpler network models produce inaccuracies inthe voltage profile (elevated or reduced levels), leading tosignificant overestimations or underestimations of operational issues (ignoring imbalances, mutual coupling or both producesan average error of 29.25%). Models N , N have unrealisti-cally low violation costs, despite exhibiting vastly differentvoltage profiles. While simpler models have benefits (speed-wise), they could be inappropriate for many applications.When the neutral wire is explicitly modelled the error dropsto approximately 1.25%. The large errors observed are notunexpected. As was observed in [17], load imbalances ofabout 50% can lead to the corresponding neutral wire beingresponsible for approximately 20% of system losses. In ourcase, some nodes have 100% imbalance, with the neutral wirecarrying very high currents. While the neutral voltages are verylow (almost always below 0.1 p.u.), the coupling effects of thelarge neutral currents create several issues. The Kron reductionis acceptable only on systems with perfectly grounded neutralwires; this is not our case, hence we observe large errors.Depending on the level of imbalance, the modelling aspectthat could be reliably dropped is the grounding.
3) Case 3:
Results for various levels of FR controllability,using both and inverters, are presented in Table VIII.Expectedly (for the case), as more flexibility is added tothe DSO’s “toolkit” the operational conditions improve. In ourcase, increased PV controllability is followed by increasedFL controllability, finally adding EV controllability into themix. The voltage profile remains relatively steady, though theupper and lower values are improved, resulting in reducedviolations costs. In addition, the available residential flexibilityalso reduces the import costs, as an increasing amount of in-system energy is re-distributed, decreased or converted (e.g.,PVs changing their pf to avoid curtailment).For the inverters case the IE price is increased 10 timesin order to motivate system self-sufficiency, discourage (asmuch as possible) interactions with the MV level and max-imize the utility of residential FRs. Even without employingany flexibility, the usage of inverters improves the voltageprofile; even for S , we observe total violations of almostzero. The usage of inverters offers tremendous amounts ofABLE VIII: Results, 1 Φ inverters vs 3 Φ inverters S S S S S S S S
1Φ 3Φ 1Φ 3Φ 1Φ 3Φ 1Φ 3Φ 1Φ 3Φ 1Φ 3Φ 1Φ 3Φ ∗
1Φ 3Φ ∗ Solution time (s) 132.03 29.30 56.06 29.30 67.73 29.30 5.94 29.30 4.92 22.54 4.75 26.63 7.20 22.05 26.77 21.11Objective function ( e ) 465 2,663 459 2,663 452 2,663 446 2,663 442 2,527 433 2,215 433 2,258 416 2,241Average system voltage (p.u.) 0.981 0.982 0.982 0.982 0.978 0.982 0.981 0.982 0.982 0.981 0.982 0.982 0.982 0.982 0.982 0.983Maximum voltage (p.u.) 1.083 1.067 1.1 1.067 1.091 1.067 1.098 1.067 1.098 1.075 1.098 1.061 1.098 1.071 1.098 1.055Minimum voltage (p.u.) 0.798 0.902 0.798 0.902 0.798 0.902 0.798 0.902 0.802 0.902 0.809 0.903 0.809 0.908 0.809 0.908 Φ case: PV pf = 1 , IE price increased tenfold to motivate system self-sufficiency. *Lower voltage limit set to 0.95 p.u.flexibility to the operator (despite the fact that we have alsoremoved the reactive capabilities of PVs), without the need toactually engage in profile alterations. A thing of note is thatwhen the connected devices are able to distribute their profilesbetween the three phases, the neutral wires carry almost nocurrent and as such most of the original active losses are nolonger observed; had the original IE cost been kept then theobjective function value (as compared to the inverters case) would have been reduced by about 55% . D. Assessing scalability1) System vs customer base expansion:
