Improving the Performance Attributes of Plug-in Hybrid Electric Vehicles in Hot Climates through Key-Off Battery Cooling

Abstract

Ambient conditions can have a significant impact on the average and maximum temperature of the battery of electric and plug-in hybrid electric vehicles. Given the sensitivity of the ageing mechanisms of typical battery cells to temperature, a significant variability in battery lifetime has been reported with geographical location. In addition, high battery temperature and the associated cooling requirements can cause poor passenger thermal comfort, while extreme battery temperatures can negatively impact the power output of the battery, limiting the available electric traction torque. Avoiding such issues requires enabling battery cooling even when the vehicle is parked and not plugged in (key-off), but the associated extra energy requirements make applying key-off cooling a non-trivial decision. In this paper, a representative plug-in parallel hybrid electric vehicle model is used to simulate a typical 24-h duty cycle to quantify the impact of hot ambient conditions on three performance attributes of the vehicle: the battery lifetime, passenger thermal comfort and fuel economy. Key-off cooling is defined as an optimal control problem in view of the duty cycle of the vehicle. The problem is then solved using the dynamic programming method. Controlling key-off cooling through this method leads to significant improvements in the battery lifetime, while benefiting the fuel economy and thermal comfort attributes. To further improve the battery lifetime, partial charging of the battery is considered. An algorithm is developed that determines the optimum combination of key-off cooling and the level of battery charge. Simulation results confirm the benefits of the proposed method.

1. Introduction

The most significant obstacle in the way of the transition from conventional vehicles to electric vehicles (EVs) is the under-developed traction battery technology [1,2]. As this transition takes place, most technology road maps and policy trends indicate that plug-in hybrid electric vehicles (PHEVs) will dominate the passenger vehicles markets in the next decade [3,4,5]. Limited experience with batteries in the automotive industry and scarce field data [6] mean that production vehicles should continue to evolve in order to fully accommodate the challenges posed by the battery [7]. Ageing of battery cells, defined as the irreversible reduction of their energy storage capacity and power [8,9], is one such challenge [10]. Lithium-ion cells, which are believed to be the most promising choice for automotive applications [11,12] have ageing mechanisms that are sensitive to real world usage conditions [13]. Cycling ageing (ageing due to the conditions of charge-discharge cycles) accelerates at high temperature, as well as with a high degree of discharge, and high charge-discharge rates [14,15]. Storage ageing (ageing due to storage conditions) increases with elevated temperature [8,16] and state of charge (SoC) [17,18]. Heat transfer from the ambient environment, the passenger cabin and other adjacent components, in addition to internal heat generation, contribute to high battery temperature [19] that can reach as high as 65 °C [20]. Therefore, some form of battery cooling system is necessary. Battery cooling is traditionally a “key-on” function, that is, it is only activated when the vehicle is switched on [21].

In hot climate conditions, the heat gain from the ambient environment alone is enough to heat the battery to extreme temperatures [22,23]. Although key-on cooling can significantly reduce the daily maximum battery temperature in most cases [24,25], extreme battery temperatures can be still experienced when vehicles are parked (“key-off”). Since typical passenger vehicles are parked for most of their lifetime, markedly short battery lifetimes have been observed in hot geographical locations [20,22].

In hot climate conditions, the only practical source of cooling power is the refrigeration circuit of the air conditioning system [26]; therefore, battery cooling systems are often integrated into the refrigeration circuit [27,28]. This integration creates two challenges. First, the compressor of the refrigeration circuit is powered by the battery itself [29,30]. Therefore, key-on cooling has a negative impact on the vehicle’s energy efficiency and electric drive range (EDR) [31,32]. Second, to cool the cabin and perform key-on battery cooling at the same time, the cooling power of the refrigeration circuit is distributed between cabin air conditioning and battery cooling. When the air conditioning load is high (i.e., when the cabin is hot), allocating part of the available cooling power to the battery causes a noticeable rise in the average cabin air temperature [33,34,35]. Therefore, key-on cooling can negatively affect the passengers’ thermal comfort [36].

Observation of short battery lifetime in hot geographical locations indicates the need for extending battery cooling beyond a key-on function. Previous research in this area have assumed that vehicles can be always plugged in when they are parked; therefore, combining key-on cooling with additional “plugged-in cooling” (cooling using the grid energy) has been proposed to enable frequent and aggressive cooling [20,22,32,34,37,38,39]. Since it uses an external energy source, plugged-in cooling is effectively ‘free’ for the vehicle. Battery lifetime improvements of approximately 1–2 years have been reported in hot climate conditions through this approach [34,37,38,39]. Enabling plugged-in cooling also benefits the energy efficiency and thermal comfort attributes of the vehicle, as the ‘free’ nature of plugged-in cooling encourages intensive precooling of the battery to reduce the need for key-on cooling during the subsequent trip [32].

A further benefit of precooling which has not received sufficient previous attention concerns the electric traction capability of the vehicle. As a safety function and to avoid damage to the battery, typical automotive battery management systems limit the power output of the battery at high temperatures [34,40,41]. This reduces the electric traction capability until the cooling system brings the temperature of the battery back into the prescribed range [42]. Reduced electric traction in plug-in parallel hybrid electric vehicles (PPHEVs) can lead to poor fuel economy on aggressive drive cycles, as the engine will be used more to fulfil the traction requirement. Precooling can help prevent the negative effect of hot battery on fuel economy.

The fact that most vehicles are not always plugged in when they are parked [43,44] limits the opportunity for plugged-in cooling. A comprehensive battery cooling strategy should therefore combine key-on and plugged-in cooling with additional cooling when the vehicle is parked and not plugged in (key-off cooling) [22,34]. However, two critical concerns have discouraged any previous consideration of key-off cooling. First, as key-off cooling is supported by the battery, it will reduce the vehicle EDR, which is equivalent to higher fuel consumption in PHEVs. Second, discharging the battery to support key-off cooling increases cycling ageing, which may adversely affect the battery lifetime.

Reducing the average SoC of the battery through optimized partial charging is another method that has been used to improve the battery lifetime [45,46,47,48]. The method typically involves using advanced knowledge of the vehicle duty cycle and charging the battery only as much as the vehicle will consume until it is plugged-in next time. However, this method is only effective when the vehicle is always plugged-in when it is parked. If the vehicle is plugged-in less frequently, as is commonly the case, limited opportunity for partial charging will be available as the battery charge should be high enough to support an extended operation. Further, the majority of the previous studies in this area ignored the benefit of lower SoC relative to lower temperature, and considered controlling the level of charge independent of controlling cooling, even when plugged-in cooling was available.

This paper, for the first time proposes the application of key-off cooling and develops an underpinning methodology for optimal control of key-off cooling in a PPHEV. For the first time, three performance attributes of fuel economy, battery lifetime, and passenger thermal comfort are considered in cooling the battery, and their trade-off is addressed. Furthermore, for the first time, this paper proposes that for improving battery lifetime in hot climates, partial charging optimisation and cooling optimisation should be integrated. A method is developed that determines the best combination of the level of charge and key-off cooling control for improved battery lifetime, while considering the fuel economy and passenger thermal comfort attributes.

