Parameter Matching of Battery–Supercapacitor Hybrid Power System for Electric Loader

Abstract

The hybrid power system formed by batteries and supercapacitors can meet the demands of electric loaders for endurance and instantaneous power. Appropriate parameter matching can optimize the operational performance of the hybrid power system. However, multiple optimization objectives and complex constraints present technical challenges for parameter matching. To address this, this paper proposes a multi-objective optimization parameter matching method for a hybrid power system based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm. First, mathematical models for the battery, supercapacitor, and DC-DC converter are established. Next, based on the performance requirements of electric loaders, objective functions and constraints for hybrid power parameter matching are defined, and an optimization model for parameter matching is developed. Finally, the optimal parameters for the hybrid power system are determined using the NSGA-II algorithm. Experimental results indicate that, compared to a single battery energy storage system, the operational energy consumption of electric loaders equipped with a hybrid power system is reduced by 3.32% and battery capacity degradation is decreased by 10.61%, with only a slight increase in costs.

1. Introduction

As a type of construction machinery frequently used for loading and unloading goods, the loader offers advantages such as mobility, high operational efficiency, and ease of use, making it widely utilized in earthworks and material handling [1,2]. Traditional loaders primarily relied on diesel as their main energy source; however, the significant emissions and low energy efficiency associated with diesel engines have led to serious environmental pollution and resource waste. To achieve sustainable development, electric loaders, characterized by zero emissions, pollution-free operation, and high energy efficiency are becoming increasingly prevalent [3,4,5].

Electric loaders typically use lithium batteries as energy storage components. These batteries are stable and reliable during operation and offer high energy density. However, their low power density limits the ability to respond quickly and efficiently to high-load demands, and sudden high current spikes can cause significant damage to the batteries [4,6]. Supercapacitors, as a novel type of electrochemical energy storage device, possess advantages such as rapid charge and discharge rates and high energy conversion efficiency. Nevertheless, their low energy density results in limited energy storage capacity [7]. Considering the dual requirements for endurance and instantaneous power in electric loaders, a hybrid power system combining batteries and supercapacitors presents a viable solution [8,9].

The parameters of batteries and supercapacitors in a hybrid power system directly influence the performance of the power system. To achieve optimal performance from the hybrid power system, rational parameter matching design becomes particularly critical [10,11]. During the parameter matching process, it is essential to comprehensively consider the cost of the power system, its efficiency, and the lifespan of the batteries, as well as the nonlinear and complex relationships between multiple optimization objectives and design parameters. Additionally, the synergies and constraints among various optimization objectives pose significant technical challenges for parameter matching in a hybrid power system.

Convex optimization methods are known for their rapid solving speed and guarantee of global optimality, making them widely used in the research of parameter matching for a hybrid power system. Nikolce Murgovski et al. proposed a method for simultaneously optimizing power management control and battery parameters, approximating the optimization problem as a nonlinear convex problem for resolution [12,13,14]. Xiaosong Hu et al. extended convex optimization to the optimization of power parameters in hybrid electric buses [15,16]. However, the convex optimization method has some limitations because of its high computational complexity and can only be applied to the optimization of convex functions.

Dynamic programming is an optimization algorithm that derives a global optimal solution through multi-stage decision-making based on predefined operating conditions. Masoud Masih-Tehrani et al. proposed an optimization approach that uses the initial costs of a hybrid power system and the ten-year replacement costs of batteries as objectives, employing dynamic programming algorithms to match power system parameters [17]. Ziyou Song et al. utilized dynamic programming methods to solve the optimal configuration problem for a hybrid power system in electric urban buses [18]. However, the dynamic programming algorithm needs a lot of memory and time to store the intermediate results, resulting in low computational efficiency.

Both convex optimization and dynamic programming are single-objective optimization algorithms. When addressing multi-objective problems, a weighted approach is typically used to obtain a set of solutions under different weights, with only one feasible solution obtained at each iteration. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a multi-objective optimization technique that can simultaneously solve all optimal parameters. The diversity of solution set and excellent convergence can be maintained by non-dominant ordering and crowding distance. Ziyou Song et al. proposed using the NSGA-II algorithm to match parameters for a hybrid power system, with optimization objectives focused on the costs of a hybrid power system and battery capacity loss [19]. However, this approach does not account for the efficiency of the power system.