Similarly to mostNLP problems, full scalability for very large problems isnot guaranteed. Nonetheless, the formulation is in generalcomputationally efficient for small/medium-size systems andsometimes even for large systems (see Section III.D.2). Forthe 18-node system, despite the already large problem sizeand the variety of available flexibility options, a solution isachieved (on average) in less than 30 seconds.The proposed formulation is applied on a partial real-life distribution feeder, composed of 180 nodes hosting 23residential customers (13% customer-to-node ratio, see [3] fororiginal), of which 5 own EVs, 8 own PVs and 10 own both(sizes of Table II). The system is assumed perfectly groundedand as such, the neutral wire can be reliably reduced. Five
L L M M M H H02505007501000125015001750200022502500 IE FLPVEVSlacks
Fig. 4: Average cost breakdown for 180-node system: 1 Φ (left)vs 3 Φ (right) inverter setups different loading conditions scenarios are examined, usingboth the 1 Φ and 3 Φ inverter setups: light (L, one third ofthe regular consumption), medium (M, regular consumption),heavy (H, triple the regular consumption) and in-betweenstates. These conditions were examined for the sake of evaluat-ing the flexibility potential of FRs under extremely challengingconditions. The results are presented in Fig. 4 and Table IX.The results illustrate some interesting points. While theproblems sizes are comparable , the 3 Φ setup has the superiorperformance. After increasing the degree of stressing of thesystem per step (indicated by the increasing objective value),an optimal solution without violations is always achieved,with the solution speed increasing, but not significantly. Con-trariwise, the 1 Φ setup cannot cope efficiently with highlevels of stress, requiring more time to reach a solution (onaverage 270% slower). In fact, to avoid infeasibility for theH case, much more residential flexibility is “activated” thanwould be reasonably expected. The 3 Φ setup provides farbetter solutions (on average 54.4% better), due to not directlyutilizing flexibility per se but rather by alleviating the negativeeffects of load imbalances through consumption redistributionamongst phases. The above are indicative of the superiority of“investing” in flexibility that is characterized by quality (phasebalancing) instead of quantity (profile alteration).In examining the coordination of a very large customerbase, the same feeder is examined under a significantly highercustomer-to-node ratio and penetration of FRs. Specifically,the customer base was expanded, composed of 120 customers,90 PVs, 75 EVs. This is a system of unrealistically highloading conditions, subject to massive stress. For this case, Remember that for loads, PVs and EVs, the variables of unconnectedphases are still present, despite being set to zero.
TABLE IX: Solution performance, 180-node system: 50Monte Carlo simulations per loading level with varying ( ± MetricLoading System voltage System voltage Solution time Solution timelevel (pu, 1 Φ ) (pu, 3 Φ ) (s, 1 Φ ) (s, 3 Φ )L 0.994 ± ± ±
11 230 ± → M 0.978 ± ± ±
16 248 ±
7M 0.961 ± ± ±
24 267 ± → H 0.944 ± ± ±
69 301 ±
20H 0.923 ± ± ±
124 329 ± ABLE X: Solution performance, 180-node system, signifi-cantly expanded customer base Φ inverters 1 Φ invertersSolution time (s) 49.6 ± ± e ) 4,220 ±
340 9,500 ± ± ± ± ± ± ± ± ± based on PF solutions, we initialize the problem and removebeforehand potentially inactive slack variables. We execute 25simulations per inverter type; results are provided in Table X.The solution is calculated faster, owing to better tailored ini-tialization and elimination of variables. Concerning the purelytechnical aspects, the 1 Φ case has higher costs, owing to to theextensive re-shaping of FL/EV profiles, as well as engagingin extensive PV curtailment and reactive power injection. The3 Φ case makes extensive use of the EV/PV inverters (almostall are utilized), avoiding the need to resort to flexibilityprocurement almost entirely. As such, the feeder is managedmuch more cheaply and efficiently. In both cases, however, theDSO has access to a very large pool of residential flexibility;the subsequent technical issues are minor and the feeder is(nearly) always operated within acceptable conditions.
