Joint Grid Topology Reconfiguration and Design of Watt-VAR Curves for DERs

1 Joint Grid T opology Reconfiguration and Design of W att-V AR Curv es for DERs Manish K. Singh, Student Member , IEEE , Sina T aheri, Student Member , IEEE , V assilis K ekatos, Senior Member , IEEE , K evin P . Schneider , Senior Member , IEEE , and Chen-Ching Liu, F ellow , IEEE Abstract —Operators can now r emotely control switches and update the control settings f or voltage regulators and distributed energy resour ces (DERs), thus unleashing the network recon- figuration opportunities to improve efficiency . Aligned to this direction, this work puts forth a comprehensiv e toolbox of mixed-integer linear programming (MILP) models leveraging the control capabilities of smart grid assets. It de velops detailed practical models to capture the operation of locally and remotely controlled regulators, and customize the watt-var DER control curves complying with the IEEE 1547.8 mandates. Maintaining radiality is a key requirement germane to various feeder opti- mization tasks. This requirement is accomplished here through an intuitive and pro vably correct f ormulation. T o the best of our knowledge, this is the first time to optimally select a feeder topology and simultaneously design DER settings while taking into account legacy grid apparatus. The developed toolbox is put into action to reconfigur e a grid for minimizing losses using real- world data on a benchmark feeder . The results corroborate that optimal topologies vary across the day and coordinating DERs and regulators is critical during periods of steep net load changes. Index T erms —W att-var control; radiality (tree) constraints; voltage regulators; IEEE 1547.8; linearized distribution flow . I . I N T R O D U C T I O N Power distribution grids, in general, operate as radial net- works connecting the substations to various customers. Often- times, these systems host normally-open switches that allow changes in the network topology and maintain radiality for protection system simplicity . The ability to switch between different topologies brings about a class of grid optimization tasks termed as distrib ution network r econfiguration (DNR). Some goals of DNR are post-outage restoration, load balanc- ing, voltage regulation, po wer loss minimization, and planned maintenance [1], [2], [3], [4]. Utilizing existing switches to enhance efficiency and re- liability of distribution systems is promising, making DNR a long pursued task [2], [5], [6]. Network reconfiguration problems are combinatorial in nature, and inevitably intro- duce integer variables when posed as mathematical programs. Howe ver , advancements in mixed-integer solvers for linear , quadratic, and second-order cone programs revi ved attempts tow ards efficient DNR reformulations [7], [8]. Meanwhile, the development of exact conic relaxations of optimal power flow and accurate three-phase linear models hav e enabled computationally scalable DNR approaches that can cater to unbalanced multiphase grids [3], [9]. The advent of distributed energy resources (DERs) such as small generators, microgrids, and flexible loads has directed recent DNR research at maximally utilizing the available infrastructure [10], [11], [12]. On the other hand, the intermit- tency introduced by DERs increases the importance of DNR tow ards maintaining voltages within safe limits [13]. Thus, attempts are being directed towards lev eraging smart grid as- sets such as dispatchable DERs, capacitor banks, and remotely controlled voltage regulators in DNR formulations [9],[3]. Nonetheless, several smart grid de vices (such as inv erter- based photov oltaics (PVs) or ener gy storage units) and legacy grid de vices alike operate based on local control rules [14], [15]. On an operational basis, these rules could be fixed (reg- ulators and capacitor banks) or reconfigured periodically [16]. This is to reduce the frequenc y in communication and optimal power flow computations [17]. Y et the outcome of DNR could be significantly af fected by inaccurate or inadequate modeling of such locally controlled devices. The attempts at proper modeling of these devices are limited and based on simplifying assumption,s such as fixed and known taps for regulators and unity po wer factor DERs [11]. Enforcing radiality is another critical aspect in grid topology reconfiguration and other optimization tasks, such as planning and topology identification [18], [19]. Popular approaches to enforce radiality include an exhausti ve loop elimination, imposing a single inflo w edge or a single parent per bus; see [20], [21], and references therein. Despite being a classical problem, the con ventional approaches for enforcing radiality fail or lack optimality guarantees in the presence of DERs [20]. The contribution of this work is threefold: i) Put forth a nov el mixed-integer linear program (MILP) model for de- signing watt-var curves for DERs that takes into account all IEEE 1547.8 standard mandates (Section III); ii) Revisit an optimization model for guaranteeing connectedness and radiality of a feeder to provide a more compact form and establish its correctness (Section IV). The model is intuitiv e, prov ably correct, and decouples radiality constraints from variables capturing actual flo ws; and iii) T o capture the ef- fect of legacy devices, we dev elop an optimization model for capturing the operation of locally controlled regulators (Section V). This is in contrast to existing schemes where regulators are either ignored or their taps are presumed known. The proposed DNR is formulated as a mixed-inte ger quadratic program (MIQP) and tested using real-world load and solar generation data on the IEEE 37-bus benchmark. The tests of Section VII corroborate that depending on the load-generation mix experienced across a day , the