Design and optimization of two-stage controller for three-phase multi-converter/multi-machine electric vehicle

2.1 Induction motor modeling

A voltage inverter operating under the control of sinusoidal-pulse width modulation (SPWM) techniques powers the employed motor, a three-phase-IM. The Krause model, a famous IM model derived from its corresponding circuit, is based on transforming the currents and magnetic fluxes of the stator to a reference frame “d–q” that revolves with the rotor [26]. All of the variables in the stationary frame to be simulated are allowed to account for DC values [27]. The three-phase parameters, which are the voltage, the current, and the flux, can be shifted along an axis such that the linking variables between the stator and rotor phases constantly vary (d–q axis stationary frame). Thus, the stator and rotor parameters can rotate at synchronous speed. Figure 2 shows electrical schematics of an IM’s per-phase corresponding circuits in a reference framework that rotates synchronously along two axes.

Figure 2

IM-equivalent circuit showing the d–q axes elements [28].

Consider the elements of the d–q axes flux linkage that correspond to the stator and rotor, Ψ d s Ψ q s and Ψ d r Ψ q r , also V d s V q s and V d r V q r which are correspond to the stator and rotor voltages, respectively; furthermore, the currents of the rotor and stator, i d s i q s and i d r i q r , respectively, as shown in Figure 2, all of these parameters will be employed throughout the analysis. During the analysis, other parameters are required, such as L s l and L r l which are denoting the stator and rotor inductances, respectively, the stator-winding resistance, R s , the rotor-winding resistance, R r , moreover, the rotation speed of the rotor, which is denoted by w r , the inductance of the magnetization, L m , and the speed of the synchronization which is abbreviated as w e , which are all shown in Figure 2. Consequently, all of the abovementioned parameters that are related to voltages are connected to each other in one expression as [28],

while the parameters that are related to currents are connected with each other as,

To determine the torque due to development in the action, T e , the rotor current, I r , can be joined with the effects that are produced from the air-gap of the flux, solving in terms of stationary q–d axis variables, revealed,

thus, the rotor’s dynamic torque is expressed intelligibly. Suppose that the EV rotors load-torque is denoted as T L and its inertia as J; then, the last expression can be reformulated in terms of T _L and J as [29],

As will be seen in the simulation and results later in this work, the primary settings to the utilized above parameter values can be previewed in Table 1, for convenience which are all related to the IM. Thus, the resistances of the rotor and the stator will be set to 0.02092 and 0.03552 Ω, respectively; the inductances are 0.000335 H, 0.0151 H, and 0.000335 H, corresponding to stator leakage, magnetizing, and rotor leakage inductances, respectively, while the frequency was 50 Hz, and finally, a total number of poles that are utilized are only two poles.

2.2 Transform of axes

To understand the axes transformation visually, consider Figure 3. Hence, in Figure 3, it is indicated that there are three-axes, as‒bs‒cs, departed by 120°. That is, these three axes’ variables should be transformed to the corresponding two stationary axes d ^s ‒q ^s and transformed to another two axes, that are stationary rotating reference-frame, d ^e ‒q ^e , and vice versa [7], as shown in Figure 3.

Figure 3

Visual transformation of as‒bs‒cs axes to the corresponding two-axes d ^s ‒q ^s .

According to the visualization in Figure 3, and as showing the two axes of the θ-angle, d ^s ‒q ^s , then, the voltages of these tow-axes, v d s s and v q s s , can be formulated mathematically in a matrix format,

Nevertheless, to transform them back, one can make use of the following expression:

To understand the last two-expressions, it is wealth to use v o s s , which is the voltage of the zero-sequence constituent. This zero-sequence voltage may be found in the real system or may be not present. Similarly, linkages that correspond to fluxes and currents can be transformed using the same procedure described above. To simplify, if the zero-sequence eliminated and by aligning the q ^s - with the as-axes, by equating the angle between them to zero, θ = 0, Equation (5) can be reduced to,

meanwhile, Equation (6) can be reduced to,

However, consider Figure 4, the d ^e –q ^e -axes are rotating at ω e radians/second with respect to the axes d ^s –q ^s , where the angle between as- and q ^s -axes is equal to ω t , as shown in Figure 4. Hence, the windings of the two-phase d ^s –q ^s -axes can be hypothetically transformed to a mounted winding over the d ^e –q ^e -axes; therefore, the d ^s –q ^s -axes’s voltages may be represented in terms of the d ^e –q ^e -axes voltages as,

Figure 4

Transformation visualization of d ^s –q ^s frame to d ^e –q ^e synchronous rotating frame.