To achieve this, a representative vehicle model is simulated, over a 24-h duty cycle composed of two trips and two park phases per day. The vehicle is subjected to hot climate conditions and undergoes a hot soak when it is parked during the day, experiencing high battery temperatures. The impact of the climate conditions on the fuel economy, battery lifetime and thermal comfort attributes of the vehicle are quantified through comparison with milder climate conditions. The need for key-off cooling of the battery in hot climate conditions is established from this comparison. Then, the optimal control of key-off cooling is defined as the minimization of a cost function composed of the weighted sum of fuel consumption, capacity loss of the battery, and a thermal discomfort index that represents the deterioration of the passengers’ thermal comfort due to key-on battery cooling. Solving the problem for different weightings of the cost function demonstrates the ability of the method to target individual or a combination of the represented performance attributes. The effect of key-off cooling for improving each performance attribute is analyzed. Later, the proposed method is expanded to include partial charging, and the benefit of integrating partial charging and key-off cooling on battery lifetime is demonstrated.

The paper is organized as follows. Section 2 defines a 24-h vehicle duty cycle, which constitutes the basis for subsequent simulations and analyses. Section 3 introduces the vehicle model developed for the purpose of this research. Section 4 discusses the result of simulating the model over the defined duty cycle and quantifies the negative effect of high battery temperature on each performance attribute of the vehicle. Section 5 defines key-off cooling as an optimal control problem. Section 6 discusses the control of key-off cooling in view of the 24-h duty cycle of the vehicle based on improving individual, and combinations, of the performance attributes. Section 7 proposes a method for integrating key-off cooling and partial charging. Section 8 discusses the applicability of the proposed methods to EVs. Section 9 provides additional discussion of the results. Section 10 outlines how the research will continue in the future. Finally, Section 11highlights the conclusions of the research.

2. The Duty Cycle

In this paper, a vehicle duty cycle is defined by three elements:

(1)

the usage case, which includes details such as the number and timing of the trips, charge frequency, etc.;

(2)

the drive cycles of the trips;

(3)

the climate conditions in which the vehicle operates.

The usage case considered in this research was a work commute, which starts at 7 a.m. with a trip (Trip 1), followed by an 8-h park (Day Park) and another trip (Trip 2), after which the vehicle is parked and plugged-in until the next day (Night Park). In Figure 1, different phases of the duty cycle are marked on the daily profile of the vehicle speed. For simplicity, Trip 1 and Trip 2 are assumed to have identical drive cycles.

The vehicle was assumed to operate in Phoenix, AZ, USA, which has hot climate conditions. In the interest of simplicity and generality, it was assumed that a year in Phoenix can be divided into a six-month warm season, from May to October, and a six-month cold season, from November to April. These will be loosely referred to as summer and winter. The daily air temperature and the solar load profiles were created for each season, as shown in Figure 2, by averaging the hourly climate conditions of the relevant months [49]. The vehicle is subjected to the above conditions during the two daily trips, and during Day Park when it is parked in the open. In the interest of simplicity, it was assumed that in Night Park, the vehicle is parked in a garage maintained at constant temperature of 25 °C in summer and 15 °C in winter.

Both trips were assumed to be on the US06 drive cycle. The aggressiveness of the US06 is representative of most real-life drive cycles [22]. The distance covered by the US06 is approximately eight miles, which is comparable with the distance covered by the current European homologation drive cycle (NEDC) [50], although it is shorter than the average commutes in the U.S. [51]. Nevertheless, the purpose of the drive cycle for the analyses that will follow is to create the necessary boundary conditions for the park phases of the duty cycle; hence, the US06 is a sufficiently representative drive cycle to showcase the methodology of this research. A summary of the duty cycle assumptions is given in

Table 1. Duty cycle assumptions.

	Phase	Duration	Plugged In		Indoor	Ambient
Phase	Trip 1	600 s (US06)	--		--	Figure 2
	Day Park	8 h	no		no
	Trip 2	600 s (US06)	--		--
	Night Park	15.5 h (approximately)	Yes 13.2 kW charger		Yes	25 °C (summer) 15 °C (winter)
Initial Conditions	Parameter			Value
	Battery SoC			95%
	Battery/cabin temperature			25 °C (summer) 15 °C (winter)

.

3. The Vehicle Model

To simulate the duty cycle defined above, a model was developed that represents a commercially available PPHEV with the high-level specifications given in

Table 2. Specifications of the target vehicle.

Specification	Value
Type	sport utility vehicle
Architecture	All-wheel-drive parallel plug-in hybrid
Test Weight (European Union)	2270 kg
Engine	3.0Lv6 diesel, 140 kW
Electric Machine	rated power	75 kW
	rated torque	420 Nm
Traction Battery	energy capacity	16.8 kWh
	voltage	430 V
	thermal capacity	190,000 J/K
	cell type	cylindrical NCA

. The layout of the model is shown in Figure 3. Details of the model and its experimental verification are explained in [52,53]. For completeness, a summary of the key attributes of the model pertinent to the problem considered here is provided below.

The battery cooling system includes a water-glycol coolant circuit that rejects heat to the refrigeration circuit of the AC system. The battery is represented by a lumped parameter thermal mass, and includes an internal heat generation model based on Joule heating [54], as well as heat transfer paths to the coolant, the ambient environment and the cabin. The electrical behavior of the battery cells is represented by a zero-order equivalent electric circuit model [55]. In this model, the open-circuit voltage and the resistance of the cells are calculated from look-up tables. The open-circuit voltage is a function of SoC, while resistance is a function of SoC and temperature. The SoC of the cells is estimated from current, based on the method of Coulomb counting [56]. The cabin model includes lumped parameter representations of the cabin interior (seats, dashboard, etc.) and the enclosed air, as well as the glass and the wall that form the shell of the cabin.

The controller block in Figure 3 includes a set of rule-based algorithms that regulate the operation modes of the powertrain, and those that set the high level set points for the AC and the battery cooling systems. These rules are developed based on the control algorithms of the target vehicle. The key parameters implemented in the rules are given in

Table 3. Control parameters adopted from the target vehicle specifications.

Parameter	Value
SoC window: charge depleting mode	95–10%
SoC window: charge sustaining mode	10–13%
Lower SoC limit for electric pull-away	11%
$T_{cabin}$ setpoint	22 °C
$T_{batt}$ setpoint	25 °C–30 °C
$T_{batt}$ overheat (power cut-off)	50 °C
$T_{batt}$ high temperature threshold (50% power cap)	45 °C

. The AC system is controlled to cool the cabin and maintain it at 22 °C during the trips. The battery cooling system is controlled in a thermostatic fashion in the temperature range of 25 °C–30 °C; and is activated during the trips (key-on cooling), as well as when the vehicle is plugged in, that is in Night Park (plugged-in cooling). The control rules include overheating thresholds: the battery power is limited at 50% when the battery temperature exceeds 45 °C and is cut off when the temperature exceeds 50 °C.

