Parameter Matching Methods for Li Battery–Supercapacitor Hybrid Energy Storage Systems in Electric Buses

Abstract

The parameter matching of composite energy storage systems will affect the realization of control strategy. In this study, the effective energy and power utilizations of an energy storage source were defined. With the miniaturization of a composite energy storage system as the optimization goal, the linear programming simplex method was employed to obtain the optimized masses of Li batteries and supercapacitors under the constraints of maximum speed, climbing gradient, acceleration time and cost-effectiveness. As the module numbers shall be integers, the matching results were modified in combination with the graphical method. Owing to the influences of parameter matching schemes on the overall performance and battery life, the critical points of constraints were analyzed and the most appropriate matching numerical points of the composite energy storage system were identified. Simulation and experimental analysis were conducted under practical urban road conditions in China. The results show that the proposed matching method delivers accurate results. Compared with conventional electric buses, the mileage and overall performance of the prototype bus are improved.

1. Introduction

The parameter design of hybrid energy storage systems (HESS) includes power capacity and energy capacity. Currently, the formula method and accurate simulation method have been widely applied. According to the power performance indexes of vehicles and power and energy requirements under typical conditions, the energy and power that the energy storage source system is supposed to provide were separated based on the designed electric transmission system. Then, the specific scheme was designed according to specific energy, specific power, cost, temperature range, reliability and safety of current energy storage source and optimized with a parameter of HESS as the optimization objective. Since the 1980s, studies of composite energy storage sources have been proposed and applied in the energy systems of earth satellites. With the development of battery and supercapacitors, researchers have committed to the matching and control strategy of composite energy storage sources consisting of supercapacitor packs and theoretical and practical explorations of applications of supercapacitors in electric vehicles. Tehrani et al. published a series of articles on supercapacitors and composite energy storage sources and summarized the principle and performance of various supercapacitor energy storages. It was pointed out that supercapacitors are suitable for cases with large power demand but short time [1,2,3,4]. Li et al. investigated simulation and control of composite energy storage sources in Advanced Vehicle Simulator (ADVISOR) and proposed general methods for parameter matching of composite energy storage sources [5,6,7]. Giannoutsos et al. achieved power distribution of the battery and supercapacitor in a composite energy source using low-pass filtering [8,9]. Composite energy storage sources with supercapacitors have been investigated [10,11]. Cao et al. connected DC/DC with a supercapacitor and in parallel with a battery, followed by parameter matching using the formula method. The results showed that the composite energy storage system can effectively provide driving power for motors [12,13]. Shen et al. used the DIRECT (dividing rectangles) method to optimize the multiobjective function based on the total mass of the HESS and the battery capacity. The simulation results showed that the mass was reduced by 66 kg after optimization [14,15]. Hu et al. investigated the impacts of price fluctuation of a single energy storage component on the overall performance of vehicles using the convex optimization method, with cost as the optimization objective [16,17]. Araujo et al. combined parameter matching and energy management based on filtering and nonlinear optimization. The results showed that the total cost was reduced by 20% and the energy utilization was improved by 7.8%. Ravey et al. applied a statistical method to parameter matching of HESSs and achieved good results [18,19].

In China, the methods and principles of parameter matching of composite energy storage systems have been investigated [20,21]. Hong Kong university proposed optimization of fuel cost-effectiveness of electric vehicles by adjusting the ratio of the total mass of energy storage source to the vehicle mass, as well as the ratio of the mass of high specific power energy storage source to the total mass of composite energy storage source, without degradations in mileage and other performances. Harbin Institute of Technology proposed matching strategies based on optimization of the mass ratio and mixing ratio of electric vehicles. A model was established to quantitatively analyze the impact of the mass ratio and mixing ratio on the overall performance of vehicles [22]. Beijing Institute of Technology proposed that other parameters can be determined based on the time constant of the supercapacitor. The number of cells in parallel and maximum output power of the battery pack were optimized with the physical parameters of the composite energy storage source and the charge/discharge rate of the battery as the optimization objectives. The results showed that the acceleration time of the electric bus from 0 to 70 km/h was reduced by 9 s, while the fluctuation range of the output current of the battery decreased significantly [23]. Chongqing University proposed linear optimization of HESS parameters from the perspective of the overall performance of the vehicle. The results showed that compared with the conventional electric vehicles, the battery was protected and the power consumption per 100 km was reduced by 8.55% [24]. Jilin University proposed parameter optimization based on the measured data of vehicles, with charging/discharging capacity and service life of the HESS energy storage component as the optimization objectives [25,26].

