Real-time control method for extended-range electric vehicle with battery and cabin heating in winter

Abstract

The invention provides a real-time control method for an extended-range electric vehicle with a battery and a cabin heated in winter. The method comprises the following steps: acquiring driving cycle data; analyzing a state variable based on the driving cycle data, and solving according to a duality principle and a Lagrange multiplier method to obtain an equivalent factor; and determining the optimal output power of the range extender, the battery and the PTC heater according to the optimal equivalent factor value. Compared with a traditional energy management strategy, the thermal management requirement of the whole vehicle in the low-temperature environment is considered, the problem is simplified and optimized, the solving process does not depend on a complex algorithm, the requirement for the algorithm and the computing power of the controller is low, and the method is suitable for an existing vehicle controller.

Description

Technical Field

[0001] The present invention belongs to the field of new energy vehicles, and in particular relates to a real-time control method for heating batteries and cabins of extended-range electric vehicles in winter. Background Art

[0002] The current automotive industry is facing dual challenges: environmental degradation and energy shortages. In response to these problems, extended-range electric vehicles have emerged and are seen by countries as an important way to promote sustainable transportation. These vehicles aim to reduce carbon emissions and improve energy utilization by combining the efficient cooperation of traditional fuel engines with electric systems. The energy management system (EMS) plays a key role in this process. It is responsible for monitoring, controlling and optimizing the vehicle's power distribution and energy use, including battery charging management, motor efficiency and driving mode selection, which directly determines the vehicle's energy efficiency and performance. An effective EMS can maximize the vehicle's energy recovery and reduce fuel consumption, thereby improving energy economy throughout the life cycle.

[0003] However, traditional energy management systems do not take into account the impact of thermal management systems, especially in winter. At low temperatures, the chemical reaction rate of the battery will slow down, causing the current flow inside the battery to slow down, which directly affects the power performance and endurance of electric vehicles. When the battery temperature is lower than the optimal range (usually 20°C to 25°C), the charging and discharging efficiency decreases, which may cause the internal resistance of the battery to increase and even cause the risk of thermal runaway. Therefore, new energy vehicles are often equipped with a battery heating system to heat the battery in winter weather and keep the battery in a suitable operating temperature range to maximize battery performance and ensure driving safety.

[0004] When the air conditioner is turned on in winter, due to the working characteristics of the extended-range electric vehicle, the PTC heater needs to be turned on frequently. Studies have shown that, especially in winter, if the heating system is used frequently, the energy consumption accounts for more than half of the total energy consumption of the vehicle, which can be as high as 50%. This is because the energy consumed by air conditioning operation is much higher than that of conventional driving. This not only reduces the driving distance on a single charge, but also has a potential impact on battery life, because frequent deep discharge is not good for battery health. Therefore, for users of extended-range electric vehicles, the rational use of air conditioning is the key to saving energy and extending battery life in cold seasons. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a real-time control method for battery and cabin heating of extended-range electric vehicles in winter, which can avoid complex iterative calculations to obtain the adaptively updated equivalent factors required in the equivalent fuel consumption calculation while satisfying the conditions of maintaining battery temperature and cabin thermal comfort, thereby obtaining the optimal range extender-battery power allocation scheme and reducing computing power and memory costs.

[0006] The technical solution of the present invention is as follows:

[0007] A real-time control method for battery and cabin heating of an extended-range electric vehicle in winter, the method comprising:

[0008] Obtain drive cycle data;

[0009] Based on the driving cycle data, the state variables are analyzed and the equivalent factors are obtained by solving them according to the duality principle and Lagrange multiplier method;

[0010] The optimal output power of the range extender, battery and PTC heater is determined according to the optimal equivalent factor value.

