A traffic-augmented macroscopic model of electric vehicle energy consumption

Abstract

2 This paper shows that the assumption of constant link speed to predict energy consumption in electric vehicle 3 routing optimisation problem significantly underestimates energy consumption. An analytical investigation 4 proves a dependency between energy consumption and vehicle acceleration variance, while a Monte Carlo 5 simulation quantifies the bias from ignoring this factor, incorporating variability in speed and slope profiles 6 through novel algorithms. A global sensitivity analysis identifies acceleration variance as the most critical input 7 factor affecting prediction accuracy. To correct the bias, a macroscopic energy consumption model is 8 developed which incorporates speed variance as an explanatory variable and links it to measurable 9 macroscopic traffic characteristics, such as link density and flow. The model is validated using laboratory and 10 experimental data, showing improved accuracy compared to the base macroscopic model. These results 11 highlight the importance of accounting for driving and traffic dynamics in energy consumption modeling for 12 electric vehicles.

14 1. Introduction 15 Accurate modelling of electric vehicles energy consumption is essential to solve an electric vehicle routing 16 optimization problem. In such problem, the vehicle energy consumption on each network segment must be 17 known in advance to design optimal routes, charging locations and schedules. In the field literature, energy 18 consumption predictions are typically provided by macroscopic models which return vehicle energy 19 consumption on each segment as a function of average inputs, such as average speed and slope on that segment 20 (Othman et al.,2019). 21 In particular, three macroscopic modelling approaches to simulate energy consumption have been adopted so 22 far: 1- assumed to be proportional to the link length through a constant coefficient (e.g., Wen et al.,2016; 23 Schiffer and Walther,2017; Bongiovanni et al.,2019; Lian et al.,2023; Su et al.,2023); 2- simulated by a 24 macroscopic data-driven model (e.g., Yi et al.,2018; Zhang et al.,2020; Pan et al.,2023); 3- simulated via a 25 simplified version of a microscopic power-based model (e.g., Masmoudi et al.,2018; Basso et al.,2019; 26 Pelletier et al.,2019; Sayarshad et al.,2020; Ma et al.,2021; Avishan et al.,2023). Generally, a microscopic 27 power-based model gives a vehicle energy consumption function of vehicle speed profile, road slope profile, 28 vehicle characteristics and environmental conditions, but since the actual speed profile is unknown when 29 solving a vehicle routing problem, a constant link speed assumption is made to derive a macroscopic energy 30 consumption model. 31 All the proposed approaches suffer from significant limitations that may jeopardise the robustness of design 32 service operations. While the first approach is unable to adequately describe the variability of consumption 33 across the network, the others lack transferability to different case studies and suffer from potential bias in 34 consumption prediction due to overly simplistic and unrealistic assumptions. In particular, the constant link 35 speed assumption adopted in the last two approaches compromises the prediction accuracy, as it implies 36 neglecting driving dynamics, i.e., the acceleration and deceleration phases. 37 Therefore, to address limitations of current modelling approaches, several contributions are provided in this 38 work: 39 1. Mathematical quantification of the inaccuracy resulting from the constant link speed assumption. 40 2. Generalisation of the theoretical results through a simulation experiment and identification of the 41 model inputs that most affect the variance of the output (i.e. of the SEC error) by means of a variance42 based global sensitivity analysis. 43 3. Proposal of two algorithms to generate speed and slope profiles in a way that emulates their real-world 44 variability. 45 4. Proposal of an augmented macroscopic energy consumption model whose aim is to eliminate the bias 46 of macroscopic models, relative to microscopic ones. 47 5. Validation of the augmented model against 1Hz real-world energy consumption data of a fleet of 48 electric minivans, as collected by Fiori and Marzano (2018).

1 2. Theoretical analysis 2 In this section the relationship between energy consumption and the standard deviation of vehicle acceleration 3 끫븜끫뢜 is examined. Analysing this aspect is necessary to understand the impact of a constant link speed 4 assumption on energy consumption prediction. The analysis is performed through mathematical steps, starting 5 with a normally distributed acceleration signal {끫뢜끫룂 }, i.e., 끫뢜끫룂 ~끫뢬. 끫뢬. 끫뢢. 끫륒(0, 끫븜끫뢜2 ), ∀끫룂 ∈ {0,1,2, … , 끫뢎}, where 끫뢎 = 6 Τ/Δ끫룂, Τ is the signal duration, and Δ끫룂 is a finite time step. The speed signal {끫룆끫룂 } results from the integral of 7 the acceleration signal. 8 According to Newton’s second law of motion, the traction force applied to the vehicle wheels is composed of 9 an inertial component, due to the applied acceleration signal, and a resistance component, due to motion 끫뢮 10 resistances, customarily modelled as a quadratic function of the instantaneous speed, 끫뢾끫룂 = ∑2끫뢮=0 끫뷺끫뢮 끫룆끫룂 .