In this study, parameter optimization was achieved using the linear programming method from the perspective of power and energy utilizations. The minimum mass of the composite energy storage system was regarded as the optimization objective, while the minimum mass of the energy storage system was regarded as the objective function. The integer solution of minimum mass was obtained in combination with the graphical method. From the perspective of requirements and evaluation indexes of power and cost-effectiveness of vehicles, the influences of characteristic factors of the evaluation indexes, mass ratio of the HESS and conditions on vehicle performances were analyzed.

2. Theories and Analysis of Parameter Matching

The basic parameters of a Cens bus (the target model) are shown in

Table 1. Vehicle performance requirements and parameters.

	System Parameters	Value
Overall vehicle performance requirements	Maximum speed (km/h)	≥80
	Maximum climbing degree (%)	≥16
	Acceleration time (s)	≤25 (0~50 km/h)
	Driving range (km)	≥250 km (40 km/h uniform speed)
Overall vehicle parameters	Size (mm)	12,000 × 2530 × 3080
	Unladen mass (kg)	12,000
	Full load mass (kg)	17,800
	Air resistance coefficient Cd	0.65
	Rolling resistance coefficient ff	0.0165
	Rolling mass conversion factor	1.08
	Windward area A (m2)	6.385
	Wheel radius (m)	0.469

. Herein, Cens ferric phosphate Li battery (CBP2450) was employed as the Li battery module unit. Its nominal voltage, specific power, specific energy, nominal capacitance and mass are 25.6 V, 1000 W/kg, 215.49 W∙h/kg, 50 A∙h and 19 kg, respectively. BMOD0165 (Maxwell, USA) was employed as a supercapacitor module unit. Its rated voltage, peak voltage, nominal capacitance, peak current and nominal mass are 48 V, 51 V, 165 F, 1900 A and 13.5 kg, respectively. The rated voltage, rated power and rated revolution speed of the motor are 300 V, 120 kW and 1860 r/min, respectively.

3. Theories and Analysis of Parameter Matching

3.1. Composite Energy Storage Device

In composite energy storage devices, two or more energy storage devices are combined so that the advantages of each energy source can be exploited, while its drawbacks can be compensated for by other energy storage devices. The combination of chemical battery and supercapacitor can overcome the low specific power of the battery and low specific energy of the supercapacitor while achieving high specific energy and high specific power [15,16]. It is assumed that the composite energy storage source consists of n (typically 2 or 3) basic energy storage devices:

$$\{\begin{array}{cc}\hfill P_{\mathrm{load}}& ={\eta }_1\times P_1+{\eta }_2\times P_2+\dots +{\eta }_n\times P_n\hfill \\ & ={\eta }_1\times {\rho }_{\mathrm{P}1}\times m_1+{\eta }_2\times {\rho }_{\mathrm{P}2}\times m_2+\dots +{\eta }_n\times {\rho }_{\mathrm{P}n}\times m_n\hfill \\ \hfill W_{\mathrm{load}}& ={\eta }_1\times W_1+{\eta }_2\times W_2+\dots +{\eta }_n\times W_n\hfill \\ & ={\eta }_1\times {\rho }_{\mathrm{W}1}\times m_1+{\eta }_2\times {\rho }_{\mathrm{W}2}\times m_2+\dots +{\eta }_3\times {\rho }_{\mathrm{W}n}\times m_n\hfill \end{array}$$

(1)

where $n=\left\{1,2,\dots ,n\right\}$; $P_{\mathrm{load}}$ and $W_{\mathrm{load}}$ represent the power and energy needs of the load, respectively; $P_n$ and $W_n$ represent the power and energy provided by nth device, respectively; ${\eta }_n$ is the efficiency of the corresponding power converter; and ${\rho }_{\mathrm{P}n}$ and ${\rho }_{\mathrm{W}n}$ are power and energy densities, respectively.

In this study, the combination of a Li battery and a supercapacitor was considered:

$$\{\begin{array}{cc}\hfill P_{\mathrm{load}}& ={\eta }_1\times P_{\mathrm{batt}}+{\eta }_2\times P_{\mathrm{capa}}\hfill \\ & ={\eta }_1\times {\rho }_{\mathrm{P}\_\mathrm{batt}}\times m_{\mathrm{batt}}+{\eta }_2\times {\rho }_{\mathrm{P}\_\mathrm{capa}}\times m_{\mathrm{capa}}\hfill \\ \hfill W_{\mathrm{load}}& ={\eta }_1\times W_{\mathrm{batt}}+{\eta }_2\times W_{\mathrm{capa}}\hfill \\ & \hfill ={\eta }_1\times {\rho }_{\mathrm{W}\_\mathrm{batt}}\times m_{\mathrm{batt}}+{\eta }_2\times {\rho }_{\mathrm{W}\_\mathrm{capa}}\times m_{\mathrm{capa}}\end{array}$$

(2)

3.1.1. Technical Indicators of Typical Battery

For electric vehicles, the battery is the main energy source and converts chemical energy into electric energy for driving. In terms of the development of battery technology, current batteries include Pb–acid batteries, Ni–Cd batteries, Ni–Fe batteries, Ni–Zn batteries, Ni–metal hydride batteries, Al–air batteries, Fe–air batteries, Zn–air batteries and Li-ion batteries.

Table 2. Comparison of characteristics of different batteries.

Battery Type	Specific Energy (W·h/kg)	Specific Power (W/kg)	Cycle Life (Times)	Price (USD/kW·h)
Pb–acid	20~100	50~400	500~2000	120~150
Ni–Cd	40~60	80~350	600~3000	250~350
Ni–Fe	50~60	80~150	1500~2000	200~400
Ni–Zn	55~75	150~300	600~1200	100~300
Ni–metal hydride	70~95	200~300	750~1200	200~350
Al–air	200~300	160	--	--
Fe–air	80~120	90	500+	50
Zi–air	100~220	30~80	600+	90~120
Li-ion	100~130	150~250	1000+	About 200

summarizes the specific energy, specific power, cycle life and price of typical batteries. Among them, the Li battery has been widely applied in electric vehicles by virtue of its excellent performance and environmentally friendly nature. However, it is limited by some drawbacks, including high price and poor low-temperature performance.

3.1.2. Parameters of Typical Supercapacitors

According to the U.S. Department of Energy, specific energy and specific power of supercapacitors used in electric and hybrid electric vehicles shall be no less than 5 W·h/kg and 500 W/kg, respectively, and the improved values should exceed 15 W·h/kg and 1600 W/kg, respectively. To date, no applicable supercapacitors that can fully meet these requirements have been reported. Nevertheless, high storage and transient large storage characteristics of supercapacitors play a dominant role in their applications, and the cost has dropped to 2.85 USD/kJ or 0.01 USD/F. The comparison of characteristics of different supercapacitors is shown in

Table 3. Comparison of characteristics of different supercapacitors.

Brand Name	Rated Voltage (V)	Rated Capacitance (F)	Series Equivalent Resistance (mΩ)	Energy Density (W·h/kg)	Power Density (W/kg)	Weight (kg)
Maxwell	2.7	3000	0.29	5.52	5400	0.55
Nesscap	2.7	5000	0.25	5.44	5063	0.93
Panasonic	2.5	2500	0.43	3.70	1035	0.395
EPCOS	2.7	3400	0.45	4.3	760	0.60
ESMA	1.7	80,000	0.5	13.38	583	2.4
Spscap	2.7	9500	0.28	7.40	5010	1.3
BMOD0115	42	145	10	2.22	2900	16
BMOD0117	14	435	4	1.82	1900	6.5
BCAP0010	2.5	2600	0.7	4.3	4300	0.525
Company A	1.6	140,000	0.3	12.0	205	5.0
Company B	2.7	20	49	9.88	1000	0.116
Company C	2.7	100	15	4.9	1677	0.021
Company D	2.7	3000	0.4	3.85	1020	0.41
Company E	1.2	1500	2.4	3.3	3800	0.36

.

3.2. HESS Constraints

3.2.1. Technical Indexes of Typical Batteries

In addition to providing power and energy for the entire system, the composite energy storage device also consumes power and energy. Herein, the proposed effective energy utilization and power utilization constraints were analyzed.

The power required by the power source to drive the motor can be obtained through the Equations (A1)–(A3) in Appendix A (1). In addition to providing power for the power-source-driven motor, batteries also consume energy during driving:

$$P_{\mathrm{batt}}={\rho }_{\mathrm{P}\_\mathrm{batt}}\cdot m_{\mathrm{batt}}$$

(3)

$${\eta }_{\mathrm{P}\_\mathrm{batt}}=\frac{P_{\mathrm{batt}}-P_{\mathrm{loss}\_\mathrm{batt}}}{P_{\mathrm{batt}}}$$

(4)

where ${\rho }_{\mathrm{W}\_\mathrm{capa}}$ and ${\rho }_{\mathrm{W}\_\mathrm{batt}}$ are energy densities of supercapacitor and battery, respectively; ${\rho }_{\mathrm{P}\_\mathrm{capa}}$ and ${\rho }_{\mathrm{P}\_\mathrm{batt}}$ are power densities of supercapacitor and battery, respectively; and ${\eta }_{\mathrm{P}\_\mathrm{capa}}$ and ${\eta }_{\mathrm{P}\_\mathrm{batt}}$ are effective power utilizations of supercapacitor and battery, respectively.

