# Utility Factor Curves for Plug-in Hybrid Electric Vehicles: Beyond the Standard Assumptions

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## Abstract

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## 1. Introduction

- The gap between standard and real UF values (Figure 1a) as well as the categories of reasons 1–3 (Figure 1b) were drawn in the “negative” direction (i.e., real UF being less than the standard UF). However, this is primarily for illustrative purposes. In reality, it is plausible for any of the three categories of reasons or the overall gap to be in either the positive (i.e., better UF than the standard rating) or negative directions.
- Each of the three main categories of reasons may include several sub-reasons; for example, category #1 (real-world attained AER) could be affected by the acceleration rate and speed driving style of vehicle owners, ambient temperature (which in turn affects both the efficiency of the electric powertrain as well as the heating/cooling power consumption for climate control of the passenger cabin), weight of passengers and cargo, gradient of the terrain (uphill/downhill), or towing load.
- It is also important to note that those three categories of reasons, while understood to be the main contributors to the UF gap, are not the only contributing reasons, nor is it necessarily true that they are linearly independent. For example, some PHEV designs may utilize electric power to warm up the battery during a cold climate, while others might utilize an alternative approach such as briefly turning on the engine, which in turn might affect the observable miles traveled in CD or EV mode.
- The (real-world) attained AER is not necessarily a static number like the nominal AER that is published by regulatory agencies such as the US EPA [14]. In fact, the attained AER can change from day to day depending on the vehicle usage conditions, and such daily variations in the attained AER can have interactions with the other two categories of reasons (charging frequency and distance traveled). Nonetheless, to avoid over-complicating the problem, secondary interactions between the reasons and “all other/unknown” reasons are often lumped with one of the three main categories of reasons.

## 2. Mathematical Model

#### 2.1. Notations and Assumptions

_{ij}, with the first index (i) referring to days of travel by a certain vehicle, while the second index j refers to a range of miles traveled per day, depending on a discretization parameter δ. For example, if δ = 0.5 miles, then j = 10 refers to the range between 4.5 and 5.0 miles per day of travel distance. In this paper, we utilize the 2010–2012 California Household Travel Survey (CHTS) dataset [16], from which the matrix D

_{ij}has been extracted. To make it easier to replicate the work in the current paper, a copy of the matrix D

_{ij}has been placed in shared/public-accessible cloud storage [17]. Assuming the discretization parameter (δ), which was chosen at 0.5 miles, provides sufficient resolution, the annual miles traveled (l

_{ij}) by a vehicle sample i on days with daily miles between (j − 1)δ and jδ is estimated (using upper bounds for daily miles interval) via Equation (1) as follows:

_{i}) and the probability density mass for a fraction of the vehicle’s miles traveled (p

_{ij}) on days with daily miles between (j − 1)δ and jδ can be calculated as:

_{i}) effectively scales it up to the population that the dataset is intended to represent (all of California in the case of the CHTS dataset). The calculation of MDIUF and FUF can thus be performed as follows:

_{i}= 1 in Equations (5)–(7)). For illustration purposes, the reference MDIUF and FUF curves (per Equations (5) and (6)) for the CHTS dataset are plotted in Figure 2 for the range of real-world attained AER (x) between 0 and 100 miles. Also shown in Figure 2 are the reference values from SAE J2841 [3] that are based on NHTS-2001. One notable observation in Figure 2 is that the reference UF curves (both MDIUF and FUF) via CHTS seem to have larger UF numbers than the reference UF curves via NHTS-2001 at any given value of x. This implies that the recorded vehicle travel in CHTS generally had fewer miles per drive day than in NHTS-2001. One plausible explanation for this could be the timing of data collection, where CHTS (data collected between 2010 and 2012) could have been affected by the 2008–2009 recession period in the US. Another plausible explanation could be due to the method of data collection, where NHTS-2001 utilized self-reported trip length data, while CHTS data utilized in this paper came from on-board device (OBD) logging of the sample vehicles. However, regardless of what dataset is used, the mathematical modeling approach in this paper could still be applied.

