**2. The mathematical modeling of the system**

#### **2.1 SynRM mathematical model**

SynRM is one of several synchronous machines and it is one in every of numerous synchronous machines, the SynRM rotor structure has synthetic without winding or magnet fabric. As an evaluation among SynRM with other sorts of reluctance Electric Vehicles (EVs) like: IM, BLDC motor and switch reluctance motor SRM. The quit result suggests that SynRM is precise in production, easy form. In addition, it has an extraordinary residence like low torque average, larger torque pulsation and occasional energy factor [6, 7].

Undoubtedly, the SynRM may additionally moreover deliver an excessive solid performance in assessment with other AC drives compared with IM. As noted earlier, the SynRM is a type of synchronous machine that does not have any winding or permanent magnet at the rotor and salient poles, it has a fragmented rotor of several barriers. The cause for made the reluctance motor rotor form laminated axially metallic it has to dominate low torque response and incorrect electricity thing, notwithstanding the truth that older variations of reluctance motor have lacked this period of producing [5–7]. The stator-winding format of SynRM is quite much like the IM. Whereas the rotor shape of SynRM is quite precise from IM, it is not caged rotor or twisting and does not have any magnetic fabric, it has only laminated obstacles which designed in a complex manner, and an optimized to have a pinnacle quadrature axis compliments and non-direct axis jealousy, once the magnetic concern goes with the flow in the stator winding and in step with the rotor structure it's far a low and high hesitation vicinity and they represent almost the magnetic poles [5–16].

The rotor layout in SynRM is rotated to reach the low reluctance regions and drifting away of the immoderate reluctance areas inside the equational time of rotation, the purpose of this work method to acquire the magnetic location synchronous speed. The stator machine to each of SynRM and IM is the same signal in the rotating frame. The SynRM does no longer want any magnetic or winding substance at the rotor form which makes the motor rugged, introduction simplicity, the most inexpensive rate of manufacturing, better torque according to unit quantity possibility, working at most high speeds functionality which makes the SynRM, and the rotor windings Failing to end result from easy control strategies, and the decline's minimization create SynRM an appealing and famous desire for several business and care programs due to all of these awesome and splendid traits [7, 12].

The earliest variations of SynRMs are used immediately a caged rotor, the most crucial cause that pristine SynRMs do no longer have starting torque attributes, however now the modern-day SynRM and using the most updated styles of inverters, area orientation manage FOC generation at the side of using Pulse Width Modulation (PWM) method deliver a convenient approach for manage, so without any rotor cage that the tool may also be initiated. The velocity variable parameters have utilized in SynRM motor system layout to correct the motor speed strength because of numerous factors like electricity conservation, manage the situation, velocity, and enhancement of the brief response traits [6, 7–12, 28, 31].

The aim of a motor tempo controller is to take a sign representing the reference tempo and to strain the motor at that reference velocity. Although, the control

machine consists of velocity that been comments from the machine, a SynRM, a voltage supply location vector inverter, a controller, and a speed placing the device [6, 7, 22]. The fundamental reason for the use of comments in one's systems is with a purpose to gain a reference-thing irrespective of any variation or exclusive problem within the traits the tool decrease returned to the reference issue. **Figure 2** is displayed the SynRM rotor flux barrier and IM motor rotor cage. In addition, the SynRM motor can be located from its d-q stationary axis the same circuits as in **Figure 3**.

The SynRM version is pretty like the induction motor IM. The difference is by way of neglecting the rotor losses from the IM equations. The SynRM's version is described via (Eqs. (1)–(4)) [5–7, 16].

$$\mathbf{V\_d = R\_s I\_d + \frac{d\lambda\_d}{dt} - \alpha\_r \lambda\_d} \tag{1}$$

$$\mathbf{V\_{q}} = \mathbf{R\_{s}}\mathbf{I\_{q}} + \frac{\mathbf{d}\lambda^{q}}{\mathbf{dt}} + \alpha\_{r}\lambda\_{d} \tag{2}$$

$$
\lambda\_{\mathbf{d}} = \mathbf{L}\_{\mathbf{d}} \mathbf{I}\_{\mathbf{d}} \tag{3}
$$

**Figure 2.** *The machinal contents of SynRM and IM.*

**Figure 3.** *The equivalent circuit of d axis and q axis for SynRM.*

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$$
\lambda^{\mathsf{q}} = \mathsf{L}^{\mathsf{q}} \mathsf{I}^{\mathsf{q}} \tag{4}
$$