$$\eta =\{\begin{array}{l}1-\frac{1}{3600{\rho }_{\mathrm{P}\_\mathrm{batt}}{\eta }_{\mathrm{T}}}\cdot (fg\cos \alpha +g\sin \alpha )\cdot v(\mathrm{Consider}\mathrm{the}\mathrm{battery})\\ 1-\frac{1}{3600{\rho }_{\mathrm{P}\_\mathrm{capa}}{\eta }_{\mathrm{T}}}\cdot (fg\cos \alpha +g\sin \alpha )\cdot v(\mathrm{Consider}\mathrm{the}\mathrm{supercapacitor})\end{array}$$

(5)

With constant slope angle (α), power utilization is related to speed change and power density, and it is a positive number between 0 and 1. Therefore, the constraints of effective power utilization can be obtained by:

$$\frac{(f\cdot g\cdot \cos \alpha +g\cdot \sin \alpha )\cdot v}{3600{\eta }_{\mathrm{T}}}-\frac{m_{\mathrm{capa}}\cdot {\rho }_{\mathrm{P}\_}{}_{\mathrm{capa}}}{(m_{\mathrm{capa}}+m_{\mathrm{batt}})}-\frac{m_{\mathrm{batt}}\cdot {\rho }_{\mathrm{P}\_}{}_{\mathrm{batt}}}{(m_{\mathrm{capa}}+m_{\mathrm{batt}})}\le 0$$

(6)

Likewise, the constraints of effective energy utilization can be obtained by:

$$\frac{(f\cdot g\cdot \cos \alpha +g\cdot \sin \alpha )\cdot S}{3600{\eta }_{\mathrm{T}}}-\frac{m_{\mathrm{capa}}\cdot {\rho }_{\mathrm{W}\_\mathrm{capa}}}{(m_{\mathrm{capa}}+m_{\mathrm{batt}})}-\frac{m_{\mathrm{batt}}\cdot {\rho }_{\mathrm{W}\_\mathrm{batt}}}{(m_{\mathrm{capa}}+m_{\mathrm{batt}})}\le 0$$

(7)

3.2.2. Constraint of Mileage

The most intuitive indicator of cost-effectiveness is single charge mileage or power consumption under cyclic conditions. Herein, the single charge mileage was employed. In this study, the Li battery in the composite power supply structure is directly connected to the brushless DC motor via the controller, and the number of battery packs in series is related to the level of the motor controller. $U_{\mathrm{motor}}$ and $U_{\mathrm{batt}}^{\prime }$ are defined as the voltage level of the motor controller and the nominal voltage of the battery unit, respectively. Hence, the number of battery packs in series can be determined by:

$$n_{\mathrm{ser}\_\mathrm{batt}}=\mathrm{int}(\frac{U_{\mathrm{motor}}}{U_{\mathrm{batt}}^{\prime }})$$

(8)

where int indicates rounding up.

Assuming that $m_{\mathrm{batt}}^{\prime }$ is the mass of a single battery, $m_{\mathrm{batt}}$ is the total mass of battery and p is the number of battery packs in parallel, which is determined according to the design,

$$m_{\mathrm{batt}}=m_{\mathrm{batt}}^{\prime }\cdot \mathrm{int}(\frac{U_{\mathrm{motor}}}{U_{\mathrm{batt}}^{\prime }})\cdot p$$

(9)

Analysis of bidirectional DC/DC efficiency shows that the working efficiency is high when the rated voltages of the high-voltage terminal and the low-voltage terminal are consistent. As a supercapacitor provides a large transient current for initiation, its terminal voltage should be slightly higher than that of the battery terminal. Assuming that $m_{\mathrm{capa}}^{\prime }$ is the mass of single supercapacitor, $m_{\mathrm{capa}}$ is the total mass of supercapacitor and q is the number of battery packs in parallel,

$$m_{\mathrm{capa}}=m_{\mathrm{capa}}^{\prime }\cdot \mathrm{int}(\frac{U_{\mathrm{motor}}}{U_{\mathrm{capa}}^{\prime }})\cdot q$$

(10)

The total mass of the HESS is:

$$m_{\mathrm{HESS}}=m^{\prime }{}_{\mathrm{batt}}\cdot \mathrm{int}(\frac{U_{\mathrm{motor}}}{U_{\mathrm{batt}}^{\prime }})\cdot p+m^{\prime }{}_{\mathrm{capa}}\cdot \mathrm{int}(\frac{U_{\mathrm{motor}}}{U_{\mathrm{capa}}^{\prime }})\cdot q+m_{\mathrm{DCDC}}$$

(11)

When the vehicle travels at a constant speed ($v_{\mathrm{ys}}$), the power demand is smooth and the energy of the driving motor is provided by the battery. Hence, the constraints of mileage can be obtained by:

$$\frac{({\rho }_{\mathrm{W}\_\mathrm{batt}}\cdot m_{\mathrm{batt}})}{(S/v_{\mathrm{ys}})}\ge P_{\mathrm{req}}$$

(12)

