Datasets:

Monketoo
/

math-docs-dataset

nonlinear estimators and linear parameters. The relationships between input-output of the MIMO model have been written in the equation (10) whereas vdc is DC voltage, idc DC current, vac AC voltage, iac AC current. q is shift operator as equivalent to z transform. f(•) and h(•) are input and output nonlinear estimators. In this case a deadzone and saturation are selected into the model. In the MIMO model the relation between output and input has four relations as follows (i) DC voltage (vdc) - AC voltage (vac), (ii) DC voltage (vdc) - AC current (iac), (iii) DC current (idc) - AC voltage (vac) and (iv) DC current(vdc)-AC voltage (vac).

$\left. \begin{aligned} v_{ac}(t) &= \frac{B(q)}{F(q)} f(v_{dc}(t-n_k)) + e(t) \\ i_{ac}(t) &= \frac{B(q)}{F(q)} f(i_{dc}(t-n_k)) + e(t) \end{aligned} \right\} \tag{10}$

$\begin{equation} \begin{split} V_{ac}(t) &= h\left(\frac{B_1(q)}{F_1(q)}f(v_{dc}(t-n_{k1})) + e(t)\right) \otimes h\left(\frac{B_2(q)}{F_2(q)}f(i_{dc}(t-n_{k2})) + e(t)\right) \\ I_{ac}(t) &= h\left(\frac{B_3(q)}{F_3(q)}f(v_{dc}(t-n_{k3})) + e(t)\right) \otimes h\left(\frac{B_4(q)}{F_4(q)}f(i_{dc}(t-n_{k4})) + e(t)\right) \end{split} \tag{11} \end{equation}$

$B_i(q) = b_1 + b_2 + \dots + b_{n_{bi}} q^{-n_{bi}+1}$

$F_i(q) = f_1 + f_2 + \dots + f_{n_{fi}} q^{-n_{fi} + 1} \quad (12)$

Where $n_{bi}$, $n_{fi}$ and $n_{ki}$ are pole, zero and delay of linear model. Where as number of subscript i are 1,2,3 and 4 which stand for relation between DC voltage-AC voltage, DC current-AC voltage, DC voltage-AC current and DC current-AC current respectively. The output voltage and output current are key components for expanding to the other electrical values of a system such power, harmonic, power factor, etc. The linear parameters, zeros, poles and delays are used to represent properties and relation between the system input and output. There are two important steps to identify a MIMO system. The first step is to obtain experimental data from the MIMO system. According to different types of experimental data, the second step is to select corresponding identification methods and mathematical models to estimate model coefficients from the experimental data. The model is validated until obtaining a suitable model to represent the system. The obtained model provides properties of systems. State-space equations, polynