Energy management in hybrid electric vehicles using optimized radial basis function neural network

Abstract

This paper deals with energy management in hybrid electric vehicles. Use of radial basis function neural network (RBFNN) for the problem of energy management gains importance in the present decade. Use of genetic algorithm (GA) and particle swarm optimization (PSO) as optimization algorithms for parameter estimation is also well known. However, none of the researchers in the area tried to use GA and PSO as training algorithms for the problem. Hence in this paper, we propose two novel methods, based on RBFNN. The difference between RBFNN-based approaches in the literature and those used in this paper is the use of GA and PSO (i.e. optimising algorithms) as training algorithm to train RBFNNs. Interestingly, it is seen that the proposed approaches of this paper outperform RBFNN-based approaches in the literature with traditional training.

Keywords::

• energy management
• hybrid electric vehicles
• radial basis function neural networks

1. Introduction

Energy management is an important aspect in the field of vehicular technology. The art of using artificial neural network (ANN) for energy management has been gaining momentum since the last two decades (Ates et al. 2009; 2010; Moreno, Ortúzar, and Dixon 2006; Gong, Li, and Peng 2009; Feldkamp, Abou-Nasr, and Kolmanovsky 2009). Use of neural networks (NNs) with wavelet (Ates et al. 2009, 2010), ultra-capacitor (Moreno, Ortúzar, and Dixon 2006) and also trip modelling NNs (Gong, Li, and Peng 2009) and use of recurrent neural network (RNN) (Feldkamp, Abou-Nasr, and Kolmanovsky 2009, Prokhorov 2008) are popular methods in energy management. However, NN-based approaches have the limitations of large complexity and also fail because of over-fitting, local optima. On the other hand, radial basis function (RBF) networks, with only one hidden layer, have the ability to find global optima.

Because of the obvious reasons, the most popular choice for the nonlinearity is the Gaussian function. The RBF network classifies the received signal according to the class of the centre closest to the received vector. The output of the RBFNNs provides an attractive alternative to multi layer perceptron and RNN (Sheikhan, Pardis, and Gharavian 2013) for energy management problems because of the following obvious reasons: the most popular choice for the nonlinearity is the Gaussian function replacing traditional sigmoidal transfer function. In addition to less computational complexity, simulations given in the literature reveal that energy management based on the RBF approaches produces superior performance as compared with other traditional ANN-based approaches. Hence the works on energy management using radial basis function neural network (RBFNN) became an established and an active area of academic research and development (Sheikhan, Pardis, and Gharavian 2013; Taghavipour, Foumani, and Boroushaki 2012; Li, Xu, and Xu 2012; Kaloko, Soebagio, and Purnomo 2011).

The performance of the RBFNN is dependent on the basis centres, which must be representative of the whole data-set. The popular K means algorithm does have a number of problems associated with it. The number of hidden nodes required, i.e. cluster centres has to be decided a priori. Two samples close to each other in the input space do not necessarily have similar outputs. The random choice of starting point influences the final cluster centres and so can only achieve a local optimal solution. Due to these shortcomings, the poor performance of an RBFNN is likely to be attributed to sub-optimal placement of the cluster centres.

However, there are still some difficulties with building RBFNNs. One of the main problems with RBFNNs is determining the number of RBFs. In general, the choice of the number of RBFs is experience oriented and selected by a trial-and-error procedure. The process of obtaining the above-mentioned terms is time consuming. Another important issue in designing an RBFNN system is the selection of the free parameters for the RBFs (the centres, spreads and connection weights).

In order to avoid existing trial-and-error methods and to improve local optimal problems, Barreto, Barbosa, and Ebecken (2002) used genetic algorithm (GA) and Feng (2006) used particle swarm optimization (PSO) to decide the centres of hidden neurons, spread and bias parameters by minimizing the mean square error (MSE) of the desired outputs and actual outputs. GA and PSO-trained RBF networks are termed here, in this paper, as GRBF and PRBF, respectively. Performance of GRBF and PRBF in energy management is yet to be established. Hence, we have taken an novel attempt in using PRBF and GRBF for the problem of energy management, and it is proved through simulation results that these approaches outperform contemporary GA, PSO and RBF-based approaches. The essence of the proposed approaches is that its performance outperforms the RBF-based approaches available in the literature.