[0011] Furthermore, the method for solving the equivalent factor is specifically as follows:

[0012] The required power of the vehicle is calculated according to the vehicle operating conditions;

[0013] Define control variables and establish state expressions for battery energy and coolant heat;

[0014] Fitting fuel consumption based on the quadratic function form of the control variables and power demand;

[0015] Taking fuel consumption as the objective function, the parameters to be optimized are determined according to the preset constraints, and a fuel economy optimization model is established;

[0016] Discretize the time interval to generate a series of discrete driving power requirements and heat requirements, and transform the continuous optimization problem into a discrete convex quadratic programming problem;

[0017] Finally, the optimal control variable sequence is obtained through the convex quadratic programming solution method, thereby obtaining the equivalent factor; the sequence provides a control strategy for achieving the required power under the condition of minimum fuel consumption, including the state changes of battery energy and coolant heat and the optimized control variable combination.

[0018] Furthermore, the formula for calculating the required power of the vehicle is:

[0019]

[0020] Where f represents the rolling resistance coefficient of the vehicle, m represents the mass of the vehicle, and ρ a Represents the air density, C D Represents the dynamic resistance coefficient of air, A D represents the windward area of ​​the vehicle, δ represents the vehicle's rotational mass conversion coefficient, which is set to 1.0, and v a Represents vehicle speed.

[0021] Furthermore, the fuel economy optimization model is:

[0022]

[0023] in, is the instantaneous fuel consumption, λ f is the fuel consumption temperature factor, P apu is the output power of the range extender, P ptc is the output power of the PTC heater, x1 is the state variable SOC, x2 is the state variable engine coolant temperature, x 1f and x 10 Indicates the termination and initial state value of SOC, x 20 is the initial value of coolant temperature, x 2f Indicates the temperature threshold for cooling liquid termination, P apumin and P apumax is the lower and upper limits of the output power of the range extender, P ptcmin and P ptcmax is the lower and upper limits of the output power of the PTC heater, x 1min and x 1max Respectively represent the lower and upper limits of SOC, x 2min and x 2max Respectively represent the lower and upper limits of the engine coolant, t0 and t f Respectively represent the start and end time of the cycle.

[0024] Furthermore, the constraint conditions are the power performance of the vehicle, the battery state of charge, and the initial state and final state of the engine coolant temperature.

[0025] Furthermore, the discrete driving power requirement is specifically:

[0026]

[0027] Among them, P req Represents the driving power.

[0028] Further, the discrete heat demand includes a discrete cabin heat demand and a discrete battery heat demand.

[0029] Further, the discrete battery heat demand is expressed as:

[0030]

[0031] Among them, i represents a specific moment after discreteness, T bat Indicates the battery temperature, k b1 to k b6 It is expressed by the following formula:

[0032]

[0033] in, is the entropy coefficient of the battery, C bat is the equivalent specific heat of the battery, M bat is the equivalent mass of the battery, Δt is the discrete time step, h b is the convection heat transfer coefficient between the battery surface and the outside world, A b is the convection heat transfer area between the battery surface and the outside world, ε r is the thermal emissivity of the battery, σ is the Stefan-Boltzmann constant, A br Represents the thermal radiation area of ​​the battery, T amb is the ambient temperature, and n is the total number of time steps in a prediction period.

[0034] Furthermore, the discrete cabin heat demand is specifically:

[0035] Q cd (i) = k c1i +k c2i T cab (i)+k c3 T cab (i+1)

[0036] Where, T cab represents the cabin temperature, k c1i , k c2i and k c3 It is expressed by the following formula:

[0037]

[0038] Among them, α r is the radiation absorption rate of the cabin wall material, A rj represents the cabin area, ζ j It represents the transmittance of sunlight through the glass to the interior of the cabin, C sf represents the solar radiation intensity, θ j is the angle between the transparent window and the horizontal direction, n r is the number of glass windows, h ld is the heat exchange coefficient between the body shell and the cab, which is related to the vehicle speed. ld represents the heat transfer area of ​​the body shell, T cab Indicates the equivalent temperature of the cab, m ven is the mass flow rate of air, which can be set as a constant value. ξ represents the external circulation air coefficient, which is set as a constant value, C cab is the equivalent specific heat of cabin air, M cab Indicates the quality of cabin air.

