A Second Order Cone Programming Model for Planning PEV Fast-Charging Stations

NOMENCLATURE

• PEV types, transportation nodes, paths and sub-paths, distribution buses and lines, time intervals, and scenarios index the coupled networks and planning model.The notation also specifies the ordering of distribution-network lines relative to the reference bus.Source: NOMENCLATURE · p. 1
• PEV and transportation inputs include driving ranges, charging times, traffic flows, distances, and charging arrival and departure times.Source: NOMENCLATURE · p. 1
• Planning parameters include station service level and costs for stations, charging spots, distribution lines, capacity expansion, energy, and unsatisfied charging power.The model also defines scenario probabilities, charging-spot power, and the one-hour time interval.Source: NOMENCLATURE · pp. 1–2
• Transportation decisions represent station investment, spot counts, PEV charging choices, and charging demand by type, node, time, and scenario.Source: NOMENCLATURE · p. 2
• Power-system variables track PEV and bus injections, unsatisfied charging power, substation expansion, line current and power flow, and nodal voltage.Source: NOMENCLATURE · p. 2

I. INTRODUCTION

• Station service models must handle heterogeneous charging demands and durations, while prior linear models miss scale effects and queueing models lack closed forms for direct planning use.The introduction also notes that prior queueing-based studies often relied on heuristics and did not account for heterogeneous required service durations.Source: §I · p. 3
• A closed-form service-level model measures the probability that arriving charging demands are met without extra waiting and supports SOCP-based station sizing for heterogeneous PEV ranges.Source: §I · p. 3
• The modified CFRLM SP captures time-varying charging demand under driving-range constraints, with fewer binary variables than CFRLM EN and extra constraints intended to improve efficiency.Dynamic OD traffic matters because distribution-system base loads also vary over time.Source: §I · p. 3
• The stochastic mixed-integer SOCP combines transportation constraints with AC power-flow constraints, enabling assessment of losses and voltage deviations in planning.The paper describes branch-and-cut solution with an off-the-shelf solver and convex SOCP relaxation.Source: §I · p. 4
• The planning perspective is a social planner with access to transportation and power-system information, targeting highway networks supplied by high-voltage distribution systems.The authors state the approach can extend to areas that also include higher-voltage transmission systems.Source: §I · p. 4

A. The Service Level Metric

• The service level is the probability that a PEV receives at least its required charging time without waiting.The metric is intended to describe service quality through users’ waiting experience.Source: A. The Service Level Metric · p. 4
• Requiring service level α or greater lets planners control the target service quality; the minimum spot count ycs depends on α and charging demand.The section develops a closed-form approximation for ycs(α) and reformulates the service-level model as a mixed-integer SOCP.Source: A. The Service Level Metric · p. 4

B. The Service Level Model with Homogeneous PEVs

• For homogeneous PEVs, the model assumes Poisson arrivals and FIFO service with new arrivals not waiting, so arrivals during charging time T determine whether a PEV receives T units of charging.If arrivals during T exceed the number of spots, at least one PEV cannot receive T units of charging; the section gives a closed-form approximation for sizing under this setup.Source: B. The Service Level Model with… · pp. 4–5

C. The Service Level Model with Heterogeneous PEVs

• PEVs with different driving ranges can have different charging behaviors and time requirements, so the model groups vehicles with similar ranges into types with required charging times Tk.The paper gives 200 km and 300 km as examples of ranges.Source: C. The Service Level Model with… · p. 5
• Under the stated Poisson-arrival and no-wait assumptions, the model relates each type’s charging-time demand to its arrival count and combines the counts across types.The aggregate count is used in the heterogeneous service-level criterion.Source: C. The Service Level Model with… · p. 5
• Approximating each type’s Poisson count by a normal distribution yields a closed-form approximation for the required spot count under the service-level criterion.The model relies on the sum of independent normal distributions being normal.Source: C. The Service Level Model with… · p. 5

D. Second Order Cone Reformulation of the Model

• The nonconvex closed-form station service constraint is reformulated as a mixed-integer SOCP solvable by branch-and-cut.The model gives a minimum required spot count for a chosen service level; its safety-stock term covers demand above the mean.Source: D. Second Order Cone Reformulation of… · pp. 5–6
• The station constraint accounts for heterogeneous PEV demands by modeling the traffic flow to be charged at a node across multiple paths.The charge-choice variable γq,i,k is binary: 1 when PEVs charge at that node and 0 otherwise.Source: D. Second Order Cone Reformulation of… · p. 5
• For a given α, constraint (7) sets the minimum spot count, combining spots for expected demand with extra safety-stock spots for demand above the mean.Higher α entails more investment and better service quality.Source: D. Second Order Cone Reformulation of… · p. 6
• The service-level criterion α approximates the probability of immediate charging; 1−α approximates the probability a PEV must wait.Under the model’s first-in-first-out assumption, designed service quality is exact; smarter station management can yield higher service quality.Source: D. Second Order Cone Reformulation of… · p. 6

