The Impact of Considering State-of-Charge- Dependent Maximum Charging Powers on the Optimal Electric Vehicle Charging Scheduling

Intelligent charging solutions facilitate mobility electrification. Mathematically, electric vehicle (EV) charging scheduling formulations are constrained optimization problems. Therefore, accurate constraint modeling is theoretically and practically relevant for scheduling. However, the current scheduling literature lacks an accurate problem formulation, including the joint modeling of the nonlinear battery charging profile and minimum charging power constraints. The minimum charging power constraint prevents allocating inexecutable charging profiles. Furthermore, if the problem formulation does not consider the battery charging profile, the scheduling execution may deviate from the allocated charging profile. An insignificant deviation indicates that simplified modeling is acceptable. After providing the problem formulation targeting the maximum possible vehicle battery state-of-charge (SoC) on departure, the numerical assessment shows how the constraint consideration impacts the scheduling performance in typical charging scenarios (weekday workplace and weekend public charging where the grid supplies up to 40 vehicles). The simulation results show that the nonlinear battery charging constraint is practically negligible: For many connected EVs, the grid limit frequently overrules that constraint. The resulting difference between the final mean SoCs using and not using accurate modeling does not exceed 0.2%. Consequently, the results justify simplified modeling (excluding the nonlinear charging profile) for similar scenarios in future contributions.

The Impact of Considering State-of-Charge-Dependent Maximum Charging Powers on the Optimal Electric Vehicle Charging Scheduling Kun Qian , Reza Fachrizal , Joakim Munkhammar, Thomas Ebel , Senior Member, IEEE, and Rebecca C. Adam Abstract-Intelligent charging solutions facilitate mobility electrification.Mathematically, electric vehicle (EV) charging scheduling formulations are constrained optimization problems.Therefore, accurate constraint modeling is theoretically and practically relevant for scheduling.However, the current scheduling literature lacks an accurate problem formulation, including the joint modeling of the nonlinear battery charging profile and minimum charging power constraints.The minimum charging power constraint prevents allocating inexecutable charging profiles.Furthermore, if the problem formulation does not consider the battery charging profile, the scheduling execution may deviate from the allocated charging profile.An insignificant deviation indicates that simplified modeling is acceptable.After providing the problem formulation targeting the maximum possible vehicle battery state-of-charge (SoC) on departure, the numerical assessment shows how the constraint consideration impacts the scheduling performance in typical charging scenarios (weekday workplace and weekend public charging where the grid supplies up to 40 vehicles).The simulation results show that the nonlinear battery charging constraint is practically negligible: For many connected EVs, the grid limit frequently overrules that constraint.The resulting difference between the final mean SoCs using and not using accurate modeling does not exceed 0.2%.Consequently, the results justify simplified modeling (excluding the nonlinear charging profile) for similar scenarios in future contributions.
Index Terms-Charging profile, electric vehicle (EV) charging, minimum charging power, public charging, workplace charging.

I. INTRODUCTION
T HE rapidly developing transformation targeting vehi- cle electrification requires intelligent charging solutions, especially for those infrastructures with limited supply power.One example is workplace charging (supported by more and Kun Qian, Thomas Ebel, and Rebecca C. Adam are with the Centre for Industrial Electronics (CIE), Department of Mechanical and Electrical Engineering, University of Southern Denmark, DK-6400 Sønderborg, Denmark (e-mail: kqian@sdu.dk;ebel@sdu.dk;rebeccaadam@sdu.dk).
Digital Object Identifier 10.1109/TTE.2023.3245332more companies to promote transportation electrification), where the grid supply power constrains the total charging power.The challenge to charging a group of electric vehicles (EVs) with a constrained grid power is to ensure the fair share of the supply energy and maximize the delivered energy to the EVs, i.e., drivers' satisfaction maximization [1].
In a practical charging process, apart from the fixed upper bound on the charging power, there is a minimum charging power constraint due to the higher power loss at low charging powers [11].According to [12], a charging current less than 6 A is not permitted.Ignoring the lower bound in the problem formulation can lead to inexecutable charging profiles, in which case the EVs will not charge.Furthermore, an accurate problem formulation requires a constraint considering that the maximum charging power depends on the increasing state-of-charge (SoC) [1], [13], [14], [15].Considering these practical constraints is theoretically mandatory to accurately model the charging scheduling problem.

