Proactive Sequential Phase Swapping Scheduling for Power Distribution Systems with a Finite Horizon

Abstract

A phase swapping system is provided for swapping phases in a power distribution system. The system includes an interface circuitry configured to receive, from sensors arranged at buses in the power distribution system, real-time states at current time step, wherein each of the real-time states are associated with a complex power flowing from an upstream bus to a downstream bus, one or more processors and a non-transitory computer-readable storage medium having stored thereon executable instructions that, when executed by the one or more processors, cause the phase swapping system to perform steps of determining a multi-step phase swapping action by using the real-time states and a trained optimal policy generating phase swapping commands based on the multi-step phase swapping action, and transmitting, via the interface circuitry, the phase swapping commands to phase swapping devices arranged at around the sensors, wherein the phase swapping devices are configured to perform the multi-step phase swapping action based on the phase swapping commands.

Claims

1. A phase swapping system for swapping phases in a power distribution system, comprising: an interface circuitry configured to receive, from sensors arranged at buses in the power distribution system, real-time states at current time step of a scheduling horizon, wherein each of the real-time states are associated with a complex power flowing from an upstream bus to a downstream bus; and one or more processors and a non-transitory computer-readable storage medium having stored thereon executable instructions that, when executed by the one or more processors, cause the phase swapping system to perform steps of: determining a multi-step phase swapping action by using the real-time states and a trained optimal policy; generating phase swapping commands based on the multi-step phase swapping action; and transmitting, via the interface circuitry, the phase swapping commands to phase swapping devices arranged at around the sensors, wherein the phase swapping devices are configured to perform the multi-step phase swapping action based on the phase swapping commands.

2. The phase swapping system of claim 1, wherein the optimal policy is pretrained offline for the scheduling horizon by using a state-action training set, wherein the state-action training set is generated for determining the multi-step phase swapping action for a training scenario using a multi-step mixed-integer non-linear programming (MINLP), wherein the training scenario is represented by using generation, load profiles for all buses and per-unit cost profiles with respect to the power distribution system.

3. The phase swapping system of claim 1, wherein the trained optimal policy is used to determine a guided set for phase swapping action corresponding to the generation, load and per-unit cost profiles for the scheduling horizon, wherein the guided set for phase swapping action defining time steps, total number of switching, and corresponding locations required for phase swapping.

4. The phase swapping system of claim 1, wherein the multi-step phase swapping action is generated by sequentially determining phase swapping action for each single time step that required phase swapping determined by the optimal policy; wherein single-step phase swapping action is obtained by solving a single-step phase swapping optimization model that formulated by minimizing weighted sum of multiple operational cost functions subject to constraints of radial load flow, substation voltage settings, phase swapping feasibility and a guided set of phase swapping action for the time step.

5. The phase swapping system of claim 4, wherein a multi-step phase swapping optimization model is used for generating single-step phase swapping action for each of multiple consecutive time steps that required phase swapping determined by the optimal policy.

6. The phase swapping system of claim 1, wherein the trained optimal policy is approximated with a deep neural network or random forest model, wherein inputs of the deep neural network or random forest model comprise profiles of power injections resulting from generations and loads with respect to wye-connected and delta-connected phases and locations, and profiles of per unit cost of the distribution system; wherein outputs of the deep neural network or random forest model include total number of phase swapping for each time step, and statuses for phase swapping for each wye-connected and delta-connected loads and generations.

7. The phase swapping system of claim 6, wherein the inputs for each time step include at least total net active and reactive power injections for all fixed-connection buses for each wye-connected phase, and each delta-connected phase pair, active and reactive power injections for each swappable-connection bus for each wye-connected location, and each delta-connected location, and per unit purchase costs for active and reactive powers.

8. The phase swapping system of claim 2, wherein the multi-step phase swapping action is determined by minimizing a cost function, wherein the cost function is expressed as a weighted sum of multiple cost components including a production cost for substation power purchase and generation and load curtailments, a penalty cost for branch current limit and bus voltage limit violations, a penalty cost for active power, reactive power, current and voltage imbalances, and a cost for phase swapping operations for all time steps within a scheduling horizon, while satisfied at least constraints of radial load flow for all time steps within the horizon.

9. The phase swapping system of claim 1, wherein each generation and load is wye-connected or delta-connected, wherein each wye-connected generation and load have three wye-connected locations and each location may assign to a wye-connected phase, wherein each delta-connected generation and load have three delta-connected locations and each location may assign to a delta-connected phase-pair.

10. The phase swapping system of claim 9, wherein delta-connected generation and load is converted to equivalent wye-connected generation and load through a delta-to-wye coversion matrix defined by assuming balanced three phases.

11. The phase swapping system of claim 8, wherein radial load flow model is used to represent operational constraints of the power distribution system by using a generic branch model; wherein the generic branch is a line segment, a voltage regulator, or a transformer; wherein the load flow for each branch is represented using a set of equations related to branch voltage drop, bus current balance or bus power balance, and power flows relating to bus voltages and branch currents or branch powers.

12. The phase swapping system of claim 11, wherein the generic branch is represented using a regulated model; where the regulated model connects a regulation component with a series branch, wherein the regulation component is modeled using a voltage amplifying matrix and a current amplifying matrix, and the series branch is modeled using a series impedance matrix and two shunt admittance matrices at its terminal buses.

13. The phase swapping system of claim 11, wherein squared branch currents and squared branch currents are used to represent branch voltage drop, bus power balance and relating expanded power flows with squared currents and squared voltages based on the regulated model; wherein a squared current is defined as a vector of phase currents times a conjugate transpose of the vector of phase current, wherein a squared voltage is defined as a vector of phase voltage times a conjugate transpose of the vector of phase voltages; wherein a expanded power flow is defined as a vector of bus phase voltages times a vector of branch phase currents.

14. The phase swapping system of claim 11, wherein branch currents and bus voltages are used to represent branch voltage drop, bus current balance and relating power flows with branch currents and bus voltages based on the regulated model.

15. The phase swapping system of claim 1, wherein phase swapping action is represented by using a matrix for assigning each wye-connected location to a phase for each wye-connected load and generation, and a matrix for assigning each delta-connected location to a phase-pair for each delta-connected generation and load.

16. The phase swapping system of claim 1, wherein phase swapping command for a wye-connected generation or load is described by a series of connected phases at previous and current time steps, wherein phase swapping command for a delta-connected generation or load is described by a series of phase-pairs at previous and current time steps.

17. A non-transitory computer-readable medium storing a phase swapping program including instructions that, when executed by a processor, causes a phase swapping system connected to a power distribution system through an interface circuitry, to: receive, from sensors arranged at buses in the power distribution system, real-time states at current time step of a scheduling horizon, wherein each of the real-time states are associated with a complex power flowing from an upstream bus to a downstream bus; determine a multi-step phase swapping action by using the real-time states and a trained optimal policy; generate phase swapping commands based on the multi-step phase swapping action; and transmit, via the interface circuitry, the phase swapping commands to phase swapping devices arranged at around the sensors, wherein the phase swapping devices are configured to perform the multi-step phase swapping action based on the phase swapping commands.

18. The non-transitory computer-readable medium of claim 17, wherein the optimal policy is pretrained offline for the scheduling horizon by using a state-action training set, wherein the state-action training set is generated for determining the multi-step phase swapping action for a training scenario using a multi-step mixed-integer non-linear programming (MINLP), wherein the training scenario is represented by using generation, load profiles for all buses and per-unit cost profiles with respect to the power distribution system.

19. The non-transitory computer-readable medium of claim 17, wherein the trained optimal policy is used to determine a guided set for phase swapping action corresponding to the generation, load and per-unit cost profiles for the scheduling horizon, wherein the guided set for phase swapping action defining time steps, total number of switching, and corresponding locations required for phase swapping.

20. The non-transitory computer-readable medium of claim 17, wherein the multi-step phase swapping action is generated by sequentially determining phase swapping action for each single time step that required phase swapping determined by the optimal policy; wherein single-step phase swapping action is obtained by solving a single-step phase swapping optimization model that formulated by minimizing weighted sum of multiple operational cost functions subject to constraints of radial load flow, substation voltage settings, phase swapping feasibility and a guided set of phase swapping action for the time step.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0018] The present disclosure is further described in the detailed description which follows, in reference to the noted plurality of drawings by way of non-limiting examples of exemplary embodiments of the present disclosure, in which like reference numerals represent similar parts throughout the several views of the drawings. The drawings shown are not necessarily to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments.

[0019] FIG. 1 shows a schematic of a method for proactive sequential phase swapping in power distribution systems over a finite scheduling horizon.

[0020] FIG. 2A depicts an IEEE 13-node test feeder represented as a one-line diagram.

[0021] FIG. 2B illustrates an unbalanced three-phase power distribution system represented by a full-phase diagram.

