Electrical Vehicle Charging Station Profit Maximization: Admission, Pricing, and Online Scheduling

I. INTRODUCTION

• Limited charging power and station capacity make waiting unavoidable, motivating admission control alongside pricing and scheduling.Source: §I · p. 1
• Queue-length admission can perform poorly, while demand-based admission ignores unknown future charging demands.Source: §I · p. 1
• JoAP jointly selects admission control, charging scheduling, and pricing to maximize average station profit after waiting-time penalties.Source: §I · p. 1

A. Charging Station Operation

• The station announces a price per unit energy, and admitted EVs are served FIFO, which is optimal for minimizing average waiting when EVs are homogeneous.Source: A. Charging Station Operation · p. 2

B. Optimization from EVs’ Perspective

• Under homogeneous-EV assumptions, each user chooses demand to maximize consumer surplus; optimal demand decreases with price and becomes zero when price is too high.Source: B. Optimization from EVs’ Perspective · p. 2

• For each arriving EV, the admission decision is binary, and the policy’s average admission probability is defined over arrivals.Source: C. Optimization from Charging… · p. 3
• Average waiting time under a policy is computed over possible arrival processes and depends on demand.Source: C. Optimization from Charging… · p. 3
• Station profit accounts for payment revenue, electricity cost, and a non-decreasing convex penalty on average waiting time.Source: C. Optimization from Charging… · p. 3

A. Admission Control and Queueing Model

• The multi-sub-process policy addresses the low utilization risk of overly conservative admission while balancing admission rate and waiting time.Source: A. Admission Control and Queueing Model · p. 3
• An EV is admitted if a sub-process can accept it, with admitted arrivals in each sub-process separated by at least threshold Tv.Source: A. Admission Control and Queueing Model · p. 3
• The policy spans complete regulation at n = 1 to admitting all EVs as n approaches infinity, allowing n and Tv to tune the waiting-time/admission-rate trade-off.Source: A. Admission Control and Queueing Model · p. 3
• Q.1 models admission as a virtual zero-buffer queue with n servers, while Q.2 is the real FIFO charging queue with m ports and service time d/α.Source: A. Admission Control and Queueing Model · pp. 3–4
• Q.2 stability requires nTv > md/α, expressed as nTv = τmd/α with τ > 1.Source: A. Admission Control and Queueing Model · p. 4

B. Admission Probability

• The paper derives closed-form steady-state probabilities for the zero-buffer M/D/C/C+K system, which prior work had analyzed numerically.The derivation uses a two-phase-process approximation of a deterministic server.Source: B. Admission Probability · p. 4
• Lemma 1 approximates state probabilities for an M/D/n/n queue with Poisson arrivals, deterministic service time, and zero buffer.State i denotes the probability that i users are being served simultaneously.Source: B. Admission Probability · p. 4
• 0.01% was the maximum gap between the admission-rate analysis and simulation, verifying Lemma 1’s accuracy.Each plotted point averaged 1000 time periods, with n = 3, 4, and 5 servers.Source: Figure 4 · p. 4, B. Admission Probability · p. 4

1) Admitted-arrival:

• The admitted-arrival inter-arrival distribution is derived from Q.1 departures by weighting the first-departure probabilities across queue states.For state i, the probability of a departure within x is 1 - ((Tv - x) / Tv)^i; differentiating the resulting CDF gives the PDF.Source: 1) Admitted-arrival · pp. 4–5

C. Average Waiting Time

• Because a general closed-form waiting-time distribution for GI/D/m is unavailable, the analysis uses its equivalent GI(m*)/D/1 waiting-time distribution and a phase-type approximation.The coordinated inter-arrival time Y is formed from m inter-arrival times of the GI/D/m queue.Source: 2) Phase-type Approximation · p. 5

2) Phase-type Approximation:

• The approximation replaces Q.2’s inter-arrival distribution with a mixture exponential matched to Y’s first and second moments, then approximates waiting-time behavior using a Ph/D/1 queue.The paper derives an approximated average waiting time for admitted EVs.Source: 2) Phase-type Approximation · p. 5
• For fixed n, the approximated average waiting time ωπn(d) is increasing and convex in demand d.Source: 2) Phase-type Approximation · p. 5
• 0.1%: the reported difference between the approximation and simulation is no more than this in the m = 4 comparison.The comparison concerns average waiting time in Fig. 5.Source: Figure 5 · p. 5, 2) Phase-type Approximation · p. 5

A. Optimization Problem Recasting

• Reparameterizing the integer decision n as P = Pπn(d) yields a concave objective that can be solved efficiently after relaxing the integer constraint.The relaxed solution gives (P^v, d^v), from which n^v is recovered using equation (6).Source: A. Optimization Problem Recasting · pp. 5–6
• The optimal integer n is one of the floor or ceiling of the relaxed n^v, with the corresponding optimal d evaluated for each candidate.The algorithm uses the better of the two candidates.Source: A. Optimization Problem Recasting · p. 6
• Lemmas 3 and 4 yield a three-step optimal-solution algorithm for Problem (11).For a fixed n, the objective is concave in d on the stated feasible domain, and it is also concave in d at the optimal n.Source: A. Optimization Problem Recasting · p. 6

