The rapid development of cloud technologies, such as Vehicle-to-Cloud (V2C) [1], provides a chance to connect and control multiple electric robots (ERs)/electric vehicles (EVs) as fleets, achieving higher revenue with a longer total operational time. Due to the high energy density feature of Li-ion batteries [2], Li-ion batteries play an essential role as the energy storage method of electric vehicles and robots. Unlike a normal car with an internal combustion engine that can be fueled in several minutes, an electric vehicle/robot with Li-ion batteries may take several hours to be fully charged. However, due to the cost of the charging infrastructure, the chargers in the charging station are usually not enough to ensure all the EVs/ERs are charged simultaneously [3]. To charge an EV/ER, the most popular charging method, the constant current constant voltage (CCCV) [4], is usually the first choice. The current profile of the CCCV charging process is nonlinear and consistent with two charging stages, constant current (CC) and constant voltage (CV). In the CC stage, the battery is charged at a constant current, aiming primarily to quickly boost its stored energy without significant degradation [5]. Once the battery voltage hits a specific threshold, known as the cut-off voltage, the process transitions to the CV stage [6]. In this phase, a constant voltage is maintained while the charging current progressively decreases, largely due to the increasing State-of-Charge (SOC) [7]. This decline in current during the CV stage can also be indirectly influenced by factors such as temperature and relative humidity [8]. This phase plays a critical role in fully charging the battery and preventing overcharging. Given identical charging setups, the charging current in the CV stage is typically much lower than in the CC stage, leading to a slower increase in SOC over an equivalent period. At the same time, because the aging conditions of a battery directly determine the depreciation and cash values of the ERs, the battery degradation loss should be minimized during operation. To maximize total revenue by optimizing the overall operational time of an EV/ER fleet, battery degradation and the nonlinear charging profile of batteries must be considered alongside the limited availability of charging stations in the EV/ER fleet management and optimization process.
Presently, various EV/ER fleet applications, such as agricultural robots [9], warehouse mobile robots [10], mining robots [11], cruiser drones [12], and construction robots [13], hold significant potential for replacing human labor in repetitive and physically demanding tasks [14]. These robots can perform a range of activities, substituting human workers in fields like agriculture, warehousing, mining, patrolling, and construction. For example, agricultural activities can be done by agricultural robots, including land preparation before planting, sowing/planting, plant treatment, and harvesting [15]. Warehouse mobile robots help to facilitate and transfer cargo to the goal point or required shelf [16]. Mining robots help miners mine mineral resources in extreme environments, including the high arctic, extreme desert climates, and deep underground, improving productivity, safety, and costs [17]. Cruiser drones patrol given areas for the tasks such as guarding, traffic monitoring, and forest fire detection [18]. Construction robots are able to move heavy building materials to designed destinations in the field [19].
These EV/ER application cases share the following common features,
• Electric mobile robots operate with a limited number of chargers at charging stations. • The EVs/ERs in the fleet make revenues by doing tasks.
• The work field is unchanged, the charging station is located at an almost fixed distance from the work field.
In this paper, we focus on an EV/ER fleet system that has the above characteristics. Besides these characteristics, similar to other electric systems, the main cost of the EV/ER fleet system is electricity cost. Thus, this study endeavors to devise an optimal fleet management system that optimizes operational profit by maximizing total work revenues while simultaneously minimizing battery degradation losses and electricity costs.
The literature on EV/ER fleet charging scheduling and optimization primarily focuses on three critical aspects: minimizing charging costs, reducing charging times, and mitigating battery degradation. These factors are crucial for optimizing fleet management, as they directly impact short-term operational costs, revenue generation, and long-term profitability of the EV/ER fleet. Numerous methods have been proposed to tackle these challenges, with linear programming (LP), dynamic programming (DP), and mixed-integer linear programming (MILP) being among the most common approaches.
LP has been widely employed to optimize EV fleet charging coordination due to its high computational efficiency and ability to find optimal solutions. Electricity prices typically fluctuate throughout the day, with off-peak periods generally featuring lower prices than peak hours [20]. By taking advantage of these price variations, linear programming-based optimal schemes have been developed to minimize the EV fleet charging cost [21]. In Ref. [22], LP is utilized to optimize electric vehicle battery charging behavior, aiming to reduce charging time using a linear approximation of the battery's behavior. However, the application of LP is limited by its requirement for a linear objective function and constraints specified using only linear equalities and inequalities. This constraint significantly restricts the method's applicability in situations where high nonlinearity is present, such as battery degradation models.