The organization of the paper is as follows: Section 2 discusses the problem statement. GRBF and PRBF-based approaches are discussed in Sections 3 and 4, respectively. Simulation results and discussion are discussed in Section 5. Finally conclusion of the paper is outlined in Section 5.

2. The problem

As it is revealed from the literature, use of RBFNNs for energy management is a non-model-based technique. However, this paper uses a simple power-based model since the proposed methods use heuristics such as GA and PSO to train RBFNNs. The models with a bit complicated power-train physics and operating constraints like that used in Moura, Stein, and Fathy (2013) will be reflected in our future work with model-based techniques. Model of the vehicle used is parallel and is explained in the Appendix.

The energy management problem can be formulated as an optimization problem, where a cost function is minimized subject to constraints. Because energy is temporarily stored and later retrieved, the optimization problem is usually defined over a time horizon instead of a single time instant.

The idea of controlling the vehicle power is initiated by the fact that energy losses in the internal combustion engine, alternator and battery change according to their operating point. Minimizing these energy losses will result in an energy management strategy that achieves higher fuel economy.

The control objective of energy management is to lower the fuel consumption and exhaust emissions while satisfying several constraints. This control problem can be formulated as a dynamic optimization problem.

If $x(k)$ are the state variables (corresponding hybrid electric vehicle (HEV) states), such as vehicle speed, engine speed and energy storage levels, and $u(k)$ are the control variables at kth instant, the control variables can be continuous, for instance, the power flow, discrete, such as engine on/off, or complementary, meaning that only one of a set of variables can be non-zero at a time, such as the gear position.

Using discrete time, the vehicle is represented by a dynamic system (Lin et al. 2001):

(1) $x(k+1)=f\left(x\right(k),u(k),k)$(1)

which has to be controlled, such that the cost criterion

(2) ${\displaystyle \sum _0^n} \gamma \left(x\right(k),u(k),k)\Delta t$(2)

is minimized, satisfying the constraints

(3) $\phi \left(x\right(k),u(k),k)\le 0,\psi \left(x\right(k),u(k),k)=0.$(3)

In this application, the only relevant state is the energy level in the battery $E_{\text{s}}$.

The energy level of the battery is given by a discrete time version of (1):

(4) $E_{\text{s}}(k+1)=E_{\text{s}}\left(k\right)+P_{\text{s}}\left(k\right)\Delta t.$(4)

Assuming the signals’ engine speed $(\omega (k))$, mechanical drive train power $(P_{\text{d}}(k))$ and electrical load power $(P_{\text{l}}(k))$ to be known, and combining the characteristics of all components given by

(5) $\begin{array}{c}{P}_{\text{b}}=P_{\text{s}}+P_{\text{loss}}\left(P_{\text{s}}\right),\\ P_{\text{e}}=P_{\text{l}}+P_{\text{b}},\\ P_{\text{g}}=g(P_{\text{e}},\omega ),\\ P_{\text{m}}=m\text{ax}(P_{\text{d}}+P_{\text{g}},P_{m min}),\\ \ddot{m}=f(P_{\text{m}},\omega ),\end{array}$(5)

where $P_{\text{b}}$ is battery power, $P_{\text{s}}$ is stored battery power, $P_{\text{e}}$ is electrical motor power, $P_{\text{g}}$ is mechanical motor power, $P_{\text{m}}$ is mechanical engine power and $\ddot{m}$ is fuel rate, the fuel rate can be expressed as a function of the battery storage power

(6) $\ddot{m}\left(w\right(k),P_{\text{d}}(k),P_{\text{l}}(k),P_{\text{s}}(k\left)\right)=\ddot{m}\left(P_{\text{s}}\right(k),k).$(6)

The cost function expresses the fuel used over the driving cycle in the time interval $t=\Delta t·[0,\dots ,n]$, so (2) becomes