The power demand for driving at a constant speed can be expressed as:

$$P_{\mathrm{req}}=\frac{v_{\mathrm{ys}}}{3600{\eta }_{\mathrm{T}}}\cdot (m\cdot g\cdot f\cdot \cos {\alpha }_{\max }+\frac{C_{\mathrm{d}}\cdot A\cdot v_{\mathrm{ys}}^2}{21.15})$$

(13)

The constraints of mileage can be expressed as:

$$\frac{({\rho }_{\mathrm{W}\_\mathrm{batt}}\cdot m_{\mathrm{batt}})}{(S/v)}\ge \frac{v_{\mathrm{ys}}}{3600{\eta }_{\mathrm{T}}}\cdot (m\cdot g\cdot f+\frac{C_{\mathrm{d}}\cdot A\cdot v_{\mathrm{ys}}^2}{21.15})$$

(14)

3.2.3. Constraint of Maximum Power of Motor

When the maximum power of the motor is known, from the perspective of energy supply by energy storage source, it can be obtained that:

$$P_{\max }-(m_{\mathrm{batt}}\cdot {\rho }_{\mathrm{P}\_\mathrm{batt}}+m_{\mathrm{capa}}\cdot {\rho }_{\mathrm{P}\_\mathrm{capa}})\le 0$$

(15)

Herein, power and energy densities of the supercapacitor are 14,080 W/kg and 3.91 W·h/kg, respectively; power and energy densities of the battery are 1000 W/kg and 215.49 W·h/kg, respectively.

3.2.4. Constraint of Supercapacitor Capacitance

The energy provided the supercapacitor module can be expressed as:

$$W_{\mathrm{UC}}=\frac{1}{2}\cdot C\cdot (U_{\max }^2-U_{\min }^2)$$

(16)

where $U_{\max }$ is the maximum discharge voltage of the supercapacitor, $U_{\min }$ is the discharge cut-off voltage and $W_{\mathrm{UCmax}}$ is the maximum energy released by the supercapacitor in road cycles and is related to the corresponding conditions. The constraint equation is:

$$\frac{1}{2}\cdot C\cdot N\cdot (U_{\max }^2-U_{\min }^2)\ge W_{\mathrm{UC}}{}_{\max }$$

(17)

4. Linear Programming Parameter Optimization Method

4.1. Establishment of Optimized Objective Function

In this study, the masses of the battery ($m_{\mathrm{batt}}$) and supercapacitor ($m_{\mathrm{capa}}$) were selected as the optimization variables. Related to energy loss and cost during operation, quality is a key factor at the early stage of design. The overall objective function was the output. The HESS device includes Li battery, supercapacitor and DC/DC. The mass of DC/DC with fixed power is ignored in the optimization process as it is constant. Hence, the optimized objective function is as follows:

$$F(m_1,m_2)=m_{\mathrm{batt}}+m_{\mathrm{capa}}$$

(18)

4.2. Parameter Optimization of Linear Programming

Determination of the optimal solution using linear programming is essentially based on the assumption that only one feasible solution is available. $m_{\mathrm{batt}\_\mathrm{main}}=2581.98\mathrm{kg}$ and $m_{\mathrm{capa}\_\min }=628.65\mathrm{kg}$. The number of units under the constraints in this study was considered. The number of battery modules was calculated to be 136. Based on the rated voltage of the driving motor and the integer principle, it was determined that 12 battery modules shall be connected in series in one branch. As energy is provided by the supercapacitor during starting acceleration, from the perspective of cost saving, 3.84 battery modules are absent with 11 branches in parallel selected, and an additional energy storage source is required to provide 74 kW power and 4.99 kW∙h energy. It was calculated that $m_{\mathrm{batt}}=2508\mathrm{kg}$.

The selected supercapacitor module had a rated voltage of 48 V, discharge cut-off voltage of 24 V and maximum released energy of $1.43\times {10}^5$ J (39.6 W∙h). Likewise, seven modules in series are required for one supercapacitor. The mass of one supercapacitor mass was 13.5 × 7 = 94.5 kg and the mass of two supercapacitors was 13.5 × 7 × 2 = 189 kg; the energy provided by one supercapacitor was 277.2 W∙h and the energy provided by seven supercapacitors was 1940.4 W∙h (1.94 kW∙h). When all constraints are satisfied, $m_{\mathrm{capa}}=661.5\mathrm{kg}$ and the mass of DC/DC was assumed to be 100 kg. In this case, the total mass of the HESS was $m_{\mathrm{HESS}}=3269.5\mathrm{kg}$.