The superscript e has been removed from the synchronously rotating frame variables for reasons of simplicity in the rest of the analysis in this article. Moreover, the stationary framework can be resolved from the rotating frame as,

where the two axes can be derived from each other using the last two expressions, Equations (14) and (15).

2.3 Current model of flux estimation

By exploiting the signals of the current and speed, it is possible to synthesize the flux components of the rotor. That is, the d ^s –q ^s corresponding circuit diagram, as indicated in the study by Bose [7], can be expressed as,

Reforming the last expression, after the addition of both sides with the ( L m R r / L r ) i d s s and ( L m R r / L r ) i q s s , respectively, the following two expressions are concluded, Equations (18) and (19), respectively,

in the last two expressions, ψ d r s can be calculated by,

and ψ q r s ψ q r s can be determined as,

Consequently, to simplify Equations (18) and (19), the last two Equations (20) and (21), will be substituted in Equations (18) and (19), reformulating them to get,

The rotor circuit time constant is represented by T r in the last two expressions, which is determined by dividing the leakage inductance of the rotor, L _r , by its resistance, R _r . Thus, the expressions in Equations (22) and (23) stand for the rotor flux in terms of speed and current of the stator. Being aware of these signals allows one to figure out the fluxes and the accompanying unit vector indicators. Nonetheless, these formulas have been described as the present flux assessment hypothesis, which was initially proposed by Blaschke and is commonly referred to as the Blaschke model. As a block diagram, Figure 5 draws the estimating approach of the cosine/sine indicators/signals alongside the other parameters, such as T _r, i _d, and i _s, and the air-gap stator’s fluxes [7].

Figure 5

Structure of current model (aka Blaschke equation) for flux estimation.

It is feasible to construct a combination model where the voltage structure is more operational at higher speed spans and effortlessly switches to the current structure at slower velocity levels. This is because the current model can be used for flux prediction at any speed, while the voltage model performs better at greater speed levels [7].

2.4 Voltage source space vector inverter modeling

To transform DC from any DC source into AC, power electronics designers employ inverters for this task. The inverted voltage may be a single, two, three phase, etc., according to the type of inverter. The primary applications of an inverter that supplies synchronous motors are those involving high-performance variable speed, variable voltage, and variable frequency. The greater DC bus voltage and fewer harmonic distortion make the SVPWM technology the most popular PWM approach. The six switches and power source that make up a three-phase voltage‒source conversion circuit are illustrated in Figure 6. Space vectors can only be used in this method inside the area defined by the output voltage vector. Figure 7 shows the six space vector sectors together with the phase voltages that correspond to them.

Figure 6

Three-phase VSI circuit connected to the IM [30].

Figure 7

Six sectors of SVPWM [31].

Hence, mathematically, the two schematics in Figures 6 and 7 can be expressed as

In the last expressions, T _s stands for the period of the sampling operation in the PWM schematic, which can be determined by doubling the switching frequency, T _z . The switching periods that correspond to the voltages V ₁, V ₂, and V _0,7 is denoted by T _a , T _b , and T ₀, respectively. In order to achieve a high switching frequency, it has been presumed that the reference space vector remains unchanged throughout a single switching session. Remember that the start-switching of the switching time T _z has been represented by the zero vectors, which is also can be given in terms of the sampling period T _s for the full-null per vector, where each duration period has width equals one-halve T ₀; therefore, the mathematical representation of the space vector can be expressed as,

Here, the phase-angle of the voltage is denoted by α, the index of the modulation denoted by m _a due to the SVPWM operation, while the DC, which is measured in Volts, is represented by V _dc as indicated in the studies by Aganah and Ojo and Kumar et al. [30,31].