(7) ${\displaystyle \sum _{k=0}^n} \gamma \left(P_{\text{s}}\right(k),k),$(7)

where $\gamma (P_{\text{s}},k)=w_1\ddot{m}\left(P_{\text{s}}\right(k),k)+w_{2}C\text{O}{}_2\left(P_{\text{s}}\right(k),k)+w_{3}C\text{O}\left(P_{\text{s}}\right(k),k)+w_{4}N\text{O}{}_x\left(P_{\text{s}}\right(k),k)+w_{5}H\text{C}\left(P_{\text{s}}\right(k),k).$

By choosing $P_{\text{s}}$ as decision variable $z$, the characteristics of all components are included in the cost function. The actual controlled input in the vehicle is $P_{\text{e}}$. Because the relation between $P_{\text{s}}$ and $P_{\text{e}}$ is known, $P_{\text{e}}$ can be computed from the optimal $P_{\text{s}}$.

The operating range of the components is limited, so bounds have to be set on the engine power, electrical power and battery power throughput. This can be done using the following constraints:

(8) $P_{m min}\le P_{\text{m}}\le P_{m max},P_{e min}\le P_{\text{e}}\le P_{e max},P_{b min}\le P_{\text{b}}\le P_{b max}.$(8)

These constraints can be translated to time-varying bounds on $P_{\text{s}}$. Combining them leads to one lower and upper bound for $P_{\text{s}}$ at each time instant:

(9) $P_{s min}\le P_{\text{s}}\le P_{s max}.$(9)

The bounds on the battery energy level $E_{\text{s}}$ can also be translated to constraints on $P_{\text{s}}$:

(10) $E_{s min}-E_{\text{s}}\left(0\right)\le {\displaystyle \sum _{i=0}^k} P_{\text{s}}\left(i\right)\Delta t\le E_{s max}-E_{\text{s}}\left(0\right)\forall k\in [0,n].$(10)

A charge-sustaining vehicle requires some kind of endpoint penalty to guarantee that the state of charge of the battery remains in a neighbourhood around a desired value. An endpoint constraint will be used here, requiring the state of energy (SOE) at the end of the cycle to be the same as at the beginning:

(11) $E_{\text{s}}\left(t_n\right)=E_{\text{s}}\left(0\right)\to {\displaystyle \sum _{k=0}^n} P_{\text{s}}\left(k\right)=0.$(11)

3. RBF in energy management

The main aim of this section is to outline the concept of energy management using RBF for the ease of the reader. The RBF network is constructed by the inherent parameter that influences energy management, so complex modelling of the vehicle can be avoided. The important problem for energy management using RBF networks is to place the optimal centres at the desired optimal states. We can find the relation between the desired states. This relation causes the centres can be composed from a data-set with smaller size. In fact, this elements set is the same as the parameter set. In other words, if the optimal states are known, the RBF network can be designed. The object of the problem can be changed to find the optimal data-set (desired output states). The energy management problem becomes a function maximization problem where target is the maximum Bayesian likelihood and variables are channel output states. Because the mathematical relationships between the desired output states and the Bayesian likelihood are too complex to be formulated or cannot be formulated when the structure of vehicle is unknown, we apply GA and PSO to solve this complex optimal problem with local minima. We have taken a novel attempt in using GA and PSO to train RBF that learns the data-set instead of centres and use the pre-known mapping to obtain the centres of the RBF network, for energy management.

In order to use RBF network in energy management, we follow three steps: clustering, training and evaluation. For the energy management problem, the inputs to the RBFNN are $x(k)=[{\omega }_{\text{g}},{\tau }_{\text{g}},{\omega }_{\text{m}},{\tau }_{\text{m}},P_{\text{b}},\gamma ]$ _. Here, ${\omega }_{\text{g}}$ and ${\tau }_{\text{g}}$ are the motor speed and torque, respectively, ${\omega }_{\text{m}}$ and ${\tau }_{\text{m}}$ are the engine speed and torque, respectively, $P_{\text{b}}$ is battery power and $\gamma$ is the instantaneous cost. The network generates the desired control actions based on the measurements of the system states and operates as the actual controller of the system and the network is trained so that the output will be the desired one.