operator has to select different topologies as well as regulator and DER settings. Albeit our results b uild on a linearized and balanced grid 2 model, they constitute a solid foundation for e xtensions to A C models and unbalanced multiphase setups. Regarding notation , lower - (upper-) case boldface letters denote column vectors (matrices). Calligraphic symbols are reserved for sets. Symbol > stands for transposition, and vectors 0 and 1 are the all-zero and all-one vectors. I I . P RO B L E M S TA T E M E N T A N D E X I S T I N G M O D E L S Suppose a utility knows the model of a feeder as well as the anticipated load and solar generation on a per-b us basis for the upcoming operating period of 4 hours or so. The utility operator would like to reconfigure the grid via remotely controlled switches to minimize ohmic losses. A key requirement is that the reconfigured topology has to remain radial at all times. In addition to switches, the operator can change the tap settings of remotely controlled voltage regulators and select the watt-v ar curves of DERs to ensure that voltages and line flows remain within specified limits. While selecting the feeder topology and optimizing the settings of regulators and PVs, the operator has to further take into account non-controllable loads and legacy devices. T o tackle this problem, this section re views optimization models for feeders, nodal and edge constraints, and voltage- dependent loads. It should be emphasized that these are existing models. They are presented here for completeness and for setting the ground of subsequent dev elopments: Section III puts forth a nov el approach for designing watt-var curves for PV generators. Section IV de vises an efficient optimization model for enforcing radiality . Section V presents models for locally and remotely controlled voltage regulators. Building on the pre vious models, Section VI formulates DNR and the numerical tests of Section VII validates the method. A. Grid Modeling, Nodal V ariables and Constraints Before commencing with the feeder models, some prelim- inaries from graph theory are in order . An undirected graph G := ( N , E ) is defined by a set of nodes N and a set of edges E , that are incident on the nodes in N . An y edge e ∈ E is defined by its incident nodes as ( i, j ) with i, j ∈ N . Nodes i and j are said to be adjacent if there is an edge ( i, j ) or ( j, i ) in E . Edges e 1 and e 2 are adjacent if they hav e a common end node. A path from node i to j is a sequence of adjacent edges, without repetition, starting from i and terminating at j , such that no node is revisited. A graph G is connected if there exists an i − j path for all i, j ∈ N . A cycle is a sequence of adjacent edges without repetition that starts and ends at the same node. A graph with no c ycles is acyclic . A connected and acyclic graph is a tr ee . A graph ˇ G := ( ˇ N , ˇ E ) is a subgraph of G if ˇ N ⊆ N and ˇ E ⊆ E . If every edge e ∈ E is assigned a direction, the obtained graph is termed directed . A single-phase distribution system with N + 1 buses can be modeled as a connected graph G ( N 0 , E ) . The nodes in N 0 := { 0 , . . . , N } correspond to buses; and its edges E to distribution lines, voltage regulators, and switches. The sub- station b us is inde xed by i = 0 , and other buses are contained in N := N 0 \ { 0 } . The assumption of a single feeder bus (substation) is without loss o f generality . The detailed gener- alization will be commented upon at various instances while declaring constraints relating to feeder bus. T opologically , a graph representing instances of multiple substations may be augmented by appending a virtual b us that is connected to all substations, thus acting as a single substation bus. Each edge e = ( i, j ) is assigned a direction from the origin node i to the destination node j . If ( i, j ) ∈ E , then ( j, i ) / ∈ E . Each bus i ∈ N is assumed to host at most one generator or load. The subset of buses hosting loads is denoted by N ` ⊆ N . This is without loss of generality because a b us with multiple loads and/or generations can be modeled as a set of single- load b uses, all connected by non-switchable zero-impedance lines. Let v i represent the voltage magnitude and p i + j q i the complex power injection on bus i . The nodal voltages and injections at all nodes in N can be stacked in the N -length vectors v and p + j q , respectiv ely . The substation voltage v 0 is assumed known and fixed. The general case of multiple substations can be handled by defining voltages independently for all substation buses connected to bus 0 . W e do not consider this scenario to keep the presentation uncluttered. A distribution grid may host dif ferent types of loads and DERs. Some e xamples include (in)elastic ZIP loads; (non)dispatchable DERs; and volt- or watt-dependent reactiv e power sources per the IEEE 1547.8 standard [14]. The con- straints on voltage and power injection for all nodes can be abstractly expressed as v 1 ≤ v ≤ ¯ v 1 (1a) p ( v ) ≤ p ≤ ¯ p ( v ) (1b) q ( v , p ) ≤ q ≤ ¯ q ( v , q ) . (1c) The individual limits are discussed next. The v oltage limits may be set to the typical operational limits: The ANSI standard dictates that service voltages should remain within ± 5% per unit (pu) [22]. Our model stops at the level of distribution transformers. Expecting a voltage deviation along the cable between a distribution transformer and the service voltage, the practice is to maintain voltages at distribution transformers within ± 3% pu; see also [23], [16]. The functions p ( v ) , ¯ p ( v ) , q ( v , p ) , and ¯ q ( v , q ) apply entry- wise, and depend on load and DER characteristics. Regarding loads, in steady-state analysis the v oltage dependence of loads is captured by the ZIP model. According to this model, each load is a composition of a