[0022] FIG. 2C depicts a modified IEEE 13 node test feeder with phase swapping devices (i.e. phase swapper) represented by a full-phase diagram.

[0023] FIG. 3 is a schematic for modeling a generic branch (i,j) using the regulated model.

[0024] FIG. 4 shows a set of normalized profiles for load demands, solar generation, wind generation, and power purchase prices.

[0025] FIG. 5A lists the comparison of summarized phase swapping schedules determined using different phase swapping strategies.

[0026] FIG. 5B lists the detailed phase connection changes in the phase swapping schedule, determined using a sequential single-step phase swapping optimization strategy.

[0027] FIG. 5C lists the detailed phase connection changes in the phase swapping schedule, determined using either a simultaneous multi-step phase swapping optimization strategy or a proactive sequential single-step phase swapping optimization strategy.

[0028] FIG. 6A depicts total operational cost variations over time within the finite scheduling horizon using no phase swapping strategy.

[0029] FIG. 6B depicts total operational cost variations over time within the finite scheduling horizon using the sequential single-step phase swapping strategy.

[0030] FIG. 6C depicts total operational cost variations over time within the finite scheduling horizon using the simultaneous multi-step phase swapping strategy.

[0031] FIG. 7A depicts substation by-phase active power variations over time within the finite scheduling horizon using no phase swapping strategy.

[0032] FIG. 7B depicts substation by-phase active power variations over time within the finite scheduling horizon using the sequential single-step phase swapping strategy.

[0033] FIG. 7C depicts substation by-phase active power variations over time within the finite scheduling horizon using the simultaneous multi-step phase swapping strategy.

[0034] FIG. 8A depicts substation by-phase reactive power variations over time within the finite scheduling horizon using no phase swapping strategy.

[0035] FIG. 8B depicts substation by-phase reactive power variations over time within the finite scheduling horizon using the sequential single-step phase swapping strategy.

[0036] FIG. 8C depicts substation by-phase reactive power variations over time within the finite scheduling horizon using the simultaneous multi-step phase swapping strategy.

[0037] FIG. 9 shows a schematic of a system for proactive sequential phase swapping in power distribution systems over a finite scheduling horizon.

[0038] While the above-identified drawings set forth presently disclosed embodiments, other embodiments are also contemplated, as noted in the discussion. This disclosure presents illustrative embodiments by way of representation and not limitation. Numerous other modifications and embodiments can be devised by those skilled in the art which fall within the scope and spirit of the principles of the presently disclosed embodiments.

DETAILED DESCRIPTION

[0039] In the following description, for purposes of explanation, numerous specific details are set forth to provide a thorough understanding of the present disclosure. It will be apparent, however, to one skilled in the art that the present disclosure may be practiced without these specific details. In other instances, apparatuses and methods are shown in block diagram form only to avoid obscuring the present disclosure.

[0040] As used in this specification and claims, the terms for example, for instance, and such as, and the verbs comprising, having, including, and their other verb forms, when used in conjunction with a listing of one or more components or other items, are each to be construed as open ended, meaning that the listing is not to be considered as excluding other, additional components or items. The term based on means at least partially based on. Further, it is to be understood that the phraseology and terminology employed herein are for the purpose of the description and should not be regarded as limiting. Any heading utilized within this description is for convenience only and has no legal or limiting effect.

[0041] The goal of phase swapping scheduling over a given time horizon is to determine phase swapping schemes for each time step within the time horizon to minimize associated operational costs while maintaining system stable and secure operation. This is a multi-step optimization problem. Generally, two different solution strategies can be applied.

[0042] The first strategy involves solving the phase swapping problem sequentially, step by step, where each time step is addressed separately to adjust phase connections for swappable loads and generations according to the previous step values. This approach can quickly obtain solutions, but they are usually locally optimal.

[0043] The second strategy involves solving the multi-step problem as a whole, with all time steps addressed simultaneously. This approach can achieve globally optimal solutions for the entire time horizon but suffers from a heavy computational burden, particularly for large-scale systems.

[0044] To address this, a novel proactive approach is disclosed in this invention. It uses imitative learning (i.e. imitation learning) of multi-period phase swapping optimization to identify the time steps with phase swapping needs and the relevant loads and generations for phase swapping based on predicted load, generation and per-unit cost profiles. Then, an optimal phase swapping scheme is solved only for the necessary time steps with determined sets of loads and generations for phase swapping. By doing so, phase swapping scheduling with better optimality can be determined with reasonably limited computational efforts.

[0045] FIG. 1 is a schematic for the method of proactive sequential phase swapping of power distribution systems over a finite scheduling horizon.

[0046] As shown in FIG. 1, the phase swapping scheduling for a finite planning horizon T includes two stages, the first stage 110 is offline training stage to model the relationship between phase swapping actions, and load, generation and per unit profile states based on imitation learning, and the second stage 120 is online scheduling stage to generate full decisions on phase connection changes and related generation schedule according to real-time load, generation and per unit cost forecasts. The imitation learning mimics a mixed-integer non-linear programming (MINLP) solver of the multi-step phase swapping optimization problem, and the policy demonstrated by the MINLP expert is approximated with a deep neural network or an ensemble random forest model.

[0047] The off-line training stage 110 includes step 130 for defining training scenarios using load, generation and per-unit cost profiles S, step 140 for generating state-action training set D by solving multi-step phase swapping using MINLP, and step 150 for determining optimal phase swapping policy : state.fwdarw.action, that is deciding is phase swapping to be taken and where phase swapping to be taken based on corresponding load, generation and per-unit cost profiles. In this invention, the state for each time step includes the total un-swappable active and reactive power injections for each wye-connected phase and delta-connected phase pair within the system, the active and reactive power injections at each wye-connected location, or delta-connected location for each swappable load/generation, and the action at each time step defines if a phase swapping is needed at the specific time step, how many total phase swaps required, and which swappable load or generation needs phase swapping.

[0048] The online phase swapping scheduling stage 120 includes step 160 for receiving system state s(t), using load, generation and per-unit cost forecasts, step 170 for determining guided action for phase swapping (t)=(s(t)) for each time t, step 180 for determining detailed phase connection changes for each swappable load/generation and substation power purchase decisions at time t, and step 190 for outputting phase swapping scheme for time t. Guided action for phase swapping provides guidance on whether phase swapping is needed at time t, how many swaps are required, and which loads and generations require swapping.

[0049] In this section, we will first propose a generic branch model using the regulated model to represent commonly used branches in power distribution systems, including line segments, voltage regulators, and transformers. Next, we will derive branch-flow-based load flow models for radial distribution systems, either in terms of squared voltages and currents or explicitly using bus voltages and branch currents. Following this, we will present two mixed-integer nonlinear programming models using different load flow formulations to generate training samples for mimicking optimal phase swapping schemes over the finite planning horizon. Finally, we will describe the proactive phase swapping approach, which applies imitation learning to tackle the multi-step phase swapping problem, achieving a good compromise between solution optimality and computational burden.

Unbalanced Three-Phase Power Distribution System

[0050] FIG. 2A shows an exemplary three-phase power distribution system 200, i.e., the IEEE 13-node test feeder, represented using a one-line diagram. As illustrated, the typical power distribution system 200 adopts a radial configuration. Power originates from the main grid (or transmission system) 210, travels through a substation bus (or feeder head) 215, and is delivered to customer loads connected to buses 225 through line segments. The system may incorporate voltage regulation devices like the voltage regulator 235 and transformer 275. It is inherently unbalanced and may consist of three-phase components (e.g., line 246), two-phase components (e.g., line 247), or single-phase components (e.g., line 248).

[0051] FIG. 2B illustrates an unbalanced three-phase power distribution system 200 represented using a full-phase diagram. The power fed from the main grid 210 is measured by sensors 211 at the feeder head 220. Based on grounding conditions, this system can be divided into two portions. The left portion, bounded by the feeder head 220 and the left side of transformer 275, is an ungrounded three-phase three-wire system. In this section, buses 230 are not grounded, and each line segment contains only phase wires. The right portion, starting from the right side of transformer 275 and extending downstream, is a grounded three-phase four-wire system. Here, each line contains a neutral wire in addition to phase wires, and each bus 270 is connected with both phase and neutral wires. Transformer 275 is grounded by connecting one of its terminals to the earth 280. Multiple groundings are commonly adopted for operational and safety needs in power distribution systems. The system includes both delta-connected loads 240, 250, and 260, and wye-connected loads 280, 290, and 295. Wye-connected loads are connected between a phase and the neutral for each phase load component, while delta-connected loads are connected between two phases of the three phases for each phase-pair load component.

[0052] FIG. 2C depicts a modified IEEE 13-node test feeder with phase swapping devices, represented by a full-phase diagram, ignoring neutral connections. This system is modified from the feeder depicted in FIG. 2A by replacing all transformers, voltage regulators, and single or double-phase line segments with three-phase line segments, and adding a solar generator at bus 684 and a wind generator at bus 680. Some loads and generations are phase swappable, while others are not.