2) Further expansions and limitations:
The problem sizeexpansion can follow two directions: system expansion (nodes)and customer base expansion (controllable FRs). As such, wemust explore the framework’s performance with respect to twokinds of expansion. Four additional feeders (originals availablein [3]) of 200, 400, 600 and 800 nodes, respectively, wereexamined (using the 3 Φ inverter configuration for PVs/EVs),hosting different sizes of customer bases. A single simulationis performed per case. The customer distribution (node/phase)is random. PF-based initialization was again employed. Dueto technical issues encountered (solver crashing without clearreason), the commercial solver KNITRO [39] is employedinstead of IPOPT. The results are presented in Table XI.As is obvious, some of the examined setups results invery highly loaded systems, requiring the “activation” ofhigh percentages of FRs, thus stressing the solution processitself. As expected, increasing the system size (nodes) subse-TABLE XI: Solution times (s) for different problem sizes Customer-to-node ratio (%)Nodes 10 20 30 40200 38.7 58.2 105.5 276.1400 191.6 318.2 491.5 802.5600 904.2 1373.4 2016.5 Int800 1617.5 2905.6 4389.9 IntPV penetration: 50%, EV penetration: 25%
Int: Solution manually interrupted after 10,000 seconds quently increases the solution time. However, if the number ofcontrollable elements remains low (lower system stress, lessvariables), a locally optimal solution is (generally) achievedwithin acceptable time-frames for day-ahead settings. Forvery high penetrations of controllable elements the system ismuch more stressed, increasing the solution time. In fact, forlarger systems hosting unrealistically high numbers of FRs,no solution is returned even after 10,000 seconds. However,it appears that the negative impact on the solution time stemson a larger part from the number of controllable elements,rather than from the size of the system (very large system withno controllable elements are simulated very fast). For mostLV distribution systems (10-30% customer-to-node ratio), asolution can be calculated within reasonable time-frames.On a final note, the authors wish to re-stress that despitethe several pros of the framework in and of itself, no claimis made on the quality of the chosen modelling/simulationtools. However, the exact actions to be taken for fine-tuningthe solution process are out of paper scope.
E. Additional cases of interest
The results so far demonstrated the capabilities of theframework for the most common applications of optimizationin LV distribution systems. However, for the sake of compre-hensiveness, we re-turn our focus to the 18-node system andexamine the impact (on the solution) of five more intricatemodelling choices:
1) Specialized load models:
While the traditional flexibleload model (17) is the one most commonly used, it isstill useful to perform an impact analysis for the expanded(generic) FL model. For that purpose, we consider FLs asthe only controllable elements. We examine three different FLmodels, where 50% of the load is fixed and the remainder50% is either fully flexible, energy-shiftable or time-shiftable.A fourth model (most generic) is also examined, where 25% ofthe load is fixed and the remainder 75% is equally distributedto the first three models. Results are presented in Fig. 5.
Objective Solution time Violations FL cost020406080100120140 M e t r i c ( F L m o d e l ) / M e t r i c ( C o m b i n a t i o n ) *100 % Fully flexibleEnergy-shiftableTime-shiftableCombination
Fig. 5: Impact analysis of different FL modelsig. 5 illustrates through various metrics the percentagedifference between using the most generic FL model anddifferent versions of it. Obviously, more refined models trans-late to higher FL costs, albeit to somewhat lower objectivefunction values and total technical violations. Specifically, thefirst three load models result in 20%, 19% and 5% moreviolations, respectively, than their combination, while the totalFL cost is 76%, 47% and 22% lower, respectively. However,the objective function values are practically the same for allFL models. Whatever small improvement is achieved comesat a cost of drastically higher solution times, with the firstthree models reaching an optimal solution 72%, 61% and 23%faster, respectively. This is an important reason why the firstFL model is the one most commonly employed. Reducingthe solution time would require specialized approximation (anecessity for much larger networks), which are out of paperscope. The reader is referred to [33] for further details.