constant-impedance (Z), a constant- current (I), and a constant-power (P and Q) component. Gi ven bus voltage magnitude v i , the power injection of load i is modeled as [23] p i ( v i ) = α p 0 + α p 1 v i + α p 2 v 2 i q i ( v i ) = α q 0 + α q 1 v i + α q 2 v 2 i with all α coefficients being non-positiv e and assumed known. Because under normal operation voltages are close to 1 pu, we can linearize the quadratic dependence of ZIP loads around the nominal v oltage to approximate v 2 i ' 2 v i − 1 ; see e.g., [24]. Then, for all buses hosting loads, the active and reactiv e power limits of (1b)–(1c) can be compactly written as [ p i ( v i ) ¯ p i ( v i ) q i ( v i ) ¯ q i ( v i )] > = α 0 + v i α 12 , ∀ i ∈ N ` . (2) 3 If load i is inelastic, then apparently p i ( v i ) = ¯ p i ( v i ) ; and p i ( v i ) ≤ ¯ p i ( v i ) otherwise. Similarly for reactive power injec- tions. Modeling of DERs is deferred to Section III. B. Edge V ariables and Constraints The edge set E can be partitioned into the set of switches E S ; regulators E R ; and fixed lines E \ ( E R ∪ E S ) . The basic network reconfiguration task aims at selecting a subset of switches to be closed. T o capture which switches are closed, let us introduce the binary v ariables y e ’ s for all switchable lines e ∈ E S . V ariable y e = 1 indicates that switch e is closed or connected; and vice versa. Let the po wer flow on edge e ∈ E be P e + j Q e . The po wer flow constraints on distribution lines may be expressed as [ P e Q e ] ≤ [ P e Q e ] ≤ [ ¯ P e ¯ Q e ] , ∀ e ∈ E \ E S (3a) y e [ P e Q e ] ≤ [ P e Q e ] ≤ y e [ ¯ P e ¯ Q e ] , ∀ e ∈ E S . (3b) If switch e is open ( y e = 0 ), constraint (3b) sets the po wer flow on e to zero. Else, box constraints on the po wer flo w are enforced and usually P e = − ¯ P e and Q e = − ¯ Q e . Although apparent power flo w limits of the form P 2 e + Q 2 e ≤ S 2 e can be added to our formulation, they result in a mixed-integer quadratically-constrained quadratic program (MI-QCQP), which does not scale as well as an MIQP . Alternativ ely , apparent power constraints on flows can be handled by a polytopic approximation of P 2 e + Q 2 e ≤ S 2 e ; see [25]. This approach is not taken here for clarity of presentation. C. P ower Flow Model T o relate power injections and flows to voltages, we build upon the linearized distribution flow (LDF) model of [2]. Albeit approximate, the LDF model has been extensiv ely employed for v arious grid optimization tasks with satisfactory accuracy [26]. By ignoring power losses, LDF postulates that the po wer injections for each bus i ∈ N are p i = X e :( i,j ) ∈E P e − X e :( j,i ) ∈E P e (4a) q i = X e :( i,j ) ∈E Q e − X e :( j,i ) ∈E Q e . (4b) If r e + j x e represents the impedance of line e : ( i, j ) ∈ E , the LDF model relates the squar ed v oltage magnitudes to power flows linearly as v 2 i − v 2 j = 2 r e P e + 2 x e Q e . Inv oking the assumption of small voltage deviations, squared voltages can be approximated as v 2 i ' 2 v i − 1 . Then, the non-squared voltages can be substituted in the LDF model to yield v i − v j = r e P e + x e Q e (5) for each line e : ( i, j ) ∈ E \ ( E R ∪ E S ) . The approximate v oltage drop model of (5) can be alternatively deri ved by linearizing the power flo w equations at the flat voltage profile [26], [27], [28]. For switchable lines in E S , the voltage drop of (5) applies only if the switch is closed, that is y e ( v i − v j − r e P e − x e Q e ) = 0 , ∀ e : ( i, j ) ∈ E S . (6) The bilinear products appearing in (6) such as y e v i are handled using McCormic k linearization , which is briefly revie wed next. McCormick linearization replaces the product of variables by their linear conv ex en velopes to yield a relaxation of the original non-conv ex feasible set [29]. If at most one of the factor variables is continuous and the rest are binary , the re- laxation becomes e xact . T ake for instance the product z = xy ov er a binary v ariable x ∈ { 0 , 1 } , and a continuous variable y bounded within y ∈ [ y , ¯ y ] . The constraint z = xy can be equiv alently expressed as four linear equality constraints xy ≤ z ≤ x ¯ y , (7a) y + ( x − 1) ¯ y ≤ z ≤ y + ( x − 1) y . (7b) T o see the equivalence, ev aluate x = 0 in (7) to get z = 0 , and ev aluate x = 1 to get z = y . For a continuous-binary bilinear product, the McCormick linearization is equiv alent to the so called big- M trick. Howe ver , particular emphasis on tight bounds y ∈ [ y , ¯ y ] in McCormick linearization tends to provide numerical superiority . All continuous-binary bilinear products encountered henceforth will be handled by McCormick lin- earization. The resulting linear inequalities of (7) will not be provided explicitly for brevity . I I I . D E S I G N I N G W AT T - V A R C O N T RO L C U RV E S F O R D E R S This section specifies the po wer injection limits of (1b)– (1c) for DERs. Con ventionally , DERs hav e been modeled as constant-power sources operating at unit po wer factor [30]. W ith smart DERs featuring enhanced sensing, communi- cation, and actuation, the IEEE 1547.8 standard mandates DERs to provide reactive po wer support [14]. According to the standard, the reactive power injection of DERs can follow four possible modes [14]: i) constant power factor; ii) voltage-dependent reactiv e po wer (v olt-var); iii) activ e po wer- dependent reactiv e power (w att-var); and iv) constant reacti ve power mode. The volt-v ar and watt-v ar dependencies are captured by control rules described by piece wise affine functions; see Fig. 1. The operator may change these rules on a daily , hourly , or near-real-time basis. T o effecti vely integrate DERs, their control rules should be decided optimally based on feeder and loading conditions. T o this end, it is henceforth assumed that DERs are operating in the watt-v ar mode and their parameters are selected and kept fixed per periods of say 4 hours. The watt-var in verter control is implemented via the piece- wise affine rules of Fig. 1. The left half applies to DERs featuring acti ve po wer absorption, such as energy storage units. T o simplify the exposition, we consider DERs operating in the right halfspace of the watt-var rule, that is DER that only inject active power to the feeder (e.g., renewable generation). Giv en the rated reacti ve power capacity ¯ q i for the i -th DER, the controllable parameters are p i, 1 and p i, 2 . The IEEE 1547.8 standard further constraints ( p i, 1 , p i, 2 ) so that 0 . 