[0053] For a phase-swappable load 299, its phase swapping is implemented through an automatic load phase swapping device, phase swapper 298. The load 299 is a single-phase wye-connected load, and its phase can switch among phase A, phase B, and phase C. It currently resides at phase B (solid line represents the initial position, and the dashed line represents candidate positions). Similarly, for a phase-swappable generation 289, its phase can be switched through a generator phase swapping device, phase swapper 288. In this example, the generator 289 is a single-phase delta-connected wind generator. It can be relocated among phase pairs CA, AB, or BC, with its initial position on phase pair CA.

[0054] For a three-phase radial distribution network, we use ={0,1, . . . , n} denoting the set of buses, in which 0 is the index the substation (or point of common coupling to transmission system), and .sup.+=.sup.\{0}. We also use denoting the set of branches connecting between a pair of buses. (i,j) represents a branch connecting an ordered pair of buses (i,j), where bus i lies between bus 0 and bus j. We use (i, j) and i.fwdarw.j interchangeably.

[0055] For ease of exposition and notation simplicity, we assume that all the buses i and branches (i,j) have three phases: a, b, c; and define .sub.: ={a, b, c} and .sub.D: ={ab, bc, ca}. For i and .sub.,

[00001]$\mathrm{let}{V}_i^{}$

denote the complex voltage on phase of bus i, and define

[00002]$V_i:={\left[\begin{array}{ccc}{V}_i^a& V_i^b& V_i^c\end{array}\right]}^T.$

For i.fwdarw.j and .sub., let

[00003]$I_{\mathrm{ij}}^{}\mathrm{and}{S}_{\mathrm{ij}}^{}$

denote the current and power flow on the phase of branch from bus i to bus j, and define

[00004]$I_{\mathrm{ij}}:={\left[\begin{array}{ccc}{I}_{\mathrm{ij}}^a& I_{\mathrm{ij}}^b& I_{\mathrm{ij}}^c\end{array}\right]}^T,\mathrm{and}{S}_{\mathrm{ij}}:=[\begin{array}{ccc}{S}_{\mathrm{ij}}^a& S_{\mathrm{ij}}^b& {S_{\mathrm{ij}}^c]}^T.\end{array}$

If any phase is missing, the corresponding voltage, current, or power flow components will be set as zero. Loads and generations at any bus can also be differentiated by their locations, and we can define .sub.: ={a, b, c} and .sub.D: ={ab, bc, ca} to represent the location sets for wye-connected and delta-connected loads or generations at a bus with full phases.

[0056] Without loss of generality, let every bus i have three wye-connected generations and loads (one on each phase, with grounded neutral at three wye-connected locations) and three delta-connected generations and loads (one across each pair of phases, ungrounded at three delta-connected locations). Let

[00005]$s_{G,Y_i}:={\left[\begin{array}{ccc}{s}_{G,Y_i}^a& s_{G,Y_i}^b& s_{G,Y_i}^c\end{array}\right]}^T\mathrm{and}{s}_{L,Y_i}:={\left[\begin{array}{ccc}{s}_{L,Y_i}^a& s_{L,Y_i}^b& s_{L,Y_i}^c\end{array}\right]}^T$

denote the complex power productions and consumptions of wye-connected generations and loads at bus i. Let

[00006]$s_{G,D_i}:={\left[\begin{array}{ccc}{s}_{G,D_i}^{\mathrm{ab}}& s_{G,D_i}^{\mathrm{bc}}& s_{G,D_i}^{\mathrm{ca}}\end{array}\right]}^T\mathrm{and}{s}_{L,D_i}:={\left[\begin{array}{ccc}{s}_{L,D_i}^{\mathrm{ab}}& s_{L,D_i}^{\mathrm{bc}}& s_{L,D_i}^{\mathrm{ca}}\end{array}\right]}^T$

denote the power productions and consumptions of delta-connected generations and loads at bus i, respectively. If a particular phase of a particular type of connection does not exist at bus i, then the corresponding element of S.sub.G,Y.sub.i, S.sub.L,Y.sub.i or S.sub.G,D.sub.i, S.sub.L,D.sub.i are set as 0. In practice, S.sub.G,Y.sub.i, S.sub.L,Y.sub.i or S.sub.G,D.sub.i, S.sub.L,D.sub.i represent the power productions and consumptions observed on the primary windings of the wye and delta-connected service transformers at bus i. Their secondary windings, which connect to loads and distributed energy resources, are ignored from our model. The delta-connected complex power injections at bus i, (S.sub.G,D.sub.iS.sub.L,D.sub.i) will be converted into wye-connected ones, A.sub.S.sub.D.fwdarw.Y(S.sub.G,D.sub.iS.sub.L,D.sub.i) by using a delta-to-wye conversion matrix A.sub.S.sub.D.fwdarw.Y.sup.33. The delta-to-wye conversion matrix is determined by assuming voltages are balanced at bus i, that is:

[00007]$\frac{V_i^a}{V_i^b}\frac{V_i^b}{V_i^c}\frac{V_i^c}{V_i^a}{e}^{j2/3},$

and set as:

[00008]$A_{S_{D.fwdarw.Y}}=\left[\begin{array}{ccc}\frac{1}{2}-j\frac{\sqrt{3}}{6}& 0& \frac{1}{2}+j\frac{\sqrt{3}}{6}\\ \frac{1}{2}+j\frac{\sqrt{3}}{6}& \frac{1}{2}-j\frac{\sqrt{3}}{6}& 0\\ 0& \frac{1}{2}+j\frac{\sqrt{3}}{6}& \frac{1}{2}-j\frac{\sqrt{3}}{6}\end{array}\right].$

[0057] Any delta-connected generation and load is converted to equivalent wye-connected generation and load through this delta-to-wye coversion matrix defined by assuming balanced three phases.

[0058] There are three typical branches in the power distribution system, including a line segment, a voltage regulator, and a transformer that can be modeled through a generic branch model, as shown in FIG. 3.

[0059] FIG. 3 is a schematic for a generic branch (i,j) modeled using a regulated model. It can be expressed as a regulator 310, modeled using voltage and current conversion matrices, connected with a series branch 320 modeled using series impedance 330 and shunt admittances 340 and 350. The regulator is connected between bus i and an intermediate bus i, and the series branch is connected between the intermediate bus i and bus j. In the figure, y.sub.Cj.sup.33 360 denotes the shunt admittance, such as capacitor banks at bus j, .sub.Uij, .sub.Dij.sup.33 denote the equivalent shunt admittance of branch (i, j) introduced by line segments, voltage regulators, or transformer branch (i,j), and z.sub.ij.sup.33 denotes the series impedance of branch (i,j). A.sub.Vi.sub..fwdarw.j.sup.33 and A.sub.l.sub.i.fwdarw.j.sup.33 denote the voltage and current conversion factor matrices between the primary and secondary sides of the voltage regulator or transformer between bus i and bus j. , and N denote the set of real, complex, and nonzero natural numbers, respectively. A.sub.V.sub.i.fwdarw.j and A.sub.l.sub.i.fwdarw.j can be regarded as a voltage amplifying matrix and a current amplifying matrix.

[0060] According to the type of branches, we can be configured associated parameter matrices for the branch accordingly. For a line segment (i,j), A.sub.V.sub.i.fwdarw.j and A.sub.l.sub.i.fwdarw.j are identity matrices, z.sub.ij, y.sub.Uij and y.sub.Dij are the series impedance between bus i and bus j and shunt admittances evenly allocated to bus i and bus j, respectively.

[0061] For a voltage regulator branch (i,j), A.sub.V.sub.i.fwdarw.j and A.sub.l.sub.i.fwdarw.j are determined based on the voltage regulator primary and secondary winding connections, and tap ratios, and used to convert voltages and currents at primary side (bus i) to ones at the secondary side (bus i). z.sub.ij is the series impedance of regulated line segment (i,j), and y.sub.Uij and y.sub.Dij are the half-shunt admittances for the regulated line segment, y.sub.cj is shunt admittance at bus j.

[0062] For a transformer branch (i, j), A.sub.V.sub.i.fwdarw.j and A.sub.l.sub.i.fwdarw.j are determined based on the transformer primary and secondary winding connections, and tap ratios. z.sub.ij and y.sub.Uij and y.sub.Dij represent the equivalent series impedance and shunt admittances of the transformer determined according to its impedance and admittance, winding connections and tap ratios.

Radial Load Flow Formulations with Generic Branch Model

[0063] For notational simplicity, we assume that all the buses and lines have three phases. To deal with missing phases on certain buses and lines, we fill zeros at the corresponding locations of current/power/voltage vectors and impedance/admittance matrices.