A. Related Work
The existing EV charging scheduling solutions often assume a relaxation on the minimum charging power, the SoC-dependent maximum charging power, or both the constraints.Table I summarizes how the related contributions combine the four resulting constraint modeling options.Note that only [1] models the combination of both the constraints.Casini et al. [4] formulated the scheduling problem to minimize the overall daily peak power while satisfying the required energy from the EV users.They developed two real-time algorithms to handle the uncertainties, one using the prior knowledge (statistics of the historical data) while the other not, and they showed the effectiveness via numerical simulations and comparisons.Gan et al. [18] also presented real-time scheduling algorithms for a similar objective, though their decentralized algorithms have advantages such as scalability.However, both the papers applied a continuous charging power range starting from 0 to express the constraint on the charging power.The generated charging profile from the scheduling algorithm can potentially be inexecutable to the charging stations if the charging powers fall below the minimum value.Similarly, this can happen in all the literature categorized in the same group above (some literature starts from a negative value since they consider discharging).Instead of investigating the practical minimum charging power constraint, the existing literature often focuses on studying other aspects of smart charging scheduling (e.g., control architecture [21], [33], control methods [18], [34], and uncertainty handling [35], [36]).To tackle the impracticality, Uiterkamp et al. [23] presented an efficient scheduling algorithm that allows only charging above a given minimum threshold.Zdunek et al. [37], in their modeling, included binary variables to indicate whether the EV is charging or not, which can potentially include the minimum charging power in modeling.Besides traditional modeling, recent literature has successfully applied model-free reinforcement learning (RL) to EV smart charging, where the action space of the RL agent often avoids charging power allocation below the minimum charging power.Sadeghianpourhamami et al. [24] applied RL to schedule the charging of a group of EVs.Their state encapsulates the timing information, the required charging time, and the parking time for those connected EVs.Suited to the state information, the action space consists of selections of different EVs to charge.For charging, they assumed a constant power during the charging process, and consequently, the charging power would never be below the infrastructure-dependent minimum charging power.Tuchnitz et al. [26] discretized the charging power into no charging, charging at half power, and charging at full power, with the possibility of having other intermediate charging levels but avoiding charging levels lower than the minimum charging power.Nevertheless, the contributions above have not studied the effects of the SoC-dependent maximum charging power, whereas, in reality, EVs have variable maximum charging powers dependent on their current SoC [31].Aziz and Oda [38] presented how the temperature affects the relationship between the charging rate and the SoC.The maximum charging power is constant when the SoC is low in a mild-temperature environment.Two scheduling strategies forbid the executed and the planned schedules from deviating: The first option is to keep the SoC level low.Keeping the SoC level low will not only circumvent the varying maximum charging power but also extend the battery lifetime [39].However, sometimes it is expected to charge the EVs as much as possible (e.g., to sufficiently cover the next trip).The second option is to consider the SoC-dependent maximum charging power in the problem formulation.Cao et al. [29] showed a nonlinear relationship between the maximum charging power and the SoC to formulate the maximum charging powers over the full SoC range.They take it into account in their rule-based EV charging scheduling.Similarly, Korolko and Sahinoglu [32] considered a smooth curve representing the relationship.They developed a cutting plane algorithm to solve the nonlinear optimization problem by solving a sequence of linear programming problems.Morstyn et al. [15] considered the nonlinear charging profile by applying the equivalent circuit model (ECM) to describe the battery charging.Furthermore, they proposed a reduced-order model which gives approximately the same form of curves as the complex model.El-Bayeh et al. [14] provided a unified mathematical expression where the parameters can change and thus can be used to approximate different battery models.However, they ignored the infrastructure-dependent minimum charging power constraint when formulating the problem.Frendo et al. [1], on the other hand, trained a regression model to predict the maximum charging powers from the historical charging processes.Even though they did not explicitly include the infrastructure-dependent minimum charging power constraint, the model training process should recognize that historically, the EVs never charged below the minimum charging power.However, their rule-based heuristic solution is not optimal, and the impact on optimal scheduling is not studied.

B. Knowledge Gap in EV Charging Scheduling
To directly apply the scheduling computation results and achieve the expected objective, modeling should consider electrical engineering constraints accurately.Ignoring the minimum charging power constraint will potentially lead to inexecutable charging profiles.Thus, this constraint should always be addressed in the problem formulation.Ignoring the SoC-dependent maximum charging power may result in deviation during scheduling execution.As summarized in Table I, most of the existing literature failed to include the minimum charging power constraint when they provided problem formulation considering the SoC-dependent maximum charging power.How considering both the constraints impacts the optimal scheduling performance requires an exploration.Particularly the following questions are currently open.
1) What is the accurate problem formulation for optimizing smart charging given practical electrical engineering constraints?2) How significant is the deviation between the formulation considering and not considering the SoC-dependent maximum charging power constraint? 3) How to systematically choose the appropriate model given the deviation?

