*3.7. Proposed MPTFC*

MPTC's cost function is generally defined as a linear combination with torque-flux errors to determining the finest voltage vector, that is defined as:

$$\mathbf{G}\_1 = \left| \mathbf{T}\_{\mathbf{e}}^{\text{ref}} - T\_{\mathbf{e},k+1}^p \right| + \lambda \mathbf{v} \left| \mathbf{Y}\_{\mathbf{s}}^{\text{ref}} - \overline{\mathbf{Y}}\_{s,k+1}^p \right| \tag{26}$$

where λΨ is stator flux's weighting factor in MPTC. Stator flux weighting factor is calibrated in actual time to achieve minimum torque ripple, as suggested in [13]. The weighting factor λΨ value has a decisive role in containing the unavoidable ripples if it properly chosen. Therefore, the λΨ value must be correctly selected, which requires an online optimization process to choose the optimum value, which results in a further increase in the microcontroller's computational burden. To avoid the use of weighting factor in MPTC, a new stator flux reference based upon IM model is suggested in [28], which is equivalent to the original reference of torque and stator flux. To pick the finest voltage vector amongs<sup>t</sup> the feasible ones, MPFC's cost function described in (27) is minimized.

$$\mathbf{G\_2} = \begin{vmatrix} \Psi\_s^{\text{ref}} - \stackrel{\rightarrow}{\Psi}\_{s,k+1}^p \end{vmatrix} \tag{27}$$

**Figure 6.** Conventional MPFC scheme.

It is clear that the weighting factor will be no longer needed, since only stator flux vector tracking error is involved. The key shortcoming of this technique is that it relies on reference flux angle estimate by using inverse-trigonometric function, which fails to estimate the angle during particular operating conditions [28].

Most often for MPTC and MPFC drives, stator flux reference Ψref s is usually set constant to its rated value as:

$$\Psi\_{\rm s}^{\rm rrd} = \begin{vmatrix} \Psi\_{\rm s}^{\rm rrd} \end{vmatrix} \tag{28}$$

In order to accurately regulate torque and stator flux, we propose the MPTFC design in Figure 7 where the predictive values of stator flux, current and torque are based on suggested AFOO in (17).

MPTFC model predicts electromagnetic torque and stator flux vector's magnitude at next step in discrete-time k + 1 as a switch position function to determine vopt at the present time step k. Prediction accuracy of MPCC and MPTFC, is main issue, as he process of prediction begins with the sampled of measured currents. Unless the measuring and sampling processes were not performed well, noise level will increase, which will then be reflected in the expected values and hence exacerbate voltage selection process, which eventually results in greater ripple content. To avoid that problem, we assume both prediction of torque-flux is dependent on stator currents estimate instead of the measured one.

Integrating estimated stator current in (17) with Euler method from t = k Ts to t = (k + 1) Ts and inserting (1) into it relates to the representation of discrete-time:

$$
\stackrel{\rightarrow}{\Psi}\_{s,k+1}^{p} = \stackrel{\rightarrow}{\Psi}\_{s,\mathbf{k}} + \mathrm{T}\_{\mathbf{s}} \stackrel{\rightarrow}{\mathbf{v}}\_{s,\mathbf{k}}(\mathbf{i}) - \mathrm{R}\_{\mathbf{S}} \stackrel{\rightarrow}{\mathbf{T}}\_{\mathbf{s}} \stackrel{\rightarrow}{\mathbf{I}}\_{s,\mathbf{k}} \tag{29}
$$

Hence, predicted stator flux appears as an estimated function.

$$\stackrel{\rightarrow}{i}\_{s,k+1}^{p} = \left(1 + \frac{\text{T}\_{\text{s}}}{\tau\_{\text{r}}}\right) \stackrel{\rightarrow}{\mathbf{i}}\_{s,\text{k}} + \frac{\text{T}\_{\text{s}}}{\tau\_{\text{r}} + \tau\_{\text{s}}} \left\{\frac{1}{\text{R}\_{\text{r}}} \left[\left(\frac{\text{k}\_{\text{r}}}{\tau\_{\text{r}}} - \text{k}\_{\text{r}}\right) \stackrel{\rightarrow}{\Psi}\_{\text{r},\text{k}} + \stackrel{\rightarrow}{\mathbf{v}}\_{s,\text{k}}(\text{i})\right] \right\} \tag{30}$$

where τr = Lr Rˆ r , kr = LmLr , ks = LmLs , Rσ = Rˆ S + Rˆ r + kr2, τσ = σ Ls Rσ , Ts is sampling-period and vs(i)correspondingto7dissimilarvoltagestatus, i isfrom0to6.