The capacity loss of the battery cells of this vehicle has been analyzed in [57,58]. These analyses give the storage capacity loss of the cells for different SoC levels and temperatures. They also give the cycling capacity loss of the cells based on the number of charge-discharge cycles at different temperatures. As a general trend, the data presented in [57,58] show that storing the battery at higher SoC or temperature increases the rate of capacity loss. Furthermore, a higher number of cycles increases the capacity loss, especially when the cells are cycled at higher temperatures. Using these data, the capacity loss of the battery and its lifetime can be calculated as the vehicle undergoes its duty cycle. To achieve this, it was assumed that the total capacity loss of a cell is equal to the sum of its storage and cycling capacity losses [59]. It was also assumed that all cells within the battery pack age at the same rate. Defining the state of the battery as $x_b=\left[\begin{array}{c}{T}_{batt} \\ \mathsf{\zeta }\end{array}\right]$, the capacity loss calculations can be notationally described as below:

$$\Delta C=\Delta C_{stor}\left(T_{batt}, \zeta \right)+\Delta C_{cyc}\left(T_{batt}, n\right)=\Delta C\left(x_b, n\right)$$

(1)

Simulating the duty cycle of the vehicle gives the variations in the temperature, the SoC; and the current of the battery. The charge-discharge cycles of the cells can be calculated from the battery current:

$$n=\frac{\int \left|i\right|dt}{2\times C_0}$$

(2)

and the total time that the battery is stored at any particular condition, as well as the number of cycles at any particular temperature can be calculated. The annual capacity loss of the battery can be then calculated as the sum of the capacity loss in different seasons of the year; and across different phases of the duty cycle. Having calculated the annual capacity loss, the battery lifetime can be determined based on the end of life (EoL) at 30% capacity loss [16,60,61,62]. It is assumed here that the vehicle undergoes the same duty cycle seven times per week. Although this assumption does not strictly apply to most vehicles, it simplifies the simulations without invalidating the subsequent analyses.

4. Baseline Simulation Results

In this section, the result of simulating the vehicle model over the assumed duty cycle will be presented. The variation of the battery temperature during the duty cycle will be demonstrated, and the effect of the battery temperature on the performance attributes of the vehicle will be quantified from the simulation results. Figure 4 shows the daily profile of the vehicle state $x_v={\left[\begin{array}{c}{T}_{cabin},\begin{array}{c}{T}_{batt} ,\mathsf{\zeta }\end{array}\end{array}\right]}^{\prime }$ in summer. These results can be explained as follows:

Trip 1: The cabin and the battery are both initially at 25 °C. During the trip, air conditioning maintains the cabin at 22 °C, while the battery gradually warms up until it reaches 30 °C and cooling initiates. The vehicle operates mostly in CD mode, discharging the battery from 95% to about 74%.

Day Park: Influenced by the ambient conditions, the vehicle undergoes a hot soak. As a result, the temperature of the cabin and the battery rise to 60 °C and 48 °C respectively, while the SoC remains constant. It is worth noting that since six-month average climate conditions are assumed, the simulations do not represent the worst-case scenario.

Trip 2: Air conditioning and battery cooling are applied to cool down the vehicle from hot conditions. Within the duration of the trip, the cabin and battery are only cooled to about 33 °C and 39 °C, respectively. The battery is discharged to about 54% SoC. Considering that the CS operation can commence at 13% SoC, the battery has approximately 40% excess charge over this duty cycle.

Night Park: When the vehicle is parked again, air conditioning terminates. Plugged-in battery cooling is applied until the battery temperature drops to 25 °C. Both the cabin and the battery continue to exchange heat with the ambient environment until they are stabilized at 25 °C. Whilst the vehicle is plugged-in at the start of Night Park, the just-in-time charging strategy [22,63] is applied (i.e., charging is delayed until just before the next trip) to minimize the adverse effect of high SoC on the battery.

4.1. The Impact on Battery Lifetime

Table 4. Battery capacity loss and lifetime calculations.

	Winter				Summer
	Trip 1	Day Park	Trip 2	Night Park	Trip 1	Day Park	Trip 2	Night Park
$\overline{\mathsf{\zeta }}$ (%)	84.4	74.7	62.6	53.4	84.5	74.6	68.8	59.0
${\overline{T}}_{batt}$ (°C)	18.5	24.9	29.6	15.0	28.0	39.0	43.0	26.0
Number of cycles	28.4	0.0	30.7	40.2	28.2	0.00	27.8	36.4
Storage $\Delta C$ per season (%)	0.00	0.64	0.00	0.86	0.00	1.07	0.00	1.17
Cycling $\Delta C$ per season (%)	0.16	0.00	0.16	0.23	0.15	0.00	0.16	0.17
Total $\Delta C$ per season (%)	2.05				2.72
Total $\Delta C$ per year (%)	4.76
Lifetime (years)	6.3

gives the time average of the battery temperature and SoC, as well as the number of its charge-discharge cycles in each phase of the duty cycle in summer and winter. The seasonal storage and cycling capacity losses are also calculated for each phase, as explained in Section 3. For example, the table shows that in summer Day Park, the battery is stored at an average SoC of 74.6% and at an average temperature of 39 °C, and it is not cycled (the vehicle is key-off and no cooling is applied). Due to the storage conditions, the battery loses 1.07% of its capacity, while the cycling capacity loss is zero. Adding the cycling and storage capacity losses for different phases of the duty cycle in each season leads to 2.72% capacity loss in summer, compared to 2.05% in winter. The total capacity loss in one year is therefore 4.76%, which indicates a battery lifetime of just over six years.

The calculated lifetime is significantly shorter than the eight-year target of most warranty schemes [64]. The calculations in Table 4 indicate that the high battery temperature in summer Day Park is a major contributor to the low battery lifetime. The battery loses more than 1% of its capacity in summer Day Park compared to 0.64% in winter Day Park. The average battery temperature is 14 °C higher in summer Day Park compared to winter Day Park, while the SoC levels are similar. A similar difference also occurs in peak temperatures (not given in Table 4). The variability of battery lifetime with geographical location can be highlighted here: if the climate conditions of the winter season, which is not particularly mild, were to apply year-round, the battery could last one year more. Key-off cooling in Day Park can be beneficial to the battery if it does not lead to a disproportionately high cycling capacity loss.

4.2. The Impact on Fuel Consumption

Allowing the battery temperature to rise above 45 °C in Day Park limits the electric traction capability of the vehicle. Given that limited time is allowed for natural cooling prior to Trip 2, and that the US06 drive cycle starts aggressively, limited electric traction capability leads to high fuel consumption in Trip 2. This can be better explained with the help of Figure 5 in which more details about the two trips of the vehicle are shown. Despite having identical drive cycles, the vehicle operates differently in the two trips. In Trip 1, the vehicle operates mostly in CD mode, depleting about 21% of the battery charge, with the engine starting several times in the high power-demanding parts of the drive cycle, as expected from a PPHEV. Furthermore, the cabin is maintained at its set point while the battery temperature gradually rises until cooling is initiated.

In Trip 2, with the battery initially overheated, the vehicle is forced to enter hybrid mode (engine is used) in the first 100 s of the trip, despite having adequate battery charge (compare the engine torque profiles in Figure 5b). Once the battery is sufficiently cooled, the engine operates in a similar fashion to Trip 1, the battery is gradually discharged and its temperature continues to fall. Therefore, due to the high battery temperature at the start of Trip 2, the vehicle consumes approximately 0.37 liters of fuel in Trip 2 (67.5 liter per season), compared to approximately 0.23 liters in Trip 1 (41.7 liter per season). Enabling key-off cooling allows key-off pre-cooling of the battery for Trip 2 or maintaining a low battery temperature will eliminate the extra fuel consumption in Trip 2.