C. Novel Contributions
This article complements the existing smart EV charging literature by the following.
1) The detailed optimal EV charging scheduling formulation considering both the infrastructure-dependent mini-Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II
NOMENCLATURE FOR BATTERY MODELING mum charging power and the SoC-dependent maximum charging power.2) Providing a method to evaluate scheduling deviation in simplified modeling.3) Providing a systematic modeling workflow for future studies.4) Concluding whether neglecting the SoC-dependent maximum power constraint constitutes a suitable simplified model in typical charging scenarios.The optimization objective is to maximize the drivers' satisfaction, i.e., to ensure the fair share of the supply energy and maximize the delivered energy to the EVs.Without loss of generality, the same approach is potentially applicable for exploring other objectives and justifying whether it is necessary to consider the SoC-dependent maximum charging power constraint.

D. Notational Conventions, Nomenclature, and Article Structure
In this article, lowercase boldfaced letters denote vectors.Furthermore, let ⊙ denote the elementwise product, ⊗ denote the Kronecker product, ⪯ denote elementwise smaller than and equal to, ⪰ denote elementwise greater than and equal to, and || • || denote the Euclidean norm.To ease the reference, the nomenclature in this article is summarized separately in Tables II and III for battery modeling and charging scheduling formulation, respectively.
The remainder of this article is structured as follows.Section II provides the battery model and the computational details for battery charging under the nonlinear charging profile.Section III provides detailed mathematical formulations and important details for scheduling implementation.Section IV first presents the battery charging process simulation and then describes the scheduling simulation scenarios.Section V presents the scheduling simulation and its analysis.Finally, Section VI presents the discussion, and Section VII concludes the work.

II. SOC-DEPENDENT MAXIMUM CHARGING POWER
Formulating the SoC-dependent maximum charging power requires modeling the battery charging process.The electrical circuit models (ECMs) can simulate the battery charging process with networks of electrical components such as voltage sources and resistors [40], [41].Fig. 1 shows a simple battery ECM consisting of a controlled voltage source in serial with an internal resistor [13], [15], [40], [41], where v oc is the open-circuit voltage (OCV), R is the impedance, i t is the charging current, and v t indicates the terminal voltage.Arranging the battery cells in M p × M s , where M p and M s denote the number of cells in parallel and series, respectively, yields an equivalent model for the battery, as shown in Fig. 2.Here v oc eq is the battery OCV, R eq is the impedance, i b is the battery charging current, v b indicates the terminal voltage for the battery, and p ac is the charging power provided by the charging station.Due to the parallel and series arrangement of the cells, these parameters read Furthermore, this article uses the linear model from [15] to express the battery cell's OCV v oc depending on the stored energy (neglecting the battery efficiency during charging) at the time index k where l 0 is the SoC at the arrival time (AT) index k 0 , E nom is the battery cell's nominal energy, τ is the time slot duration, and a v measured in mV/Wh and b v measured in V are the model coefficients.
Battery charging commonly starts with either constant current or constant power and ends with constant voltage (CC-CV or CP-CV).Currently, most EV chargers lead to a CC-CV charging process [42], [43], [44].However, with the increasing battery voltage during the charging process, the CC-CV scheme results in varying maximum power demands, with a peak power demand at the turning point from the CC mode to CV mode.This is challenging since charging stations usually have a fixed maximum power limit.Instead, the CP-CV charging scheme ensures that the battery charges with constant power until the battery reaches the maximum voltage.
Under the CP-CV charging scheme, the battery is charging with a constant power, when the terminal voltage v t is less than the maximum cell voltage V max at time index k, and the terminal voltage and the battery charging current read Following the derivation in Appendix in turn determines v b,k Substituting ( 8) in ( 1) and ( 7) in (3) and rearranging the equations yields v t,k and i t,k , respectively, and allows to update the stored energy in each cell at the end of time index k.If the determined v t exceeds V max , the battery is charging with constant voltage, i.e., v t = V max .Then the battery charging current reads Substituting ( 9) yields the battery cell charging current i t according to (3).The recursive CP-CV battery charging process requires the following steps as shown in Fig. 3. 1) Calculate v oc and v t .
2) Determine the operation mode (CP or CV).
3) Calculate mode-dependent i t and SoC.Then according to (5), the SoC determines the OCV v oc .Afterward, substituting the OCV in (6) yields the battery cell voltage v t , which indicates the battery charging mode.Note that when the charging station allocates a charging power to the EV, the EV can only take it when charging in the CP mode.
If battery charging has moved to the CV mode at time index Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
k, the real supplied power p ac,k by the charging station reads Consequently, the real charging powers depend on the SoC.