→

**Figure 7.** Proposed MPTFC diagram.

 Predicted torque could be obtained after determining the predicted current and flux with:

$$\mathrm{T}^{\mathbb{P}}\_{\mathsf{e},k+1} = \frac{\mathfrak{Z}}{2} \mathrm{P} \left[ \stackrel{\rightarrow}{\Psi}\_{\mathsf{s},k+1}^{p} \times \stackrel{\rightarrow}{i}\_{\mathsf{s},k+1}^{p} \right] \tag{31}$$

Therefore, a proper cost function would be developed for the proposed MPTF control centered on G1 and G2 as:

$$\mathbf{G} = \lambda\_{\mathrm{T}} \left| \mathbf{T}\_{\mathrm{e}}^{\mathrm{ref}} - \mathbf{T}\_{\mathrm{e},k+1}^{\mathrm{p}} \right| + \lambda\_{\mathrm{Y}} \left| \Psi\_{\mathrm{s}}^{\mathrm{ref}} - \overset{\rightarrow}{\Psi}\_{s,k+1}^{\mathrm{p}} \right| \tag{32}$$

where, torque-flux weighting coefficients are λT and λΨ respectively. All feasible inverter topologies for the seven different flux and torque estimates are applied at every sampling stage. Then these samples are being used for determining the cost function. In one sampling period, a switching state that minimizes the G value is picked to be implemented. To drive 2−level-inverter, the switching states are then becoming the output. The flow chart of the whole process is seen within Figure 8.

**Figure 8.** Proposed MPTFC flowchart with weighting factor optimization.

#### **4. Optimal Steady-State Flux for Losses Minimization Criterion**

To achieve an optimal stator flux reference for LMC, efficiency optimization is being discussed in this section. IM losses must be precisely defined to obtain a criterion from which whole losses could be minimized. Figure 1 presents the overall IM losses, which consisting of stator copper, rotor copper and core-losses; but not inverter, load, or mechanical losses.

$$\mathbf{P\_{Lo66}} = \mathbf{3} \left[ \mathbf{P\_{CuS}} + \mathbf{P\_{CuR}} + \mathbf{P\_{f0}} \right] \tag{33}$$

where PcuS, PcuR are stator and rotor copper losses and Pfe is core-loss.

$$\mathbf{P\_{cuS}} = \left| \mathbf{R\_{\&}} \mathbf{I\_{\&}} \right|^{2} = \mathbf{R\_{\&}} \left\{ \mathbf{I\_{m}}^{2} + \left[ \mathbf{I\_{I}} + \frac{\mathbf{R\_{m}}}{\omega \mathbf{L\_{m}}} \mathbf{I\_{m}} \right]^{2} \right\} \tag{34}$$

$$\mathbf{P\_{curR}} = \stackrel{\circ}{\mathbf{R\_{r}}} \mathbf{I\_{r}}^{2} = \left(\frac{\mathbf{L\_{m}}}{\mathbf{L\_{r}}}\right)^{2} \mathbf{R\_{r}} \mathbf{I\_{r}}^{2} \tag{35}$$

$$\mathbf{P\_{fo}} = \mathbf{R\_{fo}} \mathbf{I\_{fo}}^2 = \begin{array}{c} \boldsymbol{\omega}^2 \mathbf{I\_{m}}^3 \\ \hline \text{L}\_{\text{f}} \mathbf{R\_{m}} \end{array} \left[ \begin{array}{c} \text{R} \mathbf{m} \\ \boldsymbol{\omega} \text{L} \mathbf{m} \end{array} \right]^2 \tag{36}$$