4.3. The Impact on Thermal Comfort

High battery temperature also negatively affects the passengers’ thermal comfort. Figure 5d shows that in Trip 1 the average cabin temperature is maintained close to its set point, implying a thermally comfortable cabin. However, following the hot soak in Day Park, the thermal load of the cabin in Trip 2 is so high that it does not cool down to its set point within the duration of the trip. As such, allocating a part of the available cooling power of the refrigeration circuit to the battery deteriorates the thermal condition of the cabin. Figure 6 compares the conditions of the cabin of the PHEV in Trip 2 to that of an equivalent conventional vehicle in similar circumstances. Although the cabin remains hot throughout the trip even in the conventional vehicle, the cooling load of the battery in the PHEV leads to a higher cabin temperature by up to approximately 4 °C.

To quantify this impact, the discomfort index of the PHEV is defined here as the increase in the cabin average temperature due to the existence of the battery in the vehicle, as:

$$I_{discomf}={\displaystyle \sum }\left(\frac{T_{cabin}^{\begin{array}{c}PHEV\end{array}}-T_{cabin}^{\begin{array}{c}CONV\end{array}}}{T_{cabin}^{CONV}}\right)$$

(3)

In the above equation, $T_{cabin}^{\begin{array}{c}PHEV\end{array}}$ and $T_{cabin}^{\begin{array}{c}CONV\end{array}}$ are the time histories of cabin temperature in the PHEV and the conventional vehicle, respectively. $I_{discomf}$ is primarily applicable to Trip 2 since the cooling demand of the cabin in Trip 1 is low, making the impact of cooling the battery negligible. $I_{discomf}$ is a positive integer, with an ideal value of zero which indicates that the PHEV cabin is cooled at a rate similar to, or faster than, the conventional vehicle cabin. Precooling the battery for Trip 2 can help minimize the thermal discomfort by reducing the need for key-on battery cooling. In the ideal case, the battery can be precooled sufficiently to shift its cooling load to after Trip 2 (i.e., to Night Park). It is worth noting that precooling the cabin can further improve thermal comfort, but requires significantly higher energy [33] and addressing it is beyond the scope of this paper.

4.4. Summary of the Simulation Results

Table 5. Performance measures of the PHEV (summer).

Battery Lifetime	${\left[{\mathit{v}}_{\mathit{f}\mathit{u}\mathit{e}\mathit{l}}\right]}_{\mathit{T}\mathit{r}\mathit{p}1}$	${\left[{\mathit{v}}_{\mathit{f}\mathit{u}\mathit{e}\mathit{l}}\right]}_{\mathit{T}\mathit{r}\mathit{p}2}$	${\mathit{I}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{c}\mathit{o}\mathit{m}\mathit{f}}$
6.3 (years)	0.23 (L)	0.37 (L)	59

summarizes the performance attributes of the vehicle based on the results discussed above. It can be inferred from the simulation results that the benefits of key-off cooling extend beyond its potential for reducing the capacity loss resulting from the high temperature in Day Park, to attributes such as fuel economy and thermal comfort. However, allocating the energy of the battery to key-off cooling increases cycling ageing which can negate the benefit of cooling, and it reduces the EDR which can worsen the fuel economy. Therefore, determining the appropriate key-off cooling strategy is a non-trivial task that requires solving an optimal control problem considering all the implications mentioned above. The following sections of this paper address this problem. Table 5 summarizes the key aspects of the simulation results discussed so far. These values will be considered as the baseline in the forthcoming discussions.

5. Optimal Control Statement for Key-Off Cooling

For simplicity, it is assumed in this work that key-off battery cooling takes place at maximum cooling power. Therefore, to control key-off cooling, it will be sufficient to control a flag command (binary control input indicating cooling on/off). A key-off cooling flag of “1” will be translated as the refrigeration system working at maximum power and the cooling pump switched on, while a flag of “0” indicates a passive system. With the above assumption, the number of the charge-discharge cycles, $n$, in Day Park, is related to the cooling flag, $u$. Therefore:

$${\left[\Delta C\right]}_{DPrk}={\left[\Delta C\left(x_b, n\right)\right]}_{Dprk}\equiv {\left[\Delta C\left(x_b, u\right)\right]}_{Dprk}.$$

(4)

Key-off cooling of the battery can be defined as an optimal control problem in which the aim is to find the optimum cooling flag trajectory. Here, to address all aspects of the problem, the formulation will be developed in three steps, within the remainder of the current section, as well as in Section 6 and Figure 7 shows the focus of each section and its relation to the baseline simulation results presented in Section 4. In the remainder of the current section, key-off cooling is optimally controlled to minimize Day Park capacity loss, ignoring the other phases of the duty cycle. This solution is expanded in Section 6 by considering the effects of key-off cooling on the performance attributes of the vehicle in Day Park and the subsequent phases of the duty cycle. In Section 7, optimization of partial charging and its integration with optimal key-off cooling is discussed.

To enable a numerical solution for the optimum control problem, the duration of Day Park is divided to into N intervals of $\Delta t$ seconds, and the cooling flag is assumed to be piece-wise constant, changing only at instances $k\Delta t$. Hence, the total capacity loss in Day Park is the sum of the capacity losses over all N intervals:

$${\left[\Delta C\right]}_{DPrk}={\left[\Delta C\left(x_b, u\right)\right]}_{Dprk}={\displaystyle \sum }_{k=0}^N\Delta C_k \left(x_{bk}, u_k\right)$$

(5)

The trajectory of the optimum cooling flag ($u^{*})$ is the solution of a minimization problem of the following form

$$\underset{u^{*}}{min}J=min\left({\displaystyle \sum }_{k=0}^N\Delta C_k \left(x_{bk}, u_k\right)+h_N\right)$$

(6)

The first term on the right-hand side (6) of is the arc-cost, which results from progression through consecutive time intervals, while the second term depends on the state at final time and is referred to as the terminal cost. The above problem can be solved with dynamic programming (DP) which is a common method for solving optimal control problems numerically [65,66,67]. To achieve this, the terms on the right-hand side of (6).

Should be evaluated for a grid of the vehicle states. To account for every possible scenario, the following range was assumed for the vehicle state:

$$\left[\begin{array}{c}22 °\mathrm{C}\\ 20 °\mathrm{C}\\ 5\%\end{array}\right]\le x_v=\left[\begin{array}{c}{T}_{cabin}\\ \begin{array}{c}{T}_{batt}\\ \zeta \end{array}\end{array}\right]\le \left[\begin{array}{c}62 °\mathrm{C}\\ 50 °\mathrm{C}\\ 80\%\end{array}\right]$$

(7)

$T_{cabin}$, $T_{batt}$, and $\mathsf{\zeta }$ in the state grid were discretized into steps of 1 °C, 2% and 2 °C, respectively. For discretization in time, increments of 240 s (4 min) are assumed. It is worth noting that the boundaries of the state grid act as hard constraints for the solution of DP.