2.5 Modeling the dynamic vehicle load

Figure 8 shows the EV that is in equilibrium with respect to all of the forces acting on it while it is moving. However, when moving, the EV is subject to a number of forces, including these forces [32]: the aerodynamic force, F _aero, the rolling force, F _tire, and the climbing force, F _slope. The following subsections will be dedicated to discuss these forces.

Figure 8

The action of forces on a moving vehicle on an inclined road [32].

3.1 Design of hybrid RISE‒PI controller

Figure 10 shows the IM’s output equation for the hybrid RISE‒PI (first stage) controller which can be expressed as

where sat(·) is the saturation function, k 1 is suitable constant gain, u 1 ( t ) is PI controller, which is given by

where e 1 ( t ) is the error signal equal which is expressed as

where r(t) is the desired input signal, while P s 1 − r ( t ) is the flux linkage current, and u 2 ( t ) is the control signal of the RISE controller.

In contrast to other controllers, the RISE controller may be built from the error ‒ that is, the discrepancy amongst what the system produces and its original reference point tracking ‒ alone. In spite of disturbances from outside sources and mistakes in the model, the control approach aims to ensure that the real output, Y _a (t), follows the target signal, Y _d (t). Here is how the differentiation between the two signals:

Applying a robust continuous control law ensures that the error, which is defined as the difference between the predicted input and the actual output, asymptotically settles to zero, or |e1| → 0 as t → 0. This is the main objective of the controller. In order to accomplish the control goal, auxiliary error signals e _i are employed, where e _i is an element of R (real number) and i ranges from 1 to n, with n representing the system’s order. The auxiliary error is defined as follows for the three-phase EVs with n = 2 [34]:

and,

where the auxiliary error signal is taken as indicated in reference [35]. Now, we can get the RISE controller equation from the following control signal [35]:

where the controller parameters are k r 1 , k r 2 , k ar , σ , and β ∈ R . These parameters are determined by MVO, which will be explained in the next section. The e r 2 ( t o ) term in Equation (51) was used to ensure that u ( t o ) = 0 ; also, the integration in Equation (51) is equal to the summation (⅀).

3.2 PI controller design and adeline NN

The output equation of the PI controller, second stage, which is configured in Figure 10 can be expressed as,

where

and, where I qs − e is the stator current for the rotating frame. The final controller with the Adeline NN is given as

where

in which c ₁ and c ₂, are the positive selected constants that are obtained by the optimization algorithm.

6.1 Simulation results of FOC system

In order to model the control arrangement, we use the MATLAB/Simulink environment. Tables 3 and 4, respectively, list the MVO that performs better with the proposed and four PI controllers tuned parameters for a hybrid RISE‒PI and PI‒Adeline NN, using two PI controllers via the torque command and flux command, respectively.

In order to test the whole system’s step responsiveness, a fixed input signal is utilized. The proposed controller controls the IM in EV compared to the PI based on MVO. Figure 13 shows the results of the simulation, and Table 5 summarizes the time response parameters. The results show that the suggested controller based on MVO had better speed response and less oscillation, at which the maximum peak M_P is less than 0.06, than the PI controller, which shows chattering and more oscillation, at which the maximum overshot M_P is greater than 0.07 in PI.

Figure 13

Speed response with the proposed and PI based on MVO.

The proposed controller had a smoother response with less chattering compared to the PI controller, as shown in Figure 13. This indicates that the proposed controller is more effective in controlling the system and can provide better performance in terms of stability and smoothness.

But all the major components of the EV, as shown in Figure 14, are here in this study. In addition, it detailed the interconnections between them so that the complete system simulation could be obtained. The simulation provides a thorough evaluation of the suggested controller and its benefits in comparison to conventional control techniques.

Figure 14

Simulink simulation of the structured model of the EV system.

