Integration of design cycles
In a conventional vehicle for example, pressing the accelerator pedal corresponds directly to a fuel injection demand, supplied to the combustion engine, providing mechanical torque afterwards. Intuitively, the driver regulates the speed of the vehicle by pressing or releasing the accelerator pedal.
If the driver acts in a similar manner for a hybrid vehicle, management laws need to be developed to improve the powertrain’s performance. In fact, for each operation point, which can be represented by a torque demand and a given vehicle speed, diﬀerent « discrete » hybrid modes are possible (as presented in Figure 1.9). These modes are then translated into an infinite number of possibilities when using the vehicle, depending on the selected power split between the engine and the electric motor.
This choice is then oriented in order to achieve the best possible savings in fuel con-sumption, the major aim of hybridization, while considering the battery storage constraints. These are referred to as real-time power management strategies or online control strategies, implemented within the vehicle’s hybrid control unit (HCU).
There is an abundant literature for the diverse strategies that have been developped on the matter. Rule-based strategies are the first ones to be studied and remain the easiest to deploy [21, 50, 70]. They generally rely on the levels of the battery charge and requested power/torque as inputs to switch between the diﬀerent hybrid modes, as shown for instance in Figure 1.19. These strategies are still the most widely used methods in today’s hybrid car fleet.
Figure 1.19: Rule-based strategy proposed by : T t is the requested torque, v the vehicle’s speed and u the power-split ratio between the EM and ICE. The map on the right is for when the battery’s state of charge is high, and the map on the left is for when it is low Fuzzy logic controllers [71, 72, 73, 74] have also been suggested as well as the use of trained Neural Networks [75, 76]. These methods have shown they are only as eﬃcient when the driving cycle remains close to the learning base used for their development.
Other techniques such as convex optimization  have managed to improve fuel eﬃ-ciency through simplification of the powertrain model. The most eﬃcient strategies however remain those derived from optimal control theory such as -Control [78, 79], Equivalent Consumption Minimization Strategy (ECMS, ), Loss Minimization Strategy (LMS, ), Stochastic Dynamic Programming (SDP, ) and model predictive control [72, 83].
Thus, the above-mentioned strategies allow the ECU to find adequate commands quickly without the need to know for certain the future behaviour of the vehicle. On the other hand, it is clear that the absolute minimum of fuel consumption can only be achieved by knowing the speed cycle in advance. For instance, if the driver is aggressively decelerating in the near future, the potential electrical energy to be recovered through regenerative braking can be injected much earlier into the electric motor to reduce the load on the combustion engine, leading to better fuel eﬃciency.
These are referred to as optimal control strategies in the sense that the minimum fuel consumption provided, subject to the assumptions made, is the absolute minimum taking into account a maximum number of parameters, including some that are diﬃcult to know in reality such as the evolution of the vehicle speed in this case.
In the context of this research project, the comparison between the diﬀerent powertrains should be based on this value. This limits the impact of the control strategy’s bias and prevent controller conditioning, leading to suboptimal solutions for the global optimization problem. For this reason, the focus is mainly shifted towards optimal control strategies in this work.
Optimal control strategies
The purpose of optimal control strategies is to determine the system control that minimizes the fuel consumption over the whole driving cycle, which is known in advance. The battery’s state of charge (SoC) at the end of the driving cycle needs to be close to its initial value, as it is a requirement in hybrid vehicle homologation, referred to as the charge-sustaining condition. This defines the optimal control problem whose solution takes much longer to find as when compared to real-time strategies. For this reason, calculations generally need to be executed oﬄine, which explains why optimal control methods are referred to as oﬄine strategies.
Several authors relied on meta-heuristic methods to solve the optimal control problem:  used the simulated annealing method while  applied a particle swarm algorithm to coordinate the powertrain control strategy.  on the other hand turned to Marco Dorigo’s method (ant colonies) to deduce the optimal command.
The main drawback of these strategies is the excessive number of model evaluations, which increases exponentially with the number of variables involved. Since the latter in-creases with the number of time steps considered, application of these strategies is only limited to short cycles. Deterministic optimization algorithms have also been tested on numerous occasions. For instance,  used sequential quadratic programming (SQP) on short driving cycles. This limitation is due to non-convergence problems for longer driving cycles (over 200 time steps), especially in the absence of convexity or problem linearization options.
If the current homologation procedure is considered, the control needs to be optimized over the WLTC 3-b cycle (Figure 1.6). This cycle lasts 30 minutes and is discretized to the second, leading to thousands of control variables. Diﬀerent techniques that are not limited to a reduced number of variables should then be explored.  optimized parameters of a rule-based strategy for example with genetic algorithms, as shown in Figure 1.20, as a solution.
Figure 1.20: Rule-based strategy proposed by 
The most commonly used approaches for optimal power management however remain those deriving from the Calculus of Variations as in Pontryagin’s Minimum Principle (PMP, [29, 55, 87, 88, 89]) and those based on solving the Hamilton-Jacobi-Bellman equation (HJB, ) like dynamic programming (DP, [90, 91, 92]).