4 ¯ p i ≤ p i, 1 ≤ 0 . 8 ¯ p i (8a) p i, 1 + 0 . 1 ¯ p i ≤ p i, 2 ≤ ¯ p i (8b) where ¯ p i is the rated active power for DER i . These specifica- tions are set by the standard to ensure a substantial deadband and to avoid steep slopes in Fig. 1. 4 Fig. 1. Activ e power -reactive power (watt/var) DER control curve [14]. Giv en ( p i, 1 , p i, 2 ) , the reactive power injection of DER i depends on its active power injection as q i ( p i ) =    0 , 0 ≤ p i ≤ p i, 1 − ¯ q i ( p i − p i, 1 ) p i, 2 − p i, 1 , p i, 1 ≤ p i ≤ p i, 2 − ¯ q i , p i, 2 ≤ p i ≤ ¯ p i (9) The control rule of (9) induces a non-linear equality constraint between the optimization variables q i , p i, 1 , and p i, 2 . W e next capture this constraint via a novel MILP model. Let us introduce the binary v ariables ( δ i, 1 , δ i, 2 , δ i, 3 ) to pick which of the three segments in (9) is activ e each time: ( δ i, 1 , δ i, 2 , δ i, 3 ) ∈ { 0 , 1 } 3 (10a) δ i, 1 + δ i, 2 + δ i, 3 = 1 . (10b) The selection of a segment depends on the value of p i as δ i, 2 p i, 1 + δ i, 3 p i, 2 ≤ p i ≤ δ i, 1 p i, 1 + δ i, 2 p i, 2 + δ i, 3 ¯ p i . (11) Then, the rule of (9) can be expressed by the constraint q i = δ i, 1 · 0 − δ i, 2 ¯ q i ( p i − p i, 1 ) p i, 2 − p i, 1 − δ i, 3 ¯ q i . (12) Although (11) inv olves binary-continuous variable products, and can be thus handled by MacCormick linearization, that is not the case for (12). Unfortunately , the latter entails ratios or products among continuous variables. T o bypass this difficulty , the key idea here is to parameterize Figure 1 using the slope/intercept of its middle segment instead of the breakpoints ( p i, 1 , p i, 2 ) . If the middle segment of (9) is denoted by q i ( p i ) = β i p i + γ i for some negati ve ( β i , γ i ) , then (12) is equiv alent to q i = δ i, 2 ( β i p i + γ i ) − δ i, 3 ¯ q i . (13) Different from (11), constraint (13) in volv es only binary- continuous v ariable products. W e next reformulate (11) in terms of ( β i , γ i ) . Because the line q i ( p i ) = β i p i + γ i passes through the points ( p i, 1 , 0) and ( p i, 2 , − ¯ q i ) , we get that p i, 1 = − γ i β i and p i, 2 = − ¯ q i + γ i β i . (14) Plugging (14) into (11); multiplying all sides by β i < 0 ; adding γ i ; and using (10b), eventually provides δ i, 3 ( γ i − β i ¯ p i ) − δ i, 2 ¯ q i ≤ β i p i + γ i ≤ δ i, 1 γ i − δ i, 3 ¯ q i (15) which is still amenable to McCormick linearization. The control rule of (9) is equiv alent to (10), (13), and (15). W ith the help of McCormick linearization, the latter can be posed as an MILP model. The aforesaid model captures the piecewise control rule, b ut do not enforce the limitations of (8). T o capture these IEEE 1547.8 requirements, we will translate the constraints on ( p i, 1 , p i, 2 ) to constraints on ( β i , γ i ) . Sub- stituting (14) into (8) implies that ( β i , γ i ) should satisfy − 0 . 4 ¯ p i β i ≤ γ i ≤ − 0 . 8 ¯ p i β i (16a) ¯ p i β i + γ i ≤ − ¯ q i ≤ 0 . 1 ¯ p i β i . (16b) T o summarize, the control curve for DER i is optimally tuned via v ariables ( β i , γ i ) that satisfy (10), (13), (15), and (16). T o the best of our knowledge, this is the first model to optimally design the IEEE 1547 control curves for DERs. I V . E N S U R I N G R A D I A L T O P O L O G I E S Ensuring a graph is radial is of central importance to various grid optimization tasks. In grids with a single power source and no DERs, enforcing radiality entails limiting the number of edges with incoming flo w to one per bus [7]. In the presence of DERs, a bus may receiv e power from multiple edges even if the grid is radial. T o handle such networks, the model of [8] enforces an edge orientation so that each bus has a single parent bus. Despite its extensi ve use in the grid topology reconfiguration/restoration literature, counterexamples where this parent-child model produces disconnected graphs do ex- ist [21]. A dual graph-based model was suggested in [21], yet it is limited to planar graphs. For a general network, cycles can be av oided by imposing that the number of connected edges on each cycle to be less than the length of the cycle [12]. Despite its generality , this cycle-elimination approach can lead to exponentially many constraints. One of the most popular radiality model ensures connecti vity of loads to DERs via the power flo w equations, and connects DERs to the substation via flows of a virtual commodity [31]. The tightness of a linear programming relaxation for this model has also been recently reported in [20]. In this section, we advance upon the commodity flow approach and propose a more succinct model with fewer variables and constraints. Moreover , this is the first time the commodity flo w model is supported with a formal proof. Giv en the complete graph G ( N 0 , E ) , define a subgraph ˇ G ( N 0 , ˇ E ) , such that ˇ E := E \ { e : e ∈ E S , y e = 0 } . The subgraph ˇ G represents the reconfigured distrib ution network. T o capture the line infrastructure of G , define its |E | × ( N + 1) branch-bus incidence matrix ˜ A with entries ˜ A e,k :=      +1 , k = i − 1 , k = j 0 , otherwise ∀ e = ( i, j ) ∈ E . Separate the first column a 0 of ˜ A related to the substation bus 0 as ˜ A = [ a 0 A ] , to get