The definition of $h_N$ in (6) has a high significance. For an initial examination of the behavior of the optimization algorithm, consider controlling key-off cooling only to minimize the capacity loss in Day Park, (without accounting for any subsequent phase of the duty cycle nor other performance attributes). $h_N$ can be defined as a lower SoC limit, for example 40%:

$$h_N=\{\begin{array}{c}0if {\zeta }_N\ge 40\%\\ \infty if {\zeta }_N<40\%\end{array}.$$

(8)

Figure 8 shows the trajectories of the cooling flag as well as the temperature and the SoC of the battery achieved from solving (6) considering (8) and the initial condition of $x_v={\left[\begin{array}{c}21 °\mathrm{C}, 30 °\mathrm{C},75\%\end{array}\right]}^{\prime }$. Since charge is abundant, cooling is applied both to cool the battery and to spend the excess charge in order to minimize the storage capacity loss, as long as the resulting cycling ageing does not negate the benefits. Examining Figure 8 and refereeing to Figure 2 and Figure 4, three trends can be identified. First, during the morning hours when the heat gain from the ambient environment is comparatively low, a mild level of cooling is applied to maintain the battery in the range of 23 °C–25 °C. Second, cooling intensifies when the SoC drops below 70% (at approximately 8 a.m.). This is because the capacity loss in the battery cells is highest in the SoC range of 50–70%, according to the dataset implemented in the model [57,58]. Therefore, the cooling decisions are dominated by the tendency to discharge the battery below 50% SoC. Note that the lower boundary of the battery temperature in the state grid (i.e., 20 °C) is observed. Third, continuing to the afternoon hours when the heat gain from the ambient environment and cabin is high, it becomes increasingly inefficient to maintain a low temperature. Therefore, the cooling is targeted at keeping the battery temperature below 40 °C.

The impact of controlling key-off cooling as above on the battery can be inferred from

Table 6. Effect of applying key-off cooling as shown in Figure 8 on the battery. Data only represent Day Park.

	Without Key-Off Cooing	With Key-Off Cooling
$\overline{\mathsf{\zeta }}$ (%)	74.6	58.8
${\overline{T}}_{batt}$ (°C)	39.0	27.8
Cycle count (n) *	0	26.7
Storage $\Delta C$ (%) *	1.07	0.40
Cycling $\Delta C$ (%) *	0	0.14

* Per summer season.

. Key-off cooling has noticeably reduced the storage capacity loss while leading to a small cycling capacity loss. The effect of key-off cooling on the subsequent phases of the duty cycle will be discussed in the following section.

6. Optimal Control of Key-Off Cooling in View of the Daily Duty Cycle

In view of the daily duty cycle of the vehicle, improving the fuel economy, the battery lifetime and the thermal comfort can be considered as the objectives of key-off battery cooling. This can be achieved through minimizing a cost function composed of the weighted sum of three cost terms: the fuel consumption, the capacity loss and the thermal discomfort index as:

$$J={\lambda }_1{v}_{fuel}+{\lambda }_2\Delta C+{\lambda }_3{I}_{discomf}$$

(9)

in which $\lambda =\left\{{\lambda }_1, {\lambda }_2, {\lambda }_3\right\}$ is the relative weighting of the cost terms. All terms in (9) depend, among other factors, on how the vehicle is controlled during Trip 2. Since the focus here is on key-off control, it was assumed that the key-on control strategies of the vehicle were already optimized. In other words, the powertrain and cooling are optimally controlled in the trips, for any given initial conditions. This assumption enables defining the cost terms as a function of vehicle states. As shown in Figure 7c, $v_{fuel}$ can be written as:

$$v_{fuel}\equiv {\left[v_{fuel}\right]}_{Trp2}={\left[v_{fuel}\left(x_v\left(t_{Dprk}\right)\right)\right]}_{Trp2},$$

(10)

which states that the fuel consumption in Trip 2 depends on the state of the vehicle after the Day Park, $x_v\left(t_{Dprk}\right)$. Similarly, the capacity loss in Trip 2 depends on $x_v\left(t_{Dprk}\right)$. In the Night Park, given that plugged-in cooling controls the battery temperature, the capacity loss is a function of the remaining battery SoC after Trip 2, which only depends on $x_v\left(t_{Dprk}\right)$. Therefore, $\Delta C$ in (9) can be written as:

$$\Delta C={\left[\Delta C\left(x_b, u\right)\right]}_{Dprk}+{\left[\Delta C \left(x_v\left(t_{Dprk}\right)\right)\right]}_{Trp2}+{\left[\Delta C\left(x_v\left(t_{Dprk}\right)\right)\right]}_{Nprk}$$

(11)

where capacity loss in Day Park is replaced from (5). The discomfort index also depends on $x_v\left(t_{Dprk}\right)$:

$$I_{discomf}=I_{discomf}\left(x_v\left(t_{Dprk}\right)\right)$$

(12)

The problem described by (9) can be reformulated in form of (6)

$$\underset{u^{*}}{min}J=\begin{array}{c}min\end{array} \left({\lambda }_2 {\displaystyle \sum }_{k=0}^N\Delta C_k \left(x_{bk}, u_k\right)+h_N\right)$$

(13)

for which the terminal cost is derived from (10)–(12) as:

$$\begin{array}{ll}{h}_N=& {\lambda }_1 {\left[v_{fuel}\left(x_v\left(t_{Dprk}\right)\right)\right]}_{Trp2}+{\lambda }_2 {\left[\Delta C \left(x_v\left(t_{Dprk}\right)\right)\right]}_{Trp2}\\ & +{\lambda }_2 {\left[\Delta C\left(x_v\left(t_{Dprk}\right)\right)\right]}_{Nprk}+{\lambda }_{3 }{I}_{discomf}\left(x_v\left(t_{Dprk}\right)\right).\end{array}$$

(14)

Solving (13) with DP requires a priori knowledge of the terminal cost that results from each value of $x_v\left(t_{Dprk}\right)$ within the state grid. In other words, the effect of $x_v\left(t_{Dprk}\right)$ on the cost terms over Trip 2 and the Night Park should be evaluated in advance. To achieve this, Trip 2 and the Night Park were simulated for all possible values of $x_v\left(t_{Dprk}\right)$ as initial state. The cost terms were then evaluated and mapped onto $x\left(t_{Dprk}\right)$ to evaluate (14). As an example, Figure 9 shows the resulting maps for a cabin temperature of 60 °C at the end of Day Park.

Figure 9a shows the fuel consumption in Trip 2, which has a non-zero value. The fuel consumption gradually increases as the initial SoC reduces below 50% (due to entering CS operation) or as the battery temperature increases above 45 °C (due to reduced electric traction capability).

Figure 9b shows the capacity loss in Trip 2. When the initial SoC is in the range of approximately 50–80%, the capacity loss decreases with any increase in the SoC (due to lower current throughput at higher voltage), but increases with temperature. Below 50% SoC, the capacity loss map is noisy since any small change in the SoC or the battery temperature can alter the sequence of supervisory rules triggered in Trip 2, changing the current throughput and the temperature.

Figure 9c shows the discomfort index which is high when the battery is initially hot, and low when the trip starts with a low battery temperature.