11 The power signal results from the element-wise product of the traction force signal and the speed signal, while 12 the total energy consumption results by the integral of the power signal over time. 끫뢮+1 13 끫롰끫롬끫뢎 = 끫롰끫롬끫뢎끫뢬끫뢬 + 끫롰끫롬끫뢎끫뢾 = 끫룆0 끫뢎 Δ끫룂 + 끫뢰끫뢜2 Δ끫룂 2 + 끫뢎Δ끫룂끫룆0 ∑2끫뢮=0 끫뷺끫뢮 ∑끫뢎끫룂=1�∑끫룂끫뢬=1 끫뢜끫뢬 Δ끫룂^2� (1)

14 where 끫뢰 = 0.5끫뢎(끫뢎 + 1). After several mathematical steps and considering the expected value of the total 15 vehicle energy consumption, the result shows that the expected value of the total energy consumption is a 16 function of 끫븜끫뢜2 : 끫뢮 17 E[끫롰끫롬끫뢎 ] = E�끫롰끫롬끫뢎끫뢬끫뢬 � + E[끫롰끫롬끫뢎끫뢾 ] = 끫뢰끫븜끫뢜2 Δ끫룂 2 + 끫뢎끫뢎 끫룆0 ∑2끫뢮=0 끫뷺끫뢮 끫룆0 + Φ(끫븜끫뢜4 ) (2)

18 If speed dynamics are neglected, i.e., 끫븜끫뢜 = 0, (2) becomes: 끫뢮 19 E[끫롰끫롬끫뢎 ]끫븜끫뢜=0 = E�끫롰끫롬끫뢎끫뢬끫뢬 �끫븜 =0 + E[끫롰끫롬끫뢎끫뢾 ]끫븜끫뢜=0 = 끫뢎 끫룆0 ∑2끫뢮=0 끫뷺끫뢮 끫룆0 (3) 끫뢜

20 In conclusion, assuming a Gaussian white noise acceleration signal, the percentage error, i.e. the consumption 21 underestimation of assuming 끫븜끫뢜 = 0, is: E[끫롰끫롬끫뢎 ]끫븜끫뢜=0 −E[끫롰끫롬끫뢎 ] 끫뢰끫븜끫뢜2 Δ끫룂 2 +Φ(끫븜끫뢜4 ) 22 =− 끫뢮 (4) E[끫롰끫롬끫뢎 ] 끫뢎끫뷊 끫룆0 ∑2끫뢮=0 끫뷺끫뢮 끫룆0 +끫뢰끫븜끫뢜2 Δ끫룂 2 +Φ(끫븜끫뢜4 )

23 which is not negligible, especially in congested traffic. For example, for a 5.5 ton electric minivan, with a 24 payload of 2.5 ton, an average speed 끫룆0 = 10 m/s, a duration T = 100 s, an acceleration variance 끫븜끫뢜2 = 2 m2/s4, 25 road load coefficients 끫뷺0 = 0.0787, 끫뷺1 = 5.6∙10-4, 끫뷺2 = 0.5441, the vehicle energy consumption 26 underestimation ranges from 79% for a constant regenerative braking efficiency equal to 0.3, to 35%, for an 27 efficiency equal to 0.9. 28 3. Methodological framework 29 The theoretical findings presented above have been generalised through simulation. To quantify the degree of 30 underestimation of macroscopic model predictions, the consumption distributions by a macroscopic model and 31 its underlying microscopic counterpart are compared under uncertain model parameters, and uncertain inputs 32 (speed and slope profiles). 33 The impact on the variability of model prediction errors of any uncertain inputs or parameters is quantified 34 through a sensitivity analysis. Results are relevant to identify what inputs or parameters are most influential 35 on the mentioned prediction error. This analysis significantly extends the study by Fiori et al. (2021), also 36 considering uncertain speed and slope profiles. 37 The whole study is built on the methodology depicted in Figure 1. 38 In the methodology, non-parametric inputs and model parameters are sampled according to the Sobol’ design 39 in a quasi-random Monte Carlo setting, from uniform independent distributions (Table 1 lists all the uncertain 40 factors with the selected lower and upper bounds of each distribution). The speed and slope profiles are then 𝑇𝑇

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1 generated through algorithms devised to generate profiles in a way which emulate their real-world variability 2 by sampling values of 끫븎끫룆 , 끫븜끫뢜 , 끫븎끫븆 , 끫븜끫븆 and 끫롾. 3 The speed profile generation algorithm aims to generate a speed profile with a mean 끫븎끫룆 , a zero-mean first 4 derivative (끫븎끫뢜 = 0) and a given 끫븜끫뢜 . Applying the algorithm in a Monte Carlo framework produces a population 5 of speed profiles that incorporates the sought variability of 끫븎끫룆 and 끫븜끫뢜 . The slope profile generation objective 6 is to smooth a slope profile 끫븆�끫룊(끫룂)�, 끫룂 ∈ [0, 끫뢎] resulting from a Gaussian process with mean 끫븎끫븆 and standard 7 deviation 끫븜끫븆 (being 끫룊(끫룂) the longitudinal vehicle position), without altering the slope values, thus preserving 8 끫븎끫븆 and 끫븜끫븆 . Algorithms are not provided for brevity. 9 For each generated set of input profiles and parameter values, the microscopic power-based model by Fiori 10 and Marzano (2018) and its macroscopic model counterpart are applied to simulate vehicle energy 11 consumption. The result of such uncertainty propagation is a distribution of simulated energy consumption.

