Such as,
2 2 2
, , r s s r r r s r r r
L L R L R M
L L M L PM L R
Where rd s i u , , and r
: are respectively, the stator voltage, stator current, rotor flux and rotor speed. The indices d, q indicates a direct and quadrate index according to the usual d-axis and q-axis components in the synchronous rotating frame. M L L R R r s r s
, , , , and
: are respectively, stator and rotor resistance, stator and rotor inductance, mutual inductance and total leakage factor. P, J, TL and f: are respectively, the number of pole pairs, the rotor inertia, the load torque and the friction coefficient. A. Backstepping Control Design
In order to synthesize the stator voltage components
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So, the control is asymptotically stable. Then, the tracking requirements are achieved. Step 2:
In this step, we define other errors between the stator currents components and their references,
sd sdref sq sqref i i e i i e
4
3 (14) So, the (10) can be expressed as,
4 2 2 2
3 1 1 1 e M e k e e J e k e rd
(15) And the derivative of (14) is obtained as,
In this step, for calculating the actual control inputs, a second Lyapunov function based on all errors is defined as follows, By choosing the following controller,
q sqref s q d sdref s d f i e k L u f i e k L u
3 3
4 4
(18) The derivative of errors and will be as follows:
4 4 2 4
3 3 1 3 e k e M e e k e
J
e rd
(19)
So, from equations (15) and (19) we can write,
B. Rotor Flux Estimation
The field oriented control is based on Park transformation which is converted a three-phase time and speed dependent system into invariant time
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RESULTS AND OPERATION MODES DISCUSSION
To demonstrate the effectiveness of the proposed robust controller of IM drive fed by standalone PV/battery system, numerical simulations are implemented under
Simulink/Matlab. The block diagram of the proposed controller is presented in Figure 1. The parameters of the tested motor are given in Table II. In this system, six similar solar panels are connected in series and 6 batteries are connected in series, the specifications of PV panel and battery are shown in Tables III and IV, respectively.
Figures 6 and 7 illustrate successively, the power system behavior, and IM performance under fast variable irradiation. Figure 6a shows the irradiation variation profile
(1000, 800, 500 and 200 W/m2
), figure 6c shows the DC bus voltage response which demonstrates a continuity with high stability, where VDCref =515V.
In terms of trajectory tracking, simulation results are obtained for a reference flux fixed to 0.6 Wb. The tracked peak power versus time plot is shown in Figure 6b, the stator voltage output and current for a single phase are shown in
Figure 7e and Figure 7f. As seen in Figure 7a, the motor follows the speed set points using field oriented