#### 2.2. Charging Frequency Less Than Once per Drive Day

#### 2.2.1. Overview

#### 2.2.2. Special Case: Binary Charging Behavior

_{i}) can only take a value of either 0 or 1. In other words, while the overall average for the population of vehicles (λ) is between 0 and 1, this average is only attained via one set of vehicles (A) always charging (i.e., λ

_{i}= 1, i ∈ A), while some vehicles are never charging (i.e., λ

_{i}= 0, i ∉ A). This can be mathematically expressed as:

_{i}= 1) as the decision variables. The optimization problem has the form:

_{i}), with all other quantities in Equations (13)–(15) being constant or possible to pre-compute before running the optimization problem to determine the upper/lower bounds for MDIUF or FUF. However, in the case of binary charging behavior, it is an integer-linear program (per the constraint in Equation (16)). While a generic integer linear program can be challenging to solve, when the number of vehicle samples in the dataset is sufficiently large, the equality constraint in Equation (15) can be satisfied within reasonable tolerance while relaxing the optimization problem to only solve its linear program version. The results of this model (independent, upper and lower bounds) for λ = 0.5 are shown in Figure 3a.

#### 2.2.3. Special Case: All Vehicles with the Same Charging Frequency

_{i}) is exactly equal to the charging frequency of the population average (λ). In other words, this is a case where all vehicles in a population are behaving exactly the same in terms of frequency of charging. For this special case, going from vehicle UF to population MDIUF and FUF is fairly straight-forward (similar to Equations (5) and (6)) as:

_{i}= λ) requires some further modeling assumptions. For this, we consider the calculation of an expected UF value based on an independent probability distribution for which days have a charging event before them, as well as upper and lower bounds.

_{i}= λ). Under such conditions, the UF of a vehicle sample can be expressed as:

_{i}(1 − λ

_{i}) that is multiplied by the second term in Equation (19) represents the probability of the current driving day not having a charging event after the previous driving day had a charging event. We don’t consider the rest of the terms after the second term in Equation (19) as they would be multiplied by λ

_{i}(1 − λ

_{i})

^{2}(or even smaller numbers), so we consider them negligible compared to the first two terms. The equivalent UF for two days in a row after one overnight charging event is calculated as:

_{ij}is the probability density mass for vehicle i having a day with miles traveled within a certain range of miles per day:

_{i}approaches a value of either 0 or 1.

_{i}= 0.5 with uniform spacing, this means that a charging event happens exactly one per two drive days, which can maximize utilization of each charging event. The opposite, least favorable temporal distribution is when/if all the charging events happen on back-to-back drive days while leaving a long gap of days without any charging events. Furthermore, if the stacked-up charging events are occurring before drive days that have the least contribution to attained electric miles, it would represent the lower bound for UF. To compute this lower bound, we set up an integer linear programming optimization problem similar to the setup in Section 2.2.2, but with the decision variables (v

_{j}) controlling the temporal distribution of charging events.

_{i}= 0.5: in this estimate of an upper bound for UF, it is assumed that the temporal distribution of charging events is exactly evenly spaced at one charging event per two drive days, while being statistically independent from the number of miles traveled on any given day. The UF can then be calculated in a similar manner as Equation (19), but with the drive days either having a charging event or a charging event one day earlier (and no three or more drive days without a charging event):

_{i}< 0.5, we consider an upper bound for the UF via linear interpolation between zero and the upper bound value obtained from Equation (26). This corresponds to a temporal distribution of charging events where a portion of the time horizon has evenly spaced charging events at a frequency of one per two drive days, while the rest of the time horizon has no charging events. Likewise, for cases when λ

_{i}> 0.5, we consider an upper bound for the UF via linear interpolation between the value obtained from Equation (26) and the case when λ

_{i}= 1. This corresponds to a temporal distribution of charging events where a portion of the time horizon has evenly spaced charging events at a frequency of one per two drive days, while the rest of the time horizon has one charging event for every drive day. Combining the cases, the formula for the upper bound is summarized as:

#### 2.2.4. Generalized Upper and Lower Bounds

_{i}), such as the terms defined via Equation (19) or even sub-optimization problems, such as when estimating lower bounds for an individual vehicle sample via Equations (23)–(25). As such, the optimization approach we tested for solving Equations (29)–(33) is the nonlinear programming technique known as successive linear programming [21]. The calculated upper and lower bound UF curves for λ = 0.5 are shown in Figure 3c.

