When designing electric drive systems for commercial vehicles, particular challenges lie in the versatility of the operating conditions. Frequently, statements about the achievable driving performance and the consumption of electrical energy are to be made on the basis of a designed but not yet mass-produced electric machine and the associated converter. With this aim in mind, a simulation environment has been developed which contains a distance travel calculation in the time domain and calculates the occurring motor currents integrated into it in discrete time steps and, with the efficiencies of motor and converter, calculated The focus of this paper is the analytical calculation of the critical winding temperature of the electric machine. The simulation tool only needs a few main data and is therefore also suitable for continuously calculating the temperatures during operation. In addition to the data for power losses, it also provides a statement as to whether the electric machine and inverter are sufficiently dimensioned for the required task.

## Table of contents

- 1 Challenges in drive project planning1
- 2 The simulation environment
- 2.1 Goals and areas of application
- 2.2 Input data for vehicle, route, ambient conditions
- 2.3 Route simulation in the time domain
- 2.4 Efficiencies and temperature calculations2
- 2.4.1 Steady-state temperature behavior
- 2.4.2 Transient temperature behavior
- 2.4.2.1 Analogies to electrical components
- 2.4.2.2 Creation of a thermal model with three masses
- 2.4.2.3 Differential equations for three thermal masses
- 2.5 Calculation results
- 3 Summary and outlook

## 1. challenges in drive project planning

When commercial vehicles are equipped with electric drives, special requirements often have to be met, which can differ greatly from passenger cars and are more varied overall: In terms of vehicle mass, there is a very wide range for commercial vehicles. In addition, there can be considerable differences in mass between empty and loaded condition, the carrying of trailers and the operation of attachments. The routes to be traveled can either be little known, so that assumptions have to be made, or they can be relatively precisely defined, as is the case with scheduled buses or vehicles in works transport.

A particular challenge in the project planning of commercial vehicle drives is the correct dimensioning of inverter and motor. Cost, weight and installation space advantages speak for the choice of smaller units. The short-term overload capacity should be used in a targeted manner compared to the permanently possible power. However, the requirements for operational safety, performance and service life of the drive must be met in any case.

## 2. the simulation environment

### 2.1 Goals and areas of application

At ARADEX AG, a simulation environment has been created that makes the project planning of electric drives in commercial vehicles faster and safer.

Taking into account the data of the vehicle, the route profile and other operating conditions, the electrical and thermal behavior of the powertrain (Fig. 1) can be simulated. The simulation serves several purposes:

- Calculation of the efficiency of the motor and inverter in driving mode
- Determination of the energy consumption and thus the range
- Determination of the temperatures occurring in the drive (motor, inverter)
- Determination of the driving performance for a given route profile

### 2.2 Input data for vehicle, route, ambient conditions

The driving cycles known from the passenger car sector are essentially defined by value pairs of speed and time and are thus usually insufficient for commercial vehicles with their diverse areas of application. Here, it is necessary to simulate driving on real routes, which are defined, for example, by their geodata, i.e. a sufficient number of points with their geographic longitude, latitude and altitude above normal zero, and the respective assigned target or maximum speed values. The actual driving speed can only be determined in the simulation calculation, where various data can be taken into account: Slopes of the track, wind with speed and sign, characteristics of the road surface, additional forces (e.g. snow blade) or power of auxiliary units.

### 2.3 Route simulation in the time domain

A driver model with look-ahead function is used to have braking processes start in time so that the speed specifications from the route file are adhered to. In this driver model, the agility of the driving mode, i.e. the acceleration and deceleration value, can also be limited.

From the target acceleration, the powertrain model, which includes the transmission, electric machine and inverter, calculates the required current to be supplied from the inverter to the motor. This calculation follows the procedure of the ARADEX inverter, which calculates and regulates the optimum current in terms of strength and angle at each operating point of the motor, so that the total current strengths and thus the losses are minimal. Possible limitations of the current intensity due to high temperature in the motor or inverter as well as definable current limits of the battery are dynamically considered by the simulation tool.

The motor torque is determined from the controlled current in conjunction with the applied speed. The actual acceleration is calculated from this in the vehicle model. This is integrated over time so that the new velocity is calculated as the result.

The repetition frequency of the calculation can be adapted to the simulation task.