[0064] The following notations are used to formulate the load flow model. Define j={square root over (1)}. For a complex number c, Re (c) and Im(c) denote its real and imaginary parts, respectively. For n, let .sup.nn denote the space of all n-by-n Hermitian matrices. For any scalar, vector, or matrix A, let A.sup.T, A*,and A.sup.H denote its transpose, element-wise conjugate, and conjugate transpose, respectively. Further, given an n-by n matrix A, diag(A) denotes the column vector composed of its diagonal entries.

[0065] Considered most of power distribution systems installing with voltage regulation devices, such as voltage regulars and transformers, it is important for modeling this kind of devices in load flows to maintain load flow solution with sufficient accuracy, in particular, for phase imbalance problems. Unfortunately, most of conventional branch flow based load flow models ignore such kind of devices in load flow calculations. To this end, this invention extends conventional branch flow based load flow models with capability for modelling voltage regulation devices using the generic branch model as represented by FIG. 3.

[0066] For a branch (i,j) modeled using regulated model as shown in FIG. 3, bus i and bus j are the sending and receiving buses of (i,j), and bus j and bus k are the sending and receiving buses of (j, k). Through the regulators, the currents injected into the branch from bus i, I.sub.ij and voltages at bus i, V.sub.i can be regulated into related ones at intermediate bus i,

[00009]$I_{\mathrm{ij}}^{}\mathrm{and}{V}_i^{},\mathrm{where}{I}_{\mathrm{ij}}^{}=A_{I_{i.fwdarw.j}}{I}_{\mathrm{ij}},V_i^{}=A_{V_{i.fwdarw.j}}{V}_i.$

Load Flow Formulation-I: Using Squared Voltages and Currents

[0067] For branch (i, j), three sets of variables are used to represent network states, including expanded branch flows,

[00010]${\stackrel{}{S}}_{\mathrm{ij}}=V_i{I}_{\mathrm{ij}}^H{}^{33}$

representing power flowing from bus i toward bus j, squared currents,

[00011]${\stackrel{}{I}}_{\mathrm{ij}}=I_{\mathrm{ij}}{I}_{\mathrm{ij}}^H{}^{33}$

representing squared current passing through branch from bus i toward bus j, and squared voltages,

[00012]${\stackrel{}{V}}_i=V_i{V}_i^H{}^{33}$

representing squared voltages at bus i. The actual power flowing from i toward bus j can be calculated as S.sub.ij=diag({tilde over (S)}.sub.ij).

[0068] Using expanded power flows and squared voltages and currents, the power flow on branch (i, j) can be formulated as follows:

[00013]$\begin{array}{cc}& \left(1\right)\end{array}$${.Math.}_{i:i.fwdarw.j}\mathrm{diag}\left(A_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}^H-A_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i{A}_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H-{\stackrel{}{V}}_j{y}_{\mathrm{Dij}}^H+z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}^H-z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i{A}_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H+z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}^H{A}_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H-z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{I}}_{\mathrm{ij}}^H{A}_{I_{i.fwdarw.j}}^H\right)-\mathrm{diag}\left({\stackrel{}{V}}_j{y}_{\mathrm{Cj}}^H\right)+s_j={.Math.}_{k:j.fwdarw.k}\mathrm{diag}\left(S_{\mathrm{jk}}\right)$$\begin{array}{cc}{s}_j=s_{G,Y_j}-s_{L,Y_j}+A_{S_{D.fwdarw.Y}}{s}_{G,D_j}-A_{S_{D.fwdarw.Y}}{s}_{L,D_j}& \left(2\right)\end{array}$$\begin{array}{cc}& \left(3\right)\end{array}$$A_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i{A}_{V_{i.fwdarw.j}}^H-{\stackrel{}{V}}_j=A_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}^H{z}_{\mathrm{ij}}^H-A_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i{A}_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H{z}_{\mathrm{ij}}^H+z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}^H{A}_{V_{i.fwdarw.j}}^H-z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{I}}_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}^H{z}_{\mathrm{ij}}^H+z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}^H{A}_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H{z}_{\mathrm{ij}}^H-z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i{A}_{V_{i.fwdarw.j}}^H+z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}^H{z}_{\mathrm{ij}}^H-z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i{A}_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H{z}_{\mathrm{ij}}^H$$\begin{array}{cc}\left[\begin{array}{cc}{\stackrel{}{V}}_i& {\stackrel{}{S}}_{\mathrm{ij}}\\ {\stackrel{}{S}}_{\mathrm{ij}}^H& {\stackrel{}{I}}_{\mathrm{ij}}\end{array}\right]=0& \left(4\right)\end{array}$

[0069] Equation (1) enforces the complex power balance at bus j. The left-hand side (LHS) of (1) consists of the complex power leaving bus i towards bus j subtracted complex power flowing along shunt elements at bus i and bus j, power losses on the branch from bus i to bus j, and added complex power injected at bus j, while the right-hand side (RHS) of (1) represents the complex power sent to downstream buses.

[0070] Equation (2) enforces the complex power injected at bus j, S.sub.1 consists of wye-connected complex power generations and loads S.sub.G,Y.sub.j, s.sub.L,Y.sub.j, and complex power generations and loads converted from delta-connected generations and loads, S.sub.G,D.sub.j, s.sub.L,D.sub.j.

[0071] Equation (3) describes the squared voltage drop along the branch from bus i to bus j. It accounts for the voltage at bus i, the voltage drops due to the power flow and the branch series impedance.

[0072] Equation (4) calculates the magnitude of the square current in the branch based on the expanded power flow and the square voltage at the sending end of the branch.

[0073] The equations (1)-(4) can be separated into the real and reactive components to be solved.

[0074] When taking the above-described as operational constraints in phase swapping optimization problems. (1)(3) are convex linear constraints. The equation (4) is a nonconvex because of the quadratic equations. The semidefinite relation is used to obtain a convex surrogate of (4). (4) is relaxed with a positive semidefinite constraint (5) for a branch (i, j):

[00014]$\begin{array}{cc}\left[\begin{array}{cc}{\stackrel{}{V}}_i& {\stackrel{}{S}}_{\mathrm{ij}}\\ {\stackrel{}{S}}_{\mathrm{ij}}^H& {\stackrel{}{I}}_{\mathrm{ij}}\end{array}\right]0& \left(5\right)\end{array}$

Load Flow Formulation-II: Using Explicit Bus Voltages and Branch Currents

[0075] For a large-scale power distribution system, the branch flow-based model expressed in terms of squared voltages and currents has a considerably larger number of decision variables, which leads to a computational burden when modeled as constraints for phase swapping optimization. To overcome this challenge, a branch flow-based load flow model in terms of explicit bus voltages and branch currents is proposed here. For any branch i.fwdarw.j, three equations are used to describe its power flow allocation. The first equation is a voltage drop equation:

[00015]$\begin{array}{cc}{A}_{V_{i.fwdarw.j}}{V}_i-V_j=z_{\mathrm{ij}}\left(A_{I_{i.fwdarw.j}}{I}_{\mathrm{ij}}-y_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{V}_i\right)& \left(6\right)\end{array}$

where V.sub.i and V.sub.j are the complex voltages at sending bus and receiving bus of branch i.fwdarw.j, A.sub.V.sub.i.fwdarw.j is the ideal voltage conversion factor matrices to amplify voltage at sending bus, and A.sub.I.sub.ji j is the ideal current conversion factor matrix to amplify current injected from sending bus into the branch i.fwdarw.j. z.sub.ij, and y.sub.Uij are an equivalent series impedance matrix for branch i.fwdarw.j, and equivalent shunt admittance matrix at bus i.

[0076] The second equation is a current balance equation for any bus j:

[00016]$\begin{array}{cc}{.Math.}_{i:i.fwdarw.j}\left(A_{I_{i.fwdarw.j}}{l}_{\mathrm{ij}}-y_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{V}_i-y_{\mathrm{Dij}}{V}_j\right)-y_{\mathrm{Cj}}{V}_j+I_j={.Math.}_{k:j.fwdarw.k}{I}_{jk}& \left(7\right)\end{array}$

where bus j has connected with an upstream branch i.fwdarw.j and a set of downstream branches j.fwdarw.k, I.sub.ij and I.sub.jk are the complex currents flowing from bus i to bus j, and from bus j to bus k, and I is the complex injection current at bus j. y.sub.Dij is the equivalent shunt admittance matrix of branch i.fwdarw.j at bus j, y.sub.cj is the admittance matrix at bus j.

