Keywords

1 Introduction

In the last decade, the passenger vehicle market has witnessed a significant increase in its share of electrified variants. However, widespread adoption is still hindered by the relatively high price and short range of these vehicles, compared to their conventional, fossil-fuel-powered counterparts [1]. A few researchers have addressed this problem by developing holistic and system-level optimization methodologies to lower the electric vehicles’ price while preserving performance [2, 3]. Although these approaches appear intriguing, their main limitation is that the optimal design is inherently influenced by the accuracy of the models employed in the framework, causing unreliable results and a lower fidelity in the design. In this paper, we overcome the shortcomings owing to the motor model presented in a previous paper [4], where we considered the motor peak power to stretch the efficiency map linearly. Against this backdrop, we devise a low-scale, quasi-static state model of the electric motor as a function of its geometry and operations, leveraging a convex surrogate modeling approach. Ultimately, we leverage our framework to jointly design battery size and motor low-scale dimensions for three different vehicle types: a city car, a compact car, and a cross over.

Related Literature The topic of this paper relates to the design optimization of electric vehicle powertrains. An overview of system-level perspectives of this problem can be found in [5], where authors generally choose simplified models for components, together with derivative-free [6] or convex optimization [7,8,9] algorithms. Relevant advances have been made in powertrain design using more detailed component models, for instance in [10] and [11]. However, the former does not guarantee global optimality, whereas the latter makes assumptions to pre-compute the required mechanical electric motor power. Therefore, we deem the powertrain models used in these optimization frameworks to date unsuitable to represent the system behavior for different geometrical scalings and operations.

Statement of Contributions In this paper, we present a concurrent design optimization framework that jointly optimizes the size of the battery with the geometric dimensions of the electric motor for a family of electric vehicles, with global optimality guarantees. Specifically, we developed an electric motor model to estimate with a high degree of fidelity the motor’s internal power losses as a function of its dimensions (axial and radial scaling factors, \( k_{\textrm{ax}} \) and \( k_{\textrm{rad}} \) respectively), and the vehicle’s operations (motor power \( P_{\textrm{m}} \) and speed \(\omega _{\textrm{m}}\)), using a surrogate modeling approach. Finally, compared to our previous work in  [4], this framework makes it possible to consider a transmission, in this case a Fixed Gear Transmission FGT, thanks to the novel Electric Motor EM model and central drive/axle motors. In conclusion, here we contribute by jointly optimizing the design of the powertrain of a fleet of electric vehicles with global optimality guarantees, whilst including the transmission ratio and accounting for the low-scale design of the electric motor.

Organization This paper is organized as follows: Sect. 2 presents the framework’s methodology, including vehicle and powertrain component models and the optimization problem. In Sect. 3, we showcase the optimization framework with a numerical study on designing a fleet of electric vehicles. The conclusions are drawn in Sect. 4, together with an outlook on future research.

2 Methodology

This section presents a convex optimization framework to design a family of Battery Electric Vehicles BEVs, with particular attention to low-level motor geometry effects on performance and losses. First, we describe the set of equations constituting the vehicle model, then we frame the optimization problem. We finalize this section with a discussion of the assumptions and limitations of the approach.

2.1 Longitudinal Vehicle Dynamics

For every vehicle i out of N we aim to co-design, we can compute the required power at the wheels \(P_{\textrm{v},i}(t)\) as

$$\begin{aligned} P_{\textrm{v},i}(t) = v(t)\left( m_i a(t) + \frac{1}{2} \rho \; c_{\textrm{d},i} A_{\textrm{f},i} v(t)^2 + m_i g (\sin \alpha (t) + c_{\textrm{r},i} \cos \alpha (t)) \right) , \end{aligned}$$
(1)

where v(t), a(t), and \(\alpha (t)\) are the velocity, acceleration and road inclination given by the selected drive cycle as a function of the time t, respectively; \(m_i\), \(c_{\textrm{d},i}\), and \(A_{\textrm{f},i}\) are the total mass, the aerodynamic drag coefficient, and the frontal area of vehicle i, respectively. Moreover, \(\rho \) is the density of air, g is the gravitational constant, and \(c_{\textrm{r},i}\) is the rolling resistance coefficient. From this point onward, we will drop the time dependence on the variables whenever it is clear from the context.