_{i}in this case) for a nonlinear objective and/or constraints via solving a series of linear programming optimization problems. In each iteration of SLP, a linear approximation is constructed for the nonlinear objective and/or constraints (via function value and gradient at the “current solution” point). The linear approximation is then solved via linear programming techniques, with the solution of the linear programming becoming the new “current solution” point, and the process is repeated until convergence, which is typically when iterations of SLP can no longer find a better solution satisfying the problem constraints. In implementation for the current problem (solving for all λ

_{i}), the constraints (Equations (32) and (33)) are actually linear, which means that the successive iterations of SLP always retain a feasible solution. Since the results of SLP, much like gradient-following optimization techniques, can be dependent on the “starting point”, a multi-start point strategy is employed, with two special cases (from Section 2.2.2 and Section 2.2.3) included among the starting points. This ensures that the solution returned by SLP (as shown in Figure 3c) is always “outside the envelope” of either of the two special cases (shown in Figure 3a,b).

#### 2.3. Charging Frequency: More Than Once per Drive Day

_{i}can have binary (0 or 1) values and an integer linear optimization framework (similar to Section 2.2.2) can be applied to estimate upper and lower bounds, as shown in Figure 4.

#### 2.4. Summary of Modelled Cases

## 3. Discussion

_{2}/kWh (calculated via fuel mix information from the US EIA [25] and GREET model [24]) and 10,778 g-CO

_{2}/gal for E10 gasoline. We also consider scenarios for the year 2050 with projected carbon intensity for electricity at 180 g-CO

_{2}/kWh, as well as 50% reduced carbon intensity for gasoline. Using plausible FUF values (from Figure 4b) at different charging frequencies, carbon emissions offset results for the considered scenarios are shown in Figure 5.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Illustration of three categories of assumptions mismatch between standard UF curves and real-world.

Case | Discussed | Description |
---|---|---|

λ_{i} ∈ {0, 1}, μ = 0 | Section 2.2.2 | No daytime charging. Overnight charging behavior is binary; some vehicles always charge, others never charge. |

All λ_{i} = λ, 0 ≤ λ ≤ 1,μ = 0 | Section 2.2.3 | No daytime charging. The overnight charging frequency is the same for all vehicles. |

0 ≤ λ_{i} ≤ 1, μ = 0 | Section 2.2.4 | No daytime charging. Generalized case for overnight charging, where some vehicles always charge, some never charge, others somewhere in-between. |

All λ_{i} = 1, 0 ≤ μ_{i} ≤ 1 | Section 2.3 | Vehicles always charge overnight. Some vehicles also gain one additional charging event during the day. |

0 ≤ λ_{i} ≤ 1, 0 ≤ μ_{i} | Section 3, future work | Fully generalized case where overnight charging frequency for each individual vehicle can be anywhere between always and never, while at the same time, each individual vehicle may have additional one or more charging events during the day |

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**MDPI and ACS Style**

Hamza, K.; Laberteaux, K.P.
Utility Factor Curves for Plug-in Hybrid Electric Vehicles: Beyond the Standard Assumptions. *World Electr. Veh. J.* **2023**, *14*, 301.
https://doi.org/10.3390/wevj14110301

**AMA Style**

Hamza K, Laberteaux KP.
Utility Factor Curves for Plug-in Hybrid Electric Vehicles: Beyond the Standard Assumptions. *World Electric Vehicle Journal*. 2023; 14(11):301.
https://doi.org/10.3390/wevj14110301

**Chicago/Turabian Style**

Hamza, Karim, and Kenneth P. Laberteaux.
2023. "Utility Factor Curves for Plug-in Hybrid Electric Vehicles: Beyond the Standard Assumptions" *World Electric Vehicle Journal* 14, no. 11: 301.
https://doi.org/10.3390/wevj14110301