The motor torque is determined from the controlled current in conjunction with the applied speed. The actual acceleration is calculated from this in the vehicle model. This is integrated over time so that the new velocity is calculated as the result.

### 2.4 Efficiencies and temperature calculations

Losses in the inverter occur at diodes and IGBTs. Simulation tools from the manufacturers are used here to calculate them. The heat sink temperature is calculated from the coolant flow and the coolant input temperature according to the design, and the temperature of the chips results from this and the power loss. This is the limiting factor for the current that the inverter can supply.

For the electric machine, too, the temperatures resulting from the losses are limiting for torque and power. In principle, electric machines are very well suited to the operating requirements in vehicles, as they may be heavily overloaded for short periods beyond their continuous power limit. Critical with respect to overheating are the enamel insulations of the copper conductors in the coils, especially in the so-called winding heads (Fig. 2), i.e. the loops of the winding wire extending beyond the laminated core, which are the worst cooled in typical PM machines. Another limitation is due to the properties of the permanent magnets in the rotor.

The aim is therefore to ensure that the temperature limit is not exceeded at any point on the machine by limiting the level and duration of the current load. However, this should not result in a hard cut-off of the current, but rather the power should be regulated back in a way that is recognizable to the driver, so that he can adjust to this in his driving style.

Constant measurement of the temperatures leads to high construction costs and is not yet possible at all in the project planning phase. Therefore, simulation is of great importance here. In vehicle drives, the converted power is highly variable in time. The simulation task thus requires the calculation of the transient, i.e. time-dependent, temperature behavior.

Because it is to be used in project planning, the method described here must do without CAD data of the electric machine. In the field of thermodynamics, it therefore works with analytical equations derived from basic equations of heat conduction. Thus, suitable simplifications are necessary to reduce the complexity and still consider the essential effects.

- Rotor and rotor shaft are considered as one body with homogeneous temperature distribution.
- At a diameter of the stator close to the air gap, the heat input to be assigned to the stator is assumed at an imaginary cylinder shell surface. Temperature differences between the copper wires of the coils and the teeth of electrical sheet metal in between, the coil cores, are neglected. Averaged values for temperature, heat conduction and heat capacity are assumed for both together. These assumptions are supported by an exemplary FEM temperature model of a single tooth of the coil.
- Heat radiation as well as convection on the outer surface of the housing are negligible compared to heat transport via the cooling medium.

#### 2.4.1 Steady-state temperature behavior

As a basis for validating the transient calculation, the equations for calculating the steady-state temperature behavior, i.e. the steady-state temperatures, are established, because these can be verified in a relatively simple manner with test results. The basis of the investigations is a PM machine with 300 / 600 kW continuous / peak power at speeds up to 3500 min-1. Rotor losses are low for the type of electric machine considered and are not taken into account here.

The stator losses are diverted to the cooling jacket via the electromagnetically active part of the motor, the laminated core. In order to be able to calculate with a planar model, an axially homogeneous temperature distribution is initially assumed. The uneven distribution due to the end-side heat input from the winding heads is superimposed in a later calculation step. In the area of its active length, the motor resembles a multilayer tube with a rotationally symmetrical cross section (see Fig. 3).

Fig. 3

**Layer model: simplified cross-section through a cylindrically structured PM machine.**

*Shaft, material steel**Sheet package rotor, material electrical sheet (magnetic material neglected).**Air gap**Coil area with teeth, material properties averaged from copper, investment material and electrical sheet according to their mass fractions**Winding-free area of the stator, electrical sheet material**Solid area of the cooling jacket, material aluminum**Area with cooling channels, material properties averaged from aluminum and cooling medium**Bereich mit Kühlkanälen, Materialeigenschaften gemittelt aus Aluminium und Kühlmedium*

The losses occurring in the copper wires of the winding are distributed to the winding heads and the winding portion in the slots of the stack of laminations according to the proportional wire lengths or copper masses. For each winding head, there is a certain power loss which is evenly distributed over the unwound wire length of the winding head (here 160 mm, see Fig. 4) and – this is the assumption – is dissipated exclusively by heat conduction in the copper wire symmetrically on both sides into the slots.