III. MODIFIED CAPACITATED FLOW REFUELING LOCATION MODEL BASED ON SUB-PATHS

• The modified CFRLM adapts the model in to improve accuracy and represent time-varying origin-destination traffic flow.It defines the feasible set of charging-station locations under PEV driving-range constraints.Source: §III · p. 6

A. PEV Driving Range Constraint Based on Sub-paths

• Each sub-path longer than a PEV’s driving range must contain a charging station, accounting for battery state on network entry and exit.For the 100 km example, the paper identifies five sub-paths and two required stations; it uses driving-range-length sub-paths rather than half-range segments and models outbound and return trips separately.Source: A. PEV Driving Range Constraint Based… · p. 6, Figure 3 · p. 6

B. Capacitated Flow Refueling Location Model

• The CFRLM enforces service ability for every hour and scenario, requires charging at least once per sub-path, and permits charging only at station nodes.It uses hourly average traffic flow in the service constraint and limits spot counts to nodes with stations.Source: B. Capacitated Flow Refueling Location… · pp. 6–7
• The assumption is introduced because binary-variable scale grows with transportation-network size and can make the optimization intractable.The reduction depends on network structure; very short separate segments relative to a shared segment allow the two paths to be treated as one.Source: B. Capacitated Flow Refueling Location… · p. 7
• The shared-path assumption gives vehicles with the same arrival node identical charge choices along a common sub-path until their routes separate.It reduces the binary-variable scale from 2o0 + o1 + o2 to o0 + o1 + o2 for two paths; longer shared segments yield greater reductions.Source: B. Capacitated Flow Refueling Location… · p. 7, Figure 4 · p. 7

IV. CHARGING STATION PLANNING MODEL

• The complete model combines service-ability and transportation-feasibility models with AC power-flow constraints to plan station locations and sizes.It represents uncertain traffic flow and traditional base loads with finite future scenarios in a two-stage stochastic program.Source: §IV · p. 7, A. Objective · p. 7

A. Objective

• The objective accounts for station construction and charging-spot costs, distribution-line and substation upgrades, expected energy purchases, and penalties for unsatisfied charging demand.Station maximum charging power and corresponding substation capacity expansion are calculated by constraints (13) and (14).Source: A. Objective · p. 7

B. Constraints

• The transportation constraints are the CFRLM constraints (8)–(11), while the distribution network must satisfy AC power-flow constraints.The model adopts an SOCP relaxation of the AC power-flow model.Source: B. Constraints · p. 7, 2) Power Network Constraints · p. 7

2) Power Network Constraints:

• The SOCP relaxation is exact for the assumed radial high-voltage distribution network with unidirectional power flow.If higher-voltage transmission networks are included, the paper proposes modeling them with DC power flow while retaining SOCP for radial distribution networks.Source: 2) Power Network Constraints · p. 8
• Distribution-line currents and nodal voltages must remain within their permitted ranges.These limits are part of the power-network constraints alongside the SOCP relaxation.Source: 2) Power Network Constraints · pp. 7–8

3) Coupled Constraints of Transportation & Power Networks:

• At low PEV traffic, charging power at a transportation node is proportional to average traffic requiring charging; beyond service ability, some demand is unfulfilled.The model assumes a charging-load power factor of approximately 1, so tan θ = 0.Source: 3) Coupled Constraints of… · p. 8
• For transmission systems, the paper proposes DC power flow while retaining SOCP for radial distribution networks.The stated DC approximation relies on negligible line resistance relative to reactance, near-unit voltage magnitudes, and small neighboring-node angle differences.Source: 3) Coupled Constraints of… · p. 8
• The transportation case network is expanded from 25 nodes to 93 by adding auxiliary nodes so no adjacent-node distance exceeds 20 km.The paired distribution case is a 110 kV network with a 150 MVA transformer connecting node 1 to 220 kV.Source: Figure 5 · p. 8
• The model requires base loads to be satisfied and includes the coupled transportation and power constraints in an MISOCP.The formulation remains feasible because charging demand may be left unsatisfied when operating constraints, such as AC power flow, bind.Source: 3) Coupled Constraints of… · p. 8

A. Case Overview and Parameter Settings

• The test system couples a 25-node highway network with a 14-node, 110 kV distribution network and uses 24 weekday and weekend scenarios based on load and traffic profiles.The study assumes a radial network structure and generates daily origin-destination flows from node weights and arc distances.Source: A. Case Overview and Parameter Settings · pp. 8–9
• The base case assumes four equally represented PEV types with driving ranges of 200–500 km, 44 kW charging spots, and an 80% service-quality criterion.The assumptions also specify charging demand, station and grid-upgrade costs, and spare substation capacity; the authors state that parameters are illustrative and should be replaced with practical values.Source: A. Case Overview and Parameter Settings · p. 9
• Eight benchmark cases illustrate the proposed method: Case 1 is basic; Case 2 doubles daily PEV traffic; Case 3 assumes homogeneous 200-km ranges; Case 4 excludes Section III-C extra constraints.Cases 5 and 6 use the DC approximation for power-flow constraints; Cases 7 and 8 ignore electrical constraints and distribution-system upgrade costs at the planning stage.Source: A. Case Overview and Parameter Settings · p. 9