2.2 Mass

We consider the effect of weight introduced by powertrain components’ sizing in the vehicle mass \(m_i\) equation. Therefore, we account for the glider mass \(m_{\textrm{g},i}\), the driver mass \(m_{\textrm{d}}\), the payload mass \(m_{\textrm{pl},i}\), the total mass of the \(N_{\textrm{m},i}\) motors \(m_{\textrm{m},i}\), and the \(N_{\textrm{b},i}\) battery packs \(m_{\textrm{b}}\)

$$\begin{aligned} m_i = m_{\textrm{g},i} + m_{\textrm{d}} + m_{\textrm{pl},i} + m_{\textrm{m},i} + m_{\textrm{b},i}. \end{aligned}$$
(2)

2.3 Transmission

We model the FGT as an efficiency \(\eta _\textrm{gb}\), in line with common practice in the field [12]. Hence, we write the output mechanical power of each motor \(P_{\textrm{m},i}\) following the convex relaxation introduced by [13]

$$\begin{aligned} P_{\textrm{m},i} & \ge P_{\textrm{v},i} \frac{1}{\eta _\textrm{gb} N_{\textrm{m},i}}, \end{aligned}$$
(3)
$$\begin{aligned} P_{\textrm{m},i} & \ge P_{\textrm{v},i} \eta _\textrm{gb} \frac{r_{\textrm{b},i}}{N_{\textrm{m},i}}, \end{aligned}$$
(4)

where \(r_{\textrm{b},i}\) is the regenerative braking fraction, depending on whether vehicle i is equipped with a front-wheel or all-wheel drive. Hereby we assume that both motors mounted on each axle in an all-wheel drive configuration are operated identically. The transmission has a fixed ratio \(\gamma \), such that

$$\begin{aligned} \omega _{\textrm{m},i} = \gamma \frac{v}{r_{\textrm{w},i}}, \end{aligned}$$
(5)

where \(\omega _{\textrm{m},i}\) is the mechanical speed of the motor and \(r_{\textrm{w},i}\) is the radius of the wheels.

2.4 Electric Motor

We devise a model of the electric motor as a function of its dimensions and operating point, whereby we take inspiration from the surrogate modeling approach in [11] and the scaling in [14]. The design variables are the scaling factors, which uniformly scale all inner and outer dimensions of the motor in axial and radial direction, increasing the supplied current but keeping the voltage equal. Assuming a constant density of the motor, the mass of the motor \(m_{\textrm{m},i}\) is then given by

$$\begin{aligned} m_{\textrm{m},i} = {m}_{\textrm{m,o}} k_{\textrm{ax}} k_{\textrm{rad}}^2, \end{aligned}$$

where \({m}_{\textrm{m,o}}\) is the mass of the reference motor and \(k_{\textrm{ax}} \in [\underline{k}_{\textrm{ax}}, \overline{k}_{\textrm{ax}}]\) and \(k_{\textrm{rad}} \in [\underline{k}_{\textrm{rad}}, \overline{k}_{\textrm{rad}}]\) are the axial and radial scaling factors, respectively. In order to maintain convexity, we rewrite this expression as

$$\begin{aligned} m_{\textrm{m},i} k_{\textrm{sh}} \ge m_{\textrm{m,o}} k_{\textrm{rad}}^2, \end{aligned}$$
(6)
Fig. 1.
figure 1

The EM AC power NRMSE values of each sample in the design space, for both the training and test set.

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Fig. 2.
figure 2

The training and testing data and predictions, along with the model over the full design space.