V. SIMULATION RESULTS

• Simulations cover 24 hours with four charging ports, 11.5 kW charging, 40 parking lots, Poisson EV arrivals, and a linear waiting-time penalty h(ω) = cω.Unless otherwise specified, EVs share β = 0.05 and battery capacity ϕ = 100 kWh; the method is described as adaptable to varying arrival rates, penalty rates, and electricity prices.Source: §V · p. 6
• The queue-length-based admission benchmark admits an EV only when the number already admitted is below the parking-lot threshold.The simulation sets that threshold to the station’s total number of parking lots.Source: §V · p. 7
• The greedy benchmark admits an EV only when doing so increases short-run profit, without considering future arrivals.Source: §V · p. 7

A. Average Profit Evaluation

• QBA earns only 44% of JoAP’s average profit under the low penalty and makes a negative profit under the high penalty.The paper attributes QBA’s high-penalty loss to delay penalties after bulk arrivals fill parking spaces, while later EVs are rejected.Source: A. Average Profit Evaluation · p. 7
• During light traffic, JoAP’s profit is only 0.5% and 2% above greedy admission under low and high penalties; its advantage is more evident during heavy traffic.The paper says JoAP’s advantage over greedy grows with electricity price as its admission rate decreases; QBA’s profit is dominated by delay penalties.Source: A. Average Profit Evaluation · p. 7
• JoAP’s full-day average profit is 330% and 531% higher than greedy admission at waiting-time penalties of $0.4/min and $1/min, respectively.Fig. 7 compares the schemes under the two penalty rates.Source: Figure 7 · p. 7, A. Average Profit Evaluation · p. 7
• JoAP considers EVs in service and possible future arrivals when admitting vehicles, whereas greedy admission cannot balance current and future charging schedules.The paper says greedy admission can reject EVs even under very light traffic, while JoAP achieves higher profit than both benchmarks.Source: A. Average Profit Evaluation · p. 7

B. Admission Rate Evaluation

• JoAP’s admission rate is 85% under the low penalty and 80% under the high penalty, versus greedy’s 70% and 69%; QBA reaches 86% overall.QBA’s rate falls below JoAP’s in some moderate-arrival periods because it is oblivious to possible future arrivals.Source: B. Admission Rate Evaluation · p. 7

C. Impact of τ

• Optimizing τ raises average profit by 5.9% over fixing τ at 1.01 across the scenarios.The authors therefore suggest focusing on optimizing n with fixed τ in practice.Source: C. Impact of τ · pp. 7–8, Figure 9 · p. 8

VI. CONCLUSIONS

• JoAP uses multi-subprocess admission control to balance admission rate and EV quality-of-service according to arrival rate, electricity price, and delay penalty.The paper analyzes joint admission control and scheduling with a tandem queueing model and proposes an algorithm to compute the optimal solution.Source: §VI · p. 8
• The authors identify heterogeneous EVs, renewable and distributed generation, and demand-sensitive electricity pricing as future research directions.They specifically plan to extend the work to heterogeneous EVs and study renewable and distributed generation’s impact on admission control and station efficiency.Source: §VI · p. 8

APPENDIX A PROOF OF LEMMA 1

• The proof replaces each deterministic service process with a two-phase exponential process, using a rate parameter κ derived for service time d_v.The passage states that the deterministic service time equals the constant d_v.Source: Appendix A · p. 8, Figure 10 · p. 8
• The state (s1, s2) records phase-1 busy processes and total busy processes, and transitions define the generator matrix T^k for k servers.The described state can transition to at most six states; the generator is specified by cases.Source: Appendix A · pp. 8–9
• The proof derives steady-state probabilities through relations among state probabilities, then applies a boundary condition to obtain the normalized distribution.It identifies the steady-state probability for state (s1, s2) and uses the condition that probabilities sum to one at the boundary.Source: Appendix A · p. 9

• The proof approximates average waiting time using Proposition 1 for a Ph/D/1 queue, with service time S and inter-arrival time A [19].Source: Appendix B · p. 9
• Applying Proposition 1 to the tandem queue model uses the Laplace transform of Y and a2(s)=(s+λ1)(s+λ2).Source: Appendix B · p. 9
• The proof states that ψ is positive and increasing in d, then substitutes λ1 and λ2 using μY and σY to express average waiting time.Source: Appendix B · p. 9
• For fixed n, Pπn(d) is convex increasing in d, so ωπ(d) is convex, consistent with Kingman’s formula.Source: Appendix B · p. 9