Whereas Rm<sup>2</sup> (ω Lm)2, substituting (34) and (36) into (33) yields the total loss expression as:

$$P\_{Loss} = 3\left\{ \left[ \begin{array}{c} R\_s + \frac{R\_{\text{all}}}{L\_r} \end{array} \right] I\_m^{-2} + \left[ \begin{array}{c} R\_s + \left( \frac{L\_{\text{all}}}{L\_r} \right)^2 R\_r \end{array} \right] I\_r^2 + \frac{2 \ R\_{\text{all}} R\_{\text{ls}}}{\omega L\_{\text{ll}}} I\_m I\_r \right\} \tag{37}$$

where a = Lm/Lr , Lm = a Lm, Rr = a2Rr, Rfe = a ( ωLm )2 Rm = ω2Lm3 LrRm , Ife = Rm ω Lm Im and Is = 'Im<sup>2</sup> + (Ir + Ife)2. Direct FOC equations of IM that take core loss into accounts are:

$$\text{Rotorflux} \colon \quad \Psi\_{\mathbf{r}}^{'} = \mathbf{I}\_{\mathbf{m}}^{'} \mathbf{I}\_{\mathbf{m}} \tag{38}$$

Electromagnetictorque : Te = 3 P Ψr Ir (39)

$$\text{Slipfrequency}: \quad \omega\_{\text{sl}} = \, \prescript{\mathbf{q}'\_{\text{r}}}{\mathbf{1}\_{\text{r}}}\_{\mathbf{1}\_{\text{r}}} \tag{40}$$

The substitution of (38) and (39) into (37) helps us to rewrite the complete loss as:

$$\mathbf{P\_{Loss}} = 3 \left\{ \left[ \mathbf{R\_{s}} + \frac{\mathbf{R\_{m}} \, \mathbf{L\_{m}}}{\mathbf{L\_{r}}} \right] \frac{\mathbf{L\_{r}}^{2}}{\mathbf{L\_{m}}^{4}} \left| \mathbf{V\_{z}} \right|^{2} + \left[ \mathbf{R\_{s}} + \left( \frac{\mathbf{L\_{m}}}{\mathbf{L\_{r}}} \right)^{2} \mathbf{R\_{r}} \right] \frac{\mathbf{T\_{o}}^{2}}{9 \, \mathbf{P}^{2}} \left| \mathbf{V\_{z}} \right|^{-2} + \left[ \frac{2 \, \mathbf{R\_{m}} \, \mathbf{R\_{s}}}{\omega \mathbf{L\_{m}}} \right] \frac{\mathbf{L\_{r}} \, \mathbf{T\_{o}}}{3 \, \mathbf{P} \, \mathbf{L\_{m}}^{2}} \right\} \tag{41}$$

This implies that PLoss is a variable dependent on rotor flux Ψr. Assuming that under constant load torque, none of machine parameters have any dependency upon rotor flux Ψr. By setting a derivative of Equation (41) with respect to Ψr to zero, we could obtain a rotor flux that provides the minimum losses. *∂*

$$\frac{\Phi\_{\text{Loss}}}{\Phi \stackrel{\prime}{\Psi}\_{\text{tr}}} = 0\tag{42}$$

$$\frac{\partial \left| \mathbf{P\_{Loss}} \right|}{\partial \left| \Psi\_{\mathbf{r}}' \right|} = 3 \left\{ \begin{array}{ll} 2 \left[ \left. \mathbf{R\_{\delta}} + \frac{\left. \mathbf{R\_{\delta}} \left. \mathbf{I\_{m}} \right|}{\mathbf{L\_{\delta}}} \right| \right. \\ 2 \left[ \left. \mathbf{R\_{\delta}} + \left( \frac{\left. \mathbf{I\_{m}}}{\mathbf{L\_{\delta}}} \right)^{2} \mathbf{R\_{\delta}} \right. \right| \right. \end{array} \right| \frac{\left. \mathbf{V\_{r}'} \right|}{\left. \mathbf{V\_{r}'} \right|} \frac{\left. \mathbf{V\_{r}'} \right|}{\left. \mathbf{V\_{r}'} \right|} \right\} \tag{43}$$

Solving Equation (43) gives us an appropriate rotor flux Ψ- *A*r corresponding to the maximum efficiency point, where. *K* = Rˆ s Lr<sup>2</sup>+Rˆ r Lm<sup>2</sup> Rˆ s Lr<sup>4</sup> + Lm Lr<sup>3</sup> Rm 14 2 *Rm Rs ωLm* . √Lm3 P . Ψ - *A* r= K +Te(44)