III. PROBLEM FORMULATION
This section provides a detailed formulation for the optimal EV charging scheduling and covers all the three relevant cases.
1) Ideal Profile: The SoC-dependent maximum charging power constraint is not considered in this problem formulation.The ideal profile provides an upper bound for the final mean SoC.This article uses the ideal profile to systematically choose appropriate modeling by examining the deviation between the ideal and realistic profiles.2) Realistic Profile: This article dubs the profile gained by limiting the ideal allocated powers in a second step by the SoC-dependent maximum charging power as the realistic profile.3) SoC-Dependent Profile: The SoC-dependent profile determines the charging power profile gained by directly allocating powers constrained by the SoC-depending maximum charging powers.Therefore, the SoC-dependent profile constitutes both the optimal and realistic performance.Let N represent the number of charging stations.Furthermore, let H represent the scheduling horizon, T the decision time slot duration, and K = ⌊H/T ⌋ the total number of time slots.
For the charging behavior, let t a n , t d n , l 0 n , and e n represent the nth EV's AT, departure time (DT), the initial SoC, and energy demand (ED) for all the nodes n ∈ {1, . . ., N }, respectively.This work assumes homogeneous (of the same characteristics) EVs.Let P min , P max , E bat , and l max represent the minimum, maximum charging power, battery energy, and maximum SoC, respectively.Thus, e n = (l max − l 0 n )E bat .The vector describes the charging schedule for EV n ∈ {1, . . ., N }.Furthermore, η denotes the charging efficiency.The charging schedule depends on the EV connection status.For all EVs n ∈ {1, . . ., N }, the binary connection status vector reads This article presents an implementation-friendly and hence convenient matrix-vector notation by stacking the charging , and the connection time vectors and EDs (ATs, DTs, and EDs T , and e = [e 1 , . . ., e N ] T , respectively.
To express the summations over the changing schedules p n for all n ∈ {1, . . ., N } as a vector containing the total load at each time index k ∈ {1, . . ., K } this article uses the auxiliary matrix A load ∈ R K ×N K ≥0 as follows: Similarly the auxiliary matrix expresses the summations over the charging schedules p n (k) for all k ∈ {1, . . ., K } and for each n ∈ {1, . . ., N } as follows: This vector contains the total charged energy for each n ∈ {1, . . ., N }.This article targets maximizing EV driver satisfaction in terms of SoC on departure.Therefore, here the objective is to minimize the difference between the total charged energy and the EDs.To reach the maximum SoC, upon EV's arrival, the required energy is the desired energy.Thus, the objective function for the optimization problem is

A. Ideal Profile
The ideal profile scenario does not consider the SoC-dependent maximum charging powers.Thus, there are three types of constraints to consider.
1) EVs can either not charge (= 0) or charge within a defined power range (∈ [P min , P max ]) when connected.2) At each decision time slot, the total charging power allocated to EVs should be bounded within the grid limit.3) EVs can, at maximum, acquire the energy that equals their EDs.Constraint (a) means that the decision variables are bounded in two disjoint ranges, i.e., either 0 or in the range [P min , P max ].Similar to [37], this article uses extra binary decision variable s ∈ B N K ×1 to constrain charging power variables by the following equation: For constraint (b), let P grid denote the grid limit Similarly, constraint (c) uses the auxiliary matrix A energy Consequently, the optimization formulation to compute the "ideal profile," p ideal , is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Realistic Profile
Let l n (k) denote the accumulated SoC until time index k −1.Due to the maximum charging power SoC dependency, the actual charging power may deviate from the allocated charging profile.To take this possible deviation into account, the k, nth entry (∀k ∈ {1, . . ., K } and ∀n ∈ {1, . . ., N }) of the realistic power profile vector p realistic always determines the minimum value between the ideal profile values and the SoC-dependent maximum charging power p ac,n (k) With (21), the accumulated SoC up to time index k − 1 is Following the derivation in Appendix, the SoC-dependent maximum charging power in the CV operation mode reads where α n and β are constants, and α n depends on the nth EV's initial SoC.As a result, upon getting p ideal , there will be an iterative process going through each time index for each EV and comparing it to p ac,n (k), and the smaller values are indeed the real supplied charging powers, which will be stored as p realistic .