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where \(k_{\textrm{sh}} = \frac{1}{k_{\textrm{ax}}}\) is defined as the motor shortness scaling factor or, in other words, the reciprocal of the axial scaling factor. Hence, the mass relation in Eq. (6) can be rewritten as a convex second-order conic constraint [15]. In this framework, we conduct a Latin Hypercube experimental design with the electric motor design software Motor-CAD [16]. For a given transmission ratio we can pre-compute the motor speed through Eq. (5). Then, similarly to [11, 17], we create a model of the power losses \(P_{\textrm{m,loss},i}\) for multiple levels of motor speed j, which is equal to

$$\begin{aligned} P_{\textrm{m,loss},i} \ge x_{\textrm{m},i}^\top Q_{\textrm{m},j}(t) x_{\textrm{m},i} + q_{\textrm{m},j}(t) x_{\textrm{m},i} + q_{\textrm{m0},j}(t) \qquad \forall j \in \{1,...,N_\mathrm {\omega } \}, \end{aligned}$$
(7)

where \(N_\mathrm {\omega }\) is the number of motor speed fitting levels, and \(x_{\textrm{m},i}\) is a vector consisting of convex functions of the model inputs

$$\begin{aligned} x_{\textrm{m},i} = \left[ |P_{\textrm{m},i}|, P_{\textrm{m},i}^2, P_{\textrm{m},i}^4, k_{\textrm{sh}}, k_{\textrm{sh}}^2, k_{\textrm{sh}}^4, k_{\textrm{rad}}, k_{\textrm{rad}}^2, k_{\textrm{rad}}^4 \right] ^\top . \end{aligned}$$

The fitting parameters \(Q_{\textrm{m},j}(t)\), \(q_{\textrm{m},j}(t)\), and \(q_{\textrm{m0},j}(t)\) are pre-computed at each time step, whereby we linearly interpolate the fitting parameters between sampled speed levels. We ensure that the motor model is convex by imposing the following conditions on the coefficients: every \(Q_{\textrm{m},j}\) is positive semi-definite, and the elements of all \(q_{\textrm{m},j}\) and \(q_{\textrm{m0},j}\) should be non-negative [18, 19]. To the end of retaining the problem’s convexity, we relax Eq. (7) to

$$\begin{aligned} P_{\textrm{m,loss},i} \ge x_{\textrm{m},i}^{+\top } Q_{\textrm{m},j}(t) x_{\textrm{m},i}^{+} + q_{\textrm{m},j}(t) x_{\textrm{m},i}^{+} + q_{\textrm{m0},j}(t) \qquad \forall j \in \{1,...,N_\mathrm {\omega } \}, \end{aligned}$$
(8)
$$\begin{aligned} P_{\textrm{m,loss},i} \ge x_{\textrm{m},i}^{-\top } Q_{\textrm{m},j}(t) x_{\textrm{m},i}^{-} + q_{\textrm{m},j}(t) x_{\textrm{m},i}^{-} + q_{\textrm{m0},j}(t) \qquad \forall j \in \{1,...,N_\mathrm {\omega } \}, \end{aligned}$$
(9)

where

$$\begin{aligned} x_{\textrm{m},i}^{+} = \left[ P_{\textrm{m},i}, P_{\textrm{m},i}^2, P_{\textrm{m},i}^4, k_{\textrm{sh}}, k_{\textrm{sh}}^2, k_{\textrm{sh}}^4, k_{\textrm{rad}}, k_{\textrm{rad}}^2, k_{\textrm{rad}}^4 \right] ^\top , \end{aligned}$$
(10)
$$\begin{aligned} x_{\textrm{m},i}^{-} = \left[ (-P_{\textrm{m},i}), (-P_{\textrm{m},i})^2, (-P_{\textrm{m},i})^4, k_{\textrm{sh}}, k_{\textrm{sh}}^2, k_{\textrm{sh}}^4, k_{\textrm{rad}}, k_{\textrm{rad}}^2, k_{\textrm{rad}}^4 \right] ^\top , \end{aligned}$$
(11)

following a lossless epigraphic relaxation of the constraint [18]. Due to the particular problem structure, the constraint will always hold with equality for the optimal solution. In fact, assuming any value higher than the strict necessary would be sub-optimal as it entails a higher energy consumption. The electrical input power \(P_{\textrm{ac},i}\) is given by

$$\begin{aligned} P_{\textrm{ac},i} = P_{\textrm{m},i} + P_{\textrm{m,loss},i}. \end{aligned}$$
(12)

As shown in Figs. 1 and 2, we train our models on a training set and evaluate the quality of our models on a test set, which is also synthesized using a Latin Hypercube design of experiments. We obtain a \(P_{\textrm{ac},i}\) normalized root-mean-squared error (NRMSE) of 0.19% and an adjusted \(R^2 = 0.92\) for the full test set. Finally, we display the efficiency maps of the test set alongside the model’s predictions in Fig. 3.