Figure 9d shows the capacity loss in Night Park. The shape of this map is driven by the dataset used in propagating the ageing model. In calculating this map, the variation in the duration of charge with the remaining battery SoC is neglected for simplicity.

Results: Effect of Key-Off Cooling on the Performance Attributes

Figure 10 shows the result of simulating the duty cycle with the optimum key-off cooling flag achieved through solving (13) with four different choices for the weighting vector $\lambda$. For each $\lambda ,$ the 24-h profile of the battery temperature and SoC are given alongside the profile of the cabin temperature and the engine torque in Trip 2.

Figure 10a-1,a-2 are related to the scenario in which minimizing fuel consumption is the only objective. In this solution, the temperature of the battery is allowed to rise during Day Park but (key-off) cooling starts just before Trip 2, precooling the battery to approximately 42 °C. Comparing the profile of engine torque in Trip 2 with the baseline scenario (Figure 5b) suggests that this degree of precooling has eliminated the unwanted operation in hybrid mode in the initial part of the trip.

Figure 10b-1,b-2 are related to minimization of the thermal discomfort. Figure 10b-1 shows that the battery is cooled and maintained at approximately 21 °C, well before Trip 2 starts. Consequently, the key-on cooling in Trip 2 has been shifted backward by 400 s, making the cabin temperature similar to that of the conventional vehicle, suggesting improved thermal comfort. Battery cooling initiates after approximately 400 s into Trip 2, leading to a slight rise in the cabin temperature.

In Figure 10c-1,c-2 minimization of the capacity loss is considered as the only objective. In this solution, key-off cooling has been intensely applied to maintain a low battery temperature. This has consumed approximately 50% of the battery charge and has forced the vehicle to operate in CS mode for most of Trip 2, suggesting high fuel consumption. As a result of the low battery temperature in Day Park, the cabin temperature has improved in the first 250 s of the trip. Note that this solution highlights a trade-off between minimizing the capacity loss on the one hand, and minimizing fuel consumption and discomfort on the other hand.

When all three objectives are considered simultaneously, a solution similar to that shown in Figure 10d-1,d-2 can be expected. Here, key-off cooling has limited the temperature of the battery during Day Park. Furthermore, sufficient energy has been allocated to precool the battery to 21 °C, which has eliminated the unwanted operation in hybrid mode and has improved the thermal comfort, in Trip 2.

The evolution of the cost terms in the results presented in Figure 10 are given in

Table 7. Breakdown of cost terms for different solutions of Equation (13). The capacity loss and the fuel consumption figures are quoted per season.

Solution	${\left[\Delta \mathit{C}\right]}_{\mathit{D}\mathit{p}\mathit{r}\mathit{k}}$	${\left[\Delta \mathit{C}\right]}_{\mathit{T}\mathit{r}\mathit{p}2}$	${\left[\Delta \mathit{C}\right]}_{\mathit{N}\mathit{p}\mathit{r}\mathit{k}}$	Batt. EoL	${\left[{\mathit{v}}_{\mathit{f}\mathit{u}\mathit{e}\mathit{l}}\right]}_{\mathit{T}\mathit{r}\mathit{p}2}$	${\mathit{I}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{c}\mathit{o}\mathit{m}\mathit{f}}$
	(%)	(%)	(%)	(years>)	(L)	(-)
Baseline	1.07	0.16	1.364	6.3	67.5	59
Minimization of $v_{fuel}$: $\lambda =\left\{1, 0, 0\right\}$	1.07	0.17	0.98	6.8	41.7	56
Minimization of $\Delta C$: $\lambda =\left\{0, 1, 0\right\}$	0.47	0.18	0.69	8.5	121.3	21
Minimization of $I_{discomf}$: $\lambda =\left\{0, 0, 1\right\}$	0.87	0.16	0.90	7.3	41.7	6
Trade-off solution: $\lambda =\left\{0.3, 0.3, 0.3\right\}$	0.62	0.17	0.77	8.0	41.7	6

. When minimization of the capacity loss is set as the only objective, key-off cooling improves the battery lifetime to 8.5 years (from 6.3 years in the baseline scenario), although fuel consumption inevitably increases. When all objectives are accounted for, the fuel consumption and the discomfort index are minimized (to their lower bounds), and still, a significant improvement in the battery lifetime is achieved. The solutions obtained above with different weighting of the cost terms fall on the Pareto of optimality that governs the optimum trade-off between the objectives.

An important observation from Table 7 is that a part of the gain in battery lifetime that is achieved by key-off cooling is a result of the lower capacity loss in Night Park. Key-off cooling increases the cycling component of the capacity loss in Night Park, given that the battery will receive more charge to compensate for the extra spent energy. Temperature variations are negligible as plugged-in cooling is applied in Night Park. Therefore, the reduction in Night Park capacity loss is due to lower storage capacity loss enabled by the lower storage SoC. Recall from Section 4 that the vehicle has approximately 40% excess charge over its summer duty cycle. This highlights an opportunity for achieving further improvements in the battery lifetime through partial charging as it will result in lower storage ageing without the need for extra cycling. Since the ability to perform key-off cooling depends on the available battery charge, it is necessary to integrate the optimization of partial charging and key-off cooling to maximize their benefit for the battery. This will be addressed in the following section.

7. Integrated Control of Partial Charging and Key-Off Cooling in View of the Daily Duty Cycle

In previous sections, it was assumed that the battery was fully charged every day (${\mathsf{\zeta }}_{max}=95\%$). Partial charging refers to optimizing ${\mathsf{\zeta }}_{max}$. Since ${\mathsf{\zeta }}_{max}$ is equivalent to initial SoC for every repeat of the duty cycle $({\mathsf{\zeta }}_0={\mathsf{\zeta }}_{max})$, partial charging affects the fuel consumption and capacity loss in Trip 1:

$$\left({\left[v_{fuel}\right]}_{Trp1}, {\left[\Delta C\right]}_{Trp1}\right)\propto {\zeta }_{max}.$$

(15)

So they should be included when propagating the cost function (9). Therefore, to optimize partial charging and key-off cooling, a minimization of the following form is required:

$$\underset{u^{*}, {\zeta }_{\mathit{max}}^{*}}{min}J=\begin{array}{c}min\end{array} \left({\lambda }_1{\left[v_{fuel}\right]}_{Trp1}+{\lambda }_2 {\left[\Delta C\right]}_{Trp1}+{\lambda }_2 {\displaystyle \sum }_{k=0}^N\Delta C_k \left(x_{bk}, u_k\right)+h_N\right),$$

(16)

where ${\zeta }_{max}^{*}$ denotes the optimum level of charge. $h_N$ in the above equation is of the same form as (14) but since partial charging affects the cycling component of capacity loss in Night Park, ${\left[\Delta C\right]}_{Nprk}$ should be calculated as:

$${\left[\Delta C\right]}_{Nprk}=\left[\Delta C_{stor} \left(x_v\left(t_{Dprk}\right)\right)\right]+\left[\Delta C_{cyc} \left({\zeta }_{max}, x_v\left(t_{Dprk}\right)\right)\right]$$