C. SoC-Dependent Profile
Besides modeling constraints (a)-(c) like in the ideal profile, the formulation to consider the SoC-dependent maximum charging powers requires an additional constraint for nth EV at time index k Note that (24) only forms the accurate maximum charging power limits in the constant voltage mode.Combining (24) and constraint (a) will provide the accurate maximum charging power limits during the whole charging process since P max will limit the charging power in the constant power mode.
Rearranging (24) and expanding it over the scheduling horizon ∀k ∈ {1, . . ., K } results in resulting in the following matrix form: where the matrix B's diagonal elements have the value 1, and its lower left triangular elements have the value β.Stacking N parameters α n for all EVs in α = [α 1 , . . ., α N ] T and together with (25) can result in a convenient matrix-vector representation for the extra constraint formulation for all EVs over the scheduling horizon Consequently, optimization problem formulation to compute the optimal charging schedule considering the SoC-dependent maximum charging powers p SoC-dependent , referred to as "SoCdependent profile," reads D. Systematic Modeling Workflow 1) Choosing an Appropriate Model: Note that the final mean SoC from the three different profiles is related by the following inequality: The inequality indicates the lower and upper bounds for lSoC-dependent .Thus, EV scheduling practitioners can define a threshold ε for the acceptable final SoC deviation and compare it to the deviation between the "ideal" and "realistic" profiles (σ = lideal − lreal ) to decide whether simplified modeling is acceptable This simple decision rule is shown in Fig. 4.
2) Implementation Details: Note that the matrix-vector representations in (20) and ( 27) facilitate easy charging profile implementation for p ideal and p SoC-dependent , with CVX [45] and the MOSEK solver [46] in MATLAB.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 4. Simple modeling workflow for practitioners shows that simplified modeling is practically applicable for small final SoC deviations between the "ideal" and "realistic" profiles.

IV. SIMULATION SETUP
This section first declares and defines the EV battery's relevant parameters, and then shows the charging process simulation.The considered two scenarios (weekday workplace charging and weekend public charging) apply the battery parameters and other relevant parameters.Finally, a discussion on the performance evaluation illuminates the later simulation results.

A. Battery Charging Process
For further simulations, this article uses the battery parameters from [15], as summarized in Table IV.Other parameters summarized in Table V are a prerequisite to simulate the charging process.
Fig. 5 shows the battery charging process simulations based on the parameters from Tables IV and V.In Fig. 5, the charging power stays at the maximum charging power (P max ac = 11.04 kW) during the constant power charging mode, which is on the left side of the blue dash-dot line.The battery voltage increases while the charging current decreases.The charging mode changes to the constant voltage mode when the battery voltage reaches the maximum value (M s V max = 415 V), which is indicated by the blue dash-dot line.Then the battery cannot use the maximum charging power anymore.Instead, the used charging power is nonlinearly decreasing over time.Considering the practical minimum  5. Battery charging process simulation.The left side of the blue dash-dot line shows the constant power charging mode, and the mode changes to the constant voltage when the battery voltage reaches 415 V, as shown in the right side of the blue dash-dot line.The charging process will end earlier if the model includes the practical minimum charging power, as shown via the dashed lines.Fig. 6.SoC-dependent maximum charging power.The dashed line indicates that the maximum charging power decreases from 4.14 kW directly to 0 if considering the practical minimum charging power limit.charging power (P min ac = 4.14 kW), the charging process will end earlier, as shown by the dashed lines in Fig. 5. Fig. 6 shows the resulting SoC-dependent maximum charging power, and the dashed line indicates the maximum charging power for the case considering the practical minimum charging power limit.

