A Novel Fairness-Based Cost Model for Adopting Smart Charging at Fast Charging Stations

Abstract

This research introduces a cost model for application at fast charging stations (FCSs) with the aim to adopt smart charging, which can mitigate voltage fluctuation caused by the on/off status of FCS. When the operation of FCSs causes a voltage fluctuation and light flicker, the FCSs may be disconnected, as per the utility general standard practice, which results in financial loss represented by FCS downtime. However, FCS downtime can be avoided by applying the smart charging method referred to in this paper, or by installing mitigation devices that are available on the market. The proposed smart charging method provides three charging powers (or options), namely premium, regular, and economic, which consumers can select according to their needs and/or priority, whether this may be time or cost. Thus, the output power of each type is different as well as the per unit cost. The offered cost of smart charging is reliant on a ‘fairness’ policy that is, from the viewpoint of an FCS operator or investor, characterized by the value of cost for any customer at the FCS being equivalent to the customer’s value of time. For instance, customers A and B require X kwh from the FCS. When arriving at the FCS, if customer A values time the most, the premium power can be selected with the highest $/kwh cost. If the cost is more valuable to customer B, regular or economic power can be selected, but customer B will spend more time than customer A to get the same X kwh. The proposed fairness policy indicates that, to get the required X kwh, the percent of total costs saved by B (by using regular or economic power) in comparison to A is equivalent to the percent of total time saved by A (by using premium power) in comparison to B, for the same X kwh. The annual cost of applying smart charging at the FCS is estimated and compared with the annual cost of the best flicker mitigation device. The comparison reveals that distribution static compensators are considered the cheapest mitigation device, according to the cost per kVAr basis and the total annual equivalent cost. The proposed smart charging method achieves a tremendous reduction in the cost of mitigating the voltage fluctuation and light flicker. The annual cost of the proposed smart charging method is less than the annual cost of distribution static compensators by a minimum of 90% to a maximum of 99%.

1. Introduction

1.1. Background

It is predicted that electric vehicles (EVs) will share 33% of the global fleet and 55% of all new car sales, by 2040 [1]. Several factors are attributed to this increase, including: an increase in battery capacity, a reduction in battery price, incentives to EV buyers, and developing infrastructure whether in private (home charger) or public (commercial charger) places. However, as the number of EVs increases, charging stations get deployed to accommodate the EVs, which may increase the burden on the electric power systems. To alleviate the anticipated burden on the grid, smart charging has been proposed in relevant works at slow charging stations [2,3,4,5,6] and at fast charging stations (FCSs) [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The applied smart charging method includes different levels of control over the charging process, aiming to achieve several objectives related to the power grid, such as peak shaving, grid congestion management, ancillary service, and frequency control [23]. A few studies have barely investigated the impact of FCS on voltage fluctuation [24,25,26,27]. When FCS causes voltage fluctuation, which leads to light flicker, the FCS gets disconnected as per the utility general standard practice. This results in financial loss incurred by the FCS’s operator or investor, due to station’s down time. Recently, the work presented in [22] proposed a smart charging approach by which the voltage fluctuation at the FCS was mitigated and the limit of light flicker was not exceeded. The applied smart charging method consists of three levels of charging power: premium, regular, and economic. Each charging power in [22] differs in terms of rate of charge. Premium charging power has the highest charging rate, whereas economic charging power has the lowest rate of charge. Nevertheless, what motivate driver to select economic charging power, which has lowest charging rate or premium charging power, and which has highest charging rate was not illustrated in [22]. Beside using smart charging, voltage variation has been reduced in the literature using several mitigation devices such as [28,29,30,31]: thyristor switched capacitors (TSC), fixed capacitors/thyristor controlled reactors (FC/TCR), distribution static compensator (DSTATCOM), dynamic voltage restorer (DVR), unified power quality conditioner (UPQC), and fixed series capacitor (FSC).

1.2. Aim of the Work

The study aimed to fulfill the research gap by answering the following question: what solution would be preferable to mitigate the voltage fluctuation at the FCSs, by using a mitigation device or applying the smart charging method proposed in [22]? Therefore, a novel cost model was developed in this study, assuming that the proposed smart charging method in [22] is operating with a fairness policy that is (from the viewpoint of the FCS’s operator or investor) characterized by the value of cost for any customer (driver) at the FCS being equivalent to the customer’s value of time. Then, the cost of the proposed smart charging method was determined and utilized to be compared with the cost of other flicker mitigation devices available on the market.

1.3. Contribution

The work presented in this paper is devoted to developing a fairness-based cost model for adopting smart charging at the FCS, which reduces the voltage fluctuation and light flicker caused by FCSs. A per unit cost as well as a per unit time have been proposed for premium, regular, and economic charging powers from which customers can select, according to individual priorities, time or cost. A rebate is offered to customers to adopt regular and economic charging powers. The rebate is designed to compensate the increase in the charging duration by reducing the per unit cost of regular and economic charging power. The cost is modeled to preserve a balance in the required cost to charge 1 kwh using the regular or economic power and the required time to charge 1 kwh by the premium power, and vice versa. The annual cost of the proposed smart charging method is estimated and compared with the cost of the best flicker mitigation device selected as per the total annual equivalent cost.

2. Characteristics of Smart Charging at FCSs

2.1. Smart Charging Power

The proposed smart charging method in [22], applied at the FCS, encompasses three charging powers:

• Premium;
• Regular;
• Economic.

Each port can provide any of these services (charging power). The charging power is selected by the customer upon the arrival at the FCS.

• Premium charging power: can provide an EV with an output power (${\beta }^p$) ranging from $\overline{p} \%$ to $\underset{\_}{p} \%$ of the maximum output power $\beta$.
• Regular charging power: can provide an EV with an output power (${\beta }^r$) ranging from $\overline{r} \%$ to $\underset{\_}{r} \%$ of the maximum output power $\beta$.
• Economic charging power: can provide an EV with an output power (${\beta }^e$) ranging from $\overline{e} \%$ to $\underset{\_}{e} \%$ of the maximum output power $\beta$.

Once the charging power is determined, the vehicle will be charged using the selected power until the battery capacity reaches the required SOC.

2.2. Smart Charging Constraints

Charging power constraints specify limits of each charging power. Premium, regular, and economic charging powers are bounded as follows:

$$\underset{\_}{p}\cdot \beta \le {\beta }_t^p\le \overline{p}\cdot \beta$$

(1)

$$\underset{\_}{r}\cdot \beta \le {\beta }_t^r\le \overline{r}\cdot \beta$$

(2)

$$\underset{\_}{e}\cdot \beta \le {\beta }_t^e\le \overline{e}\cdot \beta$$

(3)

where ${\beta }_t^p$, ${\beta }_t^r$, and ${\beta }_t^e$ are, respectively, premium, regular, and economic charging powers at time $t$, $\beta$ is the maximum charging power per port, $\overline{p}$, $\overline{r}$, and $\overline{e}$ are factors that set the upper limits of premium, regular, and economic charging powers, and $\underset{\_}{p}$, $r$, and $\underset{\_}{e}$ are factors that set the lower limits.

Constraint (4) ensures that drawing power from the FCS at any instance in time is not zero during the charging event, whereas (5) ensures that the maximum output power of the charger is not exceeded. In addition, constraints (6)–(8) determine that the output of premium charging power is more than the output of regular charging power, which in turn is more than the output of the economic charging power.

$$\overline{p},\underset{\_}{p},\overline{r},\underset{\_}{r},\overline{e},\underset{\_}{e},>0$$

(4)

$$\overline{e}<\overline{r}<\overline{p}\le 1$$

(5)

$$\underset{\_}{p}\le \overline{p}$$

(6)

$$\underset{\_}{r}\le \overline{r}$$

(7)

$$\underset{\_}{e}\le \overline{e}$$

(8)

3. Cost of Smart Charging at FCSs

3.1. Cost of Flicker Mitigation Technologies

This section attempts to quantify and compare the costs of the various mitigation devices such as [28,29,30,31]: thyristor switched capacitors (TSC), fixed capacitors/thyristor controlled reactors (FC/TCR), distribution static compensator (DSTATCOM), dynamic voltage restorer (DVR), unified power quality conditioner (UPQC), and fixed series capacitor (FSC). For these technologies, there will be two costs involved. The first is the initial purchase price of the equipment, while the second is the maintenance costs associated with the selected equipment. The cost of power quality mitigation devices is determined based on their power rating such as $/kVar [32]. This cost represents a rough estimate of the equipment; however, the total system cost may contain additional elements [32]. The total annual equivalent cost of installing a voltage flicker mitigation device can be found in:

$${\Lambda }^y={ℂ}^y+{ℳ}^y$$

(9)

where ${\Lambda }^y$ is the total annual equivalent cost, $/year, ${ℂ}^y$ is the annual equivalent cost of capital invested, $/year, and ${ℳ}^y$ is the annual equivalent cost of maintenance, $/year. The annual equivalent capital cost is called capital recovery cost, ${ℂ}^y$.