The requirements of energy and power by the vehicle for the composite energy storage device can be described by the energy/power ratio ($R_{\mathrm{W}/\mathrm{P}}$). $R_{\mathrm{W}/\mathrm{P}}$ is also an important parameter of the composite energy storage device. A large $R_{\mathrm{W}}$ indicates that the HESS is a high-energy component, while a small $R_{\mathrm{W}}$ indicates that the HESS is a high-power component.

$$R_{\mathrm{W}/\mathrm{P}}=\frac{3.91\times 661.5+67.37\times 2508}{\mathrm{14,080}\times 661.5+1000\times 2508}=1.45\%$$

(19)

The graphic analysis method was employed to determine the constraints of the HESS. $k$ is defined as the mass ratio of the supercapacitor and Li battery; $k_1$ is the mass ratio of the Li battery and the residual mass of the vehicle. Additionally, $k_{\mathrm{p}}$ and $k_{\mathrm{w}}$ are defined as power and energy coefficients, respectively; ${\eta }_{\mathrm{P}\_\mathrm{HESS}}$ and ${\rho }_{\mathrm{P}\_\mathrm{HESS}}$ are power utilization and density of the energy storage system, respectively. Constraints are as follows:

$$\{\begin{array}{l}{k}_{\mathrm{W}}-\frac{k\cdot k_1\cdot m_{\mathrm{qt}}\cdot {\rho }_{\mathrm{W}\_\mathrm{capa}}}{(k\cdot k_1\cdot m_{\mathrm{qt}}+k_1\cdot m_{\mathrm{qt}})}-\frac{k_1\cdot m_{\mathrm{qt}}\cdot {\rho }_{\mathrm{W}\_\mathrm{batt}}}{(k\cdot k_1\cdot m_{\mathrm{qt}}+k_1\cdot m_{\mathrm{qt}})}\le 0\\ k_{\mathrm{P}}-\frac{k\cdot k_1\cdot m_{\mathrm{qt}}\cdot {\rho }_{\mathrm{P}\_\mathrm{capa}}}{(k\cdot k_1\cdot m_{\mathrm{qt}}+k_1\cdot m_{\mathrm{qt}})}-\frac{k_1\cdot m_{\mathrm{qt}}\cdot {\rho }_{\mathrm{P}\_\mathrm{batt}}}{(k\cdot k_1\cdot m_{\mathrm{qt}}+k_1\cdot m_{\mathrm{qt}})}\le 0\\ P_{\max }-(k_1\cdot m_{\mathrm{qt}}\cdot {\rho }_{\mathrm{P}\_\mathrm{batt}}+k\cdot k_1\cdot m_{\mathrm{qt}}\cdot {\rho }_{\mathrm{P}\_\mathrm{capa}})\le 0\\ \frac{({\rho }_{\mathrm{W}\_\mathrm{batt}}\cdot k_1\cdot m_{\mathrm{qt}})}{(S/v)}\ge \frac{v}{3600{\eta }_{\mathrm{T}}}\cdot (m\cdot g\cdot f+\frac{C_{\mathrm{d}}\cdot A\cdot v^2}{21.15})\\ \frac{1}{2}\cdot C\cdot N\cdot (U_{\max }^2-U_{\min }^2)\ge W_{\mathrm{UCmax}}\end{array}$$

(20)

The constraints of power and energy utilization have a dominant effect on the size and shape of the reasonably selected area. If ${\rho }_{\mathrm{P}\_\mathrm{batt}}<k_{\mathrm{P}}<{\rho }_{\mathrm{P}\_\mathrm{capa}}$ and $\frac{{\rho }_{\mathrm{P}\_\mathrm{batt}}-k_{\mathrm{P}}}{k_{\mathrm{P}}-{\rho }_{\mathrm{P}\_\mathrm{capa}}}<\frac{{\rho }_{\mathrm{W}\_\mathrm{batt}}-k_{\mathrm{W}}}{k_{\mathrm{P}}-{\rho }_{\mathrm{W}\_\mathrm{capa}}}$, Point E (intersection of the effective power utilization constraint curve and the energy feedback constraint curve) is higher than Point F (intersection of the energy feedback constraint curve and the dynamic performance constraint curve) but lower than Point A (intersection of the mileage constraint curve and the energy feedback constraint curve). The constraints obtained are shown in Figure 1. Herein, the horizontal and vertical coordinates are $k_1$ and $k$, respectively. As observed, Points B and C exhibited optimized mileage. Located at the boundary of dynamic performance and energy feedback constraints, Point B exhibited limited dynamic performance and insignificant energy feedback effect; its role as auxiliary energy storage source was also hindered. Point C was at the intersection of the effective power utilization and dynamic performance constraints, resulting in low effective power utilization of the energy storage source; a surplus was observed and the level of dynamic performance was not high. Similarly, Points A and D are at the intersection of the energy feedback constraint and mileage constraint and the intersection of the mileage constraint and effective power utilization constraint, respectively. Point A exhibited the lowest energy feedback and shortest mileage; Point D exhibited limited mileage and the lowest effective power utilization. Point E exhibited the lowest effective power utilization and energy feedback efficiency. If the mass ratio of the energy storage system corresponding to effective energy utilization constraints is higher than Point G (the intersection of the dynamic performance constraint and mileage constraint) or the mass ratio of the energy storage system corresponding to effective power utilization constraints is lower than Point F, the constraint shape developed would change.