[0039] Furthermore, the fuel economy optimization model is specifically:

[0040]

[0041] in,

[0042] B=[b p1 ,c p1 ,0,…,b pn ,c pn ,0],

[0043] Here, a pi , b pi (i=1,2,…,n) represents the fitting parameters of instantaneous fuel consumption at different time steps

[0044] A1=[-100,-100,…,-100] 1×3n ,

[0045] A2=[k 11 -a h1 ,η ptc +b h1 ,k2,…,k 1n -a hn ,η ptc +b hn ,k2] 1×3n ,

[0046]

[0047] Here a hi , b hi , c hi (i=1,2,…,n) represents the fitting parameters of the heat entering the water jacket of the engine at each time and the control variable. 1f , X 10 , X 2f , X 20 represents the alternative state variable, q bat =U ocv Q b , U ocv and Q b They represent the open circuit voltage and battery capacity of the battery respectively, k 1i (i=1,2,…,n), k2,k3 are substitution variables,

[0048] Compared with the prior art, the present invention has the following advantages:

[0049] Compared with traditional energy management strategies, the present invention takes into account the thermal management needs of the entire vehicle in a low-temperature environment. By simplifying the optimization problem, the solution process does not rely on complex algorithms, and has low requirements on the controller's algorithm and computing power, and is suitable for existing vehicle controllers. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] The accompanying drawings generally illustrate various embodiments by way of example and not limitation, and together with the description and claims, serve to illustrate the embodiments of the invention. Where appropriate, the same reference numerals are used throughout the drawings to refer to the same or similar parts. Such embodiments are illustrative and are not intended to be exhaustive or exclusive embodiments of the present apparatus or method.

[0051] Figure 1 The figure is a diagram showing a power system structure and a thermal management system structure according to an embodiment of the present invention.

[0052] Figure 2 The solution process for solving the equivalent factor is shown as an embodiment.

[0053] Figure 3 The flowchart of the real-time control method is shown in the embodiment. DETAILED DESCRIPTION

[0054] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0055] like Figure 1 As shown, a schematic diagram of a power system structure and a thermal management system structure of an extended-range electric vehicle is provided in this embodiment; the schematic diagram is mainly divided into a transmission system and a thermal management system.

[0056] In the transmission system, the mechanical power generated by the engine is transmitted to the generator through a reducer with a constant speed ratio. The mechanical energy is converted into electrical energy by the generator and then output by the range extender. The journey of electricity continues as it moves from two power sources: the range extender and the power battery, through the DC / DC converter to the drive motor. Finally, the electrical energy is converted into mechanical energy by the drive motor and transmitted to the wheels through the final reducer.

[0057] The vehicle thermal management system is a combination of the engine thermal management system, the cabin thermal management system and the battery thermal management system, including components such as the engine water jacket, PTC heater, electronic water pump, heat exchanger and three-way valve. When working, the engine generates heat, which is transported to the water cycle through the engine water jacket. The three-way valve adjusts the proportion of heat entering the cabin thermal management system and the battery thermal management system. The electrical energy output by the DC / DC converter is directly transmitted to the PTC heater to generate heat energy, which is then transmitted to the cabin thermal management system and the battery thermal management system. The water pump ensures the normal flow of coolant in the water circuit. The heat generated by the cabin thermal management system and the battery thermal management system is sent to the cabin and the battery through the heat exchanger.

[0058] like Figure 2 As shown, the solution of the equivalent factor is divided into the following steps:

[0059] Step 1: Get the predicted dynamic power demand based on the predicted data

[0060] The predicted working condition data can be calculated to obtain the dynamic power requirement of the whole vehicle, which is calculated by the following formula:

[0061]

[0062] Where f represents the rolling resistance coefficient of the vehicle, m represents the mass of the vehicle, and ρ a Represents the air density, C D Represents the dynamic resistance coefficient of air, A D represents the windward area of ​​the vehicle, δ represents the vehicle's rotational mass conversion coefficient, which is set to 1.0, and v a Represents vehicle speed.

[0063] The above formula can be converted into the input electric power of the drive motor as follows:

[0064]

[0065] Among them, η m Indicates the efficiency of the drive motor, η f Indicates the mechanical efficiency of the drive system. The power required to drive the motor is considered to be the same as the power required by the system.