Before applying Ψ- *A*r to the direct FOC of IM, the coefficient with estimated parameters must be multiplied as follows:

$$
\stackrel{\rightarrow}{\Psi}\_{r,k}^{A} = \frac{\mathcal{L}\_{\mathbf{r}}}{\mathcal{L}\_{\mathbf{m}}} \underline{\mathbf{v}}\_{\mathbf{r}}^{\prime}{}^{A} = \frac{\mathcal{L}\_{\mathbf{r}}}{\mathcal{L}\_{\mathbf{m}}} \hat{\mathcal{K}} \sqrt{\mathcal{T}\_{\mathbf{s}}} \tag{45}
$$

Corresponding suitable stator flux can also be determined based on the steady-state relationship among stator-rotor fluxes as [33]:

$$\stackrel{\rightarrow}{\Psi}\_{s,k}^{A} = \sqrt{\left(\frac{\text{L}\_{\text{e}}}{\text{L}\_{\text{m}}}\right)^{2} \stackrel{\rightarrow}{\Psi}\_{r,k}^{A} + \left(\frac{4\ \sigma\text{ L}\_{\text{e}}\text{ L}\_{\text{r}}}{3\ \text{P}\text{ L}\_{\text{m}}}\right)^{2} \frac{\text{T}\_{\text{a}}^{2}}{\left(\stackrel{\rightarrow}{\Psi}\_{r,k}^{A}\right)^{2}}}\tag{46}$$

The optimum stator flux's reference amplitude required by suggested MPTFC in LMC is provided by Equation (46). Expression of (46) updates (28) as:

$$\Psi\_s^{\text{ref}} = \begin{vmatrix} \stackrel{\rightarrow}{\Psi}\_{s,k}^A \\ \end{vmatrix} \tag{47}$$

It thus completes the data necessary to apply (29) and so the cost function (32).

In order to calculate the efficiency of IM based on the total losses calculated in Equation (41), as [47]:

$$
\eta = \frac{\mathcal{P}\_{\text{Out}}}{\mathcal{P}\_{\text{Out}} + \mathcal{P}\_{\text{Loss}}} \tag{48}
$$

where η is the efficiency and the output power is POut = ωr Te.

## **5. System Layout**

in

Complete approach layout is shown in Figure 9. Stator voltages and currents are the measured quantities, that used by suggested AFOO to obtain the estimated values of; rotor velocity ω<sup>ˆ</sup> r,k, rotor flux → Ψ ˆ r,k, stator current → ˆ i s,k, stator flux → Ψ ˆ s,k and torque T ˆ e,k to predict stator flux → Ψ ˆ *p <sup>s</sup>*,*k*+1 and corresponding predicted torque <sup>T</sup>Pe,k+1. Estimated states are fed back toward the stator flux outer

loop and inner loop of the torque to produce stator flux and torque references for cost function in Equation (32). The LMC algorithm provides optimum stator flux's amplitude; alternatively, stator flux reference could be followed through fed back flux loop. Optimum voltage vector is selected through applying cost function G on the base of the references and predicted values of both torque and stator flux to complete the suggested MPTFC of IM drive.

**Figure 9.** Proposed MPTFC layout with LMC for IM drive.

## **6. Results and Discussion**

The suggested MPTFC approach is simulated in MATLAB/Simulink environment to verify its effectiveness. For simplicity, the proposed MPTFC performance is compared to conventional MPTC that utilize cost function G1 in Equation (26) and constant stator flux reference Ψref s = 0.71 wb. MPTC technique is referred to as method I, and the proposed one of MPTFC with the modified AFOO is referred to as method II in the following phrases, respectively. Motor parameters are mentioned in Table 1.



The following hints about the simulating system should be noted; the IM has been considered nonlinear and takes the core-loss influence into account. Moreover, the proposed observer-based on the mentioned equations in the paper has been implemented using MATLAB® and Simulink® environment. Furthermore, the inverter has been built from the Simulink libraries of Simscape Electrical Specialized Power Systems library. However, no extra band-limited white noise was introduced into the signals.