the r educed branch-bus incidence 5 matrix A . Similarly , let ˇ A ∈ R |E 0 |× N represent the reduced branch-bus incidence matrix of subgraph G 0 . The next claim establishes an efficient model for imposing graph connectivity . Proposition 1. A graph ˇ G ( N 0 , ˇ E ) with reduced branch-b us incidence matrix ˇ A is connected if and only if there exists a vector f ∈ R | ˇ E | , such that ˇ A > f = 1 . (17) Pr oof. Pro ving by contradiction, suppose ˇ G ( N 0 , ˇ E ) is not con- nected, and there exists f ∈ R | ˇ E | satisfying (19). If ˇ G ( N 0 , ˇ E ) is not connected, then there must exist a connected component, that is a maximal connected subgraph ˇ G S ( N S , ˇ E S ) , such that N S ⊂ N 0 and 0 / ∈ N S . Let A S be the branch-bus incidence matrix of ˇ G S . By definition, it holds that A S 1 = 0 . The fundamental theorem of linear algebra implies 1 ∈ (range( A > S )) ⊥ or 1 / ∈ range( A > S ) . (18) By hypothesis, graph ( N S , ˇ E S ) is a maximal connected subgraph of ˇ G , and hence, there is no edge ( i, j ) ∈ ˇ E with i ∈ N S and j ∈ ¯ N S where ¯ N S := N 0 \ N S . Since the order of edges and nodes forming the rows and columns of ˇ A are arbitrary , without loss of generality , partition ˇ A as ˇ A =  A ¯ S 0 0 A S  . Heed that A ¯ S is a r educed branch-b us incidence matrix, whereas A S is a complete branch-bus incidence matrix since 0 / ∈ N S . Partitioning f conformably to ˇ A , equation (17) reads  A > ¯ S 0 0 A > S   f ¯ S f S  = 1 . The second block implies that A > S f S = 1 , which contradicts (18) and completes the proof. T o provide some circuit theoretic intuition on Proposition 1, vector f represents flows on ˇ E resulting from a unit injection at all network nodes except for node i = 0 . For this flow setup to be feasible, there must be a withdraw al of N units at node 0 , and the injection from all nodes in N must hav e a path to reach node 0 . Ha ving all nodes connected to node 0 entails a single connected component. It is worth emphasizing that v ariable f does not relate to the actual line flows and is introduced only to enforce connectivity . The condition for connectedness of Proposition 1 is defined on matrix ˇ A , which depends on the switch status v ariables y e ’ s. Notice that ˇ A is deriv ed from A by removing the rows related to open switches. Therefore, condition (17) can be expressed with respect to the original matrix A , by forcing the virtual flo ws in f to be zero for open lines. The following corollary establishes a conv enient constraint for connectedness to be used later in our network reconfiguration problem. Corollary 1. Let A be the reduced br anch-b us incidence matrix of G , and ˇ G ⊆ G be a subgraph defined by opening switches { e ∈ E S : y e = 0 } . Subgraph ˇ G is connected if and only if there exists f ∈ R |E | such that A > f = 1 , and (19a) − y e N ≤ f e ≤ y e N , ∀ e ∈ E S . (19b) Constraint (19b) implies that the virtual flows on open switches are zero, and bounds the flows on closed switches within [ − N , N ] . Once a ˇ G is ensured to be connected, the requirement of radiality can be readily enforced as X e ∈E S y e = N − |E \ E S | (20) to ensure the total number of connected edges is N . V . M O D E L I N G V O L TAG E R E G U L A T O R S In addition to the DER control settings and its topology , the voltage profile of a feeder is dependent on voltage regulators. This section de velops novel models for regulators, which are later used in our grid reconfiguration formulation. W e model voltage regulators as ideal transformers. This is without loss of generality because the impedance of a non-ideal regulator can be modeled as a line connected in series with the ideal regulator . An ideal regulator scales its secondary-side voltage by ± 10% on increments of 0 . 625% using tap positions [15]. Consider a regulator modeled by edge e : ( i, j ) ∈ E R . Its voltage transformation ratio can be set to 1 + 0 . 00625 · t e , where t e ∈ { 0 , ± 1 , . . . , ± 16 } is its tap position. W e consider two classes of regulators [23]: 1) Locally contr olled re gulators: Collect such regulators in set E L R ⊆ E R . A locally controlled regulator e : ( i, j ) ∈ E R is programmed to maintain v j within a given range [ v j , ¯ v j ] . The regulator changes its taps after a time delay until v j is brought within [ v j , ¯ v j ] , unless an extreme tap position has been reached. Ignoring the time delay , this operation is illustrated in Figure 2 and described by v j ( v i ) =    1 . 1 · v i , v i ≤ v j 1 . 1  v j , v j  , v j 1 . 1 < v i < v j 0 . 9 0 . 9 · v i , v i ≥ v j 0 . 9 . (21) The first branch relates to the case where the primary voltage v i is quite low and even with t e = +16 , the secondary voltage v j = 1 . 1 · v i remains below v j . Likewise, the third branch relates to the case where the tap has reached its minimum of t e = − 16 . Normal operation is captured by the second branch, where v j is successfully regulated within [ v j , v j ] . For a locally controlled regulator , the operator cannot fully monitor and/or control the e xact tap position. Because of this, we propose approximating the second branch of (21) by setting v j at the mid-point of the regulation range, that is v j ( v i ) = v j + v j 2 when v i ∈  v j 1 . 1 , v j 0 . 9  . Since the regulation range typically spans 2 – 4 taps or 0 . 0125 – 0 . 025 pu [15], this approximation incurs negligible modeling error . The actual and approximate models for locally controlled regulators are illustrated in Fig. 2. The re gulator operation of Fig. 2 can be modeled in a fash- ion similar to the watt-var curve of (9). 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Locally controlled regulator characteristic: The left/rightmost segments occur when regulator taps have maxed out. Within the middle green box, the secondary v oltage is successfully regulated. Lacking the actual tap position, this middle area is approximated by its midpoint (reference voltage). The approximate model for a locally controlled regulator e ∈ E L R is captured by the constraints v i ≥ 0 . 