To achieve optimal power system cost, power efficiency, and battery lifespan in the parameter design of a hybrid power system, this paper proposes a multi-objective optimization parameter matching method for a hybrid power system based on the NSGA-II algorithm. The organization of this paper is as follows. Section 2 introduces the structure of the hybrid power system and establishes mathematical models for the battery, supercapacitor, and DC-DC converter. Section 3 analyzes the performance requirements of electric loaders, sets the objective functions and constraints for hybrid power parameter matching, and develops the optimization model. Section 4 presents the optimal parameters for the hybrid power system obtained using the NSGA-II algorithm. Section 5 provides the experimental results, and finally, Section 6 concludes the paper.

2. Mathematical Modeling of the System

The hybrid power structure studied in this paper is illustrated in Figure 1. The main components include the battery, supercapacitor, and DC-DC converter. The battery is directly connected to the DC bus, and its stable voltage output enhances the system’s stability. The supercapacitor is connected to the DC bus through the DC-DC converter, which actively controls the input and output power of the supercapacitor. This configuration effectively assists the battery in meeting the peak power demands of the electric loader, thereby reducing the impact of high currents on the battery.

2.1. Battery Cell

The model of the battery cell is shown in Figure 2 [20].

In the figure, U_bo is the open-circuit voltage of the battery cell (V), R_b1 is the polarization resistance (Ω), C_b1 is the polarization capacitance (F), U_b1 is the voltage across C_b1 (V), R_b2 is the ohmic resistance (Ω), I_bc is the current of the battery cell (A), U_bc is the terminal voltage of the battery cell (V).

The terminal voltage of the battery cell can be expressed as [21]

$$U_{\mathrm{bc}}=U_{\mathrm{bo}}-U_{\mathrm{b}1}-I_{\mathrm{bc}}{R}_{\mathrm{b}2}$$

(1)

The current of the battery cell can be expressed as

$$I_{\mathrm{bc}}={\displaystyle \frac{(U_{\mathrm{bo}}-U_{\mathrm{b}1})-\sqrt{{(U_{\mathrm{bo}}-U_{\mathrm{b}1})}^2-4P_{\mathrm{b}}{R}_{\mathrm{b}2}/N_{\mathrm{b}}}}{2R_{\mathrm{b}2}}}$$

(2)

where P_b is the power of the battery pack (W), N_b is the number of battery cells.

The output voltage and current of the battery pack can be expressed as

$$\left\{\begin{array}{c}{U}_{\mathrm{b}}=N_{\mathrm{bs}}{U}_{\mathrm{bc}}\\ I_{\mathrm{b}}=N_{\mathrm{bp}}{I}_{\mathrm{bc}}\end{array}\right.$$

(3)

where U_b is the voltage of the battery pack (V), I_b is the current of the battery pack (A), N_bs and N_bp are the number of battery cells in series and parallel, respectively.

The polarization resistance R_b1 and ohmic resistance R_b2 of a single battery are related to the battery’s charging state and operating temperature, and are expressed as

$$R_{\mathrm{b}1}(t)=f(SO{C}_{\mathrm{b}}(t),T_{\mathrm{b}}(t))$$

(4)

$$R_{\mathrm{b}2}(t)=f(SO{C}_{\mathrm{b}}(t),T_{\mathrm{b}}(t))$$

(5)

where f(∙) is the table lookup function, SOC_b is the state of charge of the battery, T_b is the operating temperature of the battery (K).

The polarization resistance and ohmic resistance of the battery cell can be obtained through a lookup table. Since this study is carried out under normal temperature conditions, it is not necessary to consider the effect of temperature change on the resistance.

The state of charge of the battery at any moment can be expressed as

$$SO{C}_{\mathrm{b}}=SO{C}_{\mathrm{bo}}-{\displaystyle \frac{{\displaystyle \int I_{\mathrm{bc}}(t)\mathrm{d}t}}{Q_{\mathrm{b}}}}$$

(6)

where SOC_bo is the initial state of charge of the battery, Q_b is the total charge capacity of the battery (Ah).

2.2. Supercapacitor Cell

The model of the supercapacitor cell is shown in Figure 3 [22].

In the figure, C_co is the ideal capacitance (F), R_c is the equivalent internal resistance (Ω), U_co is the open-circuit voltage (V), I_cc is the current of the supercapacitor cell (A), U_cc is the terminal voltage of the supercapacitor cell (V).

The terminal voltage of the supercapacitor cell can be expressed as [23]

$$U_{\mathrm{cc}}=U_{\mathrm{co}}-R_{\mathrm{c}}{I}_{\mathrm{cc}}$$

(7)

The current of the supercapacitor cell can be expressed as

$$I_{\mathrm{cc}}={\displaystyle \frac{U_{\mathrm{co}}-\sqrt{U_{\mathrm{co}}^2-4R_{\mathrm{c}}{P}_{\mathrm{c}}/N_{\mathrm{c}}}}{2R_{\mathrm{c}}}}$$

(8)

where P_c is the power of the supercapacitor pack (W), N_c is the number of supercapacitor cells.