Non-parametric inputs pdfs Model parameters pdfs

Quasi-random Monte Carlo sampling

Sampled Sampled non-parametric model parameter input values values

Speed and slope profile generation

Speed and slope input Hp: profiles input averaging assumption

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Variance-based Global Sensitivity Analysis

13 Figure 1 – Methodological framework. 14 To investigate the impact of the input averaging assumption of the macroscopic model on the variability of the 15 model prediction error, a further uncertain factor is included in the experimental design, a Boolean variable 16 whose values correspond to the model structure – microscopic or macroscopic – applied in SEC computation.

1 Accordingly, a total number of 1,835,008 model simulations were run to explore the impact of the uncertain 2 factors on consumption error variability and compute the sensitivity indices (1,835,008 = 217 ⋅ (K+2), where 3 K=12 is the total number of uncertain factors). 4 To study the accuracy of energy consumption prediction, the difference between a simulated SEC and a 5 laboratory SEC is used as a measure of discrepancy (see ε in Figure 1). Such laboratory consumption is 6 computed by applying the model in Genikomsakis and Mitrentsis (2017) with calibrated parameters. 7 A variance-based global sensitivity analysis is then applied to disentangle the impact of the two sources of 8 input uncertainty (parameters and speed/slope profiles) on model error variability. This analysis provides also 9 an evaluation of the impact of the input averaging assumption in macroscopic modelling. 10 Table 1 – List of uncertain factors and corresponding lower (LB) and upper (UB) bounds of uniform 11 distributions.

Non-parametric inputs LB UB Mean speed,끫븎끫룆 [km/h] 5.0 130.0 Acceleration standard deviation,끫븜끫뢜 [m/s2] 0.0 3.50 Mean slope,끫븎끫븆 -0.05 0.05 Slope standard deviation,끫븜끫븆 0.00 0.05 Vehicle load,끫룈[kg] 0 2500 Link length,끫롾[m] 100 2000 Model parameters LB UB Rolling resistance parameter,끫뢦 0.005 0.020 Frontal section area,끫롨끫뢦 [m2] 0.70 0.90 Drag coefficient,끫롬끫뢢 0.10 0.50 Powertrain efficiency,끫븄 0.70 0.90 Regenerative braking coefficient,끫뷸 0.00 5.00

13 Based on a Sobol’s variance decomposition (Sobol,2001), the first-order sensitivity index, 끫뢌끫뢬 , and the total 14 sensitivity index, 끫뢌 끫뢬 , of each input factor 끫뢬, are computed. These indices describe the contribution to the 15 unconditional error variance of a factor, both by the factor alone (first-order effect), and by the factor in 16 interaction with all the others (total effect). 17 3.1. Uncertainty and sensitivity analysis results 18 The scatter plots in Figure 2 show the simulation errors of the macroscopic and microscopic models. For each 19 simulation 끫뢬, the error 끫븀끫뢬 has been computed as follows: 끫롰끫롬끫뢴끫뢴끫뢴끫뢴끫뢴 (끫뤬끫뤔 ,끫빮끫뤔 )−끫롰끫롬끫룀끫룀끫룀끫룀ℎ 끫뢤끫뢤 (끫뤬끫뤔 ) 20 끫븀끫뢬 = 끫롾 (5)

21 where 끫롰끫롬끫뢴끫뢴 끫뢴끫뢴 is the simulated total energy consumption on the link of length 끫롾끫뢬 by the 22 microscopic/macroscopic model fed with the model parameters 끫빮끫뤔 and the speed/slope profiles generated by 23 the proposed algorithms according to the non-parametric inputs 끫뤬끫뤔 ; and 끫롰끫롬끫룀끫룀 ℎ 끫뢤 is the laboratory energy 24 consumption computed by means of the reference model fed with the same input profiles. A positive error 25 means that the model overestimates consumption, a negative value implying an underestimation. 26 In the scatter plots, the simulation errors are plotted against each analysis factor. Results show that the 27 macroscopic model significantly underestimates consumption in most of the simulations, see the bottom 28 rightmost plot. Conversely, the microscopic model has the same probability of overestimating or 29 underestimating consumption. Given a factor, the higher the variance of light grey and yellow points over that 30 factor, the higher the influence of that factor on the variation of the average SEC error of microscopic and 31 macroscopic model, respectively.

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