8 δ e, 1 + v j 1 . 1 δ e, 2 + ¯ v j 0 . 9 δ e, 3 (22a) v i ≤ v j 1 . 1 δ e, 1 + ¯ v j 0 . 9 δ e, 2 + 1 . 2 δ e, 3 (22b) v j = 1 . 1 δ e, 1 v i + δ e, 2  v j + v j 2  + 0 . 9 δ e, 3 v i (22c) δ e, 1 + δ e, 2 + δ e, 3 = 1 (22d) ( δ i, 1 , δ i, 2 , δ i, 3 ) ∈ { 0 , 1 } 3 . (22e) Constraints (22d)–(22e) ensure that exactly one indicator vari- able gets activ ated. Constraints (22a)–(22b) capture the value of the primary voltage per region of Fig. 2. Constraint (22c) captures the behavior of the secondary v oltage per region of Fig. 2. The binary-continuous variable products in (22c) can be handled via McCormick linearization. Note that 0 . 8 pu and 1 . 2 pu are arbitrarily chosen as the extreme voltage limits for defining the range of the primary voltage. 2) Remotely contr olled r e gulators: These regulators com- prise the set ¯ E L R := E R \ E L R . If e : ( i, j ) ∈ ¯ E L R , its tap t e can be changed remotely by the operator . It hence becomes a control variable taking one of 33 possible values. These values can be encoded using 6 bits [32]. For example, the binary code 100001 corresponds to tap t e = +16 ; code 010000 corresponds to the neutral position t e = 0 ; and 000000 to t e = − 16 . Then, voltage v j relates to v i as v j = 0 . 9 + 0 . 00625 · 5 X k =0 b e,k 2 k ! v i (23a) b e,k ∈ { 0 , 1 } , k = 0 , . . . , 5 (23b) with the parenthesis being the binary encoding of tap t i . The products b e,k v i can be handled by McCormick linearization. V I . P RO B L E M F O R M U L A T I O N Having modeled the major grid assets, we can now for- mulate the optimal grid reconfiguration task. Consider an operating period of 4 hr . Before the start of this period, the operator collects minute-based data capturing the anticipated Fig. 3. The IEEE 37-bus feeder with an additional regulator , lines, and DERs. load and solar generation, and partition them into 15-min intervals. Then, from each 15-min interval, the operator selects S samples, yielding a total of T = 16 S samples for the upcom- ing 4-hr period. These samples are indexed by t = 1 , . . . , T . Instead of sampling, the operator may use the av erages per 15-min interval. The data related to sample t are collectively denoted by vector θ t . The operator would like to minimize the power losses summed up over all T instances. Each one of the T instances will be experiencing different power flo w conditions. Nonetheless, all intervals share the same feeder topology , DER curves, and regulator settings. T o capture this, we group optimization variables as ω 1 := {{ y e } e ∈E S , { β i , γ i } i ∈N \N ` , { b e,k } e ∈E R \E L R } ; and ω t 2 := { v t , p t , { q t i , δ t i,k } i ∈N \N ` , { δ t e,k } e ∈E L R , P t , Q t } , ∀ t. The ultimate goal is to determine ω 1 , that is a tree topology , in verter watt-var parameters, and regulator tap settings. The grid would then be allowed to operate autonomously using local rules per interval t yielding variables { ω t 2 } T t =1 . The grid reconfiguration task can now be posed as min X t ∈T X e ∈E \E R r e ( P 2 e,t + Q 2 e,t ) (DNR) o ver ω 1 , { ω t 2 } T t =1 s . to (1) − (6) , (10) , (13) , (15) , (16) , (19) , (20) , (22) , (23) ∀ t. The cost function approximates the ohmic losses along all lines and times [2], [7]. When computing losses, only closed lines should be considered. Howe ver (3) entails that for open switches, the power flows are zero. This enables us to write the cost in (DNR) regardless of the indicator variables y t e for switchable lines. V I I . N U M E R I C A L T E S T S The dev eloped DNR was tested on a modified version of the IEEE 37-bus benchmark feeder con verted to its single-phase equiv alent [33]; see Fig. 3. Switches include three existing and two additional lines, all denoted as dashed edges. Regulator (799 , 701) is assumed to be remotely controlled. The regu- lator added on line (704 , 720) is set locally controlled with reference voltage 1 pu and bandwidth 0 . 016 pu. Fiv e PVs of equal capacity were placed at buses { 705 , 710 , 718 , 730 , 738 } . 7 12:00 am 4:00 am 8:00 am 12:00 pm 4:00 pm 8:00 pm 12:00 am 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized aggregate active power Fig. 4. Normalized aggregate activ e load and solar generation over time. The 5 panes represent the operating periods T 1 − T 5 . Residential load and solar data were extracted from the Pecan Street dataset [34]: Minute-based load and solar generation data were collected for June 1, 2018. The tested feeder has 25 buses with non-zero load. The first 75 non-zero load buses from the dataset were aggregated every 3 and normal- ized to obtain 25 load profiles. Similarly , 5 solar generation profiles were obtained. The normalized minute-based feeder- aggregated load and solar profiles are shown in Fig. 4. The normalized load profiles for the 24 -hr period were scaled so the 80 -th percentile of the total load duration curve coincided with the total nominal spot load of the feeder . This scaling results in a peak aggregate load being 1 . 29 times the total nominal load. Since the Pecan Street data contained no reactiv e power , we synthesized reacti ve loads by scaling the actual demand to match the nominal power factors of the IEEE 37-bus feeder . The linearized ZIP parameters of (2) were found using the deriv ed (re)active load profiles for each bus and the load type from the benchmark. The motiv e of the watt-v ar control is to alleviate ov ervoltages in grids with high solar integration. Thus, solar data were scaled such that 75% of the ov erall energy consumption was met from PVs. Problem (DNR) was solved using Y ALMIP and Gurobi [35], [36], on a 2 . 