[0066] Step 2: Replace state variables and control variables

[0067] Control variable substitution:

[0068]

[0069] The state variables are represented as battery energy and coolant heat:

[0070]

[0071] In the formula, q bat and q cool They can be expressed as: q bat =U ocv Q b ,q cool =C e M e .U ocv represents the open circuit voltage of the battery, Q b Indicates the capacity of the battery. C e represents the equivalent specific heat of coolant and engine body, M e Represents the equivalent mass of coolant and engine block.

[0072] The instantaneous fuel consumption is expressed by fitting the variables:

[0073]

[0074] Where a p 、a q 、a r 、b p 、c p and d p Only the power required by the drive motor (P req ) related, a p >0,a q >0,a r >0.

[0075] The fuel economy optimization problem becomes:

[0076]

[0077] The optimization problem is transformed into a convex quadratic programming problem by replacing variables.

[0078] In order to facilitate the problem solving, it is generally necessary to discretize the optimization problem during the calculation process. In order to solve the above problem, the present application discretizes the optimization problem at time [t0, t f ]Discrete,

[0079] The discrete driving power demand sequence is:

[0080]

[0081] When only the effects of termination and initial constraints are considered, the battery energy is obtained as:

[0082]

[0083] To analyze the termination and initial value constraints of the engine coolant, the rate of change of the coolant temperature needs to be analyzed:

[0084] The heat entering the water jacket of the engine can be fitted with a linear function, which can be obtained:

[0085] Q ice =a h u1+b h u2+c h

[0086] Among them, a h , b h , c h are fitting parameters.

[0087] The output power of the PTC heater can be expressed as:

[0088] Q ptc =η ptc P ptc

[0089] Where η ptc Indicates the output efficiency of the PTC heater.

[0090] By default, during operation, most of the time, the temperature of the engine coolant will be lower than the thermostat set temperature, so the heat dissipation of the engine coolant can be ignored in the process of predictive control.

[0091] The heat demand of the cabin thermal management and the battery thermal management system in low temperature environment constitutes the heat demand of the thermal management system, so the heat demand of the thermal management system can be expressed as:

[0092] Q d =Q cd / η cl +Q bd / η bl

[0093] Among them, Q cd is the heat demand of the cabin thermal management system, Q bd is the heat demand of the battery thermal management system, η cl is the thermal efficiency of the cabin thermal management system, η bl Thermal efficiency of the battery thermal management system.

[0094] The discrete battery heat demand can be expressed as:

[0095]

[0096] Among them, i represents a specific moment after discreteness, T bat Indicates the battery temperature, k b1 to k b6 , can be expressed by the following formula:

[0097]

[0098] in is the entropy coefficient of the battery, which is only related to the SOC of the battery, C bat is the equivalent specific heat of the battery, M bat is the equivalent mass of the battery, Δt is the discrete time step, h b is the convection heat transfer coefficient between the battery surface and the outside world, A b is the convection heat transfer area between the battery surface and the outside world, ε r is the thermal emissivity of the battery, σ is the Stefan-Boltzmann constant, A brRepresents the thermal radiation area of ​​the battery, T amb is the ambient temperature, and n is the total number of time steps in a prediction period.

[0099] The sum of the heat demand of the batteries during the forecast period is:

[0100]

[0101] Among them, T bti To plan the battery temperature change value, you can choose different ways to get the temperature change value. Here, choose linear change.

[0102] The discrete cabin heat demand is expressed as:

[0103] Q cd (i) = k c1i +k c2i T cab (i)+k c3 T cab (i+1)

[0104] Where, T cab represents the cabin temperature, k c1i , k c2i and k c3 It can be expressed by the following formula:

[0105]

[0106] Among them, α r is the radiation absorption rate of the cabin wall material, A rj represents the cabin area, ζ j It represents the transmittance of sunlight through the glass to the interior of the cabin, C sf represents the solar radiation intensity, θ j is the angle between the transparent window and the horizontal direction, n r is the number of glass windows, h ld is the heat exchange coefficient between the body shell and the cab, which is related to the vehicle speed. ld represents the heat transfer area of ​​the body shell, T cab Indicates the equivalent temperature of the cab, m ven is the mass flow rate of air, which can be set as a constant value. ξ represents the external circulation air coefficient, which is set as a constant value, C cab is the equivalent specific heat of cabin air, M cab Indicates the quality of cabin air.