DP, in contrast to LP, effectively handles the nonlinearities present in EV fleet charging scheduling problems and provides global optimal solutions with the flexibility to accommodate various objective functions and constraints. By using a battery aging model that considers the impact of C-rate and Ah throughput, a DP-based EV fleet charging coordination system has been proposed to prevent battery capacity fading [23]. In Ref. [24], a DP-based optimization algorithm is introduced for charging an EV fleet represented by an aggregate battery model.
Although dynamic programming addresses the nonlinearity of the battery aging model, the computational cost is significantly high due to the "curse of dimensionality." As a result of its high computational complexity, DP may not scale efficiently for large-scale EV fleet charging coordination problems or situations involving numerous charging stations and vehicles.
MILP is characterized by linear objective functions and constraints, along with integer decision variables. This approach is specifically designed to handle integer decision variables, which are essential in EV fleet charging problems, such as the number of charging stations or charging slots. In Ref. [25], a MILP approach is used to develop an EV charging scheduling system. The goal of this system is to balance the power generated and consumed by the EV charging station while also leveling out the surplus power supplied to the utility grid. A two-layer optimization framework has been proposed in Ref. [26], which employs a mixed-integer linear program to reduce the charging costs and waiting times of shared autonomous EVs. Moreover, an electric bus fleet coordinated charging strategy is designed to minimize battery degradation and electricity cost with Vehicle-to-Grid connectivity [27]. However, MILP is not well-suited for handling non-linear constraints, such as battery degradation or charging profiles. In such cases, linear approximations or other techniques must be used to convert non-linear constraints, like the battery degradation model, into linear ones.
In the literature, there is a notable gap in addressing the nonlinearity of the battery degradation model in the optimization while considering multiple battery degradation impact factors, including Ah throughput and C-rate. Furthermore, only a few studies have taken into account the nonlinear charging profile in fleet management [28]. represents the first study that incorporates a nonlinear battery charging profile into the fleet charging scheduling problem. By specifying initial and target SOC values for charging, this approach offers a way to determine the total charging time while accounting for a nonlinear battery charging profile. Nevertheless, there is still room for improvement, as this method does not consider various charging scenarios, such as fast charging and normal charging.
In the domains of agricultural robots, warehouse mobile robots, mining robots, cruiser drones, and construction robots, only a few studies have explored fleet charging scheduling. A warehouse mobile robots operational scheduling strategy is proposed in Ref. [29], utilizing allocation methods and availability rules to ensure operational efficiency and avoid order delays. In Ref. [30], a genetic algorithm-based approach is designed to optimally schedule the operation and charging time of agricultural robots, considering the limited number of charging chargers. However, these studies have not yet addressed battery degradation and nonlinear charging profiles during charging, nor have they taken advantage of the benefits of variable electricity prices.
In summary, research gaps remain in providing an effective battery degradation linearization method and considering battery degradation and nonlinear charging profiles in both fast and normal charging scenarios. Additionally, to the best of the authors' knowledge, no existing literature has reported on an ER/EV fleet charging management system applied in the fields of agricultural robots, warehouse mobile robots, mining robots, cruiser drones, and construction robots that adequately addresses battery degradation and nonlinear charging profiles during charging.
By extending the existing research, this study addresses research gaps in the ER/EVs fleet charging management domain, where the complexity arises from the involvement of multiple ER/EVs, unique operation condition, and various control actions. In particular, the study focuses on addressing the nonlinearity of the battery degradation model and charging profile in the fleet charging optimization process. Additionally, it offers an insightful analysis of the impacts of EVs/ERs' battery aging conditions and different charging selections on the total operating profits. The contributions and novelties of this study can be summarized as follows:
• Formulation of a specific fleet charging optimization problem aimed at maximizing total work revenues while minimizing battery degradation loss and electricity cost. • Development of a linear battery degradation model that considers the impact of C-rate and Ah throughput, derived from a well-known nonlinear battery degradation model for optimization simplification. • Incorporation of a nonlinear battery charging profile, accommodating both fast and normal charging scenarios, into the fleet operation optimization problem through a well-designed linearization method. • Investigation of the impact of EVs/ERs' battery aging conditions on an optimal fleet charging management strategy. • Analysis of the influence of the choice between fast charging and normal charging on the total operating profits.
The remainder of this paper is organized as follows. Section II presents the battery electric and degradation models used in the study. Section III discusses the methodologies for solving the ER fleet management optimization problem. Section IV analyzes the simulation results of the proposed system. Section V summarizes the key conclusions of this study.