7 GHz Intel Core i5 computer with 8 GB RAM. The 24-hr interv al was partitioned into fi ve periods T 1 − T 5 ; see Fig. 4. Period T 1 extended ov er 8 hr , and the rest for 4 hr . Each period was di vided into 15 -min intervals and S = 2 samples of load and generation were randomly drawn from the minute-based data. W e then solved fiv e instances of (DNR). The results are summarized in Fig. 5. The solution times for each instance were 452, 92, 800, 268, and 50 sec. The schematics of Fig. 5 depict how the optimal topology v aries with changing load-generation mix throughout the day . Three distinct topologies were found to be optimal: one topology ov er T 1 and T 3 ; one over T 2 ; and a third one over T 4 − T 5 . Period T 1 experiences lo w loads and negligible solar gen- eration. As a result, the a verage po wer loss incurred is the minimum of all periods, and hence, its watt-v ar curves are inconsequential. Period T 2 features peaking generation and low load. Due to the large PV variation, a single tap setting cannot accomplish voltage regulation, and so PVs participate via reactive po wer absorption. V oltage regulation and loss minimization via reacti ve power control are known to be 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.2 0.4 0.6 0.8 -0.4 -0.3 -0.2 -0.1 0 T 1 AAAB9HicbVDLSgMxFL2pr1pfVZdugkVwVWaqoMuiG5cV+oJ2KJk004ZmMmOSKZSh3+HGhSJu/Rh3/o2ZdhbaeiBwOOde7snxY8G1cZxvVNjY3NreKe6W9vYPDo/KxydtHSWKshaNRKS6PtFMcMlahhvBurFiJPQF6/iT+8zvTJnSPJJNM4uZF5KR5AGnxFjJ64fEjCkRaXM+cAflilN1FsDrxM1JBXI0BuWv/jCiScikoYJo3XOd2HgpUYZTwealfqJZTOiEjFjPUklCpr10EXqOL6wyxEGk7JMGL9TfGykJtZ6Fvp3MQupVLxP/83qJCW69lMs4MUzS5aEgEdhEOGsAD7li1IiZJYQqbrNiOiaKUGN7KtkS3NUvr5N2repeVWuP15X6XV5HEc7gHC7BhRuowwM0oAUUnuAZXuENTdELekcfy9ECyndO4Q/Q5w+8H5IS T 2 AAAB9HicbVDLSgMxFL2pr1pfVZdugkVwVWaqoMuiG5cV+oJ2KJk004ZmMmOSKZSh3+HGhSJu/Rh3/o2ZdhbaeiBwOOde7snxY8G1cZxvVNjY3NreKe6W9vYPDo/KxydtHSWKshaNRKS6PtFMcMlahhvBurFiJPQF6/iT+8zvTJnSPJJNM4uZF5KR5AGnxFjJ64fEjCkRaXM+qA3KFafqLIDXiZuTCuRoDMpf/WFEk5BJQwXRuuc6sfFSogyngs1L/USzmNAJGbGepZKETHvpIvQcX1hliINI2ScNXqi/N1ISaj0LfTuZhdSrXib+5/USE9x6KZdxYpiky0NBIrCJcNYAHnLFqBEzSwhV3GbFdEwUocb2VLIluKtfXiftWtW9qtYeryv1u7yOIpzBOVyCCzdQhwdoQAsoPMEzvMIbmqIX9I4+lqMFlO+cwh+gzx+9o5IT T 3 AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsy0gi6LblxW6ENoh5JJM21oJhmTTKEM/Q43LhRx68e482/MtLPQ1gOBwzn3ck9OEHOmjet+O4WNza3tneJuaW//4PCofHzS0TJRhLaJ5FI9BlhTzgRtG2Y4fYwVxVHAaTeY3GV+d0qVZlK0zCymfoRHgoWMYGMlvx9hMyaYp635oD4oV9yquwBaJ15OKpCjOSh/9YeSJBEVhnCsdc9zY+OnWBlGOJ2X+ommMSYTPKI9SwWOqPbTReg5urDKEIVS2ScMWqi/N1IcaT2LAjuZhdSrXib+5/USE974KRNxYqggy0NhwpGRKGsADZmixPCZJZgoZrMiMsYKE2N7KtkSvNUvr5NOrerVq7WHq0rjNq+jCGdwDpfgwTU04B6a0AYCT/AMr/DmTJ0X5935WI4WnHznFP7A+fwBvyeSFA== T 4 AAAB9HicbVDLSgMxFL3xWeur6tJNsAiuykwt6LLoxmWFvqAdSibNtKGZzJhkCmXod7hxoYhbP8adf2OmnYW2HggczrmXe3L8WHBtHOcbbWxube/sFvaK+weHR8elk9O2jhJFWYtGIlJdn2gmuGQtw41g3VgxEvqCdfzJfeZ3pkxpHsmmmcXMC8lI8oBTYqzk9UNixpSItDkf1AalslNxFsDrxM1JGXI0BqWv/jCiScikoYJo3XOd2HgpUYZTwebFfqJZTOiEjFjPUklCpr10EXqOL60yxEGk7JMGL9TfGykJtZ6Fvp3MQupVLxP/83qJCW69lMs4MUzS5aEgEdhEOGsAD7li1IiZJYQqbrNiOiaKUGN7KtoS3NUvr5N2teJeV6qPtXL9Lq+jAOdwAVfgwg3U4QEa0AIKT/AMr/CGpugFvaOP5egGynfO4A/Q5w/Aq5IV T 5 AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclZmq6LLoxmWFvqAdSibNtKGZZEwyhTL0O9y4UMStH+POvzHTzkJbDwQO59zLPTlBzJk2rvvtrK1vbG5tF3aKu3v7B4elo+OWlokitEkkl6oTYE05E7RpmOG0EyuKo4DTdjC+z/z2hCrNpGiYaUz9CA8FCxnBxkp+L8JmRDBPG7P+db9UdivuHGiVeDkpQ456v/TVG0iSRFQYwrHWXc+NjZ9iZRjhdFbsJZrGmIzxkHYtFTii2k/noWfo3CoDFEplnzBorv7eSHGk9TQK7GQWUi97mfif101MeOunTMSJoYIsDoUJR0airAE0YIoSw6eWYKKYzYrICCtMjO2paEvwlr+8SlrVindZqT5elWt3eR0FOIUzuAAPbqAGD1CHJhB4gmd4hTdn4rw4787HYnTNyXdO4A+czx/CL5IW 0 . 4¯ p AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4Ckkt6LHoxWMF+4FtKJPttl262YTdjVBC/4UXD4p49d9489+4bXPQ1gcDj/dmmJkXJoJr43nfztr6xubWdmGnuLu3f3BYOjpu6jhVlDVoLGLVDlEzwSVrGG4EayeKYRQK1grHtzO/9cSU5rF8MJOEBREOJR9wisZKj55b7YaosmTaK5U915uDrBI/J2XIUe+Vvrr9mKYRk4YK1Lrje4kJMlSGU8GmxW6qWYJ0jEPWsVRixHSQzS+eknOr9MkgVrakIXP190SGkdaTKLSdEZqRXvZm4n9eJzWD6yDjMkkNk3SxaJAKYmIye5/0uWLUiIklSBW3txI6QoXU2JCKNgR/+eVV0qy4/qVbua+Wazd5HAU4hTO4AB+uoAZ3UIcGUJDwDK/w5mjnxXl3Phata04+cwJ/4Hz+AOnXkG0= 0 . 8¯ p AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4CkkV7LHoxWMF+4FtKJPttl262YTdjVBC/4UXD4p49d9489+4bXPQ1gcDj/dmmJkXJoJr43nfztr6xubWdmGnuLu3f3BYOjpu6jhVlDVoLGLVDlEzwSVrGG4EayeKYRQK1grHtzO/9cSU5rF8MJOEBREOJR9wisZKj55b7YaosmTaK5U915uDrBI/J2XIUe+Vvrr9mKYRk4YK1Lrje4kJMlSGU8GmxW6qWYJ0jEPWsVRixHSQzS+eknOr9MkgVrakIXP190SGkdaTKLSdEZqRXvZm4n9eJzWDapBxmaSGSbpYNEgFMTGZvU/6XDFqxMQSpIrbWwkdoUJqbEhFG4K//PIqaVZc/9Kt3F+Vazd5HAU4hTO4AB+uoQZ3UIcGUJDwDK/w5mjnxXl3Phata04+cwJ/4Hz+APADkHE= ¯ p AAAB7nicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOTTjJkdnaYmRXCko/w4kERr36PN//GSbIHTSxoKKq66e6KlODG+v63t7a+sbm1Xdgp7u7tHxyWjo6bJkk1wwZLRKLbETUouMSG5VZgW2mkcSSwFY3vZn7rCbXhiXy0E4VhTIeSDzij1kmtbkR1pqa9Utmv+HOQVRLkpAw56r3SV7efsDRGaZmgxnQCX9kwo9pyJnBa7KYGFWVjOsSOo5LGaMJsfu6UnDulTwaJdiUtmau/JzIaGzOJI9cZUzsyy95M/M/rpHZwE2ZcqtSiZItFg1QQm5DZ76TPNTIrJo5Qprm7lbAR1ZRZl1DRhRAsv7xKmtVKcFmpPlyVa7d5HAU4hTO4gACuoQb3UIcGMBjDM7zCm6e8F+/d+1i0rnn5zAn8gff5A5kej70= avg. P loss =0 . 0149 tap = 19 AAACFnicbVDLSsNAFJ3UV42vqEs3waK4MSS1oF0IRTcuK9gHNKFMppN26OTBzE2xhPoTbvwVNy4UcSvu/Bunj4W2HrhwOOde7r3HTziTYNvfWm5peWV1Lb+ub2xube8Yu3t1GaeC0BqJeSyaPpaUs4jWgAGnzURQHPqcNvz+9dhvDKiQLI7uYJhQL8TdiAWMYFBS2zh1gd5Dhgdda/RQbWc8lnJ0fGlbtlMqu64+tQEnSnTKbaOgnAnMReLMSAHNUG0bX24nJmlIIyAcS9ly7AS8DAtghNOR7qaSJpj0cZe2FI1wSKWXTd4amUdK6ZhBLFRFYE7U3xMZDqUchr7qDDH05Lw3Fv/zWikEF17GoiQFGpHpoiDlJsTmOCOzwwQlwIeKYCKYutUkPSwwAZWkrkJw5l9eJPWi5ZxZxdtSoXI1iyOPDtAhOkEOOkcVdIOqqIYIekTP6BW9aU/ai/aufUxbc9psZh/9gfb5A5oZnlg= avg. P loss =0 . 0192 tap = 16 AAACFnicbVDLSsNAFJ34rPEVdekmWBQ3lqSKj4VQdOOygn1AE8JkOm2HTh7M3BRLiD/hxl9x40IRt+LOv3HaZqGtBy4czrmXe+/xY84kWNa3Nje/sLi0XFjRV9fWNzaNre26jBJBaI1EPBJNH0vKWUhrwIDTZiwoDnxOG37/euQ3BlRIFoV3MIypG+BuyDqMYFCSZxw5QO8hxYNuKXuoeimPpMwOLq2SZV+UHUef2IBjJdqnnlFUzhjmLLFzUkQ5qp7x5bQjkgQ0BMKxlC3bisFNsQBGOM10J5E0xqSPu7SlaIgDKt10/FZm7iulbXYioSoEc6z+nkhxIOUw8FVngKEnp72R+J/XSqBz7qYsjBOgIZks6iTchMgcZWS2maAE+FARTARTt5qkhwUmoJLUVQj29MuzpF4u2cel8u1JsXKVx1FAu2gPHSIbnaEKukFVVEMEPaJn9IretCftRXvXPiatc1o+s4P+QPv8AZJonlM= avg. P loss =0 . 