The output voltage and current of the supercapacitor pack can be expressed as

$$\left\{\begin{array}{c}{U}_{\mathrm{c}}=N_{\mathrm{cs}}{U}_{\mathrm{cc}}\\ I_{\mathrm{c}}=N_{\mathrm{cp}}{I}_{\mathrm{cc}}\end{array}\right.$$

(9)

where U_c is the voltage of the supercapacitor pack (V), I_c is the current of the supercapacitor pack (A), N_cs and N_cp are the number of supercapacitor cells in series and parallel, respectively.

The state of charge of the supercapacitor can be expressed in terms of voltage:

$$SO{C}_{\mathrm{c}}={\displaystyle \frac{U_{\mathrm{cc}}-U_{\mathrm{cc}\_\min }}{U_{\mathrm{cc}\_\max }-U_{\mathrm{cc}\_\min }}}$$

(10)

where U_{cc_max} and U_{cc_min} are the maximum and minimum terminal voltages of the supercapacitor cell (V), respectively.

2.3. DC-DC Converter

The structure of the DC-DC converter is shown in Figure 4.

In the figure, U_L and U_H are the voltage of the low and high voltage terminals of the DC-DC converter (V), I_L and I_H are the current of the low and high voltage terminals of the DC-DC converter (A), C₁ and C₂ are the filter capacitor (F), L is the inductor (H), D₁ and D₂ are the diode, S₁ and S₂ are the insulate-gate bipolar transistor (IGBT).

The efficiency of a DC-DC converter can be expressed as [24]

$${\eta }_{\mathrm{dcdc}}={\displaystyle \frac{I_{\mathrm{out}}{U}_{\mathrm{out}}}{I_{\mathrm{in}}{U}_{\mathrm{in}}}}$$

(11)

where η_dcdc is the efficiency of the DC-DC converter, I_in and I_out are the input and output currents (A), respectively, and U_in and U_out are the input and output voltages (V), respectively.

3. Parameter Matching Optimization Model

3.1. Performance Requirements

Energy demand is a critical factor in ensuring the endurance of electric loaders, while power demand is essential for guaranteeing their instantaneous performance. The total power demand curve for a hybrid power system during one operating cycle of an electric loader is shown in Figure 5.

From Figure 5, it can be observed that the duration of one operating cycle of the electric loader is 40 s. The positive power demand indicates the driving power supplied by the power system, with a maximum value of 188.41 kW. The negative power demand represents the power fed back to the power system, with a maximum value of 13.68 kW. The integral of the positive power demand portion corresponds to the driving energy provided by the power system during one operating cycle of the electric loader, which calculates to 1389.63 kJ. Conversely, the integral of the negative power demand portion represents the energy fed back to the power system, calculated at 21.72 kJ.

3.2. Objective Functions

3.2.1. Cost Function

In electric loaders, the cost of the power supply accounts for nearly half of the total vehicle cost [25]. This high price limits the promotion and application of electric loaders. Therefore, during the parameter matching process for a hybrid power system, it is necessary to consider cost as one of the optimization objectives. The cost function for the hybrid power system can be expressed as

$$c_{\mathrm{total}}=N_{\mathrm{b}}{c}_{\mathrm{b}}+N_{\mathrm{c}}{c}_{\mathrm{c}}+P_{\mathrm{dcdc}}{c}_{\mathrm{dcdc}}$$

(12)

where c_total is the total cost of the hybrid power system (CNY), c_b is the cost of the battery cell (CNY), c_c is the cost of the supercapacitor cell (CNY), P_dcdc is the power of the DC-DC converter (W), c_dcdc is the cost of the DC-DC converter (CNY/W).

3.2.2. Energy Loss Function

The power loss of the hybrid power system is closely related to the number of series and parallel connections of the battery and supercapacitor. To maximize energy utilization efficiency during the parameter matching process, the energy efficiency of the hybrid power system should also be considered as an optimization objective. Given that the operating conditions of electric loaders are cyclical, the energy loss during one operating cycle of the hybrid power system will be examined. The energy loss function can be expressed as

$$E_{\mathrm{loss}}={\displaystyle {\int }_0^T(P_{\mathrm{b}\_\mathrm{loss}}+P_{\mathrm{c}\_\mathrm{loss}}+P_{\mathrm{dcdc}\_\mathrm{loss}})\mathrm{d}t}$$

(13)

where E_loss is the energy loss during one operating cycle of the hybrid power system (J), P_{b_loss} is the power loss of the battery (W), P_{c_loss} is the power loss of the supercapacitor (W), P_{dcdc_loss} is the power loss of the DC-DC converter (W), T is the time for one operating cycle of the electric loader (s).