In this section, the battery electric and degradation models used in the study are discussed.
In this study, we employ a first-order equivalent circuit model of a battery, also known as the First-order RC or Thevenin model. This model demonstrates sufficient performance in capturing the dynamics of battery voltage responses [31,32], and we use it to simulate the electrical behaviors of the battery. As depicted in Fig. 1, the first-order RC model comprises an open-circuit voltage (OCV, in V), an ohmic resistor (R 0 , in Ω), and an RC pair, which represents polarization resistance (R 1 , in Ω) and polarization capacitance (C 1 , in C) [33]. The voltage of the RC pair is denoted by V 1 (in V). The model's output is the terminal voltage (V T , in V). Notably, the values of the OCV, ohmic resistor, polarization resistance, and polarization capacitance are influenced by the SOC (in %), temperature, and both discharge and charge currents, as detailed in Ref. [34].
Based on the first-order RC model, the dynamic of a battery can be formulated as follows [35],
where Δt is the sampling time (in s), I is the current (in A), and C bat is the nominal capacity (in Ah).
The max storage energy (E max , in Wh) and dynamics of State-of-Energy (SOE, in %) can be obtained as follows [36],
In these equations, t SOC=0% represents the initial time when SOC is at 0%, and t SOC=100% is the finial charging time when the SOC research 100%.
The empirical degradation model of the Li-ion battery used in the study is from Ref. [37]. It describes the battery capacity loss, Q loss (in %), as a function of temperature (T, in K), current (C rate , in C), and Ah-throughput (A h , in Ah).
In above equations, R represents the ideal gas constant (in J K mol ), E a is activation energy (in J), and z is the power law factor, which is set to 0.55. The fitting parameters, a and b, are 31700 and 370.3, respectively. M(C rate ) represents a pre-exponent factor that decreases as the C-rate increases, as shown in Table 1 [38].
In this section, the formulation of an ER/EV fleet management problem is proposed in the first subsection. The linearization methods of battery degradation and nonlinear charging profile are developed to address the two nonlinear parts in the proposed optimization formula for applying Mixed-Integer Linear Programming (MILP). Then, the system operation details are discussed in the last subsection.
As mentioned in Section 1.1, this study focuses on an EV/ER fleet system with the following characteristics.
• Electric mobile robots are used with limited numbers of chargers in the charging station. • The EVs/ERs in the fleet make revenues by doing tasks. • The work field is unchanged, so the charging station has an almost constant distance from the work field.
A schematic diagram of the fleet is presented in Fig. 2. To summarize, in the fleet operation, there are four different operation status/actions: doing tasks, returning to charging stations, waiting in line for charging, and fast charging or normal charging. To highlight the actions, we italicize and bold these actions throughout this paper. An EV/ER makes revenue by doing its tasks. When an EV/ER is out of battery, it needs to return to a charging station and recharge itself. Because only a limited numbers of charging stations are available, the EV/ER must wait in line for charging if all the charging stations are occupied. Once a charging station is available, the EV/ER can choose either normal or fast charging mode to charge itself. After the EV/ER is charged to the desired energy level, it will return to the work field and continue doing tasks.
In order to maximize the operational profit within a given time horizon, our proposed methodology allows for the determination of the optimal sequential actions for each EV/ER in the fleet. This involves striking a balance between maximizing total revenues and minimizing both the battery degradation loss and electricity cost during charging. For example, it may be optimal for an EV/ER to end its charging state earlier (before reaching 100% SOE) and return to its working field to avoid the time wastage inherent in the CV stage due to the slow charging speed. Thus, to find optimal actions for the fleet, we can formulate it as an optimization problem as follows,
where d, r, w, nc, and fc are binary decision variables representing the five possible actions for an EV/ER: doing tasks, returning, waiting, fast charging, and normal charging, as shown in Fig. 2. The variable t denotes the current time step, while T signifies the prediction horizon.
The process for selecting the sampling interval and determining the length of the prediction horizon is discussed in Section 3.4. The variable v indicates an EV/ER and V is size of the fleet.
In any given time step (t), for an ER (v), the revenue (Rev(t,v), in $), battery degradation cost (Cost bat (t,v), in $), and electricity cost (Cost ele (t, v), in $) are computed using the following equations.
Cost bat (t, v) = Pb n cell C p f aging (10) Cost ele (t, v) = Pe(t) Emax v f charging (11) Here, Pa(t) represents the payment given to an EV/ER at time t (in $/h).