For the derivation of the equation, it is therefore sufficient to consider half the wire loop with the positive length coordinate from l = 0 to lanb = 80 mm and the proportional power loss PV. The heat flow Q ̇ to be conducted through thus increases linearly from the center of the wire loop to both sides with the magnitude of the length coordinate l:

(Gl. 1)

Transforming and integrating the equation yields the temperature curve in the winding head. Starting from temperature Tanb at the connection points at the transition to the groove, this corresponds to a square parabola whose vertex at l = 0 represents the temperature maximum in the center of a wire in the winding head (Fig. 4).

Within the groove area, the end-side introduction of heat from the winding heads leads to a temperature increase compared to the temperature already calculated in the layer model from the total power dissipation. For the course of this temperature increase over the longitudinal coordinate x of the machine, an analytical approach similar to that of a cooling fin can be selected for heat introduction from a winding head: Heat is transported longitudinally over a specific cross-section with a defined thermal conductivity and simultaneously dissipated laterally over the surface with a specific heat transfer coefficient. As a result of the lateral heat dissipation, the heat flow remaining in the longitudinal direction and thus also the temperature gradient between the copper wire and the cooling medium driving the lateral heat dissipation decrease. By transforming and integrating the approach, a differential equation is obtained which has the following solution for the temperature profile along the wire in the Nur:

with: TU: Ambient temperature [°C]

h: Heat transfer coefficient wire to cooling medium [W/(m2-K)].

L: half groove length [m]

U: circumference of the wire [m]

λ: thermal conductivity of the wire material (copper) [W/(K·m)]

A: Cross-sectional area of the wire [m²].

ΔT_{1} is the temperature difference between wire and cooling medium at the beginning of the groove where the winding head is tied (x = 0). The heat flow at this point is:

(Eq. 3)

Since this heat flow from the winding head has already been calculated, the temperature difference between the wire and the cooling medium at the entry to the stack of sheets can be calculated by rearranging the equation:

(Eq. 4)

The end of a cooling fin is characterized by the fact that heat conduction in the longitudinal direction is no longer possible there. In our case of the copper conductor in the groove, this must be the case in the middle of the stack of sheets, i.e. at half the active length, if the machine is symmetrically constructed, because an equally large heat flow would have to occur there from the opposite side, so that the sum is zero. We denote this location by x = L and obtain:

Fig.5 shows for the length coordinates x = 0 to x = L = 180 mm (center of the active length) the graph for the heat introduced from one half of the winding head (blue line). For reasons of symmetry, a mirror-inverted course is to be expected for the other half of the active length.

In order to obtain correct temperature values in the layer model, the heat flow from the winding heads already had to be taken into account there. The task now is to determine the temperature increase that results from the lateral introduction of this heat from the winding heads at their connection point. This is read off as the difference between the temperature at the beginning of the slot (blue line, length x = 0) and the arithmetic mean value (red line): 4.3 K. Due to the linearity of the heat flow to the temperature difference, which is valid for heat conduction, the derived heat flow is proportional to the area under the lines and thus the same for both lines. The arithmetic mean value thus corresponds to the temperature from the layer model for the warmest copper conductor in the slot, 42 °C in the example. A temperature Tanb = 46.3 °C is to be expected at the end of the slot.

In order to obtain the maximum temperature in the winding head, the previously calculated temperature range of 3.3 K within the winding head must now be added and one obtains T_{max} = 49.6 °C (Fig. 6: blue dot at the actual start of winding at radius r = 180 mm). Towards the outer radius r = 211 mm of the coil, a decrease of the winding head temperature corresponding to the temperature in the groove is expected.

#### 2.4.2 Transient temperature behavior

The calculation of the steady-state temperature behavior is only to be seen as a preliminary step to the solution of the actual task. This is: Calculation of the time-dependent, i.e. transient temperature behavior as a function of the power losses occurring in the electric machine. The transient calculation requires not only the quantities for heat transfer by conduction, convection and radiation (as far as this is not negligible) but also the consideration of heat capacities.

##### 2.4.2.1 Analogies to electrical components

Under certain conditions, a system of bodies that have a heat capacity and touch each other so that heat conduction occurs behaves similarly to an electrical system of capacitors and resistors. One of the conditions is that the thermal conductivity within each of the heat-storing bodies is very good compared to the contact resistances to the adjacent bodies.