The power loss of the battery can be expressed as

$$P_{\mathrm{b}\_\mathrm{loss}}=N_{\mathrm{b}}(R_{\mathrm{b}2}{I}_{\mathrm{bc}}^2+U_{\mathrm{b}1}^2/R_{\mathrm{b}1})$$

(14)

$$I_{\mathrm{bc}}={\displaystyle \frac{P_{\mathrm{b}}}{N_{\mathrm{b}}{U}_{\mathrm{bc}}}}$$

(15)

The power loss of the supercapacitor can be expressed as

$$P_{\mathrm{c}\_\mathrm{loss}}=N_{\mathrm{c}}{I}_{\mathrm{cc}}^2{R}_{\mathrm{c}}$$

(16)

$$I_{\mathrm{cc}}={\displaystyle \frac{P_{\mathrm{c}}}{N_{\mathrm{c}}{U}_{\mathrm{cc}}}}$$

(17)

The power loss of the DC-DC converter can be expressed as

$$P_{\mathrm{dcdc}\_\mathrm{loss}}=P_{\mathrm{c}}\left(1-{\eta }_{\mathrm{dcdc}}\right){\eta }_{\mathrm{dcdc}}^{-\mathrm{z}}$$

(18)

When the supercapacitor is in discharge mode, z takes the value 0; when the supercapacitor is in charge mode, z takes the value 1.

3.2.3. Capacity Degradation Function

Batteries often face aging issues during operation, and mitigating capacity degradation can extend their lifespan. The primary factors affecting capacity degradation include the discharge rate, depth of discharge, and ambient temperature [26]. Under certain load conditions, the discharge rate is determined by the number of battery cells connected in parallel. The greater the number of parallel cells, the lower the discharge rate of each individual cell, which consequently slows the rate of lifespan degradation. Therefore, the lifespan of the battery is considered one of the optimization objectives, represented by an empirical formula for capacity degradation over one operating cycle [27]:

$$Q_{\mathrm{loss}}={\displaystyle {\int }_0^T(a{T}_{\mathrm{b}}^2+b{T}_{\mathrm{b}}+c)\exp [(d{T}_{\mathrm{b}}+e)I_{\mathrm{rate}}]I_{\mathrm{bc}}\mathrm{d}t}$$

(19)

$$I_{\mathrm{rate}}={\displaystyle \frac{P_{\mathrm{b}}}{N_{\mathrm{b}}{U}_{\mathrm{bc}}{C}_{\mathrm{bc}}}}$$

(20)

where Q_loss is the capacity degradation of the battery during one operating cycle (Ah), I_rate is the charge–discharge rate, C_bc is the capacity of a single battery cell (Ah), and a, b, c, d, e are fitting parameters.

The values of the fitting parameters in Equation (19) are shown in

Table 1. Fitting parameter values.

Parameters	Values	Parameters	Values
a	8.61 × 10−6	d	−6.7 × 10−3
b	−5.13 × 10−3	e	2.35
c	7.63 × 10−1

[27].

In the energy loss function and the capacity degradation function, the magnitude of the target values is influenced not only by the number of series and parallel connections of the battery and supercapacitor but also by the power distribution between them. Therefore, during the parameter matching process for the hybrid power system, it is essential to clarify the power distribution values for the battery and supercapacitor. Based on the energy storage characteristics of both components, a rule-based power distribution strategy is established.

The battery has a high energy density but low power density. To avoid accelerating battery aging, the battery should discharge as much as possible, subject to a power threshold P_a. When the power demand P_r of the electric loader is between 0 and P_a, the battery supplies the entire power. When P_r exceeds P_a, the battery only provides power up to P_a.

The supercapacitor, on the other hand, has a high power density but low energy density, and its high energy conversion efficiency should be fully utilized. Therefore, when P_r is less than 0, indicating an energy recovery condition, the supercapacitor recovers all the power. When P_r exceeds P_a, the supercapacitor supplies power between P_a and P_r.