By both of each (Eqs. (3) and (4)) the over shift speed could be acquired as will detect in (Eqs. (5) and (6)) as following:

$$\frac{d\lambda\_{\rm d}}{dt} = \mathbf{V\_{d}} - \mathbf{R\_{s}}\mathbf{I\_{d}} + \alpha\_{\rm r}\lambda\_{\rm q} \tag{5}$$

$$\frac{d\lambda\_{\mathbf{q}}}{dt} = \mathbf{V\_{q}} - \mathbf{R\_{s}}\mathbf{I\_{q}} - \alpha\_{\mathbf{r}}\lambda\_{d} \tag{6}$$

From (Eq. (5)) the change-speed rate of the direct axis current could be gained in (Eq. (7)).

$$\frac{d\mathbf{I}\_d}{dt} = \frac{\mathbf{1}}{\mathbf{L}\_d} \left(\mathbf{V}\_d - \mathbf{R}\_s \mathbf{I}\_d + \alpha\_\mathbf{r} \mathbf{L}\_\mathbf{q} \mathbf{I}\_\mathbf{q}\right) \tag{7}$$

And with (Eq. (6)) the change-speed rate of the quadratic axis current could be gained in (Eq. (8)).

$$\frac{d\mathbf{I\_q}}{dt} = \frac{\mathbf{1}}{\mathbf{L\_q}} \left( \mathbf{V\_q} - \mathbf{R\_s} \mathbf{I\_q} + \alpha\_\mathbf{r} \mathbf{L\_d} \mathbf{I\_d} \right) \tag{8}$$

Besides, to obtain the torque of SynRM we have used in (Eq. (9)).

$$\mathbf{T\_e = \frac{3}{4} \ \frac{\mathbf{P}}{2} \left(\mathbf{L\_d - L\_q}\right)} \mathbf{I\_d I\_q} \tag{9}$$

The of speed rate can be acquired by (Eq. (10)).

$$\frac{\mathbf{da\_r}}{\mathbf{dt}} = \frac{\mathbf{P}}{\mathbf{J}} (\mathbf{T\_e} - \mathbf{T\_L}) \tag{10}$$

The Laplace transformation of the torque is given as follows by (Eq. (11)).

$$\mathbf{T} = \frac{\mathbf{3}}{2} \left( \mathbf{L\_d} - \mathbf{L\_q} \right) \mathbf{i\_d} \mathbf{i\_q} - \left( \mathbf{B} \mathbf{o\_r} + \mathbf{J} \frac{\mathbf{d} \mathbf{or}}{\mathbf{d}t} \right) \tag{11}$$

**Table 1** show the SynRM parameters that used in the design. These parameters are the same parameters for the IEv4 motor for the ABB company.


**Table 1.** *The main parameters of the SynRM.*

#### **2.2 Voltage source inverter**

Inverters are power electronics systems that turn the DC voltage from a battery or some other DC source into alternating current voltage, which may be singlephase, two-phase, or three-phase, depending on the inverter configure ratio. Inverters that feed synchronous motors are mainly used in variable voltage and variable frequency applications for high-performance variable speed [4, 6, 7]. SVPWM is the most widely used PWM technique due to its high output voltage, low harmonic distortion, and superior efficiency as compared to other types of inverters. The SVPWM inverter is a complex scheme that produces a high voltage fed to the generator, resulting in a low overall harmonic distortion [4, 6, 15, 16]. The aim of the various modulation schemes is to provide a variable output with a maximum fundamental factor that can generate as few harmonics as possible. The switching instants are calculated using the SVM scheme, which is based on the representation of the switching vectors in the rotating or stationary frame plane, based on the position of the output voltage vector in each sector of the space vector sectors. The mathematical model of SVPWM inverter is given as seen on (Eqs. (12) and (13)), where n represents the number of the space vector sectors.

$$\mathbf{T\_a} = \frac{\sqrt{3} \mathbf{T\_z} \mathbf{V\_{ref}}}{\mathbf{V\_{dc}}} \ast \left( \sin \left( \frac{\mathbf{n} \pi}{3} - \alpha \right) \right) \tag{12}$$

$$\mathbf{T\_b = \frac{\sqrt{3}\mathbf{T\_z}\mathbf{V\_{ref}}}{\mathbf{V\_{dc}}} \left(\sin\left(\alpha - \frac{\mathbf{n} - \mathbf{1}}{3}\pi\right)\right) \tag{13}$$

#### **2.3 Mathematical model of stationary field transformation**

The transformation from the mechanical model has completed by the usage of the d-q transformation, which is used for a vector manipulate approach of the synchronous motor machine. The base at the concept which is the windings of the stator are disbursed a d-q version is a powerful tool for simulation of all AC machines such as the SynRM [6, 7, 16]. When Three-segment balanced and altered windings and symmetry to two-segment Equation Equilibrium windings deliver rotating magnetic discipline speed Φ and cost are equatorial, the three-section windings are Equational with the two-segment windings. The d-q transformation is well-balanced three-phase Vd,Vb andVc into balanced two-phase Vd and Vq as shown in **Figure 4**.