6.2 Simulation results of the EDC subsystem

This section describes and examines the EDC platform’s validity utilizing simulation and findings applied using Matlab/Simulink software. Figure 15 also shows the EDC program version’s simulation diagram.

Figure 15

The employed diagram of the EDC system model during simulation.

The EDC equations (Equations (60)–(67) in Section 5) are the foundation upon which the simulation model is built. To mimic the EV’s actual driving conditions, the simulation model takes two parameters into account: In order to test the car on inclined roads, it is necessary to first simulate an inclined road with a varied angle of inclination. The other is the level road, which is straight to the ground. The results of the simulated model show how various road conditions affect the EV’s rear wheels – the parts of the car linked to the EDC – speed and stability. Also, the EV’s propulsion drive system design incorporates two IM mounted directly to the vehicle’s wheels, referred to as the “right and left wheels,” in order to minimize hardware reduction. A power supply, an energy distribution converter, and a VS-SVPWM inverter are represented by lithium–ion rechargeable batteries. Figure 16 also shows the correct road scenario, where an EV’s two wheels should be switched at the same speed. As depicted in Figure 17, the wheels of the EVs move at a slower pace relative to their ideal wheels after it turns directly from the curve. In contrast, as seen in Figure 18, the ideal wheels move faster than the EV’s wheels once the EV turns left from the curving street.

Figure 16

Movement results of the EV at 100 km/h velocity rate on a straight line‒road.

Figure 17

Positive-eleven degrees angle of right turning angle of the EV at 100 km/h velocity rate over the curve road.

Figure 18

Positive-eleven degrees angle of turning right of the EV at 100 km/h velocity rate.

6.3 Simulation result of driving cycles system

As depicted in Figure 19, there are three phases to the street-moving EV. In addition, at different points in the simulation, the supply of the resistive force that affects the EV varies. At each of these points in the simulation, the source of the resistive force that influences the EV changes somewhat. The following lines detail these steps.

Figure 19

EV-three stages driving cycle including straight, curved, and slope roads.

In the first phase, as illustrated in Figure 20, the EV travels down a straight route from the starting location all the way to the bridge. Stage 2: Assuming it reaches the third second, the vehicle climbs the bridge in this stage. By the third second, as seen in Figure 21, the engine load has increased since the ascending angle, β has been added. In the third phase, the EV makes a sharp right turn off the curved street and descends 10 degrees into the off-road terrain. The transition starts at the third and continues until the sixth seconds. Figure 22 shows that the vehicle swung to the right as it descended, causing the left motor to rotate at a higher angle than the right motor.

Figure 20

Two EV motors’ linear speeds resultant performance.

Figure 21

Left/right EV’s linear speed wheels.

Figure 22

EV’s right/left linear speed wheels.

Figure 20 represents the linear speed of the EV’s two motors, which started from zero and reached 2,864 rpm (rotation per minute) in the first half second to stabilize in this way. In the following step, as the vehicle spins, the linear speed of the wheel of the EV increased from zero to 2,864 rpm, as shown in Figure 22. The left wheel motor’s speed increases to around 3,000 rpm at this point due to the impact, while the right motor’s speed lowers to about 2,600 rpm.

To illustrate the EV’s motor torque, Figure 23 shows it applied to the left wheel, while Figure 24 shows it applied to the right wheel. As soon as the EV starts moving, the torque on both wheels reaches 2,500 Nm. In the first stage, the torque stabilizes at 100 Nm, but it ceases stabilizing at that point. At the third second of the third stage, the torque on the left motor reaches 1,100 Nm, and on the right motor, it reverses to −1,100 Nm. From that point on, the torque remains relatively constant at 100 Nm until the sixth second, when the direction of motion is switched, resulting in a torque of −1,100 Nm for the left motor and 1,100 Nm for the right. Figure 22 shows that after that point, the torque settles at 100 Nm, which is entirely caused by the vehicle’s motion along the track.

Figure 23

The torque at the EV’s left motor performance.

Figure 24

The torque at the EV’s right motor performance.