The power distribution criteria between the battery and supercapacitor can be expressed as

$$P_{\mathrm{b}}(t)=\left\{\begin{array}{cc}0& P_{\mathrm{r}}(t)\le 0\\ P_{\mathrm{r}}(t)& 0<P_{\mathrm{r}}(t)\le P_{\mathrm{a}}\\ P_{\mathrm{a}}& P_{\mathrm{r}}(t)>P_{\mathrm{a}}\end{array}\right.$$

(21)

$$P_{\mathrm{c}}(t)=\left\{\begin{array}{cc}{P}_{\mathrm{r}}(t){\eta }_{\mathrm{dcdc}}& P_{\mathrm{r}}(t)\le 0\\ 0& 0<P_{\mathrm{r}}(t)\le P_{\mathrm{a}}\\ (P_{\mathrm{r}}(t)-P_{\mathrm{a}})/{\eta }_{\mathrm{dcdc}}& P_{\mathrm{r}}(t)>P_{\mathrm{a}}\end{array}\right.$$

(22)

where P_r is the power demand of the electric loader (W), P_a is the power threshold (W).

3.3. Constraints

3.3.1. Battery Voltage Constraint

In the hybrid power structure, the battery is directly connected to the DC bus, and the load motor draws power directly from the DC bus. Therefore, the battery voltage should match the rated voltage of the load motor, resulting in the following constraint:

$$N_{\mathrm{bs}}={\displaystyle \frac{U_{\mathrm{m}}}{U_{\mathrm{bc}}}}$$

(23)

where U_m is the rated voltage of the load motor (V).

3.3.2. Supercapacitor Voltage Constraint

The supercapacitor controls its charge and discharge power through a DC-DC converter. The transfer efficiency of the DC-DC converter is inversely proportional to the voltage difference across its terminals. The closer the voltage of the supercapacitor is to that of the battery, the higher the transfer efficiency of the DC-DC converter. However, to prevent the supercapacitor from charging the battery, the voltage of the supercapacitor must be less than the cutoff voltage of the battery, leading to the following constraint:

$$N_{\mathrm{cs}}\le {\displaystyle \frac{N_{\mathrm{bs}}{U}_{\mathrm{bc}\_\mathrm{cutoff}}}{U_{\mathrm{cc}}}}$$

(24)

where U_{bc_cutoff} is the cutoff voltage of the battery cell (V).

3.3.3. Supercapacitor Energy Constraint

When the power demand P_r exceeds the power threshold P_a, the supercapacitor supplies power between P_a and P_r. During operation, the power demand curve of the electric loader may have multiple segments that exceed P_a. Thus, the energy stored in the supercapacitor must be greater than the maximum cumulative energy of these segments, resulting in the following constraint:

$$E_{\mathrm{c}}\ge {\displaystyle \frac{E_{\max }}{{\eta }_{\mathrm{dcdc}}}}$$

(25)

$$E_{\max }={\displaystyle {\int }_{t_1}^{t_2}(P_{\mathrm{r}}-P_{\mathrm{a}})dt}$$

(26)

where E_c is the effective output energy of the supercapacitor (J), E_max is the maximum cumulative energy of the segments where P_r exceeds P_a (J).

The effective output energy of the supercapacitor can be expressed as

$$E_{\mathrm{c}}={\displaystyle \frac{1}{2}}{N}_{\mathrm{c}}{C}_{\mathrm{cc}}(U_{\mathrm{cc}\_\max }^2-U_{\mathrm{cc}\_\min }^2)$$

(27)

where C_cc is the capacitance of a single supercapacitor cell (F).

3.3.4. Total Energy Constraint

To ensure the endurance of the electric loader, the energy storage capacity of the hybrid power system must meet the total energy demand for the required operating time. The corresponding constraint is given by

$$E_{\mathrm{total}}\le E_{\mathrm{b}}+E_{\mathrm{c}}{\eta }_{\mathrm{dcdc}}$$

(28)

where E_total is the total energy demand over the specified working period (J), E_b is the effective output energy of the battery (J).

The effective output energy of the battery can be expressed as

$$E_{\mathrm{b}}=3600N_{\mathrm{b}}{C}_{\mathrm{bc}}{U}_{\mathrm{bc}}{\eta }_{\mathrm{dod}}$$

(29)

where η_dod is the depth of discharge of the battery.

3.3.5. Maximum Power Constraint

To ensure the instantaneous burst capability of the electric loader, the discharge power of the hybrid power system must meet the peak power demand of the electric loader. The corresponding constraint is given by

$$P_{\mathrm{peak}}\le N_{\mathrm{b}}{U}_{\mathrm{bc}}{I}_{\mathrm{bc}\_\max }+N_{\mathrm{c}}{U}_{\mathrm{cc}}{I}_{\mathrm{cc}\_\max }{\eta }_{\mathrm{dcdc}}$$

(30)

where P_peak is the peak power demand of the electric loader (W), I_{bc_max} is the maximum operating current of the battery cell (A), I_{cc_max} is the maximum operating current of the supercapacitor cell (A).