B. Planning Results and Analysis

• The study solves the planning model with CPLEX, relaxing integer spot-count variables and stopping when the optimization gap falls below 0.5%.Table III reports planning results; for Cases 3, 5, and 6, values outside parentheses are recalculated under the proposed model to reflect actual operating conditions, and Fig. 7 shows Case 1 station locations and sizes.Source: A. Case Overview and Parameter Settings · p. 9, B. Planning Results and Analysis · p. 9, Figure 7 · p. 9

1) PEV population:

• Doubling PEV traffic in Case 2 raises investment and operating costs, leaves 1.89% of charging load unsatisfied, and makes distribution-line congestion more severe than in Case 1.Distribution line 2 is identified as the system bottleneck and a guide for future grid expansion.Source: 1) PEV population · p. 10, Figure 9 · p. 10

2) Heterogeneity of PEV driving range:

• Assuming homogeneous 200 km driving ranges makes planning conservative: Case 3 investments are about 100% higher than Case 1, with overestimated charging demand and 5.76% unsatisfied load.The authors attribute the overestimate to longer-range vehicles charging more often than needed; in practice, they expect underused spots and wasted investment.Source: 2) Heterogeneity of PEV driving range · p. 10

4) Modeling accuracy of AC power flow:

• The DC power-flow approximation overestimates electricity costs by about 10% in Cases 5 and 6 and yields inaccurate unmet-demand estimates, which the authors say can produce suboptimal planning.Compared with Case 1, Case 5’s actual investment and total annual costs rise by 2.5% and 0.6%, respectively; Case 6’s actual unmet-load ratio is 7.9% higher than Case 2.Source: 4) Modeling accuracy of AC power flow · p. 11

5) Necessity of considering coupled transportation and power network constraints:

• Ignoring AC power-flow constraints increases unmet charging demand, while including distribution-upgrade costs reduces total investment by avoiding myopic station-investment decisions.Case 7 leaves 0.034% of demand unfulfilled; Case 8’s unmet-demand ratio is 139.7% higher than Case 2. Under low PEV population, upgrade costs can exceed the station-investment savings from omitting them.Source: 5) Necessity of considering coupled… · p. 11

C. Performance of the Service Level Model

• The service-level-model experiments vary PEV arrival rates from 20 to 300 per hour and target service levels from 70% to 90%, then simulate station operations for 1,000 hours.The supplied passages identify figures for service-level accuracy, instant charging probabilities, and average waiting times, but do not state their outcomes.Source: C. Performance of the Service Level… · p. 11, Figure 10 · p. 11, Figure 11 · p. 11

1) Accuracy of the service level model:

• Simulations show that the proposed service-level model’s actual service levels are very close to designed values across tested arrival rates and service-level criteria.The accuracy test assumes first-in-first-out station operation.Source: 1) Accuracy of the service level model · p. 11

2) Waiting time analysis in real-time operations:

• The designed service-level criterion controls both the probability of instant charging and average waiting time, with higher criteria associated with shorter average waits.Differences between instant-service probability and the designed criterion remain stable and insensitive to arrival volumes; the authors conclude the criterion provides an intuitive service-quality measure even if the FIFO assumption may not hold in practice.Source: 2) Waiting time analysis in real-time… · p. 12
• The CFRLM SP uses fewer binary charge-choice variables than CFRLM EN, and with four PEV types it solves in under 10 minutes while CFRLM EN is intractable.For homogeneous PEVs, CFRLM EN takes about fifteen times as long as CFRLM SP.Source: 2) Waiting time analysis in real-time… · p. 12

VI. CONCLUSION

• The paper concludes that combining transportation constraints with AC power-flow constraints leads to more economical investment decisions.The proposed planning framework models time-varying charging demand under driving-range constraints and uses a stochastic mixed-integer SOCP to site and size stations.Source: §VI · p. 12
• For meshed distribution networks, the SOCP relaxation may not be exact, and planning such networks is outside this paper’s scope.The authors also identify computationally efficient solution methods for large-scale transportation and power-network scenarios as future work.Source: §VI · pp. 12–13

• The proof indexes PEVs by arrival order and denotes the PEV arriving ycs places after PEV e as PEV e + ycs.Source: Appendix A · p. 13
• Under assumption A1, PEV arrival events are independent and identically distributed because arrivals are Poisson.Source: Appendix A · p. 13
• The proof equates the probability that the ycs-th subsequent PEV arrives after time ta0 + T with the probability of fewer than ycs arrivals during the interval from ta0 to ta0 + T.Source: Appendix A · p. 13