$${ℂ}^y=\zeta \cdot {\mu }^{\prime }-\mathcal{S}\cdot {\ell }^{\prime }$$

(10)

$${\mu }^{\prime }=\left(\frac{\lambda \cdot {\left(\lambda +1\right)}^{\eta }}{{\left(\lambda +1\right)}^{\eta }-1}\right)$$

(11)

$${\ell }^{\prime }=\left(\frac{\lambda }{{\left(\lambda +1\right)}^{\eta }-1}\right)$$

(12)

$$\zeta =\Gamma \cdot \phi$$

(13)

$$\mathcal{S}=\Omega \cdot \phi$$

(14)

where $\zeta$ is the first cost of the installed flicker mitigation device ($), $\mathcal{S}$ is the estimated salvage value at the end of the device’s useful life ($), ${\mu }^{\prime }$ is the capital recovery factor, ${\ell }^{\prime }$ is the single-payment discount factor, $\lambda$ is the fixed charge rate (%), $\eta$ is the useful life in years (flicker mitigation device lifetime, years), $\Gamma$ is the cost per unit of the installed flicker mitigation device ($/kVAr), $\Omega$ is the salvage value per kVAr at the end of η ($/kVAr), and $\phi$ is the operating range of the flicker mitigation device ($\mathrm{MVAr}$). The cost of the installation ($\Gamma$) of different flicker mitigation equipment are given as follows [33,34,35,36,37]:

$${\Gamma }_{UPQC}=0.0003\cdot {\phi }_{UPQC}^2-0.2691\cdot {\phi }_{UPQC}+188.2$$

(15)

$${\Gamma }_{DSTATCOM}=0.0002478\cdot {\phi }_{DSTATCOM}^2-0.2261\cdot {\phi }_{DSTATCOM}+60$$

(16)

$${\Gamma }_{TSC}=0.0003\cdot {\phi }_{TSC}^2-0.305\cdot {\phi }_{TSC}+127.4$$

(17)

$${\Gamma }_{FC-TCR}=0.0003\cdot {\phi }_{TSC}^2-0.305\cdot {\phi }_{TSC}+127.4$$

(18)

$${\Gamma }_{DVR}=0.0015\cdot {\phi }_{DVR}^2-0.713\cdot {\phi }_{DVR}+153.8$$

(19)

$${\Gamma }_{FSC}=0.000541\cdot {\phi }_{FSC}^2-0.3902\cdot {\phi }_{FSC}+90.8$$

(20)

where

• ${\Gamma }_{UPQC}$: the per unit cost of the unified power quality conditioner, $\left(\$/\mathrm{kVAr}\right)$
• ${\Gamma }_{DSTATCOM}:$ the per unit cost of the distribution static compensator, $\left(\$/\mathrm{kVAr}\right)$
• ${\Gamma }_{TSC}:$ the per unit cost of the thyristor switched capacitor, $\left(\$/\mathrm{kVAr}\right)$
• ${\Gamma }_{FC-TCR}:$ the per unit cost of the fixed capacitors/thyristor-controlled reactors, $\left(\$/\mathrm{kVAr}\right)$
• ${\Gamma }_{DVR}$: the per unit cost of the dynamic voltage restorer, $\left(\$/\mathrm{kVAr}\right)$
• ${\Gamma }_{FCS}:$ the per unit cost of the fixed series capacitor, $\left(\$/\mathrm{kVAr}\right)$
• ${\phi }_{UPQC}:$ the operating range of the unified power quality conditioner, $\left(\mathrm{MVAr}\right)$
• ${\phi }_{DSTATCOM}:$ the operating range of the distribution static compensator, $\left(\mathrm{MVAr}\right)$
• ${\phi }_{TSC}:$ the operating range of the thyristor switched capacitor, $\left(\mathrm{MVAr}\right)$
• ${\phi }_{FCTCR}:$ the operating range of the fixed capacitors/thyristor-controlled reactors, $\left(\mathrm{MVAr}\right)$
• ${\phi }_{DVR}:$ the operating range of the dynamic voltage restorer, $\left(\mathrm{MVAr}\right)$
• ${\phi }_{FSC}:$ the operating range of the fixed series capacitor, $\left(\mathrm{MVAr}\right)$

The annual equivalent cost to maintain the flicker mitigation device (${ℳ}^y$) is calculated per kVAr per year, as in (21):

$${ℳ}^y=\gamma \cdot \Gamma \cdot \phi \cdot {\mu }^{\prime }$$

(21)

where, $\gamma$ is the maintenance cost in percent of the first cost, (%). $\lambda$ is the fixed interest rate at 6% and $\eta$ is the project lifetime of 10 years. If the salvage value is assumed to be negligible, the different mitigation methods are calculated, as in

Table 1. Maintenance cost for flicker mitigation devices.

Mitigation Device	% of the First Cost	The Source
${ℳ}_{UPQC}^y$	10	[38]
${ℳ}_{DSTATCOM}^y$	5	[39]
${ℳ}_{TSC}^y$	10	[40]
${ℳ}_{FC-TCR}^y$	10	[40]
${ℳ}_{DVR}^y$	5	[41]
${ℳ}_{FSC}^y$	1	[42]

.

3.2. Comparison of Costs of Flicker Mitigation Technologies

As per the general utility practice, the annual fixed charge rate is estimated based on the summation of the following costs: costs of capital, depreciation, taxes, insurance, and operation and maintenance expenses. However, only the capital and maintenance costs are considered in calculating the ${\Lambda }^y$ in

Table 2. Cost of voltage fluctuation mitigating devices.

	Mitigation Techniques
Parameters	UPQC	DSTATCOM	TSC	FC-TCR	DVR	FSC
Lifetime $\eta$, (years)	20	20	20	20	20	20
Charge rate $\lambda$, (%)	6	6	6	6	6	6
Maintenance cost in % of first cost $\gamma$, (%)	10	5	10	10	5	1
Reactive power range $\phi$, (MVAr)	2	2	2	2	2	2
The per kVAr cost $\Gamma$, ($/kVAr)	$187.66$	$79.55$	$126.8$	$126.8$	$153.3$	$89.97$
Salvage value per kVAr $\Omega$, ($/kVAr)	0	0	0	0	0	0
Cost of installation, $\zeta , \left(\$\right)$	$375,320$	$159,100$	$253,600$	$253,600$	$306,600$	$179,940$
Salvage at end of device lifetime, $\mathcal{S} (\$)$	0	0	0	0	0	0
capital recovery factor, ${\mu }^{\prime }$	0.0872	0.0872	0.0872	0.0872	0.0872	0.0872
Single-payment discount factor, ${\ell }^{\prime }$	0.0272	0.0272	0.0272	0.0272	0.0272	0.0272
Capital recovery cost, ${ℂ}^y$, $\left(\$/\mathrm{year}\right)$	32,728	13,874	22,114	22,114	26,735	15,691
Annual maintenance cost, ${ℳ}^y$, ($\$/\mathrm{year})$	3272	694	2211	2211	1337	157
Total annual equivalent cost, ${\Lambda }^y \left(\$/\mathrm{year}\right)$	36,000	14,568	24,325	24,325	28,072	15,848

. Additionally, the salvage value is assumed to be zero at the end of the project lifetimes.

Table 2 shows a range of costs for a number of the mitigation technologies discussed above. According to the comparison, the DSTATCOM is the cheapest mitigation technology on a cost per kVAr basis and is based on the total annual equivalent cost (14,568 $/year). In contrast, the UPQC has the highest annual cost and per kVAr cost.