2) Modelling of multi-phase lines:
While the presentedresults focused on the most commonly employed networkmodels, i.e., either all-1 Φ or all-3 Φ , the proposed frameworkis fully compatible with multi-phase networks as well. Anexample of the framework’s compatibility is the solution ofthe MP-OPF problem on the CIGRE LV feeder, modifiedto an arbitrary multi-phase system, see Fig. 6. As can beseen, while the main body of the feeder is 3 Φ , the linesdirectly leading to buildings are not. Specifically, the moreactive buildings (hosting FLs, EV, PV) are connected to themain feeder through 2 Φ lines, while 1 Φ lines are used forless active buildings. All devices are accordingly modifiedso that their phases correspond to these of their respectivelines. Modelling a multi-phase line is simply achieved bydisregarding some of the parameters R l,fθ , X l,fθ , i.e., omittingthem from the formulation after network topology processing.The phase allocation per line is completely random.The random allocation of phases and solution of the MP-OPF was performed a number of times to validate the frame-work’s capabilities. The main observation was that the MP-OPF always reached an optimal solution without issue. Itis however worth stating that, on average, the solution timewas slightly higher than that for pure 3 Φ networks. This wasexpected, since when the number of phases is reduced theFig. 6: Modified, multi-phase version of CIGRE LV feeder Fig. 7: Behavior of proposed battery model ( n C = 0 . )power is not distributed as well, resulting in higher thermalstressing for some lines. As previously discussed, when thesystem is more stressed, i.e., requires the “activation” ofadditional slack variables, the solution time is generally higher.Nonetheless, this is a natural outcome of the problem setupand is not attributed to some weakness in the framework.
3) Validation of proposed stand-alone battery model:
Theproposed stand-alone battery model is validated on the 18-node feeder. The battery, originally based on the proposed EVmodel, has a standard user-driven profile, sometimes charging( P C ) and sometimes discharging ( P D ). Contrary to commonbattery models, e.g., the one presented in [4], instead of“penalizing” all charging and discharging activities, only thedeviations from the user-driven profile are remunerated. Ascan be seen in Fig. 7, which presents the behavior of abattery at node 17, the battery originally charges at noon anddischarges during the night, heuristically designed to do so bythe user. Post-optimization, the battery charges more heavily atnoon to counter the high PV production (avoid overvoltages),discharges more heavily at night to serve the high EV demand(avoid undervoltages). The extra stored energy is distributedto early morning hours. The simulation time remains small(comparable to what would be expected from a traditionalbattery model), and the battery behaves within the lines of“normalcy”. Even after optimization, the battery exchangesthe exact same energy amount as originally designed, a strongpoint of the proposed model.
4) Wye-connections vs delta-connections:
At this point itis still useful to examine how the solution is affected bythe device connection type (remember that the frameworkis fully compatible with both connection types). For thatreason, we examine three cases: the standard case, whereonly Y connections are assumed, the uncommon case, whereonly delta connections are assumed, and an intermediate case,where both connection types are assumed. All three casesare simulated through the proposed framework, and resultsare presented in Fig. 8. Delta-connected devices generallylead to higher currents (total demand), resulting in slightlyhigher objective function values. Systems with more deltaconnections may also require more time to solve, since notonly do the constraints become more intertwined, but theincreased demand also leads to higher system stress, which, as bjective Solution time Total demand020406080100120140160180 M e t r i c ( C o nn ec t i o n ) / M e t r i c ( Y - c o nn ec t i o n ) *100 % Y-connections only-connections onlyMixed connections
Fig. 8: Impact analysis of different connection typeshas been shown, may slow down the solution time. However,such cases are rare in practice, as most residential loads arein fact wye-connected.