B. Simulation Scenarios
The numerical results in this article cover two charging scenarios: weekday workplace charging and weekend public charging.For each charging scenario.First, the decision time slot duration is constant to investigate the results from varying the number of EVs; then, the number of EVs is constant to investigate the results from varying the decision time slot duration.Consequently, this work assesses four simulation scenarios.
For the fixed decision time slot duration, this article chooses the most often used value in the literature: 15 min [7], [47], [48], corresponding to the typical communication time between the energy management units and the grid [49].The experimental setting chooses the least number of EVs with a proper final average SoC to limit the computational complexity.The final mean SoC should be smaller than the maximum SoC so that there is room for potential improvement by varying the decision time slot duration.
1) Fixed Decision Time Slot Duration: This article first considers a fixed decision time slot duration and varies the number of EVs from 10 to 40, with a step size of 5, and samples the charging behavior from the existing datasets.To compensate for the sampling randomness, this work samples J (here J = 20) times for a fixed number of EVs and computes the charging profile.The scheduling horizon extends from the first AT to last DT.Apart from the different charging behaviors, the two charging scenarios use the same parameters.
There is a fixed maximum grid limit, P grid = 22 kW.Charging stations can supply minimum 4.14 kW (P min , corresponding to three-phase 6 A) and maximum 11.04 kW (P max , corresponding to three-phase 16 A) during the charging process.The charging efficiency, η, is set to 0.88.This work assumes the same battery energy of E bat = 40.7 kWh (a result from Table IV) and a maximum SoC of l max = 0.96 (this value approximates the maximum SoC according to the simulation result shown in Fig. 5).
Table VI summarizes the simulation parameters and their values.In addition, they share the SoC-dependent maximum charging power limit.In this work, the two charging behavior settings are as follows.
Weekday Workplace Charging: Assumes that EVs arrive with an initial SoC of l(0) = 0.7, corresponding to an ED of 0.26 × 40.7 kWh = 10.58 kWh (the authors are missing appropriate data to describe the ED, and therefore roughly assumed an ED covering the round trip: workplace-home- workplace.).For the ATs and DTs, this article uses the user mobility data from the Swedish travel survey in 2006 [50], [51].The arrival and DTs of trips made by cars and other information are available in the survey, and it is considered representative for Scandinavian conditions.Randomly sampling the survey data distributions in a Monte Carlo fashion and assuming that each EV made one round trip (home-workhome) during the weekday yields the ATs and DTs used here.Fig. 7 shows the histogram of the weekday ATs and DTs at the workplace.
Weekend Public Charging: Lahariya et al. [52], based on the real-world EV sessions, have trained statistical models to generate samples of realistic EV charging behavior data which consist of ATs, DTs, and EDs.According to [52], the generated samples are statistically indistinguishable from realworld data.This work used the trained model and generated the charging behavior for the date 20th June 2015 (Saturday).The following postprocessing steps prepared the data used in the simulations.
1) The DTs are limited to a single day.2) To account for the assumed battery energy of E bat = 40.7 kWh and three-phase charging, this work accordingly tripled the generated EDs.
3) The experimental setup excludes sessions for those the refined EDs exceed the maximum battery energy, for those the refined EDs fall below the minimum ED, and for those the stay duration is not long enough to finish charging.
Fig. 8 shows the histogram of the ATs and DTs, and Fig. 9 shows the histogram of the EDs.
2) Fixed Number of EVs: The result from the above simulation decides how many EVs to include for the next simulation.This setup varies the decision time slot duration from 5 to 30 min, with a step size of 5 min.The other parameters and the charging behavior data are the same as the above simulation scenarios (with fixed decision time slot duration).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

C. Performance Evaluation
The scheduling objective is to maximize drivers' satisfaction, i.e., to ensure the fair share of the supply power and maximize the delivered energy to the EVs.The fair share is ensured by the objective function minimizing the difference between the total charged energy and the ED for each EV.This article compares the average mean final SoCs (averaging over 20 simulations for the number of simulated EVs) for the introduced formulations.

A. Scheduling Performance Comparison
This work numerically compares the final mean SoCs for scheduling via simplified modeling ("ideal") and for scheduling based on accurate problem formulation including all the constraints ("SoC-dependent").Note that the "ideal" scheduling formulation calculating p ideal covers simplified modeling as formulated in [23] and [37].
Figs. 10 and 11 show comparison of the "realistic profile" mean final SoC (lower bound) to the "ideal profile" mean final SoC (upper bound).The figures only show the "SoCdependent profile," if there are significant differences between the "realistic profile" and the "ideal profile" following (25).Showing the "SoC-dependent profile" is unnecessary, if there is no visually identifiable difference or only a slight difference.Fig. 10 shows the average mean final SoC when varying the number of EVs.There is an overall decreasing trend with the increasing number of EVs.However, neither Fig. 10(a) nor (b) shows a visually identifiable difference between the "ideal profile" and "realistic profile." Based on the simulation result, further simulations will have a fixed number of EVs but varying decision time slot duration from 5 to 30 min, with a step size of 5 min.The number of EVs is 25 and 15 for weekday workplace charging and weekend public charging, respectively.Fig. 11 shows the average mean final SoC.Fig. 11 shows a decreasing trend with the increasing decision time slot duration, which confirms the intuition that the average mean final SoC should be higher when the time resolution is higher.Furthermore, there is a slight difference in Fig. 11(a) at 5-min duration for the weekday workplace charging.However, there are significant differences for the weekend public charging at 5 and 10 min duration and a slight difference at 15 min duration, as shown in Fig. 11(b).The EVs cannot always take advantage of the charging power allocated by the charging stations due to the existence of SoC-dependent maximum charging powers in some time slots.The significant differences for the weekend public charging at 5 and 10 min require further simulating the "SoCdependent profile" according to (27), and the results are also shown in Fig. 11(b).By considering the SoC-dependent maximum charging power constraint, it is clear that the "SoC-dependent profile" achieves higher average mean final SoCs compared with the "realistic profile."