3.3. Determining the per Unit Time of the Smart Charging

When a vehicle charges via a fast charger, the service is offered based on a fixed price per hour, and is billed by the second, according to [43]. In order to determine the cost of any charging power proposed in this study, i.e., premium, regular, or economic, it is required to first determine the time the charging power it takes to charge 1 kwh in the battery. Then, that cost ($/1 kwh) is multiplied by the total energy transferred (kwh). When the customer arrives at the FCS, two parameters are shared with the FCS: the initial and the required battery SOC. Therefore, given the required battery SOC and the relation of 1 kwh per minute, the total kwh required from the FCS can be calculated, and then the required time is determined. The times it takes to charge 1 kwh by premium, regular, and economic power are given as follows:

$${\mathcal{T}}^p=\frac{1\cdot \vartheta }{{\beta }^p}$$

(22)

$${\mathcal{T}}^r=\frac{1\cdot \vartheta }{{\beta }^r}$$

(23)

$${\mathcal{T}}^e=\frac{1\cdot \vartheta }{{\beta }^e}.$$

(24)

where ${\beta }^p$, ${\beta }^r$, and ${\beta }^e$ are premium, regular, and economic charging powers, respectively. The required time to charge 1 kwh is inversely proportional to rate of the charging power. As the rated power increases, the time required to charge 1 kwh decreases, and vice versa.

3.4. Determining the per Unit Cost of the Smart Charging

After the required time to charge 1 kwh per minute is found, the per unit cost of premium power (${\Gamma }^p$), regular power (${\Gamma }^r$), and economic power (${\Gamma }^e$) can be modeled as follows:

$${\Gamma }^p=\mathcal{C}\cdot {\mathcal{T}}^p.$$

(25)

$${\Gamma }^r={\Gamma }^p\cdot \frac{{\mathcal{T}}^p}{{\mathcal{T}}^r}$$

(26)

$${\Gamma }^e={\Gamma }^p\cdot \frac{{\mathcal{T}}^p}{{\mathcal{T}}^e}$$

(27)

$$\mathcal{C}=\frac{\mathfrak{h}}{\vartheta }$$

(28)

where $\vartheta$ is a factor that converts hours into minutes and $\mathfrak{h}$ and $\mathcal{C}$ are the per hour and per minute PBEV fast charging costs.

3.4.1. The per Unit Cost of the Premium Charging Power

In Equation (25), the per unit cost for the premium charging power (${\Gamma }^p$) is a function of two factors: the premium charging power (${\beta }^p$), where (${\Gamma }^p=\mathcal{C}\cdot {\mathcal{T}}^p=\frac{\mathcal{C}.\vartheta }{{\beta }^p}$), and the per hour charging cost ($\mathcal{C}$). ${\beta }^p$ does not change and represents the maximum rated power available in the charger. Charging the battery using the maximum rated power will minimize the duration of the charge, which, in turn, saves the customer time. The other factor is the per hour charging cost, which may change during the day (on- and off-peaks). When the per hour charging cost increases, the per unit cost of the premium charging power increases as well, and vice versa. The main advantage of using premium charging power is to get the maximum charging rate and reduce the charging duration but at the highest cost, as explained in Equation (22).

3.4.2. The per Unit Cost of the Regular Charging Power

The per unit cost of the regular charging power (${\Gamma }^r$) is modeled as in Equation (26). ${\Gamma }^r$ will not utilize the maximum charging rate, which adversely increases the charging duration. The per unit regular charging cost is a function of two factors: the per unit cost of premium charging power (${\Gamma }^p$) and a scaling factor ($\frac{{\mathcal{T}}^p}{{\mathcal{T}}^r}$). The scaling factor represents a rebate offered to the customer to adopt the regular charging power. The rebate is designed to compensate the increase in the charging duration by reducing the per unit regular charging cost. The per unit time of the premium power (${\mathcal{T}}^p$) is less than the per unit time of the regular power (${\mathcal{T}}^r$). Thus, the scaling factor ($\frac{{\mathcal{T}}^p}{{\mathcal{T}}^r}$) will scale down the per unit cost of the premium power (${\Gamma }^p$) in Equation (26), which will reduce the per unit cost of the regular charging power (${\Gamma }^r$). When the premium charging power is utilized, the customer will save time but not money. In contrast, when the regular charging power is utilized, the customer will save money but not time. The rebate factor is designed to save the per unit cost of the regular charging power by the same percent that the premium charging power saves the per unit charging time.

3.4.3. The per Unit Cost of the Economic Charging Power

The economic charging power has the lowest cost but the highest charging duration according to its per unit cost and per unit charging time, respectively. The per unit cost of the economic charging power (${\Gamma }^e$.) is modeled as in Equation (27). It is a function of two factors: the per unit cost of premium charging power (${\Gamma }^p$) and a scaling factor ($\frac{{\mathcal{T}}^p}{{\mathcal{T}}^e}$). ${\mathcal{T}}^e$ is more than ${\mathcal{T}}^r$ which makes the scaling factor in ${\Gamma }^e$ is less than that in ${\Gamma }^r$; thus, the economic charging power offers more rebate. In this charging power, the rebate is modeled to reduce the per unit charging cost by the same percent that would reduce the per unit charging time if the premium charging power is utilized.

3.5. Determining the Maximum Charging Cost

The maximum cost of charge (${ℂ}^{max}$) is estimated for any vehicle $\nu \in {\mathcal{V}}^{FCS}$ as in Equation (29). The difference between the maximum and minimum state-of-charge ($So{C}^{max}$) and ($So{C}^{min}$) is multiplied by the battery capacity ${\mu }^v$ as in (29) to find the required energy in kwh. The cost of the required energy is calculated according to the per unit cost of the utilized charging power (${\mathcal{T}}^{FCS}$).

$${ℂ}^{max}={\Gamma }^{FCS}\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(29)

$$So{C}^{max}=\overline{\ell }\cdot {\mu }^v$$

(30)

$$So{C}^{min}=\underset{\_}{\ell }\cdot {\mu }^v$$

(31)

$${\mu }^v≔\{\begin{array}{ll}{\mu }^s,& if \nu \in {\mathcal{V}}_S^{FCS}\\ {\mu }^m,& if \nu \in {\mathcal{V}}_m^{FCS}\\ {\mu }^l,& if \nu \in {\mathcal{V}}_l^{FCS}\end{array}$$

(32)

$${\mathcal{T}}^{FCS}≔\{\begin{array}{ll}{\Gamma }^p,& if \nu \in {\mathcal{V}}_{FCS,p}^v\\ {\Gamma }^r,& if \nu \in {\mathcal{V}}_{FCS,r}^v\\ {\Gamma }^e,& if \nu \in {\mathcal{V}}_{FCS,e}^v\end{array}$$

(33)

where

$${\mathcal{V}}_{FCS,p}^v=\left\{{\mathcal{V}}_S^{FCS,p}{\displaystyle \cup }{\mathcal{V}}_m^{FCS,p}{\displaystyle \cup }{\mathcal{V}}_l^{FCS,p}\right\}$$

(34)

$${\mathcal{V}}_{FCS,r}^v=\left\{{\mathcal{V}}_S^{FCS,r}{\displaystyle \cup }{\mathcal{V}}_m^{FCS,r}{\displaystyle \cup }{\mathcal{V}}_l^{FCS,r}\right\}$$

(35)

$${\mathcal{V}}_{FCS,e}^v=\left\{{\mathcal{V}}_S^{FCS,e}{\displaystyle \cup }{\mathcal{V}}_m^{FCS,e}{\displaystyle \cup }{\mathcal{V}}_l^{FCS,e}\right\}$$

(36)

where ${ℂ}^{max}$ is the maximum cost of charging a PBEV from an FCS ($), ${\Gamma }^{FCS}$ is the per unit cost of fast charging ($\frac{\$}{kwh}$), $So{C}^{max}$ is the maximum allowable SOC for any PBEV ($p.u.$), $So{C}^{min}$ is the minimum allowable SOC for any PBEV ($p.u.$), $\overline{\ell }$ is a percent to determine the maximum SOC ($p.u.$), and $\underset{\_}{\ell }$ is a percent to determine the minimum SOC ($p.u.$). ${\mathcal{V}}_{FCS,p}^v$, ${\mathcal{V}}_{FCS,r}^v$, and ${\mathcal{V}}_{FCS,e}^v$ are, respectively, sets of PBEVs charging by premium, regular, and economic charging powers. ${\mu }^s$, ${\mu }^m$, and ${\mu }^l$ are the capacities of a small, medium, and large battery of a vehicle $v$.