To approximate the torque and power limits, \(\overline{T}_{\textrm{m},i}\) and \(\overline{P}_{\textrm{m},i}\), respectively, we use linear regression models that are equal to

$$\begin{aligned} P_{\textrm{m},i} & \le \beta _{\textrm{p},0} + \beta _{\textrm{p},1} k_{\textrm{sh}} + \beta _{\textrm{p},2} k_{\textrm{rad}}, \end{aligned}$$
(13)
$$\begin{aligned} P_{\textrm{m},i} & \ge - \left( \beta _{\textrm{p},0} + \beta _{\textrm{p},1} k_{\textrm{sh}} + \beta _{\textrm{p},2} k_{\textrm{rad}} \right) , \end{aligned}$$
(14)
$$\begin{aligned} P_{\textrm{m},i} & \le \omega _{\textrm{m},i} \left( \beta _{\textrm{t},0} + \beta _{\textrm{t},1} k_{\textrm{sh}} + \beta _{\textrm{t},2} k_{\textrm{rad}} \right) , \end{aligned}$$
(15)
$$\begin{aligned} P_{\textrm{m},i} & \ge - \omega _{\textrm{m},i} \left( \beta _{\textrm{t},0} + \beta _{\textrm{t},1} k_{\textrm{sh}} + \beta _{\textrm{t},2} k_{\textrm{rad}}\right) , \end{aligned}$$
(16)

where all \(\beta _{\{.\}}\) are fitting parameters. Finally, we pose performance constraints on the electric motor, so that all vehicles are able to drive at a certain maximum velocity \(\overline{v}\), to launch from standstill at a certain road inclination \(\overline{\alpha }\), and to drive at a constant velocity \(v_\alpha \) at the same road inclination. These constraints are written as

$$\begin{aligned} \overline{P}_{\textrm{m},i} \eta _\textrm{gb} & \ge \frac{1}{2} \rho \; c_{\textrm{d},i} A_{\textrm{f},i} \overline{v}^3, \end{aligned}$$
(17)
$$\begin{aligned} \overline{T}_{\textrm{m},i} \eta _\textrm{gb} & \ge m_i g \; \sin \overline{\alpha } \; \frac{r_\textrm{w},i}{\gamma _i}, \end{aligned}$$
(18)
$$\begin{aligned} \overline{P}_{\textrm{m},i} \eta _\textrm{gb} & \ge m_i g \; \sin \overline{\alpha } \; v_\alpha . \end{aligned}$$
(19)

2.5 Battery

In order to compute the battery output power \(P_{\textrm{b},i}\), we consider a fixed inverter efficiency \(\eta _\textrm{inv}\) and constant auxiliary power \(P_\textrm{aux}\), such that

$$\begin{aligned} P_{\textrm{b},i} & \ge P_{\textrm{ac},i} \frac{N_{\textrm{m},i}}{N_{\textrm{b},i} \eta _\textrm{inv}} + P_\textrm{aux},\end{aligned}$$
(20)
$$\begin{aligned} P_{\textrm{b},i} & \ge P_{\textrm{ac},i} \frac{N_{\textrm{m},i} \eta _\textrm{inv}}{N_{\textrm{b},i}} + P_\textrm{aux}, \end{aligned}$$
(21)

after convex relaxations similar as in [13]. We find the internal battery power \(P_{\textrm{i},i}\) using the the short-circuit power \(P_{\textrm{sc},i}\), representing a variable efficiency depending on the losses, approximated with a piece-wise affine model [4]

$$\begin{aligned} P_{\textrm{i},i} \ge P_{\textrm{b},i} + \frac{1}{P_{\textrm{sc},i}} P_{\textrm{i},i}^2. \end{aligned}$$
(22)