0997 tap = 15 AAACFnicbVDLSsNAFJ34rPEVdekmWBQ3lqQqtQuh6MZlBfuAppTJdNoOnUzCzE2xhPgTbvwVNy4UcSvu/Bunj4W2HrhwOOde7r3HjzhT4DjfxsLi0vLKambNXN/Y3Nq2dnarKowloRUS8lDWfawoZ4JWgAGn9UhSHPic1vz+9civDahULBR3MIxoM8BdwTqMYNBSyzrxgN5DggfdXPpQbiU8VCo9unRyTrFY8DxzYgOOtOiet6ysdsaw54k7JVk0RbllfXntkMQBFUA4VqrhOhE0EyyBEU5T04sVjTDp4y5taCpwQFUzGb+V2odaadudUOoSYI/V3xMJDpQaBr7uDDD01Kw3Ev/zGjF0LpoJE1EMVJDJok7MbQjtUUZ2m0lKgA81wUQyfatNelhiAjpJU4fgzr48T6r5nHuay9+eZUtX0zgyaB8doGPkogIqoRtURhVE0CN6Rq/ozXgyXox342PSumBMZ/bQHxifP6WFnl8= avg. P loss =0 . 0463 tap = 21 AAACFnicbVDLSsNAFJ3UV42vqEs3waK4MSRtUTdC0Y3LCvYBTSiT6aQdOnkwc1Msof6EG3/FjQtF3Io7/8bpY6GtBy4czrmXe+/xE84k2Pa3lltaXlldy6/rG5tb2zvG7l5dxqkgtEZiHoumjyXlLKI1YMBpMxEUhz6nDb9/PfYbAyoki6M7GCbUC3E3YgEjGJTUNk5doPeQ4UHXGj1U2xmPpRwdX9qWXT4rua4+tQEnSiw6baOgnAnMReLMSAHNUG0bX24nJmlIIyAcS9ly7AS8DAtghNOR7qaSJpj0cZe2FI1wSKWXTd4amUdK6ZhBLFRFYE7U3xMZDqUchr7qDDH05Lw3Fv/zWikEF17GoiQFGpHpoiDlJsTmOCOzwwQlwIeKYCKYutUkPSwwAZWkrkJw5l9eJPWi5ZSs4m25ULmaxZFHB+gQnSAHnaMKukFVVEMEPaJn9IretCftRXvXPqatOW02s4/+QPv8AY3xnlA= avg. P loss =0 . 0508 tap = 21 AAACFnicbVDLSsNAFJ34rPVVdekmWBQ3hqQqdiMU3bisYB/QhDCZTtqhkwczN8US4k+48VfcuFDErbjzb5y2WWjrgQuHc+7l3nu8mDMJpvmtLSwuLa+sFtaK6xubW9ulnd2mjBJBaINEPBJtD0vKWUgbwIDTdiwoDjxOW97geuy3hlRIFoV3MIqpE+BeyHxGMCjJLZ3YQO8hxcOekT3U3ZRHUmZHl6ZhnptV2y5ObcCxEiuWWyorZwJ9nlg5KaMcdbf0ZXcjkgQ0BMKxlB3LjMFJsQBGOM2KdiJpjMkA92hH0RAHVDrp5K1MP1RKV/cjoSoEfaL+nkhxIOUo8FRngKEvZ72x+J/XScCvOikL4wRoSKaL/ITrEOnjjPQuE5QAHymCiWDqVp30scAEVJJFFYI1+/I8aVYM69So3J6Va1d5HAW0jw7QMbLQBaqhG1RHDUTQI3pGr+hNe9JetHftY9q6oOUze+gPtM8fje2eUA== Fig. 5. Results of (DNR) for the feeder of Fig. 3. T op to bottom: Results for the operating periods identified in Fig. 4. Left to right: optimal topology , watt- var curves for five generators, average power loss, and optimal tap position. opposing goals [30]. Thus, PV generators start absorbing reactiv e po wer only when overv oltages become unav oidable. This intuition is demonstrated by the watt-v ar curves of Fig. 5 for T 2 , where reacti ve absorption begins only after PVs inject more than 0 . 8 of their capacity . While all PVs tend to absorb minimal reacti ve po wer and hence hit the limits of the watt- var curve in (8), the PV at b us 738 obtains a different curve and absorbs its maximum reacti ve power before reaching its ¯ p . During T 3 , voltages remain within limits because both load and generation are high, and so watt-v ar curves coincide with minimal reacti ve absorption. Period T 4 witnesses a steep decline in generation while the load is high. Therefore, a high tap setting of 21 is needed to av oid undervoltages after the decline in generation. Howe ver , for the tap setting of 21 , reactiv e absorption is needed to av oid overv oltages during high PV generation. Finally , period T 5 with no PV generation yields generic watt-var curves similar to T 1 , b ut higher taps and different topology from T 1 due to high load. The av erage activ e power loss for all periods follo ws the loading conditions. W e also experimented with the number of operating periods and the number of samples S . The effects are on three fronts: i) Frequency of changes in taps, topology , and inv erter settings; ii) V iolation of voltage limits o ver all minute-based data after fixing ω 1 ; and iii) T otal active power loss for all minute-based data after fixing ω 1 . Shorter periods inherently result in more frequent operations on taps, switches, and inv erter settings, be- sides the communication overhead. Longer operating periods on the other hand, may render problem (DNR) infeasible due to extreme changes in the load-generation mix. For instance, while an 8 -hr period for T 1 yields an acceptable solution for (DNR), mer ging T 3 and T 4 results in infeasibility . Further , e ven when (DNR) is feasible for a longer period, the total losses increase. Giv en a length, the periods should be chosen based on disparate load-generation levels. Further , for a fixed length, increasing S results in lo wer ov erall losses and less voltage 8 violations at the cost of higher computational burden, so S should be determined based on the anticipated fluctuations. V I I I . C O N C L U S I O N S Lev eraging the automation capabilities of forthcoming ac- tiv e distribution grids, this work has put forth an optimal DNR approach. DERs operate under watt-var control curv es to save on c yber resources. These curves are optimized jointly with the feeder topology and re gulator settings. The ap- proach uniquely integrates legac y devices and ensures radiality through intuitiv e and efficient optimization models. Numerical tests hav e corroborated: a) The optimal topology v aries with the load-generation mix; b) Coordinating DERs and regulators is critical during periods of steep transitions; and c) The trade- offs inv olved in the length of operating periods and the number of scenarios. Some open research directions are discussed ne xt. Although this work has considered a single-phase feeder , the models should be extendable to unbalanced multiphase setups. This work has adopted a linearized grid model; the operational benefits vis-a-vis the possible computational challenges of an A C grid model hav e to be explored. Although substituting watt-v ar with v olt-var curves might seem straight-forward, the related optimization and stability issues hav e to be addressed. It is worth adding that the de veloped toolbox of radiality , DER, and regulator models is applicable when coping with other grid reconfiguration or restoration tasks. R E F E R E N C E S [1] C. Chen, J. W ang, and D. 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