When calculating the thermal behavior of the electric machine, heat flow sources result from the electrical power losses. Since heat also flows through the material during a heating process, the heat flow that flows into a thermal mass is the sum of the heat that flows out and the heat required for a temperature change for a given heat capacity of the mass.

For each thermal mass, an electrical model that replicates this process consists of a current source, a capacitor as storage, and an ohmic resistor connected in parallel through which current flows to the next component or directly to “ground.”

##### 2.4.2.2 Creation of a thermal model with three masses

Transferred to the type of PM machine to be calculated, a model is created that has three thermal masses (Fig. 7): 1. rotor, 2. winding heads, 3. remainder of the stator without the winding heads: slot areas, laminated core and cooling jacket.

The summary of the copper located in the slots, the insulating and impregnating varnish in between, the sheet metal package as well as the aluminum housing with its cooling channels to a thermal mass, in short “stator mass”, is a simplification which still seems permissible because the temperature gradient within these components is relatively flat compared to the temperature difference to the winding heads (see Fig. 6). The heat capacity of the stator CSt is given by the sum of the heat capacities of the contained materials, each multiplied by the proportional masses. Its temperature TSt refers to the arithmetic mean over the active length for the warmest copper conductor in the slot, as calculated in the layer model. The thermal resistance RSt is determined from the temperature difference TSt-TK with respect to the cooling water and the heat flow Q ̇ from the total power dissipation of the winding heads and stator:

(Eq. 5)

The power loss in the stator PV_St contains portions from iron and copper losses. The winding heads with the power loss PV_Wi (only proportional copper losses) and the heat capacity CWi are connected via the thermal resistance RWi-St, for the determination of which both the temperature increase at the beginning or end of the slot, where the winding heads are connected, and the temperature difference from there to the warmest point in the middle of the individual wire loops of the winding heads are taken into account.

In order to also capture the losses in the rotor and to be able to calculate its temperature TRo, the model contains the associated power loss source PV_Ro and a mass with the heat capacity CRo. The air gap is modeled as the heat conduction resistance RL.

##### 2.4.2.3 Differential equations for three thermal masses

The thermal power P absorbed by a mass (with negative sign: dissipated) is equal to the product of heat capacity C and temporal temperature change dT/dt:

(Eq. 6)

The product of the thermal resistance R and the thermal power P flowing through is the temperature difference ΔT_R:

(Eq. 7)

The temperature vector y ⃗contains the temperatures of the winding head, stator and rotor. The input variables of the calculation in the vector u ⃗ are the power losses of the winding heads P_{V_Wi}, in the stator P_{V_St}, in the rotor P_{V_Ro} and the cooling water temperature T_{K}.

The model requires the calculation of the following differential equation system:

(Eq. 8)

The influence numbers aij and bik contain terms from the heat capacities and thermal resistances specific to the machine, which are derived from the static calculation model; some are zero. The procedure can also be extended to a larger number of thermal masses. A model with four masses is currently being developed.

### 2.5 Calculation results

Fig. 8 shows a simulation of the heating process of the example machine with the following initial temperatures: rotor: 27.0 °C, stator and winding head: 30.0 °C. The cooling water temperature is kept constant at 30.0 °C.

The power loss is zero for 100 seconds and then jumps to the following values: copper losses 2038 W, iron losses 182 W, in total 2220 W. This simple example shows how the temperature of the winding head rises faster and reaches higher values than that of the stator. Both temperatures approach the steady-state temperatures determined in the steady-state calculation (cf. Fig. 6).

The rotor temperature rises only very slowly, because a power loss of only 22.4 W was applied to the rotor (1% of the total losses) and because the air gap has a very large thermal resistance, so that hardly any heat is dissipated from the stator to the rotor.

## 3. summary and outlook

The simulation tool is developed to the point where temperature calculations can be performed in the time domain. The time-varying power loss, divided into iron and copper losses, is determined from the line drive simulation.

In the near future, several motors will be run on a load test bench with load spectra – derived from the line drive simulation – so that realistic operating conditions are available. The measurement of the relevant winding temperatures during these tests will provide data for the calibration of the simulation tool.

This will include investigating whether a model with four thermal masses can provide more accurate results than the existing model with three masses.