[0077] The third equation is a bus power injection equation defined as:

[00017]$\begin{array}{cc}\mathrm{diag}\left(V_j{I}_j^H\right)=s_{G,Y_j}-s_{L,Y_j}+A_{S_{D.fwdarw.Y}}{s}_{G,D_j}-A_{S_{D.fwdarw.Y}}{s}_{L,D_j}& \left(8\right)\end{array}$

where

[00018]$I_j^H$

is the conjugate transpose of complex injected current at bus j, S.sub.G,Y.sub.j and S.sub.L,Y.sub.j are the wye-connected complex power generations and load demands at bus j, S.sub.G,D.sub.j and S.sub.L,D.sub.j are the delta-connected complex power generations and load demands at bus j, A.sub.S.sub.D.fwdarw.Y is a complex power conversion matrix to convert powers from delta-connection to wye-connection.
(6) and (7) are linear equations, but (8) is a nonlinear equation.

[0078] Similarly, the equations (6)-(8) can also be separated into the real and reactive components to be solved.

Optimal Phase Swapping Over a Finite Horizon T

[0079] In distribution networks, the diverse and predominantly ad-hoc allocation of single-phase renewable energy and electric vehicle charging installations leads to increased phase imbalances. These imbalances have the potential to negatively impact and introduce inefficiencies into the power distribution system. The subsequent rise in operational expenditures and the potential degradation of the network infrastructure underline the need for the development and implementation of optimized scheduling strategies. Phase swapping optimization becomes crucial to enhance the reliability and operational efficiency of distribution networks, ultimately lowering the operational costs.

[0080] The goal of phase swapping is to make the feeder well-balanced with minimal number of phase swapping and minimum operation cost increase for a given load, generation and per-unit cost profile over a scheduling cycle/horizon. Assumed each cycle has T time intervals, and the length of each interval is t. For example, a daily phasing swapping scheme includes 24 time steps, and each interval lasts 1.0 hours.

[0081] The optimal phase swapping aims to minimize a weighted sum of costs related to four objectives. The set of weight coefficients (.sup.SUB, .sup.OVR, .sup.IMB, .sup.SWAP) is selected based on the operational context and regulatory requirements, indicating the relative importance and economic implications of each cost component in achieving an optimal balance between production costs, system security, power quality, and device operations within the network constraints.

[0082] We assume any bus j can have three phases, and actual phase availability is defined using a vector of binary parameters,

[00019]$A_{\mathrm{BUS},j}^{}\left(t\right),{}_Y.$

j can have three-phase wye-connected and delta-connected loads and generations allocated at three distinct wye-connected locations, and three distinct delta-connected locations.

[00020]$s_{L,Y_j}^{l_Y}\left(t\right)\mathrm{and}{s}_{G,Y_j}^{l_Y}\left(t\right)$

are used to represent the forecasted load demands and generation productions at wye-connected location l.sub. of bus j, respectively.

[00021]$s_{L,D_j}^{l_D}\left(t\right)\mathrm{and}{s}_{G,D_j}^{l_D}\left(t\right)$

are used to represent the forecasted load demands and generation productions at delta-connected location l.sub.D of bus j, respectively.

[00022]$A_{\mathrm{LOC},Y_j}^{l_Y}\mathrm{and}{A}_{\mathrm{LOC},D_j}^{l_D}$

are vectors of binary parameters representing the availability of wye-connected location l.sub., and the delta-connected location l.sub.D of bus j. The actual phase connections for a bus jare determined based on location to phase assignment matrices and corresponding availabilities of locations. The matrix for assigning wye-connected locations to phases at bus j and time step t is expressed as

[00023]$\left\{B_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t\right),{}_Y{}_Y,l_Y{}_Y\right\},$

and the matrix for assigning delta-connected locations to phase pairs at bus j and time step t is expressed as

[00024]$\left\{B_{D_j}^{l_D.fwdarw.{}_D}\left(t\right),{}_D{}_D,l_D{}_D\right\}.$

Phase Swapping Formulation-I: Using Squared Voltage and Current Based Load Flow Formulation

[0083] The optimal problem for determining phase swapping actions over the given horizon can be formulated to minimize the cumulative cost over the scheduling horizon of T time steps while satisfying operational requirements for each time step t.

[00025]$\begin{array}{cc}\mathrm{Minimize}{.Math.}_{t=1}^T{J}^{\mathrm{SQ}}\left(t\right)& \left(9\right)\end{array}$

[0084] The cost function for each time step t, J.sup.SQ (t) is defined as a weighted sum of multiple cost components, including substation production (i.e., power purchase and generation and load curtailment) costs

[00026]$C_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right),$

penalty costs for branch current and bus voltage limit violations

[00027]$C_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{SQ}}\left(t\right),$

penalty cost for load (power and current) and voltage imbalance

[00028]$C_{\mathrm{tot}}^{\mathrm{IMB},\mathrm{SQ}}\left(t\right),$

and costs for phase swapping

[00029]$C_{\mathrm{tot}}^{\mathrm{SWAP}}\left(t\right)$

for the time interval:

[00030]$\begin{array}{cc}{J}^{\mathrm{SQ}}\left(t\right)={}^{\mathrm{SUB}}{C}_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right)+{}^{\mathrm{OVR}}{C}_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{SQ}}\left(t\right)+{}^{\mathrm{IMB}}{C}_{\mathrm{tot}}^{\mathrm{IMB},\mathrm{SQ}}\left(t\right)+{}^{\mathrm{SWAP}}{C}_{\mathrm{tot}}^{\mathrm{SWAP}}\left(t\right)& \left(10\right)\end{array}$

[00031]$C_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{SQ}}\left(t\right)\mathrm{and}{C}_{\mathrm{tot}}^{\mathrm{IMB},\mathrm{SQ}}\left(t\right)$

are expressed in term of squared voltage and currents.

[0085] The cost function JSQ (t) includes the following components:

[00032]$\begin{array}{cc}& \left(11\right)\end{array}$$C_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right)=\underset{{}_Y}{.Math.}\left(C_P^{\mathrm{SUB}}\left(t\right)\mathrm{Re}\left(s_0^{}\left(t\right)\right)+C_Q^{\mathrm{SUB}}\left(t\right)\mathrm{Im}\left(s_0^{}\left(t\right)\right)\right)+\underset{j{}^{+}}{.Math.}{C}_{\mathrm{SG}}^{\mathrm{CURT}}\left(t\right)\left(\underset{l_Y{}_Y}{.Math.}{B}_{\mathrm{CURT},Y_j}^{l_Y}.Math.s_{G,Y_j}^{l_Y}\left(t\right).Math.A_{\mathrm{LOC},Y_j}^{l_Y}+\underset{l_D{}_D}{.Math.}{B}_{\mathrm{CURT},Y_j}^{l_Y}.Math.s_{G,D_j}^{l_D}\left(t\right).Math.A_{\mathrm{LOC},D_j}^{l_D}\right)+\underset{j{}^{+}}{.Math.}{C}_{\mathrm{SL}}^{\mathrm{CURT}}\left(t\right)\left(\underset{l_Y{}_Y}{.Math.}{B}_{\mathrm{CURT},Y_j}^{l_Y}.Math.s_{L,Y_j}^{l_Y}\left(t\right).Math.A_{\mathrm{LOC},Y_j}^{l_Y}+\underset{l_D{}_D}{.Math.}{B}_{\mathrm{CURT},D_j}^{l_D}.Math.s_{L,}^{l_D}\left(t\right).Math.A_{\mathrm{LOC},D_j}^{l_D}\right)$$\begin{array}{cc}{C}_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{SQ}}\left(t\right)=\underset{i.fwdarw.j}{.Math.}\underset{{}_Y}{.Math.}{C}_{}^{\mathrm{OVR}}\left(t\right).Math.{\stackrel{}{l}}_{\max ,\mathrm{ij}}^{}\left(t\right).Math.+\underset{i{}^{+}}{.Math.}\underset{{}_Y}{.Math.}\left(C_{{\tilde{V}}_{\max }}^{\mathrm{OVR}}\left(t\right).Math.{\tilde{V}}_{\max ,i}^{}\left(t\right).Math.+C_{{\tilde{V}}_{\min }}^{\mathrm{OVR}}\left(t\right).Math.{\tilde{V}}_{\min ,i}^{}\left(t\right).Math.\right)& \left(12\right)\end{array}$$\begin{array}{cc}{C}_{\mathrm{tot}}^{\mathrm{IMB},\mathrm{SQ}}\left(t\right)=\underset{{}_Y}{.Math.}\left(C_P^{\mathrm{IMB}}\left(t\right)\mathrm{Re}\left({\mathrm{IMB}}_s^{}\left(t\right)\right)+C_Q^{\mathrm{IMB}}\left(t\right)\mathrm{Im}\left({\mathrm{IMB}}_s^{}\left(t\right)\right)+C_{}^{\mathrm{IMB}}\left(t\right){\mathrm{IMB}}_{}^{}\left(t\right)+C_{\tilde{V}}^{\mathrm{IMB}}\left(t\right){\mathrm{IMB}}_{\tilde{V}}^{}\left(t\right)\right)& \left(13\right)\end{array}$$\begin{array}{cc}{C}_{\mathrm{tot}}^{\mathrm{SWAP}}\left(t\right)=C^{\mathrm{SWAP}}\left(t\right)n_{\mathrm{sw}\mathrm{ap}}\left(t\right)& \left(14\right)\end{array}$

[0086] .sup.SUB is a weight factor for total production costs including substation active and reactive power purchase costs, renewable generation curtailment costs, and load demand curtailment costs.