Based on the parameters of the model vehicle selected, $k$ and $k_1$ should not be over-large with all constraints satisfied. In this study, Point O (0.251, 0.264) was identified, which is consistent with the results of previous linear programming. In order to analyze the correlation of added auxiliary energy storage component ($m_{\mathrm{capa}}$) with $m_{\mathrm{HESS}}$ and the relative relationship between the composite energy storage system and the mass of other components, the mass ratio $k_{\mathrm{zlb}}$ and the mixing ratio $k_{\mathrm{hhb}}$ were introduced. In this study, $k_{\mathrm{zlb}}=20.23\%$ and $k_{\mathrm{hhb}}=27.44\%$.

An energy storage source provides energy consumed by itself and power and energy required for the entire vehicle during operation:

$$P_{\mathrm{batt}}\cdot {\eta }_{\mathrm{P}\_\mathrm{batt}}+P_{\mathrm{capa}}\cdot {\eta }_{\mathrm{P}\_\mathrm{capa}}>P_{\mathrm{other}}$$

(21)

$${\rho }_{\mathrm{P}\_}{}_{\mathrm{batt}}{m}_{\mathrm{batt}}{\eta }_{\mathrm{P}\_\mathrm{batt}}+{\rho }_{\mathrm{P}\_}{}_{\mathrm{capa}}{m}_{\mathrm{batt}}{\eta }_{\mathrm{P}\_\mathrm{capa}}>k_{\mathrm{p}}{m}_{\mathrm{other}}+\frac{C_{\mathrm{d}}A{v}^3}{\mathrm{76,140}{\eta }_{\mathrm{T}}}$$

(22)

The quality relationships between each other were introduced:

$$k_1>\frac{k_{\mathrm{P}}+\frac{C_{\mathrm{d}}A{v}^3}{\mathrm{76,140}{\eta }_T{m}_{\mathrm{ch}}}}{{\eta }_{\mathrm{P}\_\mathrm{batt}}-k_{\mathrm{P}}}$$

(23)

As observed, the mass of the Li battery was reduced by 73.98 kg. For the selected Li cell module unit, the mass of the Li battery was reduced by 228 kg from the perspective of the equivalent total terminal voltage of a single series battery pack. The capacitance was reduced by 50 A·h if one battery branch was removed. Due to the addition of the supercapacitor, $R_{\mathrm{W}/\mathrm{P}}$ decreased from 21.55% to 1.45% and the proportion of HESS as a functional component increased. In subsequent simulations, it was demonstrated that the component parameters of the selected driving system and energy storage system can reflect the overall performance of the vehicle; the accuracy and validity of the matching results were verified. The mass ratio of HESS and energy storage source and the reasonable range of the mass ratio of entire vehicle and components other than the energy storage source that satisfy the constraints were determined using the proposed parameter matching method. Additionally, the utilization can be improved by reasonable selection of the energy storage source unit if the initial matching conditions are satisfied.

5. Modeling and Experimental Verification of Electric Buses

5.1. Simulations of Consumption under Cycle Conditions

With power demand as the input signal, a dynamic model of the vehicle was established, as shown in Figure 2. In CRUISE, the internal electrical and mechanical parts of the composite energy storage system were connected.

In the full-load climbing experiment, the climbing performance was added to the Project-Task Folder during simulations. The maximum first-gear climbing slopes of the proposed electric buses and conventional electric buses with battery as single energy storage source were 29.4% and 18%, respectively, indicating a significant improvement in the climbing performance. It was significantly higher than the preset requirement on climbing performance (see Figure 3).

The mileage was calculated. Specifically, a vehicle traveling at a constant speed of 40 km/h was added in CRUISE and the corresponding driving distance when the SOC of the Li battery dropped 90% to 20% was defined as the mileage. Compared with conventional electric buses, the mileage of the proposed electric bus increased from 265 to 373 km, as shown in Figure 4.

As shown in Figure 5, the acceleration times of 0~50 km/h of conventional electric buses and those with HESS were 23.8 and 15.88 s, respectively. Therefore, it can be concluded that the speed changing rate of conventional electric buses is small and cannot satisfy the large transient energy demand.

As shown in Figure 6, the target speed and the actual speed almost coincide under the standard single road condition and the deviation was far less than 5%, demonstrating that the driving capacity of the motor and the energy provided by the energy storage source meet the performance requirements of buses owing to the parameter matching of the energy storage system.

