Five climatic wind tunnel (CWT) trials were performed at test facilities using a Fiat 500e. Trials 1, 3, and 4 were done at one wind tunnel facility, and trials 2 and 5 at another. The settings for each of these five trials are shown in Table 1. For each trial, a temperature, vent setting, and moisture level is specified. Trials 1 and 4 run the fresh distribution setting for 60 min with a car velocity of 50 km h⁻¹ before switching the car (and therefore the HVAC) off for 120 min. Trials 2, 3, and 5 were conducted with a car velocity of 100 km h⁻¹ with the recirculation distribution setting on for 30 min before switching to fresh distribution for another 30 min. The car is then switched off for 120 min in trials 2 and 5, and 30 min in trial 3 (note this is the only trial with moisture added into the cabin).

A total of 39 time series variables, measured at either 1 or 10 s intervals depending on the test facility, are divided into control (Table 2) and state (Table 3) vectors. Control variables are considered to be exogenous inputs to the simulation and are either controllable (blower amperage), uncontrolled but measurable (car velocity), or indirectly controlled (vent temperatures). State variables include personal temperatures to allow estimation of thermal comfort (e.g., via ISO 7730 or ISO 14505) and estimation of windshield fogging. Rolling means of window 50 and 5 were applied to the 1 and 10 s interval data respectively and then the 1 s data was subsampled to 10 s intervals.

Figures 1, 2 show the locations for the vents and cabin sensors respectively.

The machine learning model aims to provide a simulator that is derived without knowledge of the physical system but rather is entirely obtained from measurement data. When formulated as differential equations, thermal systems are (mainly) linear with respect to their inputs. Given a simple thermal system that involves a body (such as a container full of water) of temperature y(t), an external environment that maintains a uniform temperature y ₀, and an insulating barrier (the outer wall of the container) of coefficient k, the rate of change of temperature of the body d y d t is proportional to the difference between the inside and outside temperature d y d t = − k y − y 0 . (1)

This is known as Newton’s model and forms the basis for ‘lumped’ thermal models (models where the components parts, or lumps, are considered to have a single uniform temperature). Note that the coefficient k might be expanded to consider the surface area and the unit thermal resistivity of the dividing layer. A transient simulation, given the current state y(t), must identify y(t + Δt) for some small increment in time Δt (say 1 s). An Euler simulation is a numerical approximation that assumes for a small Δt, Δ y Δ t ≈ d y d t = − k y − y 0 . (2)

Given this approximation, y t + Δ t ≈ y t + Δ y Δ t ⋅ Δ t (3) = y t − k Δ t ⋅ y t + k Δ t y 0 (4) = 1 − k Δ t ⋅ y t + k Δ t y 0 (5) which shows that y(t + Δt) is a linear function of y(t). Therefore, assuming Δt is small, a simple linear regression between y(t) and y(t + Δt) would yield the key coefficients in what otherwise appears to be a complex relationship between the internal temperature, the external temperature, and the thermal resistivity of the dividing wall. In summary, the simple thermal problem can be simulated using a linear correspondence between the current state y(t) and the next state y(t + Δt). Finding the coefficients for such a dynamical system is termed system or model identification.

Although a simple linear correspondence may be sufficient for a simple system as the complexity of the model increases, non-linearities will appear. Furthermore, some effects, such as radiative heat transfer, are proportional to the fourth power of the difference in temperatures and thus seem to demand a more flexible modeling method. As suggested by work on non-linear autoregressive network systems, some form of neural network (NN) or recurrent neural network (RNN) may be appropriate (Ng et al., 2014a; Engel et al., 2019).

A key insight in this work is the realisation that many physical systems can be predicted using only the current state and control inputs. In some cases, the prior state is also needed (e.g., as a proxy for the velocity of a moving object where the state contains just its position). Therefore, the structure of the simulator is a transfer function of the form x t + 1 = f x t , u t , x t − 1 . corresponding to a network structure as shown in Figure 3.

Having preprocessed the data, random hyperparameter search is used to identify the best neural network or multilayer perceptron (MLP) structure (Aureélien, 2019). The parameter space that we have chosen to search over are:

• Number of hidden layers (0–4).

The final activation layer is chosen to be linear as this is a regression problem with real-valued outputs.