The transformation matrix "T" to transfer voltage or current vector from abc reference frame into dq reference frame as seen on (Eq. (14)):

**Figure 4.** *The direct and quadratic voltage transformation.*

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$$
\begin{bmatrix} V\_d \\ V\_q \\ V\_o \end{bmatrix} = \mathbf{T} \begin{bmatrix} V\_a \\ V\_b \\ V\_c \end{bmatrix} \tag{14}
$$

The transformation matrix can be used in synchronous frame as explained in (Eqs. (15) and (16)),

$$\mathbf{T} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & \frac{-1}{2} & \frac{-1}{2} \\ 0 & \frac{\sqrt{3}}{2} & \frac{-\sqrt{3}}{2} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \tag{15}$$

$$\mathbf{T}^{-1} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & 0 & \frac{1}{\sqrt{2}} \\ \frac{-1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{2}} \\ \frac{-1}{2} & \frac{-\sqrt{3}}{2} & \frac{1}{\sqrt{2}} \end{bmatrix} \tag{16}$$

Where T�<sup>1</sup> represents the inverse transformation matrix.

$$
\begin{bmatrix} \mathbf{V\_d} \\ \mathbf{V\_q} \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 \ \frac{-1}{2} \frac{-1}{2} \\\\ 0 \ \frac{\sqrt{3}}{2} \frac{-\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} \mathbf{V\_a} \\ \mathbf{V\_b} \\ \mathbf{V\_c} \end{bmatrix} \tag{17}
$$

The zero-axis voltage is neglected, and the power is the same in both the three phase and the two-phase transformation as seen on (Eq. (17)).

#### **3. Motor control**

#### **3.1 Traditional cascade PID controller**

Traditional PID controllers are considered the most largely used controllers in the control-process industry because of their simple structure and easy parameter settlement. The major cause of user feedback is a very substation in systems is to be able to attain a set-point regardless of disturbances, noise, or any variation in characteristics [5–16, 25]. A PID controller determines an "error" value as the difference between an obtained process variable and a reference setpoint. **Figure 5** shows the general cascade control system scheme. The controller tries to make the error signal as small as possible by manipulating the process control inputs. The calculation "algorithm" of the PID controller contains three separate constant parameters, and it is mainly called the proportional, integral, and derivative values [5–16, 29]. The differential equation of a PID controller is given by (Eq. (18)).

$$\mathbf{u}(\mathbf{t}) = \mathbf{k}\_{\mathbf{p}} \mathbf{e}(\mathbf{t}) + \mathbf{k}\_{\mathbf{i}} \int \mathbf{e}(\mathbf{t}) \mathbf{d}\mathbf{t} + \mathbf{k}\_{\mathbf{d}} \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{e}(\mathbf{t}) \tag{18}$$

**Figure 5.**

*The General Cascade control system scheme.*

#### **Figure 6.**

*Traditional PID controller.*


#### **Table 2.**

*The characteristics of the tradition controller.*

Moreover, the transfer function is given by (Eq. (19)).

$$\mathbf{G}\_{\rm PID}(\mathbf{s}) = \mathbf{k}\_{\rm p} + \frac{\mathbf{k}\_{\rm i}}{\mathbf{S}} + \mathbf{S}.\mathbf{k}\_{\rm d} \tag{19}$$

**Figure 6** shows the block diagram with the parameters of the PID controller. The response to the error is connected to the proportional value, the sum of mistakes that were recent are the integral's assignment to determined and the reaction is being represented by the derivative to the rate at which the error has been shifting. The controller parameters KP, KI and Kd on traditional closed-loop systems as shown in **Table 2** give comprehensive effects of each parameter, in some control processing it may require only one or two parameters to present a suitable control system. Setting other parameters to zero is the method for correcting the PID controller to two or one activity. A PID controller will be named a PI, PD, P or I controller when neglecting one of the respective control actions [5, 6, 12]. PI controllers are common, since the activity is sensitive to sound change, whereas if there is absolutely no term that is integral the system may be avoided by it from reaching its target value due to the control activity [11–16].