PMP solves a dual problem with the same solution as the initial problem. The dual problem is simplified by introducing a Lagrange multiplier. Optimal command is found afterwards at each time step through a much simpler minimization problem.
Along with its simple implementation, PMP’s main advantage is that the total calcu-lation time is only linear with the number of steps. One of the main drawbacks of PMP however is the minimization process, often leading to local optima, as well as the diﬃculty of taking other constraints into account, such as the limitations of the powertrain components.
Meanwhile, DP’s process can be easily explained through Bellman’s principal, illustrated in Figure 1.21: if the optimal trajectory from point A to point B minimizing a certain cost function passes through a third point C, then the portion of the path from C to B is optimal as well considering the same cost function.
Figure 1.21: Example illustrating Bellman’s principle
In simple terms, DP comes down to decomposing the original control optimization problem throughout the entire cycle into easier-to-solve subproblems, where the command is only optimized from a given state at a given time step until the end of the driving cycle, and storing the solution every time so that each subproblem is only solved once. However, building an end to end solution in reasonable times using this procedure is only feasible by discretizing the number of possible state values at each time step and passing through these later on.
When an adequate discretization is selected, a solution is always provided. Moreover, the required time only evolves, similarly to PMP, in a linear fashion with respect to the number of time steps considered. Nevertheless, improving the solution naturally requires a higher number of discrete values, as this leads to a greater number of paths to explore. This in turn results in exponentially higher calculation times, the main disadvantage of DP.
Both PMP and DP have been frequently compared in literature, achieving close perfor-mance in terms of fuel consumption and computation time [83, 87]. As such, they should be confronted later on with each other for the studied application. These strategies are explored in greater detail in Chapter 2.
Once a control strategy is selected, how the machine is used at each time step is imposed. Optimizing the design over the entire rolling cycle means taking into account hundreds (or even thousands) of operation points for each considered component (as shown in Figure 1.22). Hence, a direct sizing approach using cycle performance as criteria will not find an optimal solution in a reasonable time.
Figure 1.22: Example of EM’s eﬃciency mapping and its operation points over WLTC 3-b
One of the possibilities would be to substitute the machine performance over the cycle to simply evaluating its performance over a reduced number of representative points. In order to achieve acceptable accuracy, various methods have been proposed in the literature and applied to numerous fields including the automotive industry such as the sizing of renewable energy farms, data mining, railway applications, etc.
The single point method is the most widely used in the industry and is based on expert rules to determine a unique critical point to be optimized during the conception process.
On the other hand,  proposed the Random Sampling method where a reduced num-ber of points is randomly selected from the initial pool of operation points using a statistical law. This method can be used to halve the initial number of points without a significant loss in precision . However, its use for an even further reduction (less than a dozen points of interest) remains to be explored.
The histogram method has also been used. It is based on the statistical distribution of operational points into diﬀerent intervals. The centre of each interval is then considered for the calculation of losses over the cycle, as illustrated in Figure 1.23 for the example of electric machine design.  used this method for the sizing of micro wind turbines. It is clear nevertheless that this method is only suited to other applications with correlated outputs (correlation between torque and speed in the case of the studied application).
Figure 1.23: Application of the histogram method. The center of each interval is highlighted.
Meanwhile, [45, 95, 96] relied on the barycenters method to find the optimal design of an electric machine for electric vehicles. The method relies, in a similar manner as the histogram method, in dividing the torque-speed characteristic of the machine into several zones. The operation points within each area are then reduced to their barycenters, as pre-sented in Figure 1.24.
Diﬀerent alternatives have been suggested afterwards to estimate the machine’s cycle losses, from the values calculated at the selected barycenters. In , the losses are weighted to the number of operation points in each interval while  proposed a diﬀerent formula for each type of machine losses, based on the hypothesis of torque/current and speed/voltage correlations.
Figure 1.24: Application of the barycenters method. The barycenters are highlighted.
Another method worth exploring, mainly used in data science to reduce the size of in-formation to be analyzed, is Clustering . It refers to a statistical operation that divides a group of variables into a limited number of « clusters » or segments. In contrast with the barycenter or histogram method, these clusters are not defined in advance. This approach seeks to assemble objects sharing similar characteristics, with the intention of achieving inter-nal homogeneity (similarity within the same cluster) and external heterogeneity (distinction between diﬀerent clusters), as seen in Figure 1.25.
Figure 1.25: Application of the Clustering technique. The centroïds of each cluster are highlighted.
Several algorithms are called upon to achieve this division, such as the k-means or k-medoïd method. The centres of these clusters, called centroïds, represent the points of interest to be retained for the rest of the study. For electric machine sizing, the two variants presented to estimate the machine’s performance through the barycenters method can be adapted for Clustering as well.
As such, these techniques diﬀer in their implementation and their adaptability to the conception problem. Another important criteria to consider in hybrid powertrain design is the number of interest points required to achieve acceptable accuracy, which directly impacts the overall calculation time of the optimization process. Since literature on the comparison between these methods is scarce, an assessment over the common benchmark of the hybrid powertrain application is required.