Pb is the price of a battery (in $/Ah), n cell is the number of cells in an EV/ ER, and C p (in Ah) is the capacity of a cell. The function f aging illustrates the percent of battery capacity loss within the time step due to battery degradation. Pe(t) is the electricity price (in $/kWh) at time t. Emax v (in kWh) is the maximum storage energy of an ER (defined in Eq. ( 4)), and f charging represents the percentage of SOE change within a time step. The linearization and formulation of the functions f aging and f charging are discussed in Sections 3.2 and 3.3, respectively. The objective function, J, is constrained by the battery dynamic and state constraints.
where E r and E d are the energy consumption of an EV/ER when it is in the state of returning (r v ) and doing tasks (d v ). The energy consumption of an EV/ER is assumed as zero while it is waiting in the charging station.
To ensure the safe operation of an EV/ER and avoid battery degradation due to over-charging and over-discharging, the SOE of an EV/ER is constrained under the desired range as follows,
To mimic the operation scenario of an EV/ER, its operational states are constrained by following statements/equations. An EV/ER can only stay in one single state at time, t.
As shown in Fig. 3, an EV/ER cannot skip returning state when it travels between the charging station and the working field.
The number of chargers limit the numbers of EVs/ERs that can be charged simultaneously.
where M is the numbers of the chargers in the charging station.
By discretizing and differentiating the formula of the battery degradation equation shown in Section 2.2, the discretized degradation model is expressed as,
where Δt is the sampling time. The delta capacity loss in a sample time depends on the battery temperature, C-rate, and previously accumulated battery capacity loss.
During an ER/EV operation, the thermal management system regulates temperature to avoid safety issues and battery degradation. Thus, the temperature contribution to the battery degradation could be removed by assuming the temperature of battery packs is constant, 25 • .
In addition to temperature and C-rate, the degradation rate of battery cells also depends on their current state of life [39]. As depicted in Fig. 4, with a constant battery temperature of 25 • and a C-rate fixed at 1C, the delta capacity loss of a battery per second decreases as the battery ages. This decline can be attributed to the stabilization of the Solid Electrolyte Interface (SEI) film formation as the battery approaches it EOL [40]. This implies that the cost due to battery degradation in a sampling time will decrease progressively as batteries age. Therefore, it is important to take the aging condition of batteries into account when developing an optimal control strategy. Since battery degradation is a very slow process, we assume that the battery aging condition of each EV/ER remains unchanged within a short prediction horizon. In each sample time, the proposed system will generate an optimal management strategy based on updating the state-of-health (SOH) of battery packs in each EV/ER.
As shown in Fig. 5, Fig. 6, and Fig. 7, according to the discretized battery degradation model mentioned above, the blue lines represent the ΔQ loss of a battery in 1s vs. C-rate across three different SOH conditions. SOH = 99.9% corresponds to a brand-new battery known as begin-of-life (BOL). According to the convention, SOH = 90% represents a battery in middle-of-life (MOL). SOH = 80% means 20% capacity loss, defined as end-of-life (EOL). We can see that the three blue lines are almost linear. It allows us to fit the battery degradation curve under different aging conditions using a linear curve based on the C-rate only. The three red lines shown in Figs. 5, Fig. 6, and Fig. 7 are fitted lines using linear regression. They reasonably match the nonlinear curves. Hence, the function f aging , discussed in Section 3.1 and representing the percentage of battery degradation over a given time gap and C-rate, can be approximated as follows,
where k aging (in % s C
) is the fitted parameter based on the SOH of each EV/ ER, C P is the nominal capacity of batteries, and f i is the function that returns the value of the current based on the states of the batteries during charging. The formulation of f i is discussed in Section 3.3.
In this subsection, the method to formulate the nonlinear charging profile in a linear way is discussed.