B. Example Charging Profile
Fig. 12 shows the example charging profiles, together with the EVs' connection status.As seen from Fig. 12, there are a few time slots when the EVs cannot fully take the allocated charging power by the charging stations, highlighted by the two ellipses.The "realistic profile" is below the "ideal profile" at those time slots.Finally, the "ideal profile" and "realistic profile" have an average mean final SoC at 95.86% and 95.18%, receptively.The "SoC-dependent profile" has a different shape than the "ideal profile," and it achieves an average mean final SoC at 95.72%.

C. Analysis
In both the weekday workplace and weekend public charging simulations, there is no visually identifiable difference between the "ideal profile" and the "realistic profile" with the fixed decision time slot duration at 15 min.The reason is that the simulations start with the number of EVs at 10, and due to the grid limit (P grid = 22 kW), the allocated charging powers are mainly at the lower end.Thus, the SoC-dependent maximum charging power constraint is almost inactive during the scheduling horizon, resulting in nearly no difference.Fig. 13 further demonstrates the reasoning by showing an example charging process for one EV.The vertical dashed lines indicate the arrival and DT for the EV.As shown, the ideal charging powers are consistently below the SoC-dependent maximum charging powers, and consequently, the "realistic profile" will be the same as the "ideal profile."The reason the optimizer allocates small charging powers is that the grid limit P grid is 22 kW while there are many EVs connected.In such cases, the SoC-dependent maximum charging power constraint is overruled by the grid limit, and thus it is never active.
Then, with the fixed number of EVs and varying decision time slot duration, the average mean final SoC is the highest at the duration 5 min.There are two reasons: first, since the generated ATs and DTs are timestamps, a smaller decision time slot duration may result in a longer scheduling horizon (e.g., the scheduling horizon starting point for the EV arriving at 06 04 will be 06 : 05 and 06 : 10 for the duration at 5 and 10 min, respectively); second, with a higher resolution, the optimization solver has more options to assign the charging powers, resulting in a potentially higher average mean final SoC.However, a better "ideal profile" does not always ensure a better "realistic profile," as seen from Fig. 11(b).The reason is that when computing the "realistic profile" based on the "ideal profile," the SoC-dependent maximum charging power constraint will become active with the increasing SoC level along the scheduling horizon.To cope with the constraint, the problem formulation requires including it, resulting in the "SoC-dependent profile."The resulting average mean final SoC is higher than the "realistic profile" but still below the "ideal profile" since the "ideal profile" ignores the SoC-dependent maximum charging power constraint when computing the profile.Fig. 14 demonstrates the reasoning by showing an example charging process for one EV.As seen from Fig. 14(a), with the increasing SoC, the SoC-dependent maximum charging power constraint becomes active, and the EV cannot use those allocated powers that exceed the constraint.On the other hand, p SoC-dependent ensures that the allocated charging powers are always below the constraint.
Another point worth noting is that the solution from the "ideal profile" is not unique.For example, an EV charging with 4.14 kW first and then 11.04 kW or vice versa would bring the same result according to the objective function.Nevertheless, the former allocation may cause a higher deviation when computing the "realistic profile."The SoC level increases during charging, meaning the maximum charging power will start decreasing at some point.At the beginning of the charging process, the allocation of high charging power has a higher chance of delivering the planned power.
Fig. 11(b) shows the improvement in the average mean final SoC by considering the SoC-dependent maximum charging power constraint.However, the improvement is around 0.2%, which may extend the driving range by approximately 1 km.Compared with the extra 1 km, the saving on the computational resources by excluding the SoC-dependent maximum charging power constraint may be of more interest.Besides, Fig. 10 shows no visually identifiable difference between the "ideal profile" and "realistic profile" with the fixed decision time slot duration at 15 min, which is the most often used value in the literature.