Figure 10 shows starting response from standstill to (100 rps) for the both methods. Stator flux is first developed using preexcitation, and torque is restricted to 100 percent rated value (20 Nm) during the accelerating stage. An external load with 50% of the nominal value (10 Nm) is suddenly applied at (t = 0.7 s) to the tested IM. In MPTC, actual speed is compared with speed reference unlike in the proposed system which utizes estimate speed.

**Figure 10.** Starting response simulation from standstill to 100 rps for: (**a**) method I of the conventional MPTC; (**b**) method II of the proposed MPTFC.

Curves shown in Figure 10, through top to bottom are velocity, torque, stator flux and stator current, respectively. It is obviously seen that in low-speed operation, proposed MPTFC works well and exhibits high robustness regarding load disturbance. It can also be observed that method I has similar dynamic performance, but it produces much higher flux, current and torque ripples.

A more extensive steady-state torque, current and stator flux waveforms with (50%) of rated torque (10 Nm) is shown in Figure 11. The ripples of torque, current and stator flux in method II are shown to be much smaller than in method I, demonstrating the efficiency of suggested approach of MPTFC.

Conventional and suggested drive responses are shown in Figure 12a,b throughout speed reversals. It could be said that the motor accelerates rapidly from (50 rps) to (−50 rps) for three full cycles of forward and reverse speed to test the drive performance in low and reverse speed operation. The load torque is held constant at (5 Nm) during the whole test period. The conventional MPTC works at reverse speed operation while it suffers from unwanted ripples in the flux, torque and the stator currents when compared to the proposed method, as shown in Figure 12a. Moreover, an acceptable error between actual, estimated velocity and measured one is shown in Figure 12b. Furthermore, the error between the estimated stator current and actual ones through our proposed drive is shown in Figure 12b, which validates the proposed observer's operation. The actual and estimate values of stator and rotor fluxes throughout the reverse speed operation are also shown in Figure 12b. Figure 12c shows the comparison of the response of the stator currents and stator flux with the operation of speed reversal at (50 rps) between the conventional MPTC and the proposed MPTFC. The figure proves that the response of the flux and stator currents with the proposed MPTFC is better than those of the conventional MPTC.

**Figure 11.** Steady-state response simulation at 100 rps for: (**a**) method I of the conventional MPTC; (**b**) method II of the proposed MPTFC.

**Figure 12.** *Cont*.

**Figure 12.** Performance with the operation of speed reversal at 50 rps: (**a**) Simulation responses of speed reversal at 50 rps for conventional MPTC; (**b**) Simulation responses of speed reversal at 50 rps for proposed MPTFC; and (**c**). Comparison of the response of the stator currents and stator flux with the operation of speed reversal at 50 rps between; (**a**) method I of the conventional MPTC; (**b**) method II of the proposed MPTFC.

 torque.

Furthermore, the drive responses of the conventional and proposed method under step change of the load disturbance are shown in Figure 13a,b at a very low speed operating range of (10 rps). A load torque of (5 Nm) is applied at starting, then load torque is increased to (10 Nm) at (t = 2 s) then decreased to (3 Nm) at (t = 4 s) and finally back to (5 Nm). Figure 13a shows a similar dynamic performance of MPTC at load disturbance; nevertheless, it produces greatly higher flux, current, and torque ripples when comparing it to the proposed MPTFC. Actual values of stator and rotor fluxes are very robust in the suggested drive, as they were compensated by the estimated value of the stator flux, as shown in Figure 13b. Moreover, both speed and current error can be seen as a rather pleasant feature, demonstrating high robustness against external load disturbance of the proposed MPTFC. It is also obvious that the both conventional and proposed method currents in Figure 13a,b seem to be quite sinusoidal in shape. Moreover, Figure 13c displays the comparison of the response of the stator currents, stator flux, and torque with the operation of load disturbance at (10 rps) between the conventional MPTC and the proposed MPTFC. The figure verifies that the response of the flux, stator currents and torque with the proposed MPTFC is better than those of the conventional MPTC. Inthesecondcase,theIMbeingcontrolledunderthesuggestedMPTFCwithapplyingofLMC,

where stator flux's reference is adjusted to optimal or appropriate flux → Ψ *A <sup>s</sup>*,*k* which is determined in (46) to obtain the highest efficiency or lowest losses. Figure 14a indicates total steady-state losses of IM; the conventional method is represented by dashed lines, while the proposed method is represented by solid lines at various values of speed. Overall loss operated throughout suggested method is certainly smaller than that of the conventional method with constant flux at each motor speed. It is obvious that, as can be seen in Figure 14b, the rotor flux where the losses are minimal depends upon load torque. In most other words, rotor flux that provides maximum IM efficiency stands as a functionofload