[00033]$C_{\mathrm{SG}}^{\mathrm{SUB}}\left(t\right)\mathrm{and}{C}_Q^{\mathrm{SUB}}\left(t\right)$

are the per unit active and reactive power purchase costs at the substation for the duration of time step t. It is assumed that phases are equally priced.

[00034]$s_0^{}\left(t\right)$

is the complex power fed from the substation at phase (and time step t and determined as:

[00035]$\begin{array}{cc}{s}_0^{}\left(t\right)={.Math.}_{0:0.fwdarw.i}\mathrm{diag}\left({\stackrel{}{S}}_{0i}^{}\left(t\right)\right),{}_Y& \left(15\right)\end{array}$

[00036]$s_0\left(t\right)={\left[s_0^a\left(t\right),s_0^b\left(t\right),s_0^c\left(t\right)\right]}^T.Math.{\stackrel{}{S}}_{0i}^{}\left(t\right)$

is the expanded complex power flowing through the substation at bus 0 to downstream bus i.

[00037]$C_{\mathrm{SG}}^{\mathrm{CURT}}\left(t\right)\mathrm{and}{C}_{\mathrm{SL}}^{\mathrm{CURT}}\left(t\right)$

are the per unit costs for generation and load curtailments for the duration of time step t.

[00038]$B_{\mathrm{CURT},Y_j}^{l_Y}\left(t\right)\mathrm{and}{B}_{\mathrm{CURT},D_j}^{l_D}\left(t\right)$

are binary variables to represent the cut-off statuses of wye-connected location l.sub. and delta-connection location l.sub.D of bus j and time step t. If a connection is cut off, the connected generation and load at the location will be turn off, and generation and load curtailment costs are incurred. The feasibility of curtailments are constrained by the correspondingly availability of locations:

[00039]$\begin{array}{cc}{B}_{\mathrm{CURT},Y_j}^{l_Y}\left(t\right)=\left\{0,1\right\},B_{\mathrm{CURT},Y_j}^{l_Y}\left(t\right)A_{\mathrm{LOC},Y_j}^{l_Y},l_Y{}_Y,j{}^{+}& \left(16\right)\end{array}$$\begin{array}{cc}{B}_{\mathrm{CURT},D_j}^{l_D}\left(t\right)=\left\{0,1\right\},B_{\mathrm{CURT},D_j}^{l_D}\left(t\right)A_{\mathrm{LOC},D_j}^{l_D},l_D{}_D,j{}^{+}& \left(17\right)\end{array}$

.sup.OVR is a weight factor for total penalty costs for operational limit violations including branch current limits, bus maximum and minimum voltage limits.

[00040]$C_I^{\mathrm{OVR}}\left(t\right),C_{V_{\max }}^{\mathrm{OVR}}\left(t\right)\mathrm{and}{C}_{V_{\min }}^{\mathrm{OVR}}\left(t\right)$

are the per unit penalty costs for squared current limit violation, maximum squared voltage limit violations, and minimum squared voltage limit violations for the duration of time step t.

[00041]${\stackrel{}{I}}_{\max ,\mathrm{ij}}^{}\left(t\right)$

is the amount of squared current limit violation on branch (i, j), defined as:

[00042]$\begin{array}{cc}\mathrm{diag}\left({\stackrel{}{I}}_{\mathrm{ij}}\left(t\right)\right){\stackrel{}{I}}_{\max ,\mathrm{ij}}+{\stackrel{}{I}}_{\max ,\mathrm{ij}}\left(t\right),\left(i,j\right)& \left(18\right)\end{array}$

where

[00043]${\stackrel{}{I}}_{\max ,\mathrm{ij}}={\left[{\left(I_{\max ,\mathrm{ij}}^a\right)}^2,{\left(I_{\max ,\mathrm{ij}}^b\right)}^2,{\left(I_{\max ,\mathrm{ij}}^c\right)}^2\right]}^T,$${\stackrel{}{I}}_{\max ,\mathrm{ij}}={\left[{\stackrel{}{I}}_{\max ,\mathrm{ij}}^a,{\stackrel{}{I}}_{\max ,\mathrm{ij}}^b,{\stackrel{}{I}}_{\max ,\mathrm{ij}}^c\right]}^T,I_{\max ,\mathrm{ij}}^{}.$

is the limit of current on phase of branch (i,j).

[00044]${\stackrel{}{V}}_{\max ,i}^{}\left(t\right),\mathrm{and}{\stackrel{}{V}}_{\min ,i}^{}$

are the amounts of maximum and minimum squared voltage limit violations at phase of bus i and time step t, and determined as:

[00045]$\begin{array}{cc}{\stackrel{}{V}}_{\min ,i}-{\stackrel{}{V}}_{\min ,i}\left(t\right)\mathrm{diag}\left({\stackrel{}{V}}_i\left(t\right)\right){\stackrel{}{V}}_{\max ,i}+{\stackrel{}{V}}_{\max ,i}\left(t\right),i{}^{+}& \left(19\right)\end{array}$

where

[00046]${\stackrel{}{V}}_{\max ,i}={\left[{\left(V_{\max ,i}^a\right)}^2,{\left(V_{\max ,i}^b\right)}^2,{\left(V_{\max ,i}^c\right)}^2\right]}^T,\mathrm{and}$${\stackrel{}{V}}_{\min ,i}={\left[{\left(V_{\min ,i}^a\right)}^2,{\left(V_{\min ,i}^b\right)}^2,{\left(V_{\min ,i}^c\right)}^2\right]}^T,V_{\max ,i}^{}\mathrm{and}{V}_{\min ,i}^{}$

are the upper and lower limits of voltages on phase of bus i,

[00047]${\stackrel{}{V}}_{\max ,i}={\left[{\stackrel{}{V}}_{\max ,i}^a,{\stackrel{}{V}}_{\max ,i}^b,{\stackrel{}{V}}_{\max ,i}^c\right]}^T,\mathrm{and}$${\stackrel{}{V}}_{\min ,i}={\left[{\stackrel{}{V}}_{\min ,i}^a,{\stackrel{}{V}}_{\min ,i}^b,{\stackrel{}{V}}_{\min ,i}^c\right]}^T.$

[0087] .sup.IMB is a weight factor for total penalty costs for phase imbalances including substation active and reactive power phase imbalances, substation current phase imbalances, and system maximum voltage phase imbalances. The penalty costs for phase imbalances are not directly related to real-time operation/dispatch, but phase swapping we would like to impose in our objective function implication of a constraint that along each phase, we want the magnitude of power, current or voltage to be as close as possible to a one third of total power, current or voltage.

[00048]$C_P^{\mathrm{IMB}}\left(t\right)\mathrm{and}{C}_Q^{\mathrm{IMB}}\left(t\right)$

are the per unit costs for active power and reactive power imbalance penalty for the duration of time step t.

[00049]${\mathrm{IMB}}_s^{}\left(t\right)$

is the complex power imbalance on phase at time step t, and determined as:

[00050]$\begin{array}{cc}{\mathrm{IMB}}_s^{}\left(t\right)=s_0^{}\left(t\right)-\frac{1}{3}{.Math.}_{{}_Y}{s}_0^{}\left(t\right),{}_Y& \left(20\right)\end{array}$

[00051]$C_{}^{\mathrm{IMB}}\left(t\right)$

is the per unit cost of squared current imbalance for the duration of time step t, and

[00052]${\mathrm{IMB}}_{}^{}\left(t\right)$

is squared current imbalance for phase at time step t defined as:

[00053]$\begin{array}{cc}{I}_{sq,0}\left(t\right)=\mathrm{diag}\left({.Math.}_{0:0.fwdarw.i}{\stackrel{}{I}}_{0i}\left(t\right)\right),{}_Y& \left(21\right)\end{array}$$\begin{array}{cc}{\mathrm{IMB}}_{\tilde{I}}^{}\left(t\right)=.Math.I_{\mathrm{sq},0}^{}\left(t\right).Math.-\frac{1}{3}{.Math.}_{{}_Y}.Math.I_{\mathrm{sq},0}^{}\left(t\right).Math.,{}_Y& \left(22\right)\end{array}$

[00054]$I_{sq,0}\left(t\right)={\left[I_{\mathrm{sq},0}^a\left(t\right),I_{\mathrm{sq},0}^b\left(t\right),I_{\mathrm{sq},0}^c\left(t\right)\right]}^T,I_{\mathrm{sq},0}^{}\left(t\right)$

is the squared current flowing on phase at the substation.