3.4. Optimization Model

The parameters of the battery cells and supercapacitor cells studied in this paper are presented in

Table 2. Parameters of the battery cell.

Parameters	Values	Parameters	Values
Nominal capacity	20 Ah	Maximum operating current	40 A
Nominal voltage	3.2 V	Energy density	163.06 Wh/kg
Discharge cut-off voltage	2.5 V	Power density	0.32 kW/kg
DC internal resistance	1.83 mΩ	Unit price	CNY 65

and

Table 3. Parameters of the supercapacitor cell.

Parameters	Values	Parameters	Values
Nominal capacity	3000 F	Maximum peak current	2406 A
Nominal voltage	3 V	Energy density	7.30 Wh/kg
Minimum terminal voltage	1.5 V	Power density	7.35 kW/kg
DC internal resistance	0.29 mΩ	Unit price	CNY 168

, respectively. The relevant parameters of the objective function and constraints are shown in

Table 4. Relevant parameters in the objective function and constraints.

Parameters	Values	Parameters	Values
Power of the DC-DC converter	120 kW	Cost of the DC-DC converter	0.3 CNY/W
Efficiency of the DC-DC converter	0.95	Operating temperature of the battery	298 K
Power threshold	100 kW	Rated voltage of the motor	540 V
Depth of discharge of the battery	80%	Endurance time of the electric loader	5 h
Peak power demand	188.41 kW	Total energy demand during endurance time	625,333.5 kJ

.

From Table 2, the terminal voltage of the battery cell is 3.2 V. According to Table 4, the rated voltage of the motor is 540 V. Therefore, using Equation (23), the required number of battery cells in the series is calculated to be 169. From Table 2, the cutoff voltage of the battery is 2.5 V, and from Table 3, the terminal voltage of the supercapacitor cell is 3 V. To ensure the efficiency of the DC-DC converter, the number of supercapacitor cells in the series is calculated to be 140 using Equation (24).

Based on the above analysis, the number of series-connected battery and supercapacitor cells has been determined. Consequently, the number of parallel-connected cells will be treated as an optimization variable, leading to the establishment of a parameter matching optimization model for the hybrid power system.

$$\begin{array}{l}\mathrm{find} X=\left[N_{\mathrm{bp}},N_{\mathrm{cp}}\right]\\ \left\{\begin{array}{l}\min \left\{c_{\mathrm{total}},E_{\mathrm{loss}},Q_{\mathrm{loss}}\right\}\hfill \\ s.t. 1\le N_{\mathrm{bp}}\le 40\hfill \\ \hspace{1em} 1\le N_{\mathrm{cp}}\le 4\hfill \\ \hspace{1em} {\displaystyle \frac{1}{2}}{N}_{\mathrm{c}}{C}_{\mathrm{cc}}(U_{\mathrm{cc}\_\max }^2-U_{\mathrm{cc}\_\min }^2)\ge E_{\max }/{\eta }_{\mathrm{dcdc}}\hfill \\ \hspace{1em} E_{\mathrm{total}}\le 3600N_{\mathrm{b}}{C}_{\mathrm{bc}}{U}_{\mathrm{bc}}{\eta }_{\mathrm{dod}}+{\displaystyle \frac{1}{2}}{N}_{\mathrm{c}}{C}_{\mathrm{cc}}(U_{\mathrm{cc}\_\max }^2-U_{\mathrm{cc}\_\min }^2){\eta }_{\mathrm{dcdc}}\hfill \\ \hspace{1em} P_{\mathrm{peak}}\le N_{\mathrm{b}}{U}_{\mathrm{bc}}{I}_{\mathrm{bc}\_\max }+N_{\mathrm{c}}{U}_{\mathrm{cc}}{I}_{\mathrm{cc}\_\max }{\eta }_{\mathrm{dcdc}}\hfill \end{array}\right.\end{array}$$

(31)

4. Parameter Matching Results

Multi-objective optimization is a mathematical and computational method used to solve problems involving multiple conflicting objectives. These objectives are often competing, so there is no single solution that can optimize all objectives simultaneously. The goal of multi-objective optimization is to find a set of solutions that form the Pareto front, allowing for trade-offs and choices between different objectives. NSGA-II is an important algorithm in the field of multi-objective optimization. Its core idea is to classify individuals in a population into different layers through non-dominated sorting, thereby constructing the Pareto front. The specific implementation process of the NSGA-II algorithm is as follows:

1. Initialize a population containing multiple individuals randomly or through other methods.

2. Perform non-dominated sorting on the individuals in the population, dividing them into different Pareto levels.

3. Generate the next generation population from the parent population using selection, crossover, and mutation operations.