3.1 GA-trained RBF and GRBF

GA (Goldberg 1989) inspired by the evolutionary biology is a popular method for finding approximate solutions for optimization and search problems. In the GA, a population of strings called chromosomes which encode candidate solutions to an optimization problem evolves towards better solutions. The evolution usually starts from a population of randomly generated individuals and occurs in generations. In each generation, the fitness of every individual in the population is evaluated, multiple individuals are stochastically selected from the current population based on their fitness and are modified by recombination and possibly random mutation to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations have been produced, or a satisfactory fitness level has been reached for the population. If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached. The three genetic operations selection, crossover and mutation of the main aspects of GA evolve the optimal solution from an initial population.

• GA for clustering

To avoid using any particular underlying distribution of the data-set, and to avoid sub-optimal clustering, we use GA for clustering as mentioned in Maulik and Bandyopadhyay (2000). We used string representation, population initialization, fitness computation, selection, crossover and mutation as per Maulik and Bandyopadhyay (2000).

• GA as training algorithm for RBF networks

Barreto, Barbosa, and Ebecken (2002) used the real-code GA to decide the centres of hidden neurons, spread and bias parameters by minimizing the MSE of the desired outputs and actual outputs. Because the RBF network has only one hidden layer, the task of applying the GA algorithm for optimizing the architecture is simplified. Furthermore, the number of data vectors in the training set defines an upper bound on the number of hidden nodes. As discussed earlier, optimally determining the basis centres is problematic since all methods have their pitfalls; also, determining the number of basis centres is generally accomplished by trial and error. Therefore, this encourages a hybrid approach where the GA optimizes the basis centres and architecture and the second stage continues to utilize a supervised training method (using, for example, singular value decomposition).

Steps for the training algorithm used in this paper can be outlined as follows:

1. Initialize the distance factor randomly;

2. Construct the RBFN structure

3. Update output layer weights

4. Evaluate GA fitness

5. If criteria met, then Stop evolution,

Else, produce a new distance factor by GA evolution and go to step ii.

The literature on training of NNs and in particular RBFNNs using GA, termed in this paper as GRBF, goes back to 1990s and has been reviewed in Harpham, Dawson, and Brown (2004). GRBF has also been used successfully in radar or satellite image (Leung, Dubash, and Xie 2002), photovoltaic panels (Zhang and Bai 2005), system design (Hatanaka, Kondo, and Uosaki 2006), modelling (Wu et al. 2007), terrain classification (Kurban and Beşdok 2009), function approximation (Awad 2010), time series prediction (Gan, Peng, and Dong 2012), etc. We have taken a novel attempt using GRBF for energy management. In this paper, simulations parameters of GA were used, inspired from Mohanty (2008), and is outlined in Table 1.

Table 1 Simulation parameters for GA.

Parameter	Value	Parameter	Value
Number of iterations	1000	Number of individuals	50
Selection type	Roulette	Mutation type	Uniform
Mutation ratio	0.03	Crossover type	Two point
Crossover ratio	0.9

3.2 PSO-trained RBF and PRBF

The PSO first introduced by Kennedy and Eberhart (1995) is a swarm optimization method that optimizes a problem by iteratively trying to improve candidate solutions called particles. The improvement of candidate particles with $D$ dimension in the PSO algorithm is dependent on the particle's local best solution, ${\displaystyle P_{\text{l}}^t}=[{\displaystyle P_{l1}^t},{\displaystyle P_{l2}^t},\dots ,{\displaystyle P_{\text{l}D}^t}]$ (up to the point of evaluation) and the swarm's global best solution ${\displaystyle P_{\text{g}}^t}=[{\displaystyle P_{g1}^t},{\displaystyle P_{g2}^t},\dots ,{\displaystyle P_{\text{g}D}^t}]$ at the iteration $t$. Every particle has a fitness value, which is evaluated by the fitness function for optimization, and a velocity which directs the trajectory of the particle. The D-dimensional position for particle $i$ can be at the iteration $t$ represented as ${\displaystyle x_i^t}=[{\displaystyle x_{i1}^t},{\displaystyle x_{i2}^t},\dots ,{\displaystyle x_{iD}^t}]$. Like the position, the velocity of particle $i$ can be described as ${\displaystyle v_i^t}=[{\displaystyle v_{i1}^t},{\displaystyle v_{i2}^t},\dots ,{\displaystyle v_{iD}^t}]$. The movements of particles $i$ at the $t+1$ iteration are given by