[00055]$C_{\tilde{V}}^{\mathrm{IMB}}\left(t\right)$

is per unit cost of squared voltage at time step t.

[00056]${\mathrm{IMB}}_{\tilde{V}}^{}\left(t\right)$

is the system maximum squared voltage imbalance on phase at time step t, and defined as:

[00057]$\begin{array}{cc}{\mathrm{IMB}}_{\tilde{V}}^{}\left(t\right)=\underset{j{}^{+}}{\max }.Math..Math.{\stackrel{}{V_j}}^{}\left(t\right).Math.-\frac{1}{{.Math.}_{{}_Y}{A}_{{\mathrm{BUS}}_j}^{}}\underset{{}_Y}{.Math.}.Math.{\stackrel{}{V_j}}^{}\left(t\right).Math.A_{{\mathrm{BUS}}_j}^{}.Math.,& \left(23\right)\end{array}$${}_Y,$

[00058]$A_{{\mathrm{BUS}}_j}^{}$

is a binary parameter representing the phase availability of phase at bus j.

[00059]${\stackrel{}{V_j}}^{}\left(t\right)$

is the squared voltage of phase of bus j.

[0088] .sup.SWAP is a weight factor relating to the total costs for phase swapping numbers. Since each phase swapping operation is associated with certain costs on lineman expenses, maintenance expenses, and planned outage duration if phase swapping is implemented manually, or recovered investment cost, and expected outage cost if implemented automatically. We could not afford to swap phases for all number at all the time. The number of phase swapping n.sub.swap (t) for time step t need to be compromised between benefits and costs. The cost per phase swapping at time step t is pre-determined as C.sup.SWAP(t). n.sub.swap (t) is defined as:

[00060]$\begin{array}{cc}{n}_{\mathrm{sw}\mathrm{ap}}\left(t\right)={.Math.}_{j{}^{+}}{n}_{\mathrm{swap},j}\left(t\right),& \left(24\right)\end{array}$$\begin{array}{cc}{n}_{\mathrm{swap},j}\left(t\right)=\left({.Math.}_{l_Y{}_Y}{A}_{\mathrm{LOC},Y_j}^{l_Y}-{.Math.}_{l_Y{}_Y}{B}_{\mathrm{CURT},Y_j}^{l_Y}\left(t\right)-{.Math.}_{{}_Y{}_Y}{.Math.}_{l_Y{}_Y}{B}_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t\right)B_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t-1\right)\right)+\left({.Math.}_{l_D{}_D}{A}_{\mathrm{LOC},Y_j}^{l_D}-{.Math.}_{l_D{}_D}{B}_{\mathrm{CURT},D_j}^{l_D}\left(t\right)-{.Math.}_{{}_D{}_D}{.Math.}_{l_D{}_D}{B}_{D_j}^{l_D.fwdarw.{}_D}\left(t\right)B_{D_j}^{l_D.fwdarw.{}_D}\left(t-1\right)\right),& \left(25\right)\end{array}$$j{}^{+}$$\begin{array}{cc}{B}_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t\right)=\left\{0,1\right\},{}_Y{}_Y,l_Y{}_Y,j{}^{+}& \left(26\right)\end{array}$$\begin{array}{cc}{B}_{D_j}^{l_D.fwdarw.{}_D}\left(t\right)=\left\{0,1\right\},{}_D{}_D,l_D{}_D,j{}^{+}& \left(27\right)\end{array}$

[00061]$B_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t\right)$

is a binary variable representing status for assigning phase .sub..sub. to location l.sub..sub. at bus j at time step t.

[00062]$B_{D_j}^{l_D.fwdarw.{}_D}\left(t\right)$

is a binary variable representing status for assigning phase-pair .sub.D.sub.D to location l.sub.D.sub.D at bus j at time step t. Those two sets of variables are used to represent phase swapping actions.

[0089] The technical constraints must be satisfied for each time step t include load flow equations, substation voltage settings, and phase swapping feasibility constraints.

[0090] Considering the impacts of phase swapping actions, the squared voltage and current based load flow equations at time step t are expressed as:

[00063]$\begin{array}{cc}{.Math.}_{i:i.fwdarw.j}\mathrm{diag}\left(A_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}\left(t\right)A_{I_{i.fwdarw.j}}^H-A_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i\left(t\right)A_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H-{\stackrel{}{V}}_j\left(t\right)y_{\mathrm{Dij}}^H+z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}\left(t\right)A_{I_{i.fwdarw.j}}^H-z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i\left(t\right)A_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H+z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}^H\left(t\right)A_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H-z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{I}}_{\mathrm{ij}}^H\left(t\right)A_{I_{i.fwdarw.j}}^H\right)-\mathrm{diag}\left({\stackrel{}{V}}_j\left(t\right)y_{\mathrm{Cj}}^H\right)+s_j\left(t\right)={.Math.}_{k:j.fwdarw.k}\mathrm{diag}\left(S_{\mathrm{jk}}\left(t\right)\right),& \left(28\right)\end{array}$$j{N}^{+}$$\begin{array}{cc}{s}_j^{}\left(t\right)=\underset{l_Y{}_Y}{.Math.}{B}_{Y_j}^{l_Y.fwdarw.}\left(t\right)\left(s_{G,Y_j}^{l_Y}\left(t\right)-s_{L,Y_j}^{l_Y}\left(t\right)\right)+\underset{{}_D{}_D}{.Math.}{A}_{S_{D.fwdarw.Y}}^{,{}_D}\underset{l_D{}_D}{.Math.}{B}_{D_j}^{l_D.fwdarw.{}_D}\left(t\right)\left(s_{G,D_j}^{l_D}\left(t\right)-s_{L,D_j}^{l_D}\left(t\right)\right),& \left(29\right)\end{array}$${}_Y,j{}^{+}$$\begin{array}{cc}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i\left(t\right)A_{V_{i.fwdarw.j}}^H-{\stackrel{}{V}}_j\left(t\right)=A_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}\left(t\right)A_{I_{i.fwdarw.j}}^H{z}_{\mathrm{ij}}^H-A_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i\left(t\right)A_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H{z}_{\mathrm{ij}}^H+z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}^H\left(t\right)A_{V_{i.fwdarw.j}}^H-z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{I}}_{\mathrm{ij}}\left(t\right)A_{I_{i.fwdarw.j}}^H{z}_{\mathrm{ij}}^H+z_{\mathrm{ij}}{A}_{I_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}^H\left(t\right)A_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H{z}_{\mathrm{ij}}^H-z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i\left(t\right)A_{y_{i.fwdarw.j}}^H+z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{S}}_{\mathrm{ij}}\left(t\right)A_{I_{i.fwdarw.j}}^H{z}_{\mathrm{ij}}^H-z_{\mathrm{ij}}{y}_{\mathrm{Uij}}{A}_{V_{i.fwdarw.j}}{\stackrel{}{V}}_i\left(t\right)A_{V_{i.fwdarw.j}}^H{y}_{\mathrm{Uij}}^H{z}_{\mathrm{ij}}^H,& \left(30\right)\end{array}$$\left(i,j\right)E$$\begin{array}{cc}{\stackrel{}{V}}_i\left(t\right){\stackrel{}{I}}_{\mathrm{ij}}\left(t\right)-{\stackrel{}{S}}_{\mathrm{ij}}\left(t\right){\stackrel{}{S}}_{\mathrm{ij}}^H\left(t\right)0,\left(i,j\right)& \left(31\right)\end{array}$

where

[00064]$A_{S_{D.fwdarw.Y}}={\left(A_{S_{D.fwdarw.Y}}^{,{}_D}\right)}_{33},\mathrm{and}{A}_{S_{D.fwdarw.Y}}^{,{}_D}$

is the element of A.sub.S.sub.D.fwdarw.Y coresponing to row and column .sub.D.

[0091] The substation voltage is fixed, and expressed as:

[00065]$\begin{array}{cc}{\stackrel{}{V}}_0\left(t\right)=V_0^{\mathrm{ref}}\left(t\right){\left(V_0^{\mathrm{ref}}\left(t\right)\right)}^H& \left(32\right)\end{array}$

where

[00066]$V_0^{\mathrm{ref}}\left(t\right)$

is the vector of given substation three-phase voltages at time step t.

[0092] The assignment matrix for wye-connected loads or generations must satisfy that each location is assigned to at most one phase, and each phase is assigned to at most one location:

[00067]$\begin{array}{cc}{B}_{\mathrm{CURT},Y_j}^{l_Y}+{.Math.}_{{}_Y{}_Y}{B}_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t\right)=A_{\mathrm{LOC},Y_j}^{l_Y},l_Y{}_Y,j{}^{+}& \left(33\right)\end{array}$$\begin{array}{cc}{.Math.}_{l_Y{}_Y}{B}_{Y_j}^{l_Y.fwdarw.{}_Y}\left(t\right)1,{}_Y{}_Y,j{}^{+}& \left(34\right)\end{array}$

[0093] Where parameter

[00068]$A_{\mathrm{LOC},Y_j}^{l_Y}$

indicates the availability of load/generation at location l.sub. at bus j.