Based on the definition of SOE, the amount of SOE change (ΔSOE, in %) in a time step at different SOE values can be obtained by using the following function, method, is plotted in the blue line in Fig. 8. As shown in Fig. 8, the two ranges in g(SOE) from 5% SOE to the CV stage turning point (SOE t ), and SOE t to 95% SOE are almost linear. By fitting the function g(SOE) with a piece-wise linear regression, an approximated function, ĝ(SOE), is obtained as,
where k and b are the linear fitting parameters for an ER with different SOH and charging methods. As presented in Fig. 8, the red and yellow lines represent the fitting results, ĝ, in the CC and CV ranges, respectively. However, ĝ(SOE) is still nonlinear, which prevents us from using the MILP to solve the problem directly. It requires a further linearization. To do this, we need to obtain the exact values of ĝ(SOE) at three specific points: SOE equal to 5%, SOE equal to SOE t , and SOE equal to 95%. These values can be defined as follows, ĝ5% ≜ĝ(5%) (23) ĝSOEt ≜ĝ(SOE t )
ĝ95% ≜ĝ(95%)
As a piece-wise function with three break points, ĝ(SOE) is further linearized and represented as function g,
In this equation, w 1 , w 2 , and w 3 are continuous weighting factors between 0 and 1 that constrain the values of gˆand SOE. The value of SOE can be expressed as,
To ensure that the weighting factors are within the proper range and that the binary variable z is used correctly, they must satisfy the following constraints,
where binary variable z determines whether the battery is in either CC or CV stage. As mentioned in Section 3.1, f charging represents the Δ SOE increased during normal or fast charging mode over a time gap, Δt. Thus, it can be expressed as,
where the functions g fc and g nc are g in the scenarios of fast charging and normal charging, respectively. The function introduces new nonlinear terms due to the multiplication of state variables, g fc and g nc with decision variables, fc and nc. However, since fc and nc are binary variables, the nonlinearity of f charging can be easily addressed. f charging can be reformulated as,
The average current in one time step at a particular SOE value can be calculated using the following equation, Ĩ(SOE) =
As illustrated in Fig. 9, the average current profile of a battery in the CC and CV is also almost linear in each stage. Similar to the functions ĝ, a linearly fitted piece-wise function, î, is introduced to capture the nonlinear current profile of a battery during charging,
The average current for SOE equal to 5%, SOE equal to SOE t , and SOE equal to 95% can be obtained and defined as î5% , îSOEt , and î95% .
Because î shares the same break points as g, a linear approximated function i can be expressed in a same way,
where the three weighting factors are the same as the weighting factors in g. Because î5% and îSOEt are both equal to b i1 , which is the constant current in CC stage, we have,
Following the same method as f charging , the current function, f i , mentioned in Section 3.2, can also be obtained as,
where x fc 1 , x nc 1 , x fc 2 , and x nc 2 can be also further linearized as,
Fig. 9. Current profile of BOL battery using normal charging conditions.
As shown in Fig. 10, during operation, EVs/ERs update their states, including SOE and applied actions, to the cloud servers or control center for planning every 5 min. With future electricity prices provided, the controller/online optimization system generates optimal reference control actions based on a prediction horizon of 180 min. This horizon length is sufficient to prevent short-sighted decisions, yet not too long to consume unnecessary computational resources. Considering that Model Predictive Control (MPC) is a proven tool for handling model errors and system uncertainties [41,42], the proposed controller executes the online optimization at each sampling interval to manage these factors. As a standard MPC controller, the proposed controller applies only the first computed control action of each EV/ER to the fleet system while disregarding the subsequent ones. This procedure is iteratively performed in each subsequent time step.
In the optimization, v, mentioned in (8) can index not only a single EV/ER but also a group of EVs/ERs. This allows the proposed system to handle more extensive fleet charging management problems without significantly increasing computational complexity. By classifying the entire EV/ER fleet according to their SOH conditions, multiple classes can be identified. Each of these classes can then be further partitioned into several sub-groups of the same size. The size of each group is the greatest common divisor of each class's size and the number of chargers. The total number of groups equals the total number of EVs divided by the size of a group. For example, if there are 20 chargers in the charging station and 300 ERs equally distributed among BOL, MOL, and EOL conditions, the ratio of ERs in BOL, MOL, and EOL aging conditions to the number of chargers is 100:100:100:20. These 300 ERs can be partitioned into 15 groups, each with a size of 20. In this case, v represents one of the 15 groups. The ERs in the same group share the same characteristics. Additionally, the charging station has chargers to serve an integral number of groups simultaneously. Therefore, we may treat each group of ERs as a single ER, v. In operation, the ERs in the same group follow the same actions based on the planning results. Because the ERs in the same group, v, have the same characteristics and follow the same actions, we can assume that their states, such as SOE, are also the same.
Considering the operational differences between EVs and ERs, there are two types of EVs. The first type is the autonomous EV, similar to ERs, capable of operating continuously for 24 h. The second type involves EVs driven by people, which primarily operate during the day and are typically parked at charging stations overnight. Operators can then manage their charging, leveraging the proposed system to optimize charging schedules based on varying electricity prices, minimizing energy costs and reducing battery degradation. In such scenarios, we assume that at nighttime, there are only three actions available for minimizing the charging cost while also reducing battery degradation: normal charging, fast charging, and waiting for lower electricity prices or available chargers.