5.2. Experimental Analysis

A test setup was established according to the similarity principle to investigate the performance of a matched HESS. In the test of the driving part, the power demand of the vehicle during operation was simulated by pair trawling of the brushless DC motor and the electric dynamometer. A brushless DC motor with a rated power of 5 kW and rated voltage of 48 V was employed as the motor to be tested. A DC dynamometer with a rated power of 7.5 kW was employed as the loading motor. Communication was established between controllers of the two motors through USB-to-CAN. Meanwhile, a vehicle demand model was developed in LabVIEW according to the actual vehicle speed demand in typical conditions to calculate the actual power demand. Additionally, the power was distributed between the Li battery and supercapacitor in the composite energy storage system according to the control algorithms. The experimental setup is shown in Figure 7.

5.2.1. Constant-Speed Climbing Performance

With motor speed of 970 rpm, the corresponding vehicle speed was calculated to be 8.9 km/h. For the dynamometer, the load resistance at the loading point increased from 2 N·m by 1 N·m/s Newton resistance. Figure 8 shows the measured waveform. As observed, the current increased and fluctuated at each turning point of slope increase. The current increased from 7.6 to 21.4 A as the slope increased. The simulated slope angle corresponding to the maximum torque was about 40%, which satisfies the requirements of dynamic performance.

5.2.2. Accelerating Capability

As shown in Figure 9, the vehicle speed increased from 8.3 to 17.5 km/h and the motor speed increased from 907.62 to 1907.33 rpm in simulations according to the set point. During acceleration, the vehicle delivers a driving force greater than the resistance; torque, speed and output current of the driving motor exhibit upward trends.

5.2.3. Regenerative Braking Feedback of Electric Vehicles

In this test, a standard urban deceleration road in China was employed for simulations. The motor speed dropped from 2940.5 (vehicle speed = 27.0 km/h) to 272.3 rpm (vehicle speed of 2.5 km/h). Forty sampling points were arranged and the relevant parameter information of one point was recorded every 1 s.

Table 4. Data of regenerative braking test.

Output Voltage (V)	Output Current (A)	Input Torque (N·m)	Input Power (W)	Output Power (W)	Efficiency (%)
53.39	−26.994	−6.634	1860.56	−1441.42	77.47
53.62	−26.856	−6.613	1866.72	−1440.23	77.15
53.77	−26.235	−6.587	1857.65	−1410.85	75.95
53.87	−26.172	−6.576	1838.85	−1409.88	76.67
53.93	−25.784	−6.565	1823.40	−1390.71	76.27
54.99	−25.744	−6.559	1811.09	−1390.08	76.75
54.03	−25.496	−6.549	1800.30	−1377.77	76.53
54.08	−24.907	−6.527	1746.41	−1347.17	77.14
54.12	−24.893	−6.524	1748.61	−1347.22	77.05
54.12	−24.404	−6.516	1736.84	−1320.76	76.04
54.14	−24.380	−6.512	1720.43	−1320.04	76.73
54.16	−24.187	−6.503	1709.27	−1309.97	76.64
54.18	−24.194	−6.505	1704.14	−1310.94	76.93
54.15	−23.273	−6.484	1654.57	−1260.36	76.17
54.17	−23.284	−6.482	1644.83	−1261.37	76.69
54.10	−21.677	−6.450	1541.27	−1172.73	76.09
54.11	−21.682	−6.449	1534.07	−1173.21	76.48

shows the experimental data obtained. The braking feedback test simulates the deceleration and braking of vehicles. During the test, the driving motor is decelerating or braking. Since two-way loading is possible for the dynamometer, the dynamometer drives the motor to simulate the transient process of deceleration upon braking. Besides decreasing speed, current and power of the motor fed back to the battery also decrease accordingly.

5.2.4. Continuous Road Condition Simulation

At the beginning of the test, the driving motor was started in the positive direction and the dynamometer provided a positive loading resistance to the motor. At the next sampling time, the dynamometer applied a reverse resistance torque to the motor (so that the driving operated). When the motor was exposed to reverse driving, it automatically entered the braking state; the dynamometer drove the motor until the speed was stabilized at 10 m/s. In the braking feedback test, three consecutive sampling points were set for reverse loading, and driving points were inserted between every two braking points. Figure 10 shows the trend of current on the DC side in the test. As observed, the output waveform of battery current measured by the current probe was positive during driving and negative during braking feedback.

6. Conclusions

With the minimum mass of the energy storage system as the objective function, a method combining linear programming and graphical method for parameter matching of a composite energy storage system is proposed. From the perspective of evaluation indexes and based on the rounding principle of unit quantity, the proportion of HESS as a power component increases to meet the energy demand for frequent starting and stopping of buses and buses; this method provides a novel solution to parameter matching of composite energy storage systems. Therefore, the proposed method is of great significance both theoretically and practically.