[0094] Similarly, the assignment matrix for delta-connected loads or generations must satisfy that each location is assigned to at most one phase-pair, and each phase-pair is assigned to at most one location:

[00069]$\begin{array}{cc}{B}_{\mathrm{CURT},D_j}^{l_D}+{.Math.}_{{}_D{}_D}{B}_{D_j}^{l_D.fwdarw.{}_D}\left(t\right)=A_{\mathrm{LOC},D_j}^{l_D},l_D{}_D,j{}^{+}& \left(35\right)\end{array}$$\begin{array}{cc}{.Math.}_{l_D{}_D}{B}_{D_j}^{l_D.fwdarw.{}_D}\left(t\right)1,{}_D{}_D,j{}^{+}& \left(36\right)\end{array}$

where

[00070]$A_{\mathrm{LOC},D_j}^{l_D}$

indicates the availability of load/generation at location l.sub.D at bus j.

[0095] In summary, the optimal phase swapping problem can formulate as a mixed integer nonlinear programming problem (37), according to:

[00071]$\begin{array}{cc}\left(1\right)& \\ \mathrm{Minimize}{.Math.}_{t=1}^T{J}^{\mathrm{SQ}}\left(t\right)& \left(37a\right)\end{array}$$\mathrm{Subject}\mathrm{to}:$$\begin{array}{cc}\mathrm{constraints}\mathrm{expressed}\mathrm{as}\left(10\right)\left(36\right),t=1,2,.Math.,T& \left(37b\right)\end{array}$

Phase Swapping Formulation-II: Using Explicit Voltage and Current Based Load Flow Formulation

[0096] Similarly, we can also use explicit voltage and current based load flow model to formulate phase swapping optimization over the finite horizon.

[0097] The cost function to be minimized can be expressed as:

[00072]$\begin{array}{cc}\mathrm{Minimize}{.Math.}_{t=1}^T{J}^{\mathrm{EXP}}\left(t\right)& \left(38\right)\end{array}$

[0098] J.sup.EXP(t) is the cost function for each time step t, and can be expressed based on explicit voltages and currents as:

[00073]$\begin{array}{cc}{J}^{\mathrm{EXP}}\left(t\right)={}^{\mathrm{SUB}}{C}_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right)+{}^{\mathrm{OVR}}{C}_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{EXP}}\left(t\right)+{}^{\mathrm{IMB}}{C}_{\mathrm{tot}}^{\mathrm{IMB},\mathrm{EXP}}\left(t\right)+{}^{\mathrm{SWAP}}{C}_{\mathrm{tot}}^{\mathrm{SWAP}}\left(t\right)& \left(39\right)\end{array}$

[0099] This cost function is defined as a weighted sum of four different groups of cost components

[00074]$C_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right),C_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{EXP}}\left(t\right),C_{\mathrm{tot}}^{\mathrm{IMB},\mathrm{EXP}}\left(t\right)\mathrm{and}{C}_{\mathrm{tot}}^{\mathrm{SWAP}}$

with a pre-given weight parameter set {.sup.SUB, .sup.OVR, .sup.IMB, .sup.SWAP}.

[0100] The first cost group,

[00075]$C_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right)$

is associated to operational costs for generating powers met demands, including total active and reactive power purchase costs at the substation, and load and generation curtailment costs at all buses. It is defined as:

[00076]$\begin{array}{cc}& \left(40\right)\end{array}$$C_{\mathrm{tot}}^{\mathrm{SUB}}\left(t\right)=\underset{{}_Y}{.Math.}\left(C_P^{\mathrm{SUB}}\left(t\right)\mathrm{Re}\left(s_0^{}\left(t\right)\right)+C_Q^{\mathrm{SUB}}\left(t\right)\mathrm{Im}\left(s_0^{}\left(t\right)\right)\right)+\underset{j{}^{+}}{.Math.}{C}_{\mathrm{SG}}^{\mathrm{CURT}}\left(t\right)\left(\underset{l_Y{}_Y}{.Math.}{B}_{\mathrm{CURT},Y_j}^{l_Y}.Math.s_{G,Y_j}^{l_Y}\left(t\right).Math.A_{LOC,Y_j}^{l_Y}+\underset{l_D{}_D}{.Math.}{B}_{\mathrm{CURT},D_j}^{l_D}.Math.s_{G,D_j}^{l_D}\left(t\right).Math.A_{\mathrm{LOC},D_j}^{l_D}\right)+\underset{j{}^{+}}{.Math.}{C}_{\mathrm{SL}}^{\mathrm{CURT}}\left(t\right)\left(\underset{l_Y{}_Y}{.Math.}{B}_{\mathrm{CURT},Y_j}^{l_Y}.Math.s_{L,Y_j}^{l_Y}\left(t\right).Math.A_{LOC,Y_j}^{l_Y}+\underset{l_D{}_D}{.Math.}{B}_{\mathrm{CURT},D_j}^{l_D}.Math.s_{L,D_j}^{l_D}\left(t\right).Math.A_{\mathrm{LOC},D_j}^{l_D}\right)$

[0101] The active and reactive power purchase costs are calculated based on complex power fed from the substation for each time step t,

[00077]$s_0^{}\left(t\right),$

.sub. and corresponding per-unit cost coefficients for active and reactive power purchase,

[00078]$C_P^G\left(t\right)\mathrm{and}{C}_Q^G\left(t\right).$

[00079]$\begin{array}{cc}{s}_0^{}\left(t\right)={.Math.}_{0:0.fwdarw.i}{V}_0^{}\left(t\right){\left(I_{0i}^{}\left(t\right)\right)}^H,{}_Y& \left(41\right)\end{array}$

[00080]$I_{0i}^{}\left(t\right)$

is the complex current flowing through the substation at bus 0 to downstream bus i on phase ,

[00081]$V_0^{}\left(t\right)$

is the complex voltage at the substation on phase . The load/generation curtailment costs for any bus j are calculated based on per-unit cost coefficients for load/generation curtailments,

[00082]$C_{\mathrm{SL}}^{\mathrm{CURT}}\left(t\right)/C_{\mathrm{SG}}^{\mathrm{CURT}}\left(t\right).$

The full load or generation at location l.sub. or l.sub.D is cut off if corresponding wye or delta connection curtailment status,

[00083]$B_{\mathrm{CURT},Y_j}^{l_Y}\mathrm{or}{B}_{\mathrm{CURT},D_j}^{l_D}$

is determined as 1. The feasibility constraints of load/generation curtailment variables for bus j at time step t are expressed as:

[00084]$\begin{array}{cc}{B}_{\mathrm{CURT},Y_j}^{l_Y}\left(t\right)=\left\{0,1\right\},B_{\mathrm{CURT},Y_j}^{l_Y}\left(t\right)A_{\mathrm{LOC},Y_j}^{l_Y},l_Y{}_Y,j{}^{+}& \left(42\right)\end{array}$$\begin{array}{cc}{B}_{\mathrm{CURT},D_j}^{l_D}\left(t\right)=\left\{0,1\right\},B_{\mathrm{CURT},D_j}^{l_D}\left(t\right)A_{\mathrm{LOC},D_j}^{l_D},l_D{}_D,j{}^{+}& \left(43\right)\end{array}$

[0102] The curtailment statuses are limited to the availability statues of connection location l.sub., and l.sub.D,

[00085]$A_{\mathrm{LOC},Y_j}^{l_Y},A_{LOC,D_j}^{l_D}.$

[0103] The second cost group,

[00086]$C_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{EXP}}\left(t\right)$

quantifies the penalizing costs for branch current violating its thermal threshold or bus voltage violating its lower or upper thresholds, and defined as:

[00087]$\begin{array}{cc}\begin{array}{c}{C}_{\mathrm{tot}}^{\mathrm{OVR},\mathrm{EXP}}\left(t\right)=\underset{i.fwdarw.j}{.Math.}\underset{{}_Y}{.Math.}{C}_I^{\mathrm{OVR}}\left(t\right)I_{\max ,\mathrm{ij}}^{}\left(t\right)\\ +\underset{i{}^{+}}{.Math.}\underset{{}_Y}{.Math.}\left(C_{V_{\max }}^{\mathrm{OVR}}\left(t\right)V_{\max ,i}^{}\left(t\right)+C_{V_{\min }}^{\mathrm{OVR}}\left(t\right)V_{\min ,i}^{}\left(t\right)\right)\end{array}& \left(44\right)\end{array}$

[00088]$C_I^{\mathrm{OVR}}\left(t\right)$