(12) ${\displaystyle v_{id}^t}=[{\displaystyle v_{id}^{t-1}}+c_1{r}_1({\displaystyle P_{id}^t}-{\displaystyle x_{id}^t})+c_2{r}_2({\displaystyle P_{gd}^t}-{\displaystyle x_{id}^t}\left)\right];d=1,2,\dots ,D,$(12)

(13) ${\displaystyle x_{id}^t}=[{\displaystyle x_{id}^{t-1}}+{\displaystyle v_{id}^t}];\hspace{0.5em}d=1,2,\dots ,D,$(13)

where $c_1$ indicates the cognition learning factor, $c_2$ indicates the social learning factor, and $r_1$ and $r_2$ are random numbers between [0, 1].

As mentioned earlier, in order to avoid trial-and-error methods and improve local optimal problems, PSO is proposed as the learning algorithm. This will allow the centre and spreads of each RBF to self-generate. Consequently, the connection weight of the RBFs causes the extracted RBFNs system to have a robust ability to automatically approach the desired system response.

Feng (2006) designed the parameters of centres, the spread of each RBF and the connection weights as the particle, and then applied the PSO algorithm to search for the optimal solution for constructing the RBF network for classification. Since then, several kinds and modified versions of PSO (Kang et al. 2008; Sun et al. 2009; Tsekouras 2013) were used to train RBF network and successfully applied in classification (Korurek and Dogan 2010; Qasem and Shamsuddin 2011) and forecasting (Vilovic and Burum 2011; Yang, Pan, and Hongyi 2013). This paper uses PRBF for energy management.

Steps for the training algorithm used in this paper can be outlined as follows:

1. Initialize swarm of N particles. Each particle defines a network and the associated centres and bandwidths. Set the number of iterations as MaxIteration. Set count = 0.

2. Decode each particle into a network. Compute the connection weights between the hidden layer and the output of the network by the pseudo-inverse method. Compute the fitness of each particle.

3. Update pi for each particle and pg for whole swarm.

4. Update the velocity of each particle according to Equation (10). Limit the velocity within the range.

5. Update the position according to Equation (11).

6. Set count = count+1;

7. If count < MaxIterations, go to step ii, Otherwise, end.

For the simulations, parameters of PSO are outlined in Table 2.

Table 2 Simulation parameters for PSO.

Parameter	Number of iterations	Number of particles	Coefficient C1	Coefficient C2
Value	1000	50	0.7	0.7

4. Simulations

The vehicle model that is used for simulations is based on Ford Mondeo with the 42 V power net built in 2001, with a 2.0 l spark ignition engine and a five gear manual transmission. The 42 V power net consists of a 5 kW alternator and a 36 V AGM lead–acid battery with a capacity of 27.5 A h, which corresponds to an energy capacity of 4 MJ. The power net is equipped with a programmable electric load on top of the electric loads already present in the vehicle. (Details of power flow, drive train, parameter values for the simulation of this vehicle are provided in the Appendix.)

Simulations are done for the New European Driving Cycle (NEDC). For the electric power request, simulations were carried out for the loads of 500, 1000 and 2000 W.

The battery has an energy capacity of $E_{\text{cap}}=4\times 10{}^6$ J and is operated around 70% SOE, because the efficiencies for both charging and discharging in this range are acceptable. The battery losses are approximated as quadratic with the stored power, such that

$P_{\text{b}}=P_{\text{s}}+b{\displaystyle P_{\text{s}}^2}.$

Parameter $b$ is chosen at a value of $5\times 10{}^{-5} \text{W}{}^{-1}$, which gives an energy efficiency of 95% at 1000 W and 90% at 2000 W. When the drive train power is negative and the clutch is closed, the drive train power is partly delivered by the internal combustion engine (which has a negative drag power), by the alternator and by the brakes. Because regenerative braking delivers electrical power without extra fuel use, it is expected that it will be used as much as possible. The brakes are only used when the desired deceleration power is larger than the maximum negative power that can be taken up by the engine and the alternator.