VI. DISCUSSION
This work highlighted two electrical engineering constraints often ignored in EV charging scheduling formulation: the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.infrastructure-dependent minimum charging power and the SoC-dependent maximum charging power.To generate a practically applicable charging scheduling, it is necessary to always include the infrastructure-dependent minimum charging power.As a result, the constraint for the charging power is in two disjoint ranges instead of a continuous range, which leads to the problem falling into the mixed-integer model and slightly increased problem complexity.
As for the SoC-dependent maximum charging power, this article applied the reduced ECM model from [15] for the battery.For simplicity, the proposed method assumed the same battery type for all the EVs considered in the scheduling problem.A more accurate model can potentially compute more practically applicable scheduling.This would further increase the problem's complexity and result in a longer computation time.Thus, evaluating the deviation can provide insight into whether it is necessary to include this constraint in the optimization problem.However, the evaluation still requires modeling the battery charging process.The battery Fig. 14.Example charging process for one EV: (a) computed p ideal , i.e., ideal charging profile, violates the SoC-dependent maximum charging power constraint, and the final realistic SoC level (94.19%) of this EV is smaller than planned (95.59%) and (b) SoC-dependent maximum charging power constraint is considered when computing p SoC-dependent , and thus, the allocated powers are always below the constraint, resulting in the SoC level at 94.99%.parameters differ from one battery type to another.Besides, other practical aspects such as battery degradation will also affect charging process modeling.One direction for future work is to model the battery charging behavior from the historical charging processes, which can not only simplify deviation evaluation but also improve the applicability of the accurate model when the deviation is significant.
Section III gradually presented how to compute the ideal, realistic, and SoC-dependent charging profiles.With the defined weekday workplace and weekend public charging scenarios, there are no or only slight deviations when directly applying the ideal profile.Consequently, as proposed in Fig. 4, instead of formulating the problem as in (27), the simpler version as in (20) can be used.Thus, in some existing literature where the SoC-dependent maximum charging powers are not considered, their resulting charging scheduling may be perfectly fine.What they are missing is a justification.Practical applications or future research can also directly take advantage of the result and method from this work to justify their simplified modeling.
The results are valid for the defined simulation scenarios.Nonetheless, the weekday workplace and weekend public charging are typical real-life charging scenarios.The objective, on the other hand, can vary in different studies.In addition, the extra constraint posed by the grid can change.Future studies can further explore the effect of the SoC-dependent maximum charging power constraint in different simulation settings.

VII. CONCLUSION
This article presents the mathematical formulation for optimal EV charging scheduling considering practical charging power constraints, especially the upper bound depending on the SoC.The simulations show that the scheduling performance increases by taking into account the SoC-dependent maximum charging power constraint with a small decision time slot duration.However, the increased final SoC will not extend the driving range significantly.By setting the decision time slot duration to 15 min, which is the typical value in the literature, there is no deviation when executing the schedule from simplified modeling.Further studies or practical scheduling implementations in a similar situation may exclude the SoC-dependent maximum charging power constraint in their formulation to reduce the complexity.

APPENDIX DERIVATION OF EQUATION (8)
Determining i b,k from ( 7) and substituting i b.k by ( 30) in (6) yields v b,k = v oc eq + R eq η p ac.k v b,k .
Multiplying both sides of (31) with v b,k and rearranging the equation in a way that completing the square can be applied v 2 b,k − v oc eq v b,k = R eq ηp ac,k .
Completing the square Then it is straightforward that the solution for v b,k is (8) v b,k = + 1 2 v oc eq ± R eq ηp ac,k + 1 4 v 2 oc eq .

Fig. 3 .
Fig.3.Flowchart of computing the parameters when charging the battery under the CP-CV scheme.

Fig. 7 .
Fig. 7. Histogram of the weekday ATs and DTs at the workplace from the survey data.

Fig. 8 .Fig. 9 .
Fig. 8. Histogram of the ATs and DTs for the weekend public charging scenario.

Fig. 10 .
Fig. 10.Average mean final SoC when varying the number of EVs: (a) weekday workplace charging and (b) weekend public charging.

Fig. 11 .
Fig. 11.Average mean final SoC when varying the decision time slot duration: (a) weekday workplace charging and (b) weekend public charging.

Fig. 12 .
Fig. 12. Example profiles and EVs' connection status from one simulation.

Fig. 13 .
Fig.13.Example charging process for one EV (the allocated charging powers are always below the SoC-dependent maximum charging powers)."Ideal" shows the allocated charging powers over the scheduling horizon."SoC level" is the SoC at the beginning of each time index."SoC-dependent max" shows the SoC-dependent maximum charging powers over the scheduling horizon.
Manuscript received 17 October 2022; revised 29 January 2023; accepted 12 February 2023.Date of publication 14 February 2023; date of current version 13 September 2023.This work was supported in part by the IE Industrial Elektronik Project which has received European Union co-financing from the European Social Fund under Grant SFD-17-0036 and in part by the Swedish Strategic Research Program STandUP for Energy.(Corresponding author: Kun Qian.)

TABLE III NOMENCLATURE
FOR CHARGING SCHEDULING FORMULATION Fig. 1.Battery cell model.

TABLE V PARAMETERS
FOR BATTERY CHARGING PROCESS SIMULATION Fig.

TABLE VI COMMON
SIMULATION PARAMETERS FOR THE CHARGING SCENARIOS