Cost Function

The objective of the SDP optimisation is to choose actions in order to minimise a future anticipated cost. This cost is calculated by the use of a cost function. The cost function is designed to either reward positive operating conditions or (more frequently) penalise negative conditions. The cost function can be customised in order produce a controller with the desired performance characteristics.

There are two main ways to include operating conditions in the cost function. The first is to use realistic figures for each state in order to optimise a single variable and weighting factors to account for multi-variable optimisation. For example, the hydrogen consumption of the fuel cell can be calculated for each state/action, and using this alone, the optimisation process will produce a controller which provides the optimal performance with regard to fuel economy. Using weighting factors for multiple variables allows each to be “traded off”

against each other. For example, the number of gear shifts could be used to optimise the drive-ability. A weighting factor, on either the fuel economy or number of gear shifts, could then be chosen to provide the best compromise between fuel economy and drive-ability.

The second manner in which the cost function can be used is to provide “soft” con-straints on the optimisation in order avoid certain conditions. This is often how the limits on the battery SoC are taken into account. Constraints are particularly useful for this pur-pose due to the fact that it doesn’t matter what the SoC is at any point, as long as it is within the acceptable range for the battery pack. By introducing very high penalties to excessively high or low battery SoC and no penalty in between, the controller will attempt to completely avoid these conditions. As long as the SoC stays within the constraints, the cost will not be affected. Theoretically, soft constraints could be used on their own, but usually it is more useful to use them in conjunction with a variable to be optimised. This would, for example, allow the fuel consumption to be optimised whilst keeping the battery SoC within an acceptable range.

As was found in the literature review, the vast majority of previous authors focus on optimising the fuel consumption of the vehicle and use this as the basis for the cost func-tion. Secondary considerations, such as battery SoC maintenance, fuel cell degradation and drive-ability concerns are usually accounted for by constraints on the optimisation.

The research for fuel cell vehicles lags behind that for Internal Combustion Engine (ICE) based hybrid vehicles in this regard. More recent work on ICE hybrid vehicles investigates the trade-off between fuel consumption and emissions, or the trade-off between the fuel consumption and drive-ability.

A major concern for fuel cell vehicles is the reliability of the fuel cell itself and therefore it is proposed that this should be included in the optimisation process. A number of authors have proposed strategies that combat fuel cell degradation, mainly focussed on two major causes; the reduction of transient loading, and prevention of reactant starvation. Although there is a great deal of research in the literature into specific degradation methods, no pre-vious work has been found to incorporate this research into SDP controller development. It is therefore proposed that a quantitative analysis of a variety of major degradation methods should be included in the optimisation process and that these could be weighted against the fuel consumption by using their respective monetary values. This would allow the overall running cost of the fuel cell to be minimised by the optimisation process.

The primary cost function developed for this work is made up of three main parts. The controller will be optimised in order to minimise the overall running cost, made up of the

fuel cost, Γf and the proportional fuel cell degradation cost, ΓF C. Finally, soft constraints are used in order to maintain the battery SoC, resulting in a penalty cost, ΓV. Each of these costs is described in detail in the following sections.

5.2.2.1 Fuel Consumption

The fuel consumption of the vehicle is already calculated by the vehicle model developed in Chapter 3 and is based on the operating load of the fuel cell. The output of the model is the fuel consumption, mf, in mg which can easily be converted into a monetary cost, Γf, by multiplying by the value of hydrogen fuel, γf, see Equation 5.2.4.

Γ_f = γ_fm_f (5.2.4)

5.2.2.2 Fuel Cell Degradation

A number of degradation methods have been identified from the literature review. Al-though some degradation methods, such as the purity of the fuel, are due to circumstances beyond the control of the EMS, many degradation methods will be directly affected by the operating conditions of the fuel cell and hence can be limited by optimisation of the EMS.

A list of potential EMS strategy actions was identified for each method and shown in Ta-ble 2.1. There is a lot of overlap in these actions, which can be summarised by four main operating conditions that should be avoided.

1. Low power operation, especially open-circuit

2. High power operation, especially beyond the reactant supply, product removal or heat rejection capability of the stack

3. Transient loading 4. On/Off Cycling

Ideally the voltage degradation rate under each operating condition should be quantified by extensive testing of fuel cell stacks; however, this would be extremely costly and time-consuming and therefore has been deemed out of the scope of this work. Fortunately, there is enough data available in the manufacturer’s datasheet [106] and previous literature [13], to make a reasonable estimate. The manufacturer states an expected degradation rate of approximately 11.6 µV/h per cell for the stack at full load, with essentially no degradation below approximately 80 % full load. No low power degradation rates are given; however, these have been obtained from the literature for a similar fuel cell and scaled to match the full load data given by the manufacturer. An estimate for the transient loading degradation has also been obtained from the literature, and the stop-start cycle voltage degradation has been obtained from the manufacturer’s specification. These figures are given in Table 5.2.