The RBFNNs are causal as they do not need any prediction. But, The GA and PSO strategies require knowledge of the entire driving cycle. Hence, the PRBF and GRBF do require knowledge of the current $\omega (k),P_{\text{d}}(k),P_{\text{l}}(k)$ and $E_{\text{s}}(k)$. This is achieved by using the following PI controller:

$\lambda (k+1)={\lambda }_0+K_{\text{p}}\left\{E_{\text{s}}\right(0)-E_{\text{s}}(k\left)\right\}+K_{\text{I}}\left\{E_{\text{s}}\right(0)-E_{\text{s}}(k\left)\right\}\Delta t,$

where ${\lambda }_0$ is an initial guess for the Lagrange multiplier, and the value depends on the electric load. Here, we have chosen the value of the Lagrange multiplier resulting from the global optimization, i.e. ${\lambda }_0=2.7$. The other average values chosen are $K_{\text{p}}=6.7\times 10{}^{-7};K_{\text{I}}=3.3\times 10{}^{-4}$.

The difference in SOE is accounted for in the fuel consumption using the initial value of $\lambda$:

$m_{\text{c}}={\displaystyle \sum _{k=1}^n} \ddot{m}(k)-{\lambda }_0\{E_{\text{s}}(n)-E_{\text{s}}(0)\}.$

The fuel consumption and emissions are evaluated with the control model, using the nonlinear fuel and alternator map and the quadratic battery losses. When the cost function represents only the fuel consumption, it turns out that for this case the CO₂, CO and NO _x emissions are also reduced significantly. However, the emission of HC increases. Therefore, a weighted sum of fuel and HC emission is used as cost function. The weighting factors in (7) are ${\omega }_1={\omega }_5=1$ and ${\omega }_2={\omega }_3={\omega }_4=0$. This time, all emissions are reduced, at the cost of a slight decrease in fuel reduction. Fuel consumption by different strategies is provided in Table 3. It is seen that as per fuel consumption is concerned, the methods developed in this paper is more effective than GA, PSO and RBF-based approaches.

Table 3 Fuel consumption.

$P_{\text{l}}$	500 W	1000 W	2000 W
Strategy	Fuel use (g m)	Fuel use (g m)	Fuel use (g m)
GA	557.261	590.442	662.141
PSO	556.157	589.862	661.464
RBF	546.524	581.140	655.460
GRBF	545.26	580.510	655.112
PRBF	544.913	579.821	654.002

The resulting trajectories of $P_{\text{e}}$ for the PRBF and GRBF strategies with $P_{\text{l}}=1000 W$ are shown in Figure 1. As can be seen, the optimization anticipates on regenerative braking phases and generates less in between. Figure 2 shows the SOE for all strategies. All trajectories of SOE show a similar behaviour. The variation in SOE is small, because of the large capacity of the battery. This justifies that for this simulation, the battery efficiency is chosen independently of $E_{\text{s}}$. The trajectories of the adaptive $\lambda$ for the PRBF and GRBF strategies are also shown in Figure 3. The value of $\lambda$ varies slightly around its initial value.

Figure 1 Electrical alternator power for $P_{\text{l}}$ = 1000 W.

Figure 2 Battery SOE.

Figure 3 Adapted $\lambda$ for $P_{\text{l}}$ = 1000 W.

The simulations show that the strategies, PRBF and GRBF, are effective, as they succeed in lowering the fuel consumption and the exhaust emissions. PRBF and GRBF methods are better than existing methods as the results are for the entire driving cycle, NEDC. The results might still be improved by fine-tuning the weighting factors of the cost function. Most of the profit comes from regenerative braking, which delivers a certain amount of energy for free.