3.6. Determining the Maximum Charging Duration

When a vehicle $\nu \in {\mathcal{V}}^{FCS}$ arrives at the FCS with a minimum battery SOC ($So{C}^{min}$) and needs to charge the battery from the FCS to the maximum allowable limit ($So{C}^{max}$), this duration of the charge represents the maximum charging time if $\nu$ utilizes the maximum per unit charging power ${\mathcal{T}}^{FCS}$. The maximum duration of the charge is calculated in (37); it is a function of a battery capacity ${\mu }^v$ as in (32) and a type of charging power ${\mathcal{T}}^{FCS}$.

$${\mathbb{T}}^{max}={\mathcal{T}}^{FCS}\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(37)

$${\mathcal{T}}^{FCS}≔\{\begin{array}{ll}{\mathcal{T}}^p,& if \nu \in {\mathcal{V}}^p\\ {\mathcal{T}}^r,& if \nu \in {\mathcal{V}}^r\\ {\mathcal{T}}^e,& if \nu \in {\mathcal{V}}^e\end{array}$$

(38)

where ${\mathbb{T}}^{max}$ is the maximum time it takes a PBEV to be charged, from its minimum to its maximum state-of-charge, using FCS, $\left(min.\right)$, and ${\mathcal{T}}^{FCS}$ is the per unit time it takes to charge a PBEV, from its minimum to its maximum state-of-charge, using FCS, $\left(\min ./\mathrm{kwh}\right)$.

4. FCS Annual Energy Profile

4.1. FCS Data

The number of vehicles that charge from an FCS may be influenced, to a large extent, by the location of that FCS. Therefore, the number of charging events may vary from location to location, leading to a variation in their demands or impacts on the electric power grid. In order to estimate the demand of the FCS, multiple FCSs in different locations are considered to generate an averaged energy profile representing the demand of the FCS on the grid. In this section, real data from nine FCSs are extracted and utilized to estimate the cost of smart charging. The reason for considering the average of these nine FCSs is to reduce the distinct variations of their annual profiles due to their locations. These FCSs are in Canada and share the same characteristics, such as their rated power and the number of ports, but are located in different facilities such as workplaces, retail locations, and malls [44]. The typical data depicts vehicles’ energy consumption from the FCSs. Each energy demand profile consists of a number of charging events per each FCS distributed over a period of one year and the required energy demand in kwh from the FCS for each charging event.

4.2. Modelling of Per Event Energy Demand from the FCS

To construct the average of the FCS demand profile, five values from each FCS typical data are utilized. These values are the maximum value $u{l}_i^{max}$, the minimum value $u{l}_i^{min}$, and the three sample quartiles (${\underset{\_}{q}}_i,q_i,{\overline{q}}_i$), which represent the 25th, 50th, and 75th percentiles for the FCS $i$. Given a cumulative distribution function $F_i$ for the ith FCS, percentiles can be calculated using the inverse of the cumulative distribution function as follows:

$${\underset{\_}{q}}_i=q_{25}=F_i^{-1}\left(25\right)$$

(39)

$$q_i=q_{50}=F_i^{-1}\left(50\right)$$

(40)

$${\overline{q}}_i=q_{75}=F_i^{-1}\left(75\right)$$

(41)

$$u{l}_i^{max}=q_{100}=F_i^{-1}\left(100\right)$$

(42)

$$u{l}_i^{min}=q_1=F_i^{-1}\left(1\right)$$

(43)

where $i$ is index for FCS, $f$ is total number of considered FCSs, ${\underset{\_}{q}}_i$, $q_i$, and ${\overline{q}}_i$ are the 25th, 50th, and 75th percentiles of charging energy per charging event of the FCS $i$ in a year ($kwh$), and $u{l}_i^{max}$ and $u{l}_i^{min}$ are the maximum and minimum of charging energy levels, per charging event of the FCS $i$ in a year. The maximum values in the constructed FCS profile $U{L}^{max}$ is calculated by finding the average of all maximum values extracted from each FCS profile as shown in Equation (44). Similarly, the minimum value of each FCS profile is collected and then averaged to construct the minimum value ${\mathrm{UL}}^{max}$ in the constructed FCS profile as in Equation (45). The three sample quartiles ($\underset{\_}{Q},Q,\overline{Q}$) in the constructed FCS profile are determined by collecting the first, second, and third quartiles of each FCS and then independently finding their average as in Equations (46)–(48) below:

$$U{L}^{max}=\frac{{{\displaystyle \sum }}_{i=1}^{f}u{l}_i^{max}}{f}$$

(44)

$$L{L}^{min}=\frac{{{\displaystyle \sum }}_{i=1}^{f}l{l}_i^{min}}{f}$$

(45)

$$\underset{\_}{Q}=\frac{{{\displaystyle \sum }}_{i=1}^f{\underset{\_}{q}}_i}{f}$$

(46)

$$Q=\frac{{{\displaystyle \sum }}_{i=1}^f{q}_i}{f}$$

(47)

$$\overline{Q}=\frac{{{\displaystyle \sum }}_{i=1}^f{\overline{q}}_i}{f}$$

(48)

where $\underset{\_}{Q}$, $Q$, and $\overline{Q}$ are the average value of the first, second, and third quartiles of charging energy per charging event in a year, for $f$ number of FCSs ($\frac{kwh}{event}$), and $u{l}_i^{max}$ and $u{l}_i^{min}$ are the maximum and minimum of charging energy levels per charging event in a year for $f$ number of FCSs ($\frac{kwh}{event}$).

4.3. Modelling of Annual Energy Demand from the FCS

In this section, annual energy consumption from the FCS profile is determined and utilized to estimate the cost of smart charging. The annual energy demand from the FCS is calculated using two factors: the kwh per charging event and the average annual total number of charging events ($\Xi$). The kwh per charging events is calculated in the previous section and it consists of five values: $\underset{\_}{Q},Q,\overline{Q},{\mathrm{UL}}^{max},\mathrm{and} {\mathrm{UL}}^{min}$. The annual number of charging events at an FCS is influenced by the FCS’s location. Therefore, the annual number of charging events is represented by taking the average of all the annual charging events at all nine FCSs. The average annual total number of charging events is calculated as follows:

$$\Xi =\frac{{{\displaystyle \sum }}_{i=1}^f{\epsilon }_i }{f}$$

(49)

where $\Xi$ is the average number of annual charging events in $f$ number of FCSs ($\frac{event}{year}$) and ${\epsilon }_i$ is the number of annual charging events in the FCS $i$. After the annual number of charging events and the energy demand for each charging event are determined, the annual energy demand from the FCS is estimated as follows:

$$\underset{\_}{\mho }=\underset{\_}{Q}\cdot \Xi$$

(50)

$$\mho =Q\cdot \Xi$$

(51)

$$\overline{\mho }=\overline{Q}\cdot \Xi$$

(52)

$${\mho }_{max}=U{L}^{max}\cdot \Xi$$

(53)

$${\mho }_{min}=L{L}^{min}\cdot \Xi$$

(54)

where $\underset{\_}{\mho }$, $\mho$, and $\overline{\mho }$ are the estimated 25th, 50th, and 75th percentile values of annual charging energy for an FCS ($\frac{kwh}{year}$), whereas $u{l}_i^{max}$ and $u{l}_i^{min}$ are the are the estimated maximum and minimum values of annual charging energy for an FCS ($\frac{kwh}{year}$). The meaning of ${\mho }^{max}$, for instance, is illustrated as follows: given the annual total number of charging events $\Xi \left(event\right)$, if ${\mathrm{UL}}^{max}$ ($\frac{kwh}{event}$) is the maximum kwh required from an FCS per any given charging event, then the total maximum kwh required from that FCS in that year is ${\mho }^{max}$ and equals ${\mathrm{UL}}^{max}.\Xi$. The value of ${\mho }_{max}$ represents the total annual demand from the FCS if all vehicles require ${\mathrm{UL}}^{max}$ for each charging event that occurs in that year.

5. Computing the Cost of Smart Charging

The annual cost of the proposed smart charging in this thesis can be divided into the rebate paid to customers for using regular and economic charging powers, and the revenue for energy sold to customers when using the FCS.