In this section, the system set-up of the simulations is discussed in Section 4.1. Then, the simulations are conducted to test the proposed method with analyzing in Section 4.2.
In the simulation, the total stored energy for a new ER is about 50 kWh (similar to Tesla Model 3). To simulate the high-power consumption of an ER working on high-intensity tasks and the lower power consumption when it just moves from one place to others, the power consumptions for doing tasks and returning are assumed to be 25 kW (roughly 0.5 C discharge current) and 5 kW (around 0.1 C discharge current), respectively. The fast-charging power of Tesla is set to about 1.2C [43]. In the study, the fast charging and normal charging current in the CC stage are set to be 1.2 C and 0.6 C, respectively. These values were selected arbitrarily and are used just for illustration purposes in this study. In the CV stage, the charging current will keep dropping until it reaches the cut-off limit, whatever fast charging or normal charging mode is used in the CC stage. Thus, a case with a current lower than 0.6 C in the CV stage is classified as the normal charging condition as well.
Because ERs in a fleet may have different battery SOH levels, in the simulation, three battery aging conditions, including BOL, MOL, and EOL, are assigned to tested ERs. While a battery is aging, its capacity and resistance will change. According to Ref. [44], the capacity and resistance in the simulation are set by the following equations,
where C P initial and R initial are the initial capacity and resistance of a new battery. The parameter, α, is set to be 4.545 in this study and is the fitted parameter that describes the relationship between the capacity loss and the resistance growth.
Based on the capacity and resistance of batteries at different aging conditions, the parameters of an ER with different aging conditions are calculated and shown in Table 2. As a battery ages, its capacity decreases, leading to a loss in Emax. However, its aging speed, k aging , from equation ( 20) also slows down. When using the fast charging mode, the high charging current causes the cut-off voltage to be reached earlier compared to the normal charging mode. This results in smaller SOE values of CV turning points. Namely, for an ER with the same SOH condition, its SOE fc t is smaller than SOE nc t , as mentioned in Section 3.3. As the State of Health (SOH) of a battery decreases, its internal resistance increases. This increase in resistance also contributes to smaller SOE values of CV turning points in both fast charging and normal Fig. 10. Frame and flow chart of the proposed system. charging modes. Consequently, this leads to a smaller CC range for charging, meaning that the battery transitions from the CC stage to the CV stage earlier in the charging process. The battery price is set to be 0.5244 $ Ah [45]. To take advantage of the volatility of electricity prices or the Direct Access program [46], we assume the charging stations directly participate in this highly volatile market. As our paper primarily focuses on control, we assume known electricity prices. This assumption is predicated on our system's ability to participate in a Time-of-Use rate plan [47] provided by a utility company like PG&E, or to use accurately predicted energy market prices. For the Time-of-Use rate plan, electricity costs during peak, partial-peak, off-peak, and super off-peak hours of the day are pre-determined. Regarding energy market price prediction, several studies have demonstrated highly accurate results [48,49]. As depicted in Fig. 11, electricity prices are represented by the locational marginal prices on 09/28/2022, provided at 5-min intervals in New England [50]. The historical electricity market prices utilized in our study are used strictly for the demonstration and analysis of system performance. In practical applications, our proposed method would operate based on provided Time-of-Use rate plan or forecasted electricity prices, thereby accommodating real-time scheduling scenarios. We selected electricity prices from 5:00 a.m. to 8:00 a.m., as highlighted in Fig. 11, for testing. This time range encompasses both off-peak and peak-hour prices and aligns with the system prediction horizon, effectively demonstrating the performance of the proposed system. The zoom-in plot of the electricity prices is shown in Fig. 12.
At the beginning of the simulation, the state of all ER s set to doing tasks, and the SOEs are initialized at 50%. At the end of the simulation, two terminal constraints are included to ensure the states are reset for the next working cycle. Specifically, in the final step, all ERs should have already returned to their work field, and their final SOE should be at least 50%.
In this section, we focus on demonstrating the performance of the proposed method in simulations while analyzing the impacts of the battery aging conditions and charging method selections on optimal fleet management.
This subsection investigates the impact of battery aging conditions on an optimal fleet management strategy.