4. Combine the parent population and the offspring population to form a larger population.

5. Perform non-dominated sorting and crowding distance calculation on the combined population.

6. Use the elite strategy to retain the best individuals and generate a new parent population.

7. Check if the termination condition is met, such as reaching the maximum number of iterations or finding a satisfactory solution. If the condition is met, the algorithm ends; otherwise, return to step 3.

By following these steps, NSGA-II aims to find a set of Pareto-optimal solutions that balance the trade-offs between the conflicting objectives in the problem.

Based on the MATLAB 2022 platform, the NSGA-II algorithm is selected to solve the hybrid power system parameter-matching optimization model. The main parameter settings of the NSGA-II algorithm are shown in

Table 5. Main parameters of NSGA-II algorithm.

Parameters	Values	Parameters	Values
Initial population size	100	Selection operator	Roulette
Crossover operator	0.8	Mutation operator	0.05
Maximum number of iterations	5000	Error	1 × 10−6

.

In the NSGA-II algorithm, input the range of the number of batteries and supercapacitors in parallel, input the three optimization objectives of power cost function, power energy loss function, and battery capacity degradation function, input the three constraints of supercapacitor energy constraint function, total power energy constraint function, and total power constraint function, and then perform multi-objective optimization for the parameters of the hybrid power system. The Pareto front of the obtained hybrid power system parameters is shown in Figure 6.

As shown in Figure 6a, the distribution of Pareto optimal solutions among various optimization objectives is uniform, indicating the absence of local convergence and ensuring solution diversity. Figure 6b illustrates that the power system cost and battery capacity degradation are mutually constrained. When the number of parallel batteries is increased, the current through each individual battery decreases, which delays capacity degradation; however, the overall cost of the power system rises due to the increased number of batteries. The trend of the Pareto front in Figure 6c is similar to that in Figure 6b; as the number of parallel batteries increases, the total internal resistance of the batteries decreases, reducing energy loss, which similarly demonstrates a mutually constraining relationship with power system cost. In Figure 6d, capacity degradation is positively correlated with energy loss, indicating a synergistic interaction between the two objectives.

The Pareto solution set for the parameter matching of the hybrid power system is presented in Figure 7.

From Figure 7, it can be observed that the number of parallel-connected supercapacitors primarily clusters around 1, while the number of parallel-connected batteries is more evenly distributed between 20 and 40. Given the high cost of batteries, which significantly impacts the selling price of the electric loader, this paper focuses on reducing power system costs. The final results of the hybrid power system parameter matching are summarized in

Table 6. Parameters of the hybrid power system.

Energy Storage Component	Number of Series Units	Number of Parallel Units	Voltage	Capacity
Battery	169	22	540.8 V	440 Ah
Supercapacitor	140	1	420 V	21.43 F

.

5. Experiment and Analysis

To verify the performance of the hybrid power system after multi-objective optimization, experiments were conducted to compare a single battery system with the hybrid power system. The experimental platform for the electric loader hybrid power system is illustrated in Figure 8. The test setup primarily includes the hybrid power cabinet and the electric loader (968EV, XCMG, Xuzhou, China). Key components consist of the battery (BAT540V, WLDPE, Shanghai, China), supercapacitor (SUP400V, CAS-SCAP, Chongqing, China), and DC-DC converter (DC800V, WLDPE, Shanghai, China). The battery used is a lithium-ion battery with lithium iron phosphate material, featuring a voltage range of 420 to 613 V and an operating temperature range of −20 to 60 °C. Communication is achieved through CAN, and the battery management system equipped can monitor the voltage, temperature, and SOC of the battery pack in real-time, ensuring maximum utilization of the battery’s stored energy while maintaining safety. The supercapacitor has a working temperature range of −40 to 65 °C and a cycle life of up to 1 million times. Communication is also achieved through CAN, and the capacitor management system equipped can monitor the voltage, temperature, and SOC of the supercapacitor pack in real-time, ensuring safe and stable operation of the supercapacitor.

Through experimentation, the output power of the battery and supercapacitor in the hybrid power system during one operating cycle of an electric loader is obtained, as shown in Figure 9. Additionally, the current of the single battery and the battery in the hybrid power system is depicted in Figure 10.