5.1. Computing the Rebates

In this section, the rebate of using smart charging is computed. The proposed rebate is assumed to be instant, and it is a function of the duration of the charge. When a customer uses the FCS, the customer can choose one of three charging powers: premium, regular, or economic charging power. When the premium charging power is chosen, the customer would not get a rebate. The rebate is paid instantly upon using regular or economic charging power. The annual rebate is computed as follows:

$${\underset{\_}{\complement }}^{Reb}=\underset{\_}{\mho }\cdot \left[\left(𝓇\cdot \left({\Gamma }^p-{\Gamma }^r\right)\right)+\left(\left(1-𝓇\right)\cdot \left({\Gamma }^p-{\Gamma }^e\right)\right)\right]\cdot \frac{{\eta }^{sm}}{100}$$

(55)

$${\complement }^{Reb}=\mho \cdot \left[\left(𝓇\cdot \left({\Gamma }^p-{\Gamma }^r\right)\right)+\left(\left(1-𝓇\right)\cdot \left({\Gamma }^p-{\Gamma }^e\right)\right)\right]\cdot \frac{{\eta }^{sm}}{100}$$

(56)

$${\overline{\complement }}^{Reb}=\overline{\mho }\cdot \left[\left(𝓇\cdot \left({\Gamma }^p-{\Gamma }^r\right)\right)+\left(\left(1-𝓇\right)\cdot \left({\Gamma }^p-{\Gamma }^e\right)\right)\right]\cdot \frac{{\eta }^{sm}}{100}$$

(57)

$${\complement }_{max}^{Reb}={\mho }_{max}\cdot \left[\left(𝓇\cdot \left({\Gamma }^p-{\Gamma }^r\right)\right)+\left(\left(1-𝓇\right)\cdot \left({\Gamma }^p-{\Gamma }^e\right)\right)\right]\cdot \frac{{\eta }^{sm}}{100}$$

(58)

$${\complement }_{min}^{Reb}={\mho }_{min}\cdot \left[\left(𝓇\cdot \left({\Gamma }^p-{\Gamma }^r\right)\right)+\left(\left(1-𝓇\right)\cdot \left({\Gamma }^p-{\Gamma }^e\right)\right)\right]\cdot \frac{{\eta }^{sm}}{100}$$

(59)

where $𝓇$ is a uniform random number, ${\eta }^{sm}$ is the share of PBEV that utilizes smart charging power at the FCS, ${\underset{\_}{\complement }}^{Reb}$, ${\complement }^{Reb}$, and ${\overline{\complement }}^{Reb}$ are the 25th, 50th, and 75th percentile annual rebates paid to the customer for using smart charging power ($\frac{\$}{year}$), and ${\complement }_{max}^{Reb}$ and ${\complement }_{min}^{Reb}$ are the maximum and minimum annual rebates paid to customers for using smart charging power ($\frac{kwh}{year}$). If ${\Gamma }^r={\Gamma }^e={\Gamma }^p$ in Equations (55)–(59), then ${\eta }^{sm}=0$. It means that all customers utilize the premium charging power because the share of customers using regular and economic charging powers (${\eta }^{sm}$) is zero. Therefore, the (maximum, minimum, 25th percentile, 50th percentile, 75th percentile) annual rebates in Equations (55)–(59) are zero. In that case, the FCS operator would not pay any rebate to customers and the annual cost is represented only by revenue from selling the energy to customers who use the premium charging power. If, in Equations (50)–(54), ${\Gamma }^r\ne {\Gamma }^e$ and/or ${\Gamma }^e\ne {\Gamma }^p$, then ${\eta }^{sm}\ne 0$. It means that regular and/or economic charging power have been utilized in that year. The share of customers utilizing these two charging powers (regular and economic) is determined by the factor ${\eta }^{sm}$. Multiple smart charging penetration levels (${\eta }^{sm}$) are considered in this regard. The penetration level is varied from 0% up to 100% by 10%. The factor $𝓇$, in Equations (55)–(59), is utilized to determine the percent of customers who used regular charging power and the percent of customers who used economic charging power. The factor $𝓇$ is randomly generated using a uniform random number and repeated multiple times. In Equation (50), for instance, when $𝓇=1$, it means that the share of vehicles that use the regular charging power is ${\eta }^{sm}$; thus, the economic charging power has not been utilized. In contrast, when $𝓇=0$, it means that share of vehicles that use the economic charging power is ${\eta }^{sm}$; thus, the regular charging power has not been utilized. When $𝓇=0.5$, it means that 50% of ${\eta }^{sm}$ uses the regular charging power while the economic charging power is utilized by the other 50% of ${\eta }^{sm}$. Therefore, $𝓇$ is a shaping factor utilized to assign, based on the penetration level of ${\eta }^{sm}$, the percent of vehicles that use the regular charging power and the percent of vehicles that use the economic charging power. Equations (55)–(59) can be written in a long form as follows:

$${\underset{\_}{\complement }}^{Reb}=\left[\frac{1}{f}\cdot {\displaystyle \sum }_{i=1}^f{\underset{\_}{q}}_i\cdot {\epsilon }_i\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\left[\left(𝓇\cdot \left(1-\frac{{\beta }^r}{{\beta }^p}\right)\right)+\left(\left(1-𝓇\right)\cdot \left(1-\frac{{\beta }^e}{{\beta }^p}\right)\right)\right]\left[\left(\frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(60)

$${\complement }^{Reb}=\left[\frac{1}{f}\cdot {\displaystyle \sum }_{i=1}^f{q}_i\cdot {\epsilon }_i\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\left[\left(𝓇\cdot \left(1-\frac{{\beta }^r}{{\beta }^p}\right)\right)+\left(\left(1-𝓇\right)\cdot \left(1-\frac{{\beta }^e}{{\beta }^p}\right)\right)\right]\left[\left(\frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(61)

$${\overline{\complement }}^{Reb}=\left[\frac{1}{f}\cdot {\displaystyle \sum }_{i=1}^f{\overline{q}}_i\cdot {\epsilon }_i\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\left[\left(𝓇\cdot \left(1-\frac{{\beta }^r}{{\beta }^p}\right)\right)+\left(\left(1-𝓇\right)\cdot \left(1-\frac{{\beta }^e}{{\beta }^p}\right)\right)\right]\left[\left(\frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(62)

$${\complement }_{max}^{Reb}=\left[\frac{1}{f}\cdot {\displaystyle \sum }_{i=1}^{f}u{l}_i^{max}\cdot {\epsilon }_i\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\left[\left(𝓇\cdot \left(1-\frac{{\beta }^r}{{\beta }^p}\right)\right)+\left(\left(1-𝓇\right)\cdot \left(1-\frac{{\beta }^e}{{\beta }^p}\right)\right)\right]\left[\left(\frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(63)

$${\complement }_{min}^{Reb}=\left[\frac{1}{f}\cdot {\displaystyle \sum }_{i=1}^{f}u{l}_i^{min}\cdot {\epsilon }_i\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\left[\left(𝓇\cdot \left(1-\frac{{\beta }^r}{{\beta }^p}\right)\right)+\left(\left(1-𝓇\right)\cdot \left(1-\frac{{\beta }^e}{{\beta }^p}\right)\right)\right]\left[\left(\frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(64)

5.2. Computing the Revenue

The revenue of selling energy to vehicles from the FCS is determined in this section. The FCS makes revenue when customers utilize it to charge their vehicles, regardless of which charging power is utilized. There are three charging powers available in the FCS: premium, regular, and economic charging power. The revenue is a function of the charging duration. Therefore, annual charging energy is utilized to estimate annual revenues. The annual revenue is computed as follows:

$${\underset{\_}{ℛ}}^{Rev}=\underset{\_}{\mho }\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\cdot {\Gamma }^p\right)}{100}\right)+\left(\left(\left(𝓇\cdot {\Gamma }^r\right)+\left(\left(1-𝓇\right)\cdot {\Gamma }^e\right)\right).\frac{{\eta }^{sm}}{100}\right)\right]$$

(65)

$${ℛ}^{Rev}=\mho \cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\cdot {\Gamma }^p\right)}{100}\right)+\left(\left(\left(𝓇\cdot {\Gamma }^r\right)+\left(\left(1-𝓇\right)\cdot {\Gamma }^e\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]$$

(66)

$${\overline{ℛ}}^{Rev}=\overline{\mho }\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\cdot {\Gamma }^p\right)}{100}\right)+\left(\left(\left(𝓇\cdot {\Gamma }^r\right)+\left(\left(1-𝓇\right)\cdot {\Gamma }^e\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]$$

(67)

$${ℛ}_{max}^{Rev}={\mho }_{max}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\cdot {\Gamma }^p\right)}{100}\right)+\left(\left(\left(𝓇\cdot {\Gamma }^r\right)+\left(\left(1-𝓇\right)\cdot {\Gamma }^e\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]$$

(68)

$${ℛ}_{min}^{Rev}={\mho }_{min}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\cdot {\Gamma }^p\right)}{100}\right)+\left(\left(\left(𝓇\cdot {\Gamma }^r\right)+\left(\left(1-𝓇\right)\cdot {\Gamma }^e\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]$$