The network is learnt using TensorFlow (Abadi et al., 2015) and Keras (Chollet, 2015) with the Adam optimizer (Kingma and Ba, 2014) aiming at minimising the mean square error. From the hyperparameter space given above, 200 variants are selected at random (with uniform distribution for number of nodes). Input and outputs are rescaled to a unit range using min–max rescaling.

Two considerations are made when performing the hyperparameter search. First, the data comes in the form of groups due to the presence of five different CWT trials with varying settings. Second, shuffling in cross validation is not appropriate for time series data. In order to address these, we make use of a Time Series Group Split cross-validator. This requires that each successive training set is a superset of those before it for each of the five groups.

Model training is done in open loop, meaning the true output rather than the predicted one is used for the next step prediction (Menezes and Barreto, 2008). In order to measure the performance of our model, we utilise the mean squared error (MSE) loss function.

Alongside the more complex MLPs, linear (least-squares) regression (LR) is also be considered. LR is a closed form solution to the least-squares problem whereas stochastic gradient descent (SGD) used for the MLP is iterative and approximate and thus can have a performance advantage. As with the perceptron, LR uses a minimal set of coefficients to model the system. This means that overfitting is unlikely although there may be some possibility of underfitting (not providing sufficient flexibility in the model). The simplicity of LR tends to make it robust to measurement noise although noise in sensor readings used as the independent variables causes LR to underestimate the gradient. Time series noise reduction is used to reduce this effect.

Alongside the ML model, we use AMESim version 17 software (Siemens, 2018) to model the North America Fiat 500 BEV climate system. It is a programming environment developed for the object-oriented modeling of complex physical systems. The main libraries used are: thermal, thermo-hydraulic, two-phase flow, heat, and air-conditioning.

The modeling has been carried out using the bottom-up approach: starting from the basic components, the different subsystems have been assembled and then connected to each other until reaching the final system configuration. The climate system includes the following sections:

1. HVAC system, which includes the evaporator, air ducts, and Positive Temperature Coefficient electric heater, and is connected to the cabin.

Each component model (i.e., compressor, evaporator, condenser, etc.) has been made filling the AMESim template with the geometrical data available from the datasheet. The template has some parameters that are calibrated using the data test recovered from the datasheet itself (i.e., the heat exchange coefficient derives from the Nusselt number calibration among the power exchanged by the heat exchanger). The developed cabin model consists of several thermal masses corresponding to the main inertial masses that interact between them by conduction and radiation and with the air by convection. The heat exchange by convection with the air depends on the air speed on the single surface. The heating and cooling modes have different airflow distributions, tri-level and only vent respectively, therefore, separate models are realized for these two functional modes. The heat exchange coefficients on surfaces are calibrated using measured data collected during the standard procedure of cabin warm-up and cooldown. Along the cabin air path (from the HVAC to the cabin), no pressure drop is considered than the airflow is imposed at the inlet of HVAC. The airflow rate is variable according to the thermal strategy adopted. The model is suitable to evaluate the electrical power consumption of the different actuators (blowerend fan on the low voltage net; the compressor and the Positive Temperature Coefficient on the HV net).

The top performer is LR with MSE 0.000 047 ± 0.000 005 7 for next step (10 s) predictions of min-max rescaled data. Due to the rescaling required for the neural network, the MSE is already normalised. Thus, it translates to a NRMSE of 0.000 24 100 1 % = 0.68% over 10 s. Note that the prediction error for several hours of simulation is likely to be larger.

Figure 4 provides an overview of the results from the hyperparameter search, detailed in Section 2.3. The main results shown here are that for those network structures that perform poorly, dropout helps considerably. Dropout is a regularisation technique that disables connections in the network with a fixed probability. This approach is often helpful in dealing with large, highly correlated inputs by making the network less reliant on individual inputs and thus more robust.

The top 10 performers of the hyperparameter search are shown in Table 4. The MSE is given as mean ± standard deviation over the Time Series Group Split cross-validation. One of the top performers is a simple perceptron system (single layer of linear activation). This is equivalent to a linear function between inputs and outputs, which thus suggests that linear regression (LR) may also be effective.

Linear regression does not require rescaled data but, for the purposes of comparison, cross validation LR on rescaled data gave an MSE of 0.000 047 ± 0.000 005 7 which outperforms the top performing MLP. Due to LR being a more robust model and the fact that thermal systems are mainly linear, LR is chosen as the model for this data.