It is necessary to remember that when road cycles are taken into account during the sizing of the powertrain, the problem addressed has a double complexity: powertrain design opti-mization is based on fuel consumption, which is in turn strongly impacted by the powertrain control strategy, as explained in the previous section.
Thus, the hybrid powertrain optimization problem, as studied in this work, belongs to the Plant/Controller optimization class of problems. This type of optimization problems can be solved using diﬀerent schemes: sequentially, iteratively, using a bi-level approach or simultaneously.
The sequential approach is where the design is optimized first for a certain control strat-egy, before optimizing the command afterwards, as depicted in Figure 1.26. It is the simplest to implement amongst the four studied approaches and the most intuitive, explaining its wide use in the industry. Still, it does not guarantee system optimality, as demonstrated by .
Figure 1.26: Sequential approach applied for hybrid powertrain optimization.
The iterative approach on the other hand improves the solution of the sequential method by reoptimizing the design following significant changes in the controller’s command, before reiterating again until convergence. System optimality is aﬀorded in this manner for certain cases . Application on the studied problem is shown in Figure 1.27.
Meanwhile, the bi-level approach, also referred to as nested approach, finds the optimal control for each proposed design by the top level algorithm, as presented in Figure 1.28. The objective is to select the powertrain with the best possible fuel gains at the end of the optimization loop. In this way, the required optimality conditions can still be ensured .
Figure 1.27: Iterative approach applied for hybrid powertrain optimization.
Thus, both the iterative and bi-level approaches allow for the partitioning of the orig-inal problem into design and control optimization problems. By doing so, optimal control strategies, presented in the previous section, can be used to eﬃciently solve the controller optimization problem and greatly reduce the complexity of the complete problem.
Figure 1.28: Bi-level approach applied for hybrid powertrain optimization.
However, if the nested approach is exploited extensively, implementations of the iterative scheme are quite rare. The latter should be analyzed further as diﬀerent alternatives can be explored. One of the proposed options relies on the coupling with cycle reduction tech-niques, presented in section 1.3.3, as shown in figure 1.29. This can lead to faster iterations and the ability to use higher fidelity models.
Table of contents :
1 Context and state of the art
1.1 Context of the research project
1.1.1 Air pollution consequences
1.1.2 Greenhouse gas emissions
1.1.3 Decarbonization and policy
1.1.4 Automotive strategies
1.1.5 Hybrid families
1.2 Importance of systemic design and work positioning
1.2.1 Scientific issues
1.2.2 Lab positioning of the research project
1.2.3 National positioning of the research project
1.2.4 Global positioning of the research project
1.3 Integration of design cycles
1.3.1 Power management
1.3.2 Optimal control strategies
1.3.3 Cycle reduction
1.4 Plant/Controller optimization
1.5 Multidisciplinary Design Optimization
1.5.1 Decomposition-based strategies
1.5.2 Monolithic optimization
1.5.3 Distributed optimization
2 Systemic design approaches on drive-cycle
2.1 Optimal control and design problems formulation
2.2 Hybrid railway power substation application
2.3 Optimal control strategies
2.3.1 Direct Optimization
2.3.2 Collaborative Optimization
2.3.3 Pontryagin’s Minimum Principle
2.3.4 Dynamic Programming
2.3.5 Link between PMP and DP
2.3.6 Comparison and analysis
2.4 Systemic design approaches
2.4.1 Simultaneous approach
2.4.2 Bi-level approach
2.4.3 Iterative approach
2.4.4 Comparison and analysis
2.5 Alternative design approaches
2.5.1 Approach based on the simultaneous scheme: A1
2.5.2 Approach based on the bi-level scheme: A2
2.5.3 Approach based on the iterative scheme: A3
3 Hybrid Electric Vehicle application
3.1 System presentation
3.2 Vehicle representation
3.3 Powertrain components
3.3.3 Electric machine
3.3.5 Internal combustion engine
3.4 Optimization constraints
3.4.1 Geometric constraints
3.4.2 Performance constraints
3.4.3 Process constraints
3.4.4 Mechanical constraints
3.4.5 Thermal constraints
3.4.6 Demagnetization constraints
3.4.7 Torque ripple constraints
3.4.8 Inverter constraints
3.5 Problem definition
4 Hybrid Electric Vehicle optimization
4.1 Systemic design approaches
4.2 HEV power management
4.2.1 Optimal control strategies
4.2.2 Application and comparison
4.2.3 Analysis of optimal command
4.2.4 Comparison with simulation platform
4.2.5 Application on different powertrains
4.3 HEV cycle reduction
4.3.1 Studied cycle reduction techniques
4.3.2 Comparison and analysis
4.3.3 Mirroring technique
4.4 HEV case study
4.5 Screening study
4.6 Optimization results
4.6.1 Optimization over 4 design variables
4.6.2 Optimization over 10 design variables
4.6.3 Design optimization without considering performance constraints
4.7 Comparison of optimal designs