The energy dynamic of the battery is influenced by \(P_{\textrm{i},i}\) through

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}E_{\textrm{b},i} = -P_{\textrm{i},i}. \end{aligned}$$
(23)

Hence, the energy capacity limit of the full battery pack \(\overline{E}_{\textrm{b},i}\) scales with the size \(S_{\textrm{b}}\) and number \(N_{\textrm{b},i}\) of battery modules respect to the reference battery energy capacity \(\overline{E}_{\textrm{b,o}}\) as

$$\begin{aligned} \overline{E}_{\textrm{b},i} = \overline{E}_{\textrm{b,o}} S_{\textrm{b}} N_{\textrm{b},i}. \end{aligned}$$
(24)

We set limits on the State-of-Charge SOC \(\xi \) by

$$\begin{aligned} E_{\textrm{b},i} \in \left[ \underline{\xi } \overline{E}_{\textrm{b},i}, \overline{\xi } \; \overline{E}_{\textrm{b},i} \right] , \end{aligned}$$
(25)

where \(E_{\textrm{b},i}\) is the battery State-of-Charge SOE, \(\underline{\xi }\) and \(\overline{\xi }\) are the minimum and maximum SOC levels, respectively. The mass of the full battery pack is equal to

$$\begin{aligned} m_{\textrm{b},i} = m_\textrm{b,o} S_{\textrm{b}} N_{\textrm{b},i}, \end{aligned}$$
(26)

where \(m_\textrm{b,o}\) is the mass of the reference battery. Since we consider a variable efficiency of the battery through \(P_{\textrm{sc},i}\), we include a constraint to ensure that the operations are conducted around the half-capacity level of the battery by averaging the maximum capacity at the start and the minimum capacity at the end of the driving cycle

$$\begin{aligned} E_{\textrm{b},i}(0) + E_{\textrm{b},i}(T) = S_{\textrm{b}} \left( \overline{\xi } + \underline{\xi } \right) \overline{E}_\textrm{b,o} N_{\textrm{b},i}, \end{aligned}$$
(27)

where T is the total duration of the drive cycle. Additionally, we require that the range d of the vehicle is larger than the minimum requirement \(\underline{d}_{\textrm{r},i}\) on one fully-charged battery by

$$\begin{aligned} \Delta E_{\textrm{b},i} \le \left( \overline{\xi } - \underline{\xi } \right) \overline{E}_{\textrm{b},i} \frac{d}{\underline{d}_{\textrm{r},i}}. \end{aligned}$$
(28)

Finally, we write the objective function of the optimization problem as

$$\begin{aligned} J_\textrm{o} = \sum _{i = 1}^{N} \left( E_{\textrm{b},i}(0) - E_{\textrm{b},i}(T)\right) , \end{aligned}$$

where \(J_\textrm{o}\) is the family energy consumption during the driving cycle.

2.6 Optimization Problem for a Family of Vehicles

Summarizing, to obtain the optimal powertrain component sizing for a family of electric vehicles minimizing energy consumption, we solve a convex optimal design problem, considering given \(\gamma _i\), \(N_{\textrm{m},i}\), \(N_{\textrm{b},i}\), and \(r_{\textrm{b},i}\). The design variables of the modules, shared by the whole family, are \(p = \{k_{\textrm{sh}}, k_{\textrm{rad}}, S_{\textrm{b}} \}\), whereas the only state variable is \(x = E_{\textrm{b},i}\), which is different for every vehicle.

Problem 1 (Electric Vehicle Family Design): The optimal powertrain design for a family of electric vehicles to minimize the total energy consumption is the solution of

$$\begin{aligned} \begin{aligned} &\!\min _{x,p} & J_\textrm{o} & \\ & \text {s.t. } & (1)-(6),(10) - (28), &\quad \forall i\in \{1,..,N\},\\ & &(8),(9), & \qquad \forall i\in \{1,..,N\} \quad \forall j \in \{1,...,N_\mathrm {\omega } \},\\ & & p = \{k_{\textrm{sh}}, k_{\textrm{rad}}, S_{\textrm{b}} \}.& \end{aligned} \end{aligned}$$
Fig. 3.
figure 3

The efficiency maps of the test set, both from Motor-CAD, denoted with “Sample”, and the predictions, denoted with “Pred.”