(69)

where ${\underset{\_}{ℛ}}^{Rev}$, ${ℛ}^{Rev}$, and ${\overline{ℛ}}^{Rev}$ are the 25th, 50th, and 75th percentile annual revenues from customers for using smart charging power ($\frac{\$}{year}$), whereas ${ℛ}_{max}^{Rev}$ and ${ℛ}_{min}^{Rev}$ are the maximum and minimum annual revenues from customers for using smart charging power ($\frac{kwh}{year}$). In Equation (65), for example, ${\eta }^{sm}$ represents the share of vehicles that utilize smart charging (regular and economic charging powers). If ${\eta }^{sm}=0$, it means that only premium charging power was utilized, and the annual revenue is calculated accordingly. It also means that the annual revenue of using regular or economic charging powers is zero. In contrast, when ${\eta }^{sm}=100$, revenue of using premium charging power is zero because the premium charging power was not utilized by any vehicles. Different penetration levels of ${\eta }^{sm}$ are considered to compute annual revenue of the FCS. In Equation (65), for instance, the factor $𝓇$ is used to define the share of customers that use regular charging power from those who use economic charging power. The factor $𝓇$ is generated using a uniform random number and it is effective only when ${\eta }^{sm}>0$. The function of $𝓇$ in Equation (65) is as illustrated in the previous section. For instance, when ${\eta }^{sm}=10\%$ and $𝓇=0.3$, according to Equation (65), it means that:

• The percent of vehicles that use premium charging power is $100-{\eta }^{sm}=90\%$,
• The percent of vehicles that use regular charging power is $𝓇\cdot {\eta }^{sm}=0.3\ast 10\%=3\%$,
• The percent of vehicles that use premium charging power is $\left(1-𝓇\right)\cdot {\eta }^{sm}=0.7\ast 10\%=7\%$.

Therefore, the (maximum, minimum, 25th percentile, 50th percentile, and 75th percentile) annual revenue, in Equations (65)–(69), is determined according to deterministic and stochastic factors:

• The deterministic factor is penetration level ${\eta }^{sm}$, which is varied from 0% up to 100% by 10%.
• The stochastic factor is $𝓇$, which is generated using a uniform distribution.
• The annual charging energy ($\underset{\_}{\mho }, \mho , \overline{\mho }, {\mho }_{max}, {\mho }_{min}$) is another stochastic factor which represents the vehicle demand from the FCS per year.

Equations (65)–(69) can be written in a long form as follows:

$${\underset{\_}{ℛ}}^{Rev}=\left[\frac{1}{f}{\displaystyle \sum }_{i=1}^f\left({\underset{\_}{q}}_i\cdot {\epsilon }_i\right)\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\right)}{100}\right)+\left(\left(\left(𝓇\cdot \frac{{\beta }^r}{{\beta }^p}\right)+\left(\left(1-𝓇\right)\cdot \frac{{\beta }^e}{{\beta }^p}\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(70)

$${ℛ}^{Rev}=\left[\frac{1}{f}{\displaystyle \sum }_{i=1}^f\left(q_i\cdot {\epsilon }_i\right)\right]\left[\frac{\mathcal{C}\cdot \vartheta }{{\beta }^p}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\right)}{100}\right)+\left(\left(\left(𝓇\cdot \frac{{\beta }^r}{{\beta }^p}\right)+\left(\left(1-𝓇\right)\cdot \frac{{\beta }^e}{{\beta }^p}\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(71)

$${\overline{ℛ}}^{Rev}=\left[\frac{1}{f}{\displaystyle \sum }_{i=1}^f\left({\overline{q}}_i\cdot {\epsilon }_i\right)\right]\left[\frac{\mathcal{C}\cdot \vartheta }{{\beta }^p}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\right)}{100}\right)+\left(\left(\left(𝓇\cdot \frac{{\beta }^r}{{\beta }^p}\right)+\left(\left(1-𝓇\right)\cdot \frac{{\beta }^e}{{\beta }^p}\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(72)

$${ℛ}_{max}^{Rev}=\left[\frac{1}{f}{\displaystyle \sum }_{i=1}^f\left(u{l}_i^{max}\cdot {\epsilon }_i\right)\right]\left[\frac{\mathcal{C}.\vartheta }{{\beta }^p}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\right)}{100}\right)+\left(\left(\left(𝓇\cdot \frac{{\beta }^r}{{\beta }^p}\right)+\left(\left(1-𝓇\right)\cdot \frac{{\beta }^e}{{\beta }^p}\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(73)

$${ℛ}_{min}^{Rev}=\left[\frac{1}{f}{\displaystyle \sum }_{i=1}^f\left(u{l}_i^{min}\cdot {\epsilon }_i\right)\right]\left[\frac{\mathcal{C}\cdot \vartheta }{{\beta }^p}\cdot \left[\left(\frac{\left(\left(100-{\eta }^{sm}\right)\right)}{100}\right)+\left(\left(\left(𝓇\cdot \frac{{\beta }^r}{{\beta }^p}\right)+\left(\left(1-𝓇\right)\cdot \frac{{\beta }^e}{{\beta }^p}\right)\right)\cdot \frac{{\eta }^{sm}}{100}\right)\right]\right]$$

(74)

5.3. Comparison of Rebate and Revenue

In this section, the ratio of revenue to rebate per maximum charging duration using regular or economic charging power is explained. When a customer charges a vehicle at the FCS, this makes revenue for the FCS. A rebate is given for the charging event if that customer uses regular or economic charging power. Therefore, the per event charging cost can be utilized to estimate the revenue and rebate given to customers if they charge their vehicles via regular or economic charging power. The maximum cost of charge was determined previously as follows:

$${ℛ}^{Rev,p}={\Gamma }^p\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{mic}\right)$$

(75)

$${ℛ}^{Rev,r}={\Gamma }^r\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(76)

$${ℛ}^{Rev,e}={\Gamma }^e\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(77)

The maximum rebate per charge is estimated similarly as follows:

$${\complement }^{Reb,p}=\left({\Gamma }^p-{\Gamma }^p\right)\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(78)

$${\complement }^{Reb,r}=\left({\Gamma }^p-{\Gamma }^r\right)\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(79)

$${\complement }^{Reb,e}=\left({\Gamma }^p-{\Gamma }^e\right)\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)$$

(80)

The ratio of the revenue to the rebate of the regular charging power, for example, per the maximum charging duration is estimated using Equations (76) and (79) as follows:

$$\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}=\frac{{\Gamma }^r\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)}{\left({\Gamma }^p-{\Gamma }^r\right)\cdot {\mu }^v\cdot \left(So{C}^{max}-So{C}^{min}\right)}=\frac{{\Gamma }^r}{\left({\Gamma }^p-{\Gamma }^r\right)}$$

(81)

where ${ℛ}^{Rev,r}$ and ${\complement }^{Reb,r}$ are the revenue and rebate from or paid to a customer when the regular charging power is utilized ($). From Equation (26), the per unit cost of the regular charging power ${\Gamma }^r$, is:

$${\Gamma }^r={\Gamma }^p\cdot \frac{{\mathcal{T}}^p}{{\mathcal{T}}^r}$$

(82)

Substituting (82) into (81), we get:

$$\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}=\frac{{\Gamma }^p\cdot \frac{{\mathcal{T}}^p}{{\mathcal{T}}^r}}{\left(\mathcal{C}\cdot {\mathcal{T}}^p-{\Gamma }^p\cdot \frac{{\mathcal{T}}^p}{{\mathcal{T}}^r}\right)}=\frac{{\Gamma }^p\cdot {\mathcal{T}}^p}{\left(\mathcal{C}\cdot {\mathcal{T}}^p\cdot {\mathcal{T}}^r-{\Gamma }^p\cdot \frac{{\mathcal{T}}^p\cdot {\mathcal{T}}^r}{{\mathcal{T}}^r}\right)}=\frac{{\Gamma }^p}{\left(\mathcal{C}\cdot {\mathcal{T}}^r-{\Gamma }^p\right)}$$

(83)

Substituting Equation (25), the per unit cost of the premium charging power ${\Gamma }^p$, into (83), we get:

$$\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}=\frac{\mathcal{C}\cdot {\mathcal{T}}^p}{\left(\mathcal{C}\cdot {\mathcal{T}}^r-\mathcal{C}\cdot {\mathcal{T}}^p\right)}=\frac{{\mathcal{T}}^p}{{\mathcal{T}}^r-{\mathcal{T}}^p}$$

(84)