Optimal rotor flux for maximum operating efficiency that calculated in (45), is drawn in Figure 15 for various driving conditions. It is influenced by motor speed since the core-loss is taken into account. This implies that rotor flux drops to decrease the core-loss, which raises through the excitation frequency further than the other loses at high speeds. Corresponding optimal stator flux for minimum losses that calculated in (46), is drawn in Figure 15b.

IM steady-state efficiency with constant rotor flux is plotted with dash lines and also solid lines represents optimal suggested method, that shown in Figure 15c. It clearly shows that suggested method here achieves higher efficiency over large range of load torques owing to adjusting rotor flux rendering to torque. Thus, it can be said that the proposed system efficiency with LMC has improved significantly in excess of the classical one.

To investigate dynamic performance, load torque will varied as in case of load disturbance in Figure 13 and IM speed is kept constant with very low amount (10 rps). Stator flux's reference is held at its rated value (0.71 wb) for MPTC and suggested estimated value for MPTFC without the LMC. For the suggested MPTFC with LMC, stator flux's reference is adjusted on line through optimal value, which is also presented in Figure 16.

The steady-state optimal flux, which achieves minimum steady-state losses, varies with the operating conditions. To achieve balance among copper and iron losses, it rises with torque and decreases as rotor speed increases.

In the proposed MPTFC with LMC, the optimal stator flux, that calculated in Equation (46), achieves minimum losses, and varies with the operating conditions. To achieve balance among copper and iron losses, it rises with the torque and decreases as rotor speed increases. It can also be seen from Figure 16, in which a comparison of the three control procedures is demonstrated, that proposed MPTFC (without and with LMC) exhibits better dynamic behavior compared to MPTC technique. In addition, it should be noted that losses are effectively minimized during light loading when following the LMC approach and consequently, IM drive efficiency is improved, as clearly seen in Figure 16.

To clarify the speed response throughout low-speed region, that considered among the most major aspects of this study, actual speeds is compared for MPTC and suggested MPTFC (without and with LMC) approaches in Figure 17. Step change to reference speed at (t = 0.05 s) from (0 rps) to (10 rps)is applied with constant load torque(5 Nm). Figure 17 shows a step response of the conventional MPTC and the proposed one of MPTFC with and without considering LMC. To clarify these values more precisely, the over shot, rise time, and settling time were calculated in Table 2. The results show that the overshot of proposed one of MPTFC with LMC is better than those of the conventional one and the proposed MPTFC without LMC. Moreover, the figure shows an increasing in the rise time and settling time of the MPTFC with LMC. Moreover, in general, the speed response of the three schemes is stable as shown in time domine analysis of Figure 17.

**Figure 13.** *Cont*.

**Figure 13.** Drive responses of the conventional and proposed method under step change of the load disturbance (**a**) Simulation responses of load disturbance at 10 rps for conventional MPTC; (**b**) Simulation responses of load disturbance at 10 rps for proposed MPTFC. (**c**) Comparison of the response of the stator currents and stator flux with the operation of load disturbance at 10 rps between; (**a**) method I of the conventional MPTC; (**b**) method II of the proposed MPTFC.

**Figure 14.** Steady-state overall losses versus: (**a**) load-torque with different rotor speeds; (the dash line is stand to the conventional method while the solid line is for the proposed one); (**b**) rotor flux with different load-torque.

**Figure 15.** Steady-state IM efficiency: (**a**) appropriate rotor flux map; (**b**) appropriate stator flux map; (**c**) efficiency map.

**Figure 16.** Total IM losses through three control procedures with actual and optimal fluxes.

**Figure 17.** Actual speed response of MPTC and MPTFC (without and with LMC).

**Table 2.** Values of the over shot, rise time, and settling time for MPTC and suggested MPTFC (without and with LMC) approaches.