From Figure 9, it can be observed that in the hybrid power system, the battery primarily responds to positive power below a certain power threshold, while the supercapacitor mainly handles power above this threshold and negative power. The hybrid power system leverages the supercapacitor’s ability to “peak shave” and “valley fill,” reducing the battery workload during high power demands and enhancing energy recovery efficiency during negative power demands. From Figure 10, compared to a single battery system, the hybrid power system reduces the charging and discharging currents of the battery, demonstrating that the hybrid power system effectively utilizes the supercapacitor high power density to minimize the battery peak currents.

The experiments yielded the battery SOC curves, total energy consumption curves, and battery capacity degradation curves for both the single battery and hybrid power system during a single operating cycle of the electric loader, as shown in Figure 11, Figure 12 and Figure 13, respectively.

From Figure 11, at the end of a working cycle of the electric loader, the battery SOC of the single battery and the hybrid power system are 0.99833 and 0.99861, respectively, and the hybrid power system increases by 0.028% compared with the single battery system. From Figure 12, it can be observed that the energy consumption for the single battery and hybrid power system is 1504.81 kJ and 1454.92 kJ, respectively. The hybrid power system reduces total energy consumption by 3.32% compared to the single battery system. This reduction is attributed to the higher efficiency of the supercapacitor. In the single battery system, the battery must output a large current to provide the peak power demand of the machine, leading to significant internal resistance losses. In contrast, the hybrid power system utilizes the supercapacitor to absorb the peak power of the electric loader, resulting in higher energy storage and utilization efficiency.

Figure 13 indicates that as the operating time of the electric loader increases, the battery capacity degradation for both energy storage systems gradually increases. At the end of one operating cycle, the capacity degradation of a single battery is 9.99 × 10⁻⁵ Ah, while that of the hybrid power system is 8.93 × 10⁻⁵ Ah, representing a 10.61% reduction in degradation compared to the single battery. Generally, a battery is considered to have reached the end of its life when its capacity has decayed to 80%. If we calculate based on the degree of battery capacity decay during one operating cycle of the electric loader mentioned above, the number of operating cycles for a single battery and the battery in the hybrid power system can reach 8.8 × 10⁵ and 9.85 × 10⁵, respectively. Compared to a single battery, the hybrid power system increases the number of operating cycles by 11.93%. This improvement arises because in the single battery system, the electric loader’s power demand is solely supplied by the battery, resulting in high charge and discharge currents, elevated temperatures, and accelerated capacity degradation. Conversely, the hybrid power system benefits from the high power density of the supercapacitor, allowing it to rapidly absorb and release transient high power, thereby reducing the discharge rate of the battery and delaying its capacity degradation.

The performance indicators for both the single battery system and hybrid power system are summarized in

Table 7. Performance metrics of a single battery system vs. hybrid power system.

Energy Storage System	Cost (CNY)	Energy Consumption per Operating Cycle (kJ)	Capacity Degradation per Operating Cycle (Ah)
Single battery system	2.64 × 105	1504.81	9.99 × 10−5
Hybrid power system	3.01 × 105	1454.92	8.93 × 10−5

.

Table 7 reveals that, compared to the traditional single battery power system, the cost of the parameter-matched hybrid power system has increased by CNY 37,000. However, the energy consumption per operating cycle has decreased by 3.32%, and battery capacity degradation has been reduced by 10.61%. Thus, while incurring a modest increase in cost, both operational energy consumption and battery lifespan indicators have been optimized, further validating the feasibility and effectiveness of the proposed parameter matching method for the hybrid power system.

6. Conclusions

To achieve optimal performance of the hybrid power system, this study proposes a multi-objective optimization parameter matching method for a hybrid power system based on the NSGA-II algorithm. The main conclusions are as follows:

(1) The objective functions of power system cost, power system efficiency, and battery lifespan were established, and the constraint conditions of battery voltage, supercapacitor voltage, supercapacitor energy, total power supply, and maximum power system supply were set. The optimization model of hybrid power system parameters for the electric loaders was formed with the number of parallel batteries and supercapacitors as optimization parameters.

(2) Utilizing the NSGA-II algorithm, the optimal parameter-matching results for the hybrid power system were obtained. The number of series-connected and parallel-connected batteries are 169 and 22, respectively, while the number of series-connected and parallel-connected supercapacitors are 140 and 1, respectively. Experimental results indicate that, compared to a single battery system, the optimized hybrid power system reduces the energy consumption per operating cycle of the electric loader by 3.32% and decreases battery capacity decay by 10.61%, all while only slightly increasing costs.