In the test, the ratio of the ERs in BOL, MOL, and EOL aging conditions and the numbers of chargers is set to be 1:1:1:1. As discussed in Section 3.4, ERs in each aging group are assumed to be identical with exactly the same energy consumptions. Only one group of these robots can be charged at the same time. The robots with BOL, MOL, and EOL aging conditions are labeled as Robot 1, Robot 2, and Robot 3. Additionally, the payment for performing tasks of an ER group/robot is set to be a constant value of $100/h.
As shown in Fig. 13, the actions of three robots are plotted. The labels d, r, w, nc, and fc, in the y-axis, represent the five actions: doing tasks, returning, waiting, normal charging, and fast charging. We can see that these robots strictly follow the rules/constraints discussed in Section 3.1. Besides, the robots with three different aging conditions also show different behaviors. For example, Robot 1 and 2 returned from doing tasks twice, but Robot 3 only returned once. The SOE plots of three robots are presented in Fig. 14. The changes in the status of each robot due to the action selections are more directly shown. With the terminal constraint, the SOEs of the robots settle at 50%. Besides, it shows that all the robots have been charged just up to the CV stage with fast charging.
Due to charger limitations at the charging station, it is crucial to manage each ER's charging schedules effectively. Otherwise, improper usage may lead to long wait time, resulting inefficient operation and decreased total profits [51]. The proposed system addressed this issue well. The average charging C-rates of each robot are presented in Fig. 15.
We can see that, except for the first 10 min, there is always one robot in a charging state. In other words, robots almost always occupy the charger without any time gap. Moreover, Fig. 15 also demonstrates how the average charging current decrease when the SOE of batteries reaches the CV stage. In Fig. 13, one remarkable thing is that at time 115 min, the system switches the fast charging condition of Robot 1 to waiting but allows Robot 3 charge at this time using normal charging mode. As shown in Fig. 12, the electric price is the highest at the time. Thus, as presented in Fig. 15, the optimal control strategy assigns the lowest charging current to robots. The step electricity cost of each robot is presented in Fig. 16, which reflects the cooperated impact of charging current and electricity prices at a given time. As shown in Fig. 16, at the time 115 min, the system has the lowest instantaneous electricity cost. The total electricity cost is reduced by charging with a high C-rate when the electricity price is low and charging with a low C-rate when the electricity price is high.
The results also reveal that robots with different battery aging conditions have different preferences on the action selections. Fig. 17 and Table 3 show the cumulative time spent on each action of the three robots. Based on the summarized plot in Fig. 17, Robot 2 receives the most time to charge and spends the most time doing tasks relative to the other two robots. It demonstrates the impacts of battery SOH on the optimal charging strategy. Compared with Robot 2, the batteries in Robot 1 are brand new. As discussed in Section 3.2, charging a battery with higher SOH leads to higher battery degradation costs. Fig. 18 shows the cumulative cost, revenue, and profits gained by each robot. As shown in Fig. 18, Robot 1 has the highest cumulative battery cost than others. On the other hand, Robot 3 is also less desired than Robot 2 for energy storage because the batteries in Robot 3 are at EOL. EOL batteries have much higher resistance and lower capacity than BOL and MOL batteries. These characteristics eventually lead to low charging speed and low total stored energy. While charging a robot with EOL batteries, the CV mode will be reached in a short time. The lower charging current CV model decreases its charging speed. Besides, the total energy stored in the batteries is low, increasing the risk of running out of charge while doing tasks. It requires the robot to return to the charging station more frequently and leads to additional energy consumption on returning to the charging station.
Given the limited number of chargers, the system prioritizes charging the ER, which can yield higher returns and lower costs. This means that an ER with higher profitability is given priority to be charged, while other ERs are either assigned to doing tasks, returning, or waiting to be charged. As depicted in Fig. 17, optimal fleet management prefers a robot with MOL batteries over robots with BOL and EOL batteries. Notably, Robot 3, equipped with EOL batteries, spends the most time in the waiting state. Due to Robot 3's highest resistance and lowest capacity, the online optimization system allocates more charging time to Robots 1 and 2. As Robot 3 does not receive sufficient charging time but still needs to maintain its SOE at 50% at the end of the optimization horizon, it is staged in waiting conditions while other robots are being charged.
The revenues in Fig. 18 are calculated by multiplying the total time spent doing tasks and the payment price. The profit is equal to the revenues minus costs. As a result, the robot with MOL batteries also brings the highest total profit to the system.
This section demonstrates that the fast charging mode is economically preferable to normal charging. This result is arguably different from the conventional consensus.