5) Evaluation of the V2G capability:
Though not as com-mon, some EVs also have the V2G capability. This capabilityhas limited impact during weekdays: since the EV is notpresent for a big part of the day (owner is away), this cripplesthe battery’s potential for providing flexibility, as there areno clear opportunities for it to charge and serve any possibleneed for self-consumption during nighttime. On the otherhand, the consumption re-distribution capability has limitedpotential during weekends, where the limited need for anycharging severely constrains the potential of 3 Φ inverters.The aforementioned observations are confirmed by simulationswith both inverter configurations. The best case scenario wouldbe to consider an EV with a 3 Φ inverter configuration and theV2G capability, thus this setup is rare itself.To make a fair comparison, we examine an idealized setupwhere the EV must charge its usual amount, but withouttemporal restrictions, i.e., the EV may interact with the grid atall times. We compare the performance of the V2G capability(1 Φ configuration) against the consumption re-distributioncapability (3 Φ configuration). We also set the exporting priceto 20 e /kW to motivate system self-sufficiency (see also Case3). As can be seen in Fig. 9, the 3 Φ inverter configuration beatsthe V2G capability by a clear margin. Under the former, theaverage per-phase voltage level is generally more elevated andfluctuates more closely to unity. In addition, the remunerationcost with the 3 Φ inverter configuration is significantly lower.When the V2G capability is employed, the phase to whichthe EV is connected must engage in drastic action (chargingalteration and discharging), utilizing two different flexibilityservices (thus driving up the cost) and only partially managingthe problem. With the 3 Φ inverter configuration, the EV mustonly alter its charging profile. By constantly balancing itsoperation between the three phases, the flexibility cost is kept Time period (h) V o l t ag e m ag n i t ud e ( pu ) V a (V2G)V b (V2G)V c (V2G) V a (3 )V b (3 )V c (3 ) Fig. 9: 18-node feeder, average voltage magnitude per phase:V2G vs power re-distribution comparisonlower. No single phase disproportionately affects the others,as the 3 Φ inverter configuration ensures that the interactionsbetween phases is more balanced and more harmonious.IV. C ONCLUSIONS
The authors have constructed a versatile MP-OPF frame-work that serves two key functions: it proposes and com-pares the performance of state-of-the-art device models forunlocking the flexibility potential of smart distribution gridsand it provides up-to-date guidelines with respect to howeach of the most commonly employed (or ignored) modellingchoices (concerning the loads, the network and the degreesof controlability) affect the quality and reliability of the solu-tion. The reported results can guide researchers into pickingproper equipment models depending on their respective needs.The formulation also scales well for larger systems (underproper conditions). As such, it can serve as a solid basis forapproaches aiming specifically at scalability; its results canalso be safely contemplated by the DSO for hours-ahead use.The impact of different versions of common FR devices wasanalyzed, based on novel and realistic models. Namely, theauthors proposed flexible ZIP load models (profile alterationaffects Z, I, P components), both and (balancing)versions of PVs with realistic associated costs, reactive ca-pabilities and PCC curves and both and (balancing)versions of EVs with the added novelty of building on theoriginal customer-desired profile, rather than determining anoriginal profile altogether. The proposed models can be usedin devising new approaches for unlocking the full potential ofFRs to manage violated constraints in distribution systems.In the future, the authors plan to extend their MP-OPFframework to to address the issues of uncertainty, an area were,due to the huge computational challenge, the research is scarceand often limited to small systems.R EFERENCES[1] H. Farhangi. “The Path of the Smart Grid”.
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Iason-Iraklis Avramidis (S’ 17) received the Dipl.- Ing degree in Electricaland Computer Engineering from the National Technical University of Athens(NTUA), Greece, in 2017, and the M.Sc. degree in Energy Science andTechnology from ETH Zurich, Switzerland, in 2019. He is currently withEnvironmental Research and Innovation (ERIN) at the Luxembourg Instituteof Science and Technology (LIST), while pursuing the Ph.D. degree with theElectrical Energy & Computer Architecture (ELECTA) Group, KU Leuven,Belgium. His research interests include the planning and operation of smartdistribution grids, smart sustainable buildings and power markets.
Florin Capitanescu received the Electrical Power Engineering degree fromthe University “Politehnica” of Bucharest, Romania, in 1997 and the Ph.D.degree from the University of Li`ege, Belgium, in 2003. Since 2015, hehas been with ERIN at LIST as a researcher. His main research interestsinclude the application of optimization methods to operation of transmissionand active distribution systems, particularly optimal power flow approaches,voltage instability, and smart sustainable buildings.