From Equations (22) and (23), the per unit time of the premium and regular charging powers ${\mathcal{T}}^p$ and ${\mathcal{T}}^r$ are:

$$\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}=\frac{\frac{\vartheta }{{\beta }^p}}{\left(\frac{\vartheta }{{\beta }^r}-\frac{\vartheta }{{\beta }^p}\right)}=\frac{1}{\left(\frac{{\beta }^p}{{\beta }^r}-\frac{{\beta }^p}{{\beta }^p}\right)}=\frac{1}{\left(\frac{{\beta }^p}{{\beta }^r}-1\right)}$$

(85)

From Equations (1) and (2), the premium and regular charging powers ${\beta }^p$ and ${\beta }^r$ are functions of the maximum rated power of the charger $\beta$, and are bounded as follows:

$$\underset{\_}{p}\cdot \beta \le {\beta }^p\le \overline{p}\cdot \beta$$

(86)

$$\underset{\_}{r}\cdot \beta \le {\beta }^r\le \overline{r}\cdot \beta$$

(87)

In addition, from Equations (4) and (5), we have:

$$\overline{p}\le 1$$

(88)

$$\underset{\_}{p},\overline{p}>\overline{r}$$

(89)

Therefore,

$${\beta }^p>{\beta }^r$$

(90)

If

$$p^{op}\cdot \beta \in \left[\underset{\_}{p}\cdot \beta ,\overline{p}\cdot \beta \right]$$

(91)

$$r^{op}\cdot \beta \in \left[\underset{\_}{r}\cdot \beta ,\overline{r}\cdot \beta \right]$$

(92)

then

$$p^{op}>r^{op}$$

(93)

$$p^{op},r^{op}\in \left(0,1\right]$$

(94)

where $p^{op}$ and $r^{op}$ are optimal factors that set the limit of premium and regular charging powers. Substituting Equations (91) and (92) in Equation (85), we get:

$$\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}=\frac{1}{\left(\frac{p^{op}\cdot \beta }{r^{op}\cdot \beta }-1\right)}=\frac{1}{\left(\frac{p^{op}}{r^{op}}-1\right)}$$

(95)

Equation (95) means that, as the charging duration from the FCS increases, the rebate increases as well; however, the revenue increases much more by a factor of $\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}$. This factor $\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}$ is shaped by the ratio of $\frac{p^{op}}{r^{op}}$. The rebate is increased when the rated power of regular charging is decreased. The rated power of the regular charging is controlled by $r^{op}$, as Equations in (87) and (92). Therefore, decreasing $r^{op}$ results in an increase in the ratio of $\frac{p^{op}}{r^{op}},$($p^{op}\gg r^{op}$) making the $\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}$ decrease. In contrast, when the $r^{op}$ is increased it results in a reduction in the ratio of $\frac{p^{op}}{r^{op}},$($p^{op}\approx r^{op}$), which makes the $\frac{{ℛ}^{Rev,r}}{{\complement }^{Reb,r}}$ in (95) increase. Similarly, the optimal factor sets the limit of economic charging power, $e^{op}$, as follows:

$$e^{op}\cdot \beta \in \left[\underset{\_}{e}\cdot \beta ,\overline{e}\cdot \beta \right]$$

(96)

$$p^{op}>e^{op}$$

(97)

Then, the ratio of the revenue to the rebate of the economic charging power per the maximum charging duration is estimated as follows:

$$\frac{{ℛ}^{Rev,e}}{{\complement }^{Reb,e}}=\frac{1}{\left(\frac{p^{op}}{e^{op}}-1\right)}$$

(98)

where ${ℛ}^{Rev,e}$ and ${\complement }^{Reb,e}$ are the revenue and rebate from or paid to a customer when the economic charging power is utilized ($).

6. Results and Discussions

6.1. Per Unit Time and per Unit Cost of the Smart Charging

Figure 1 and Figure 2 depict the required time and cost to charge vehicles with 1 kwh and 25 kwh using premium, regular, and economic charging powers. It is noted that premium charging power achieves the shortest time to charge a vehicle with 1 kwh, but it has the highest per unit charging cost. Additionally, it is noted that regular charging power requires less time to charge 1 kwh but a higher cost than economic power. The proposed smart charging method preserves a balance between the required time and cost to charge 1 kwh by the selected charging power. For example, referring to Figure 1, the required time to charge 1 kwh for premium power is 1.333 min whereas it is 1.447 min for regular power. The proposed cost to charge 1 kwh by premium power is 0.3777 $ whereas it is a 0.347 $ to charge by regular power. The ratio of time required by premium power to the time required by regular power is $\frac{1.333}{1.447}=0.92$, whereas the ratio of the cost incurred by using premium power to regular power is $\frac{0.377}{0.347}=1.08$. This means that when premium power is utilized, the required time to charge 1 kwh is decreased by 8% in comparison to the required time to charge the same energy by regular power. However, the incurred cost when premium power is utilized is increased by 8% in comparison to the incurred cost when the regular charging power is utilized. Similarly, the ratio of premium power to regular is found to be $\pm$13%. It is obvious that the proposed smart charging method preserves a balance between the time and cost to charge 1 kwh with different charging powers.

6.2. Effect of Charging Power on Charging Duration and Charging Cost

The maximum charging duration and the corresponding charging costs are determined based on the following assumptions:

• The minimum state-of-charge of vehicles is $So{C}^{min}=0.2p.u.$ [45,46].
• Given the battery safety considerations, the maximum state-of-charge using FCS is $So{C}^{max}=0.8p.u.$, as in [47].
• Given $So{C}^{max}$ and $So{C}^{min}$, the maximum charging duration is determined as in Equation (37).
• The charging efficiency is 90%, as in [48,49].
• The Nissan Leaf is utilized to represent a vehicle with a small battery capacity, ${\mu }^s=45\mathrm{kwh}$.
• The Chevy Bolt is utilized to represent a vehicle with a small battery capacity, ${\mu }^m=66\mathrm{kwh}$.
• The Tesla Model S is utilized to represent a vehicle with a small battery capacity, ${\mu }^l=110\mathrm{kwh}$.

The maximum per unit charging duration is determined by taking the difference between $So{C}^{max}$ and $So{C}^{min}$. Given the battery capacity ${\mu }^s$, ${\mu }^m$, or ${\mu }^l$ and the charging type ${\Gamma }^p$, ${\Gamma }^r$, or ${\Gamma }^e$, maximum charging durations and costs are obtained as in (29) and (37). Figure 3, Figure 4 and Figure 5 visualize the effect of the proposed smart charging method on vehicles with different battery capacities. It is noted that the charging duration and charging costs increase when the battery capacity increases.

6.3. Descriptive Statistics of Annual Data for Multiple FCSs

The per event annual energy profile for FCSs are obtained as in [44] and are described by maximum value, minimum value, and 25th, 50th, and 75th percentiles, as presented in Figure 6. The annual number of charging events occurs in FCSs, as presented in Figure 7. A clear variation among these annual profiles is depicted in Figure 6 and Figure 7.

6.4. Average Annual Energy Required from FCS

Given the per event energy profiles and the annual charging events for FCSs, the average energy profile and the average number of charging events for FCSs are obtained, as in Equations (44)–(49). Figure 8 depicts the per event energy profiles using the following values: the maximum and the minimum values as well as the three quartiles $(Q_1=25\mathrm{th},Q_2=\mathrm{median},$ and $Q_3=75\mathrm{th})$ percentiles. As explained previously, the average number of charging events ($\Xi$) represents the annual number of charging events that may occur at the FCS. Given $\Xi$, the energy required by vehicles for each charging event from the FCS is not determined.

In order to determine the annual energy required from the FCS in both extreme directions, the maximum (${\mathrm{UL}}^{max}$) and minimum (${\mathrm{UL}}^{min}$) values are utilized. The percentiles are utilized to quantitatively describe the annual energy required from the FCS. The average annual energy required from the FCS is obtained, as shown in Equations (50)–(54).

The reason for generating the average energy required from the FCS is to compute the annual costs of rebate and revenue. The annual cost of the rebate is a function of the smart charging penetration levels. Given the annual energy required from the FCS, different penetration levels of Ξ are considered, as presented in

Table 3. Maximum, minimum, and quantile annual energy required by different penetration levels of the proposed smart charging.