Operating Conditions Degradation Rate Low Power Operation 10.17µV/h High Power Operation 11.74µV/h Transient Loading 0.0441µV/kW Start/Stop 23.91µV/cycle

Table 5.2: PEM Fuel Cell Degradation Rates (per cell)

As shown in Chapter 3, these degradation methods have been incorporated into the vehicle model. The degradation of the fuel cell, DF C, is calculated by the model in mV. This can be divided by the voltage drop at which the fuel cell is said to be fully degraded, Dmax, to calculate the percentage degradation of the stack. If the degradation rates are assumed to be constant, which should be valid for the short term at least, then the estimated cost, ΓF C, associated with this degradation can be calculated as the percentage degradation multiplied by the monetary value of the stack, γF C.

Γ_{F C} = γ_{F C} D_{F C}

D_max (5.2.5)

5.2.2.3 Battery SoC Maintenance

In addition to minimising the total running cost of the fuel cell, the EMS is also responsible for managing the battery. The Microcab H4 does not have an integrated battery charger, and therefore the strategy has been designed to be charge-sustaining. This means there is no need to have a specific final SoC target as long as the SoC stays within the acceptable range to protect the battery from deep-discharge and overcharge throughout any journey.

Any deviation in SoC within these bounds will not detriment the performance of the vehicle and hence should not be penalised. Soft constraints are therefore ideal for this purpose.

Battery SoC sustenance can be accomplished by setting constraints on the battery volt-age and will prevent the battery from becoming over-charged or deeply discharged. By using the battery voltage, rather than state of charge directly, the battery will also be pro-tected from voltage spikes due to sudden reduction in load from the motors and from voltage drops during periods of high current demand, such as acceleration. In fact, the degradation methods identified in the literature review are more closely related to the cell potential than the SoC directly, and therefore voltage limits will tend to provide better protection than SoC constraints. This soft constraint has been achieved by assigning a cost, α, to extreme cell potentials (above Vmaxand below Vmin), see Equation (5.2.6).

Γ_V =







αR (V_min− V_bat)dt, if Vbat < V_min αR (V_bat− V_max)dt, if Vbat > V_max

0, otherwise

(5.2.6)

Note that the integral of the voltage difference is used in the penalty function. This has been used to ensure that if the battery voltage limits are exceeded, then there is a cost incentive for the controller to perform actions which will return the voltage to an acceptable range. If the vehicle components are correctly sized for the application, and the initial SoC of the battery is acceptable, then this detail will have little effect on the results. However, if the control authority is too low (perhaps due to an undersized fuel cell), or if the initial SoC is outside the acceptable range (perhaps due to self-discharge while the vehicle is unused), then the controller will always attempt to return the battery SoC to the acceptable range.

This behaviour has been implemented to improve the robustness of the controller under non-ideal circumstances.

A value of 10⁴has been chosen for α. This at least is three orders of magnitude higher than any cost that is likely to occur as a result of hydrogen consumption or voltage degra-dation of the fuel cell. This means that the voltage management of the battery will over-ride any optimisation if the constraints are exceeded. As a result, the optimisation process will attempt to completely avoid any states outside of the voltage limits of the battery. This is known as a “soft” constraint due to the fact that it is still possible for the constraint to be exceeded if otherwise unavoidable. It would be equally acceptable to choose an even higher value, for example 10⁶, however increasing the number too far may lead to computational problems during optimisation. This is because double precision floating point numbers used in the MATLAB algorithm are accurate to approximately 15 significant figures. Given a cost penalty of 10¹⁰ or higher, the rounding of the floating point numbers, could cause significant errors (in the range 10^-2or $0.01) to accumulate over typical duty cycle lengths of around 10³seconds (approximately 15 minutes).

5.2.2.4 Final Cost Function

The final cost function (Equation 5.2.7) is the sum of the three individual components; the monetary cost accumulated due to fuel consumption, the proportional monetary cost due to degradation, and a high penalty due to the battery voltage constraints. This has the net effect of minimising the running cost of the vehicle including fuel cost and degradation, whilst attempting to completely avoid exceeding the voltage limits of the battery.

Γ_t = Γ_f + Γ_{F C} + Γ_V (5.2.7)