(17)

leading to:

$$\begin{array}{ll}{h}_N=& {\lambda }_1 {\left[v_{fuel}\left(x_v\left(t_{Dprk}\right)\right)\right]}_{Trp2}+{\lambda }_2 {\left[\Delta C \left(x_v\left(t_{Dprk}\right)\right)\right]}_{Trp2}\\ & +{\lambda }_{3 }{I}_{discomf}\left(x_v\left(t_{Dprk}\right)\right)+{\lambda }_2 \left[\Delta C_{stor} \left(x_v\left(t_{Dprk}\right)\right)\right]\\ & +{\lambda }_2\left[\Delta C_{cyc} \left({\zeta }_{max},x_v\left(t_{Dprk}\right)\right)\right].\end{array}$$

(18)

For every choice of ${\mathsf{\zeta }}_{max}$, the minimization in (16) reduces to optimal control of key-off battery cooling described by (13). which can be solved with DP, as done previously. By iterating over different choices for ${\mathsf{\zeta }}_{max}$, the combination of the level of charge and the key-off cooling flag trajectory that solves (16) can be determined. It is worth noting that solving (16) only requires one evaluation of the arc-cost term for the grid of the vehicle states in Day Park, since the arc-cost term is independent of the initial conditions of Day Park (hence it is independent of ${\mathsf{\zeta }}_{max}$). Therefore, the computation is inexpensive.

The simulated annealing (SA) algorithm [68], which has been regularly applied to parametric optimization of HEV control strategies [69,70], is used to find ${\zeta }_{max}^{*}$. Figure 11 shows the flowchart of the process.

Results: Effect of Partial Charging and Key-Off Cooling on the Performance Attributes

Figure 12 shows the progress of the SA algorithm when (16)–(18) have been weighted with $\lambda =\left\{0.2, 0.5, 0.3\right\}$. The search space is again limited to $40\%\le {\mathsf{\zeta }}_{max}\le 95\%$ (with resolution of 0.5%) to accelerate convergence. The high weights assigned to fuel economy and thermal comfort make it imperative that the battery is sufficiently charged to deliver maximum CD operation in both trips, as well as an intense key-off precooling. This effectively limits the window of SoC that is available for partial charging, so the choice is between higher charge that enables more key-off cooling, or lower charge that only supports the CD operation and precooling.

Figure 12a shows the values of ${\mathsf{\zeta }}_{max}$ for each iteration, suggesting that the search space has been thoroughly sampled. The algorithm has converged to ${\mathsf{\zeta }}_{max}=89\%$ after 93 iterations. Figure 12b shows that the best achieved (normalized) value of $J$ which is first attained at Iteration 50 (${\mathsf{\zeta }}_{max}=89\%)$. For Iterations 1, 50, 59, 71, and 75 (with respective ${\mathsf{\zeta }}_{max}$ of 95%, 89%, 70%, 57%, and 42%) the breakdown of the normalized components of $J$ are compared with the baseline scenario (full charging and no key-off cooling, see Section 4) in Figure 12c. Note that the fuel consumption in trip 1 has remained unchanged in these iterations, due to the $40\%$ limit set on the minimum level of charge. Besides, the capacity loss in the two trips is small and relatively unaffected by the level of charge, as expected. Figure 12d shows the histogram of the battery temperature in Day Park for Iterations 1, 50, 59, 71, and 75 with the baseline. The battery temperature is higher when the level of charge is lower, since less key-off cooling is applied. Therefore, in Iteration 50 (89% charge, optimum solution), the battery temperature is higher than in iteration 1, but it is still much lower than the baseline.

Figure 13 shows the result of simulating the duty cycle with the optimum level of charge and the key-off cooling flag trajectory achieved through the process explained above. Comparing the profile of the battery temperature in Figure 13 with the case of key-off cooling for the fully charged battery (Figure 10d-1,d-2) shows that temperatures above 30 °C have been more frequently allowed in Day Park, as less key-off cooling has been possible with a partially charged battery. The profiles of engine torque and cabin temperature have been unaffected by partial charging since the battery is precooled to 21 °C, as was the case with the fully-charged battery.

More details about the result shown in Figure 13 are given in

Table 8. Breakdown of cost terms for the solution of Problems (13) and (16). Due to imposing a minimum charge of 40%, the impact of charge optimization on the vehicle in Trip 1 is negligible, and the associated cost terms are not quoted in the table.

Row	Solution	${\mathit{\zeta }}_{\mathit{m}\mathit{a}\mathit{x}}$	${\left[\Delta \mathit{C}\right]}_{\mathit{p}\mathit{r}\mathit{k}}$	${\left[\Delta \mathit{C}\right]}_{\mathit{t}\mathit{r}\mathit{p}2}$	${\left[\Delta \mathit{C}\right]}_{\mathit{n}\mathit{p}\mathit{r}\mathit{k}}$	Batt. EoL	${\left[{\mathit{v}}_{\mathit{f}\mathit{u}\mathit{e}\mathit{l}}\right]}_{\mathit{t}\mathit{r}\mathit{p}2}$	${\mathit{I}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{c}\mathit{o}\mathit{m}\mathit{f}}$
		(%)	(%)	(%)	(%)	(years)	(L)	--
1	Baseline: without key-off cooling or partial charging	95	1.07	0.16	1.34	6.3	67.5	59
2	optimized key-off cooling (Figure 10d-1,d-2)	95	0.62	0.17	0.75	8.0	41.7	6
3	optimized partial charging +key-off cooling (Figure 13)	89	0.61	0.17	0.70	8.2	41.7	6
4	optimized partial charging +key-off cooling $\lambda =\left\{0.5, 0.5, 0\right\}$	68	0.47	0.18	0.61	8.7	41.7	49
5	lowest possible charge without key-off cooling	55	0.58	0.17	0.58	8.5	67.5	59

(Row 3), where they are compared with the baseline results (Row 1) and the results of key-off cooling optimization which was shown in Figure 10d-1,d-2 (Row 2). Integrated partial charging and key-off cooling, through reducing the capacity loss in Day Park and Night Park, has enabled a longer battery lifetime compared to key-off cooling alone. This comparison confirms the benefits of the proposed method. The fact that despite the higher battery temperature, partial charging has reduced the capacity loss in Day Park indicates that partial charging compensates for the higher temperature. Furthermore, as was previously seen in Figure 13, partial charging has not negatively affected the thermal comfort and fuel economy improvements that key-off cooling enables.

Table 8 also includes details of two additional cases:

• Row 4 represents the solution of (16) considering only the fuel economy and the battery lifetime (i.e., $\lambda =\left\{0.5,0.5,0\right\})$, which leaves a wider widow of SoC for partial charging, at the expense of poor thermal comfort. Therefore, the battery is only charged to 68%. This still allows some key-off cooling, although less than was seen in Figure 13. As Table 8 suggests, the combination of lower battery charge and some key-off cooling improves the battery lifetime to 8.7 years.
• Row 5 represents the case in which the battery is minimized (to 55% which is the limit of the excess charge) but key-off cooling is not applied. In this case, the battery lifetime is improved to 8.5 years, which is less than the 8.7 years that can be achieved with partial charging and key-off cooling if thermal comfort is neglected. Note that lack of key-off cooling leads to high fuel consumption and thermal discomfort in this case.