		Smart Charging Penetration Levels
		0%	10%	20%	30%	40%	50%	60%	70%	80%	90%	100%
Annual Energy Required in kwh	${\mathrm{UL}}^{min}$	0	312.5	625	937.5	1250	1562.5	1875	2187.5	2500	2812.5	3125
	$\underset{\_}{\mho }$	0	1812.5	3625	5437.5	7250	9062.5	10,875	12,687	14,500	16,312	18,125
	$\mho$	0	2887.5	5775	8662.5	11,550	14,437	17,325	20,212	23,100	25,987	28,875
	$\overline{\mho }$	0	4187.5	8375	12,562	16,750	20,937	25,125	29,312	33,500	37,687	41,875
	${\mathrm{UL}}^{max}$	0	8187.5	16,375	24,562	32,750	40,937	49,125	57,312	65,500	73,687	81,875

.

6.5. Fast Charging Station Annual Revenue and Rebate

Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 present the annual rebate and revenue considering the different charging energy required from the FCS:

• The minimum annual charging energy (Figure 9) obtained as in (54);
• The 25th percentile annual charging energy (Figure 10) obtained as in (50);
• The 50th percentile annual charging energy (Figure 11) obtained as in (51);
• The 75th percentile annual charging energy (Figure 12) obtained as in (52);
• The maximum charging energy (Figure 13) obtained as in (53).

In each figure, the rebate represents the annual cost paid to the customer for using smart charging with regular and/or economic charging power. The annual rebate is shown in the right y-axis of each figure. The revenue is shown in the left y-axis of each figure and represents the costs obtained from selling the energy to customers when customers charge via premium ${\beta }^p$, regular ${\beta }^r$, and/or economic ${\beta }^e$ charging power. Each bar represents a result for a specific smart charging penetration level. For example, results of the first bar in Figure 9 is obtained as follows:

• The minimum annual charging energy as shown in Figure 8 is $\underset{\_}{Q}=2.461\mathrm{kwh}/event$. The average annual charging energy as presented in Figure 8 is $\Xi =1250event$;
• Smart charging penetration is $the$
• The costs of premium, regular, and economic charging powers as shown in Figure 1 are ${\Gamma }^p=0.378\$/\mathrm{kwh}$, ${\Gamma }^r=0.348\$/\mathrm{kwh}$, and ${\Gamma }^e=0.329\mathrm{thewh}$;
• The uniform random number $𝓇$ determines the share of regular and economic power for each penetration level;
• The optimal factors for the premium $p^{op}$, regular $r^{op}$, and economic $e^{op}$ power are 1, 0.925 and 0.875.

then,

• the average minimum annual charging energy required from the FCS is $\underset{\_}{Q}.\Xi =3076.25\mathrm{kwh}$,
• the annual rebate and revenue are calculated as in Equations (55) and (65),
• the revenue of the smart charging is zero because ${\eta }^{sm}=0$, as well as the rebate ${\underset{\_}{\complement }}^{Reb}=0$,
• thus, the revenue ${\underset{\_}{ℛ}}^{Rev}$ is calculated from selling the energy from using the premium power only, ${\underset{\_}{ℛ}}^{Rev}=3076.25\ast 0.378=1162.82\$/\mathrm{year}$,
• the ratio of the revenue to the rebate of using the smart charging is 9 and obtained by averaging optimal factors, $r^{op}$ and $e^{op}$, and substituting the average into Equations (95) or (98).

By increasing ${\eta }^{sm}$, the rest of the bars in Figure 9 can be obtained. When ${\eta }^{sm}\ne 0$, the revenue and rebate of using regular and economic charging power are determined by randomly assigned $𝓇$. The revenue and rebate of using smart charging are assigned by taking their average, individually, after multiple iterations (350 iterations) as recommended in [46]. From Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, it is noted that:

• The annual revenue changes slightly as the smart charging penetration levels increases. This means that as the revenue of using the premium power decreases; it is compensated by using the regular and economic charging powers. Therefore, the proposed smart charging method preserves the annual revenue.
• The annual revenue is increased as the energy required from the FCS is increased (i.e., increased from $\underset{\_}{\mho }$ to $\overline{\mho }$ as presented in Figure 9 and Figure 13).
• The rebate is increased as the smart charging penetration levels is increased (i.e., ${\eta }^{sm}$ increased considering the same in Figure 9).
• The amount of the annual rebate is increased as the annual energy required from the FCS is increased (i.e., increased from ${\mho }_{min}$ to ${\mho }_{max}$ as shown in Figure 9 and Figure 13).

6.6. Cost Comparison of the Proposed Smart Charging Method and DSTATCOM

As per the utility general standard practice, if flicker complaints result from the operation of the customer’s equipment, the customer’s equipment shall be disconnected. From the viewpoint of the investor, the main benefit of flicker mitigation is to avoid the service disconnection from the facility. From the viewpoint of the operator, the benefit is to protect their equipment as well as nearby customers. When the operation of fast charging stations causes a voltage fluctuation and light flicker, the FCSs may get disconnected resulting in financial losses. The financial losses are represented by the FCS downtime. The downtime is the total amount of time that the FCS is not accessible. The FCS downtime and its losses can be interpreted as the cost of voltage flicker incurred by the FCS’s owner. In order to avoid FCS downtime, voltage fluctuation and light flicker should be mitigated. In Section 4, several flicker mitigation devices are analyzed and compared, based on different criteria. It was found that the cheapest mitigation device is the DSTATCOM, according to the cost per kVAr basis and the total annual equivalent cost. The total annual equivalent cost of DSTATCOM is 14,568 $/year. The proposed smart charging method in this study can also be utilized to mitigate the voltage fluctuation and light flicker. To make customers adopt the proposed smart charging method, an instant rebate is paid to the customer for each charging event from the FCS. The annual rebate can be interpreted as the annual cost incurred by the FCS to mitigate the light flicker, thereby avoiding financial losses due to the FCS downtime. The annual rebate is a function of the annual smart charging penetration level. Figure 14 shows the annual rebates of different penetration levels of the proposed smart charging method.

As presented in Figure 14, the rebate is not only a function of the smart charging penetration level, but is also a function of the minimum, maximum, 25th, 50th, and 75th percentiles of annual energy required from the FCS. To compare the annual cost of smart charging with the annual cost of DSTATCOM, the annual average rebate of using smart charging is calculated. The annual average rebate is estimated by averaging the values of Equations (60) and (63). From Figure 14, it is noted that the average rebate increases from nearly $140 to $1400 as the smart charging penetration level increases from 10% to 100%. This means that the minimum annual cost incurred by the FCS when the proposed smart charging is applied is $140, whereas the maximum annual incurred cost by the FCS is $1400. Figure 15 represents the cost percentage decrease when the proposed smart charging method is utilized, in comparison to the cost of DSTATCOM. The annual cost of the proposed smart charging method is reduced by a minimum of 90% to a maximum of 99% of the cost of the DSTATCOM, in comparison to the annual cost of DSTATCOM. Figure 15 clearly shows that the proposed smart charging method achieves a tremendous reduction in the cost of mitigating voltage fluctuation and light flicker.

7. Conclusions

A novel smart charging approach has been introduced lately in the literature, aiming to mitigate the impact of FCSs on voltage fluctuation and light flicker. The proposed charging method encompasses three different levels of charging powers, namely economic, regular, and premium. The research presented in this work has been devoted to proposing a cost of each charging power. The per unit regular and economic charging cost is modeled to be a function of the per unit cost of premium charging power and a scaling factor. A scaling factor is proposed to represent a rebate offered to the customer to adopt the regular and economic charging powers. The rebate is designed to compensate the increase in the charging duration by reducing the per unit cost of regular and economic charging power. The cost is modeled to preserve a balance in the required cost to charge 1 kwh by the regular or economic power to the required time to charge 1 kwh by the premium power. The ratio of time required by premium power to the time required by regular power is 0.92, whereas the ratio of the cost incurred by using premium power to regular power is 1.08. Similarly, the ratio of premium power to regular was found to be ±13%. Multiple penetration levels of smart charging, varying from 0% up to 100%, were considered and their corresponding annual rebate or revenue was estimated. Additionally, costs of several mitigation devices were quantified and compared. The results reveal that DSTATCOM is nominated, based on the cost per kVAr, to achieve the minimum total annual equivalent costs incurred by an FCS operator or investor to mitigate variation of voltage at the FCS. The annual cost of DSTATCOM is found to be 14,568 $/year. When the smart charging penetration levelis 10%, the annual cost incurred by an FCS operator or investor, by applying smart charging, is less than the annual cost of installing DSTATCOM by 99%. Nevertheless, even when smart charging penetration level is 100%, the annual cost incurred by an FCS operator or investor, by applying smart charging, is less than the annual cost of installing DSTATCOM by 90%. In future work, the time-of-use price can be included to design the proposed cost model of this study.