In this test, the ratio of the numbers of ERs in BOL, MOL, and EOL aging conditions and the numbers of chargers in the charging station is set to be 1:1:1:3. As in the previous section, the robots with BOL, MOL, and EOL aging conditions are labeled as Robot 1, Robot 2, and Robot 3. With enough chargers, each robot group does not need to share the chargers with others.
When the amount of payment is set to be $100/h as in the previous section, the actions, SOE, charging C-rate, and cumulative times spent on each action of three robots are plotted in Fig. 19, Fig. 20, Fig. 21, and Fig. 22. As shown in Table 4, with enough chargers, we can see that only a few times normal charging have been assigned to Robot 1 and Robot 3. Moreover, these cases with normal charging are not for saving the battery degradation cost. Robot 1 switches its fast charging to normal charging at time 145 min because it does not want to have a remaining SOE higher than 50% at the ending time. Robot 3 is set to the normal charging mode two times simply because the current dropped to 0.6 C while charging the batteries to a high SOE range in the CV stage.
People may argue that the high payment of doing tasks is the reason for selecting fast charging as the optimal charging strategy. Thus, four different amounts of payments for doing tasks have been tested to understand the optimal control actions based on the impact of the amount of payment for doing tasks. As shown in Fig. 23, crossing the different payment amounts, the total times spent on normal charging based on the optimal strategies of different robot groups (BOL, MOL, and EOL) are unchanged. Although, the payment could be set to an even lower value, such as lower than the electricity and battery degradation costs. In this case, the optimal control strategy will just let all the robots stay in the waiting state and refuse to do any task to avoid any costs. However, lower pay than 20$/h does not make sense in the real world, so it is not further discussed here. Thus, we can conclude that if the payment for doing tasks higher than 20$/h, it does not impact optimal actions. For a more comprehensive understanding of why the optimal strategy favors fast charging, we've plotted the profiles of Robot 1's total cumulative battery degradation, electricity costs, revenue, and operating profit (based on a $20/h task payment rate) in Fig. 24. The simulation results indicate that the degradation and electricity costs incurred during fast charging lead to a decline in cumulative profit (denoted in red). However, once the robot transitions into doing tasks mode, the cumulative profit rapidly rebounds (marked in green). The time saved via the fast charging mode enables the robot to perform a greater number of economically valuable tasks, thereby offsetting the initial dip in profit caused by the costs associated with fast charging.
This study presents an ER fleet operation problem formulation aimed at maximizing total operational profit while considering the nonlinear charging profile, battery degradation loss, and electricity cost. To address the nonlinear elements in the battery degradation model and charging profile, two linearization methods are proposed for applying MILP. The simulation results demonstrate that the proposed system effectively optimizes the charging schedule for the ER fleet, with a Table 3 Time Spent On each Action of Each Robot with One Charger. limited number of chargers in the charging station. Furthermore, the impact of SOH on optimal fleet management is investigated, leading to the conclusion that ERs with MOL batteries are more economically desirable than those with BOL and EOL batteries. This is primarily attributed to the relatively low degradation rate and high energy storage efficiency of MOL batteries. Additionally, the study finds that, in most cases, fast charging is more advantageous than normal charging when the saved time can be allocated to more valuable tasks. These insights highlight the importance of taking battery nonlinear charging profiles and battery aging conditions into account when developing optimal fleet management strategies for ERs and other battery-powered systems. By doing so, more efficient and cost-effective solutions can be achieved, ultimately enhancing the overall performance and lifespan of these systems.
The generalizability of our method allows for easy adaptation to other optimization problems involving batteries, such as energy management strategy design for electric city buses [52] and electric ships [53]. In addition to applications in agricultural robots, warehouse mobile robots, mining robots, cruiser drones, and construction robots, the proposed fleet management system could also be applied to other commercial and residential fleet systems that utilize Li-ion batteries as their energy source, such as electric taxi fleets [54]. In future studies, the application of reinforming learning [55] is an interesting direction to further improve the performance of a fleet management system. By constructing the complex fleet system as a training environment and training the optimal policy with deep neural networks, reinforcement learning methods can effectively handle multiple inputs and nonlinearities. Table 4 Time Spent On each Action of Each Robot with Three Chargers.
Robot 1 (BOL) Robot 2 (MOL) Robot 3 (EOL) Time spent on each action (mins) Doing tasks 100 100 95 Returning 20 20 20 Waiting 0 0 0 Normal charging 5 0 10 Fast charging 55 60 55
J.Shi et al.