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Fig. 4.
figure 4

The efficiency maps of the optimized motor obtained with the high fidelity tool Motor-CAD [16] (left) alongside the model predictions (right).

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Since the constraint set and the cost function are convex, we ensured that our problem is fully convex and therefore we can compute the globally optimal solution [18] with standard nonlinear programming methods.

2.7 Discussion

A few comments are in order. First, the electric motor model has been trained on samples for design variables in a range of \(\pm 20\%\) with respect to the reference motor. Outside this interval, we cannot ensure an accurate representation of the losses. Second, while the identification process has been based on an interior permanent magnet synchronous machine, the methodology can be readily adapted to other technologies. Third, we do not consider any thermal effects, as they have little influence on the motor operations of conventional vehicles [20]. Moreover, we scale the battery size only by acting on the number of cells in parallel, thus changing its energy without altering the battery voltage. Concerning the numerical solution of Problem 1, we use standard nonlinear programming methods, even though the problem is a semi-definite program (convex) due to the fact that standard convex solvers are not able to work with the higher-order input functions of our electric motor model. Finally, it is important to underline that, in our framework, only scaling factors are optimization variables, whilst the modules’ multiplicities are given parameters. This limitation could be readily overcome by solving a sequence of problems in a combinatorial manner, yet this is beyond the scope of the present paper.

3 Results

In this section, we showcase our framework on the simultaneous design of a city car, a compact car, and a cross over. The general parameters are reported in Table 1, while the vehicle-specific parameters are given in Table 2. In our case study, we consider the Class 3 Worldwide harmonized Light-duty vehicles Test Procedure (WLTP), discretizing with a sampling time of 1 s and the forward Euler method. Finally, we solve the problem using the nonlinear solver IPOPT [22] in 120s after parsing it with CasADi [23] in 58s. The results of our case study show the design of single-sized components able to serve a diverse family of vehicles. We can observe the optimization output and the vehicles’ performance in Table 3. In Fig. 4, we compare the efficiency maps of the motor with optimized axial and radial geometrical dimension obtained through the high-fidelity modeling software Motor-CAD [16] (left) and our model’s predictions (right). In this specific case, the motor map of the optimal geometrical design presents two high-efficiency regions, virtually introducing higher efficiencies in the low-torque low-speed region. Since this region contains operating points common to all the vehicles, a map of this type is favored in the simultaneous design over the ones with a single high-efficiency region located in the middle shown in Fig. 3. However, this behavior could be avoided by adding regularization terms in the identification of the fitting parameters \(Q_{\textrm{m},j}(t)\), \(q_{\textrm{m},j}(t)\), and \(q_{\textrm{m0},j}(t)\) at the cost of a slight reduction in overall accuracy. Compared to the high-fidelity software, the NRMSE for the motor losses \(P_{\textrm{m,loss}}\) is 5.7%, while it equals 0.33% for the overall electrical power \(P_{\textrm{ac},i}\).

Table 1. General Parameters.
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Table 2. Vehicle Parameters [21].
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Table 3. Optimization results for the concurrent design.
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4 Conclusions

This paper presented a design optimization framework featuring a geometric model of the EM that allows for joint design of the EM low-scale dimensions and battery sizing with global optimality guarantees, for a family of battery electric vehicles. We identified the fitting parameters with an average normalized root mean squared error of 0.19% and an adjusted \(R^2\) value of 0.92 for the full test set. Finally, we determined the optimal joint design of the motor and battery from the perspective of using the components on every vehicle in the family, composed of a city car, a compact car and a cross over. The results display a normalized root mean squared error of 0.33% for the motor power in the motor map of the optimal geometrical design found.

Future Work: This work opens the field for the following extensions: We aim to sophisticate the model by incorporating the gear ratio and powertrain topology (layout of components) among the design variables, and jointly optimizing the multiplicity of components within each vehicle.