Comparing the above cases proves that combining partial charging and key-off cooling is more beneficial than applying partial charging or key-off cooling alone, unless a significant compromise on fuel consumption is acceptable.

Solving (16) with alternative weighting vectors does not necessarily lead to distinct results. Solving (16) for fuel consumption alone ($\lambda =\left\{1, 0, 0\right\})$ has non-unique solutions. This is because any sufficiently high level of charge (approximately above 57%) combined with any key-off cooling flag trajectories that sufficiently pre-cools the battery for Trip 2 will lead to equal fuel consumption. Similarly, solving (16) for thermal discomfort alone ($\lambda =\left\{0, 0, 1\right\})$ does not have a unique solution. On the other hand, solving (16) for the capacity loss alone ($\lambda =\left\{0, 1, 0\right\})$ leads to the lowest possible level of charge at the expense of high fuel consumption, as can be inferred from the results of Iteration 75 in Figure 12.

8. Applicability to EVs

To adapt the methods proposed in this paper for EVs, it will be sufficient to choose a cost function that matches the characteristics of a fully electric powertrain. That is, fuel consumption in Equation (9) should be replaced with measures of acceleration and distance travelled (equivalent to the EDR in EVs). For any drive cycle, the difference between the achieved and demanded acceleration and distance, respectively denoted as $\Delta a$ and $\Delta d$, can be used to form the following cost function:

$$J=min\left({\lambda }_1\left(\Delta a+\Delta d\right)+{\lambda }_2 \Delta C+{\lambda }_3 I_{discomf}\right).$$

(19)

Equation (19) can be implemented in the form of Equations (13) and (14) for the key-off cooling problem, and in the form of Equations (16)–(18) for the integrated partial charging and key-off cooling problem. It is worth noting that given the generality of other assumptions in this paper and the similarity of battery technologies commonly used in these vehicles, the overall benefit of key-off cooling and partial charging will be unaffected by the vehicle type.

9. Discussion

The analyses presented in this paper demonstrate the ability of the proposed methods in controlling key-off cooling and partial charging, and maximizing their effectiveness in mitigating the negative impacts of hot climate conditions on the battery of electrified vehicles. Application of key-off cooling that is proposed in this paper is an important enhancement to the current state-of-the-art of combining key-on and plugged-in cooling [20,34,39]. Key-off cooling expands the scope of battery cooling, and enables maintaining a low battery temperature throughout the day, as well as performing precooling regardless of whether an external power source is available. Further, integration of partial charging and key-off cooling that is proposed in this paper enhances the current state-of-the-art of optimized charging without considering its benefit relative to cooling.

The battery lifetime improvement that is enabled by key-off cooling and partial charging serves the greater fulfilment of warranty targets across wider geographical locations, resulting in off-cycle credit for the manufacturer [71]. The improvements achieved in the fuel economy through key-off precooling can be similarly evaluated. Any improvement in the passengers’ thermal comfort through key-off precooling contributes to higher acceptability of electrified vehicles. Realizing the benefits of the methods proposed here will require the implementation of additional control algorithms in the vehicle. Nevertheless, this will be more cost effective than any alternatives such as using advanced insulation technologies and over-sizing the battery [72] which need to be applied to the whole fleet while the issue only affects the subset of the fleet that operate in hot climate conditions.

The opportunity for applying key-off cooling and partial charging without affecting the electric range of the vehicle depends on the availability of excess battery charge over the duty cycle. The assumptions of this work represent the common subset of all battery size and duty cycle combinations that leave some excess battery charge. When little or no excess charge is available, defining the relative importance of fuel economy, battery lifetime, and thermal comfort in controlling key-off cooling becomes more critical. Addressing such circumstances requires a separate analysis, but it is evident that the methodology proposed in this work will produce the ‘optimum’ results.

The methodology proposed in this paper relies on a priori knowledge of the ambient conditions and the duty cycle. When such a priori knowledge is absent, the relevant functionalities can be implemented through a rule-based algorithm inferred from offline simulation of a range of ambient conditions and duty cycle combinations. In a more advanced setting, weather forecasts, route planning technologies and interaction with parking management systems can provide the information needed for online optimal control of key-off cooling and partial charging.

10. Future Work

The simulation results presented in this paper are promising and suggest that applying key-off cooling through the proposed methodology can enable significant improvements in the performance attributes of PHEVs and EVs in hot climates. One way to enhance the confidence in the analyses is to employ a comprehensive battery ageing model to calculate the increase in the internal resistance of the cells, as well as their capacity loss, and update the battery lifetime calculations. This will be carried out in the feature using the model presented in [73]. Furthermore, experimental validation of the proposed methodology will be carried out in the future to establish its effectiveness in real life conditions.

Since the purpose of this research is to show the ultimate benefit of key-off battery cooling, practical considerations such as the additional cost of electrical energy, component wear, and noise pollution resulting from the operation of the cooling system were neglected. The possible impact of such considerations on the key-off cooling strategy will be analyzed in the future.

11. Conclusions

The key conclusions that can be drawn from the analyses presented in this paper are as follows:

(1)

Three performance attributes of PPHEVs, namely their fuel economy, battery lifetime, and passengers’ thermal comfort, can be negatively affected by the high battery temperature that results from hot climate conditions.

(2)

Key-off cooling can be applied to reduce this effect, hence reducing the unwanted variability in the performance attributes of the vehicle with climate conditions.

(3)

The control method proposed in this paper maximizes the benefits of key-off cooling, by accounting for and containing the risks associated with using the battery to support key-off cooling, i.e., extra fuel consumption and disproportionately high cycling capacity loss. The formulation proposed for the key-off cooling control problem can be tuned to target individual, as well as combinations of, the affected performance attributes.

(4)

Integration of partial charging and key-off cooling enhances their individual benefits for the battery lifetime. The method proposed for integrated optimization of partial charging and key-off cooling can determine the combination of the level of charge and key-off cooling that maximizes the battery lifetime. This can be also tuned to achieve a trade-off between fuel economy, battery lifetime and thermal comfort.

Within the context of the assumptions of this paper, the following observations were made:

• Key-off cooling reduced the seasonal capacity loss of the battery in the Day Park phase by approximately 42% and this helped improve the battery lifetime from 6.3 years to 8 years.
• Key-off precooling eliminated the impact of battery overheating on the electric drive capability of the vehicle. This reduced the seasonal fuel consumption by approximately 23%.
• Key-off precooling improved the passengers’ thermal comfort by shifting the key-on cooling loads in Trip 2 by 400 s, reducing the thermal discomfort index from 59 to 6.
• Integrating partial charging and key-off cooling enabled further improvements in battery lifetime. By partially charging the battery (to 89% instead of 95%) and applying some key-off cooling, the battery lifetime was improved by 2.5% compared to when only key-off cooling was applied (8.2 years, compared to the eight years). When the window of excess charge was widened by neglecting the thermal comfort attribute, combining partial charging to 68% and key-off cooling improved the battery lifetime by 9%, compared to when only key-off cooling was applied (8.7 years compared to eight years). This was more than the improvements partial charging or key-off cooling can individually enable, unless a significant compromise in fuel consumption is acceptable.