Both PSO and GA do not find the global optimum of the original nonlinear optimization problem. The GRBF network uses the original nonlinear cost criterion, but restricts itself to a grid, whereas the PRBF network finds the global optimum of a quadratic approximation of the original problem. The small difference between GRBF and PRBF for fuel use and CO₂ indicates that these terms in the nonlinear cost function are approximated adequately by a PRBF problem, and that the chosen grid is not too restrictive for the GRBF problem.

A comparison between the methods previously discussed with respect to their suitability for solving energy management problems reveals the following:

The energy management problem is an optimization problem, for which GRBF and PRBF seem the most suitable method, though it takes a tolerable longer runtime as evident from Table 4. An advantage is that the cost function can be nonlinear and non-convex. A disadvantage is the large computation time, especially if there are multiple states. In GRBF, the optimization problem can be rewritten as a static optimization problem.

Table 4 Simulation runtime using MATLAB.

Strategy	GA	PSO	RBF	GRBF	PRBF
Runtime (s)	22	23	24	25s	28s

5. Summary and future work

This paper presented two novel strategies for energy management of the electrical power net, PRBF and GRBF, to reduce the fuel consumption and exhaust emissions over a driving cycle.

Simulations show that applying energy management on the vehicle power net is effective. With the degree of freedom considered here and the component characteristics used, a fuel reduction of 2% can be obtained, while at the same time reducing the emissions even more. Application of energy management on the vehicle power net does not require changes to the drive train and is therefore cheap to implement.

The strategies can also be applied to a mild hybrid electric vehicle with an integrated starter generator (ISG). The only difference is that the lower bound on the alternator power is negative instead of zero. The approach can be extended to vehicle topologies with more degrees of freedom. The approach can also be extended for online implementation by exploiting the non-convexity of the cost function and does not require a prediction of the future.

This paper also paves a way for researchers to work on models with a bit complicated power-train physics and operating constraints that will be reflected in our future works.

Appendix

Power flow

The drive train of a parallel HEV, used in this paper, is based on a conventional vehicle, where the alternator is replaced by an ISG that can also be used for propulsion. The clutch can be located before or after the power split for the ISG. This is shown in Figure A1.

The power flow in the vehicle starts with fuel that is injected in the combustion engine. The resulting mechanical power $P_{\text{m}}$ splits into two directions: one part $P_{\text{d}}$ goes to the drive train for vehicle propulsion, whereas the other part $P_{\text{g}}$ goes to the ISG. The ISG provides electric power $P_{\text{e}}$ for the electric loads $P_{\text{l}}$ but also takes care of charging the battery $P_{\text{b}}$. Contrary to the other components, the power flow of the battery can be positive as well as negative. In the end, all power, except for losses, is used for vehicle propulsion and for electric devices connected to the power net. The drive train block contains all drive train components including clutch, gears, wheels and vehicle inertia. The alternator is connected to the engine by a belt with a fixed gear ratio.

Drive train

The drive train consists of clutch, transmission, final drive, wheels and inertia. They are not modelled in detail, as only the relationship between vehicle speed, engine speed and drive train torque is of interest. For a given vehicle speed profile $v(t)$, road slope $\alpha (t)$ and selected gear ratio $g_{\text{r}}(t)$, the corresponding engine speed and torque needed for propulsion can be calculated.

When the engine speed drops below idle speed, the clutch is opened, the drive train torque becomes zero, and the engine keeps running at idle speed. The engine power becomes equal to the alternator power and the drive train power becomes equal to the brake power. In Table A1, the parameters are explained and their values as used in simulations are given.

Table A1 Vehicle parameters used in simulations.

Quantity	Symbol	Value	Unit
Mass	M	1400	kg
Frontal area	A d	2	m^2
Air drag coefficient	C d	0.3	–
Rolling resistance	C r	0.015	–
Wheel radius	w r	0.3	m
Final drive ratio	f r	4.0	–
Gear ratio	g r	3.4–2.1–1.4–1.0–0.77	–
Idle speed	${\omega }_i$	73.3	°/s
Air density	$\rho$	1.2	kg/m^3
Gravity	g	9.8	m/s^2

Figure A1 Model of the vehicle used in the paper.