[0107] The heat demand of the battery during the forecast period can be expressed as:

[0108]

[0109] Among them, T ctj For the planned cabin temperature change value, different methods can be selected to obtain the temperature change value. Here, linear change is selected.

[0110] Combining the heat demand of the battery and cabin, the total heat demand during the forecast period is expressed as:

[0111]

[0112] Among them, k 1i , k2, k3 can be expressed as:

[0113]

[0114] In summary, the initial and final constraints on the heat of the engine coolant can be expressed by the following equations:

[0115]

[0116] where a hi , b hi , c hi It can be obtained by interpolating the data table.

[0117] Step 3: Planning the optimization problem after discretization

[0118] Combining the above analysis, the fuel economy optimization problem is planned as follows:

[0119]

[0120] Among them, the matrix in the above formula can be expressed as:

[0121] B=[b p1 ,c p1 ,0,…,b pn ,c pn ,0],

[0122] A1=[-100,-100,…,-100] 1×3n ,

[0123] A2=[k 11 -a h1 ,η ptc +b h1 ,k2,…,k 1n -a hn ,η ptc +b hn ,k2] 1×3n ,

[0124]

[0125] Step 4: Solve the simplified optimization problem:

[0126] According to the duality principle and Lagrange multiplier method, we can finally get the solution of the equivalent factor as follows.

[0127] according to Figure 3 As shown, the present invention proposes a real-time control method to solve the real-time output power of the range extender, the battery output power and the PTC heater output power.

[0128] The basic process of the real-time control method is described as follows:

[0129] Using GPS and ITS information, the driving cycle acquires data within the prediction range and transmits it to the vehicle's master controller. Based on the expected driving cycle data, the equivalent factors are determined by analytical solution methods and updated regularly. The equivalent factors are solved as follows Figure 2 As shown in the figure. Initially, the state variables (i.e., SOC and engine coolant temperature) are analyzed during the solution process. Subsequently, the Lagrangian function is formulated based on the driving conditions and the real-time SOC and engine coolant temperature. Finally, the optimal output power of the range extender, battery, and PTC heater is determined using the periodically updated optimal equivalent factor values, thereby formulating the control strategy for each cycle.

[0130] The Hamiltonian function calculated in real time can be expressed as follows:

[0131]

[0132] Among them, T cl represents the coolant temperature, k is the specific driving time, T cl_tgt is the coolant termination temperature threshold, Q apu and Q bat are the equivalent friendliness of the battery and engine calculated after discretization, where Q apu The optimal range extender working curve can be used to obtain Q bat Obtained by discrete PTC heater output power.

[0133] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A real-time control method for extended-range electric vehicles with battery and cabin heating in winter, characterized in that: The method comprises: Obtain drive cycle data; Based on the driving cycle data, the state variables are analyzed and the equivalent factors are obtained by solving them according to the duality principle and Lagrange multiplier method; The optimal output power of the range extender, battery and PTC heater is determined according to the optimal equivalent factor value.

2. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 1, characterized in that: The method for solving the equivalent factor is specifically as follows: The required power of the vehicle is calculated according to the vehicle operating conditions; Define control variables and establish state expressions for battery energy and coolant heat; Fitting fuel consumption based on the quadratic function form of the control variables and power demand; Taking fuel consumption as the objective function, the parameters to be optimized are determined according to the preset constraints, and a fuel economy optimization model is established; Discretize the time interval to generate a series of discrete driving power requirements and heat requirements, and transform the continuous optimization problem into a discrete convex quadratic programming problem; Finally, the optimal control variable sequence is obtained through the convex quadratic programming solution method, thereby obtaining the equivalent factor; the sequence provides a control strategy for achieving the required power under the condition of minimum fuel consumption, including the state changes of battery energy and coolant heat and the optimized control variable combination.

3. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 1, characterized in that: The formula for calculating the required power of the vehicle is: Where f represents the rolling resistance coefficient of the vehicle, m represents the mass of the vehicle, and ρ a Represents the air density, C D Represents the dynamic drag coefficient of air, A D represents the windward area of ​​the vehicle, δ represents the vehicle's rotational mass conversion coefficient, which is set to 1.0, and v a Represents vehicle speed.

4. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 2, characterized in that: The fuel economy optimization model is: in, is the instantaneous fuel consumption, λ f is the fuel consumption temperature factor, P apu is the output power of the range extender, P ptc is the output power of the PTC heater, x1 is the state variable SOC, x2 is the state variable engine coolant temperature, x 1f and x 10 Indicates the termination and initial state value of SOC, x 20 is the initial value of coolant temperature, x 2f Indicates the temperature threshold for cooling liquid termination, P apumin and P apumax is the lower and upper limits of the output power of the range extender, P ptcmin and P ptcmax is the lower and upper limits of the output power of the PTC heater, x 1min and x 1max They represent the lower and upper limits of SOC, respectively, 2min and x 2max Respectively represent the lower and upper limits of the engine coolant, t0 and t f Respectively represent the start and end time of the cycle.

5. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 4, characterized in that: The constraints are the vehicle's power performance, battery charge state, and the initial state and final state of the engine coolant temperature.

6. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 2, characterized in that: The discrete driving power requirement is specifically: P req_list =[P req (1),P req (2),…P req (n)] Among them, P req Represents the driving power.

7. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 2, characterized in that: The discrete thermal demands include a discrete cabin thermal demand and a discrete battery thermal demand.

8. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 7, characterized in that: The discrete battery heat demand is expressed as: Among them, i represents a specific moment after discreteness, T bat Indicates the battery temperature, k b1 to k b6 It is expressed by the following formula: in, is the entropy coefficient of the battery, C bat is the equivalent specific heat of the battery, M bat is the equivalent mass of the battery, Δt is the discrete time step, h b is the convection heat transfer coefficient between the battery surface and the outside world, A b is the convection heat transfer area between the battery surface and the outside world, ε r is the thermal emissivity of the battery, σ is the Stefan-Boltzmann constant, A br Represents the thermal radiation area of ​​the battery, T amb is the ambient temperature, and n is the total number of time steps in a prediction period.

9. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 7, characterized in that: The discrete cabin heat demand is specifically: Q cd (i)=k c1i +k c2i T cab (i)+k c3 T cab (i+1) Where, T cab represents the cabin temperature, k c1i , k c2i and k c3 It is expressed by the following formula: Among them, α r is the radiation absorption rate of the cabin wall material, A rj represents the cabin area, ζ j It represents the transmittance of sunlight through the glass to the interior of the cabin, C sf represents the solar radiation intensity, θ j is the angle between the transparent window and the horizontal direction, n r is the number of glass windows, h ld is the heat exchange coefficient between the body shell and the cab, A ld represents the heat transfer area of ​​the body shell, T cab Indicates the equivalent temperature of the cab, m ven is the mass flow rate of air, ξ represents the external circulation air coefficient, C cab is the equivalent specific heat of cabin air, M cab Indicates the quality of cabin air.

10. The real-time control method for extended-range electric vehicles with battery and cabin heating in winter according to claim 2, characterized in that: The fuel economy optimization model is specifically as follows: in, B=[b p1 ,c p1 ,0,…,b pn ,c pn ,0] Among them, λ f is the fuel consumption temperature factor, a pi , b pi Represents the fitting parameters of instantaneous fuel consumption at different time steps; A1＝[-100,-100,…,-100] 1×3n ， A2=[k 11 -a h1 ,η ptc +b h1 ,k2,…,k 1n -a hn ,η ptc +b hn ,k2] 1×3n a hi , b hi , c hi represents the fitting parameters of the heat entering the water jacket of the engine at each time and the control variable, X 1f , X 10 , X 2f , X 20 represents the alternative state variable, q bat =U ocv Q b , U ocv and Q b Respectively represent the open circuit voltage and battery capacity of the battery, η ptc Indicates the output efficiency of the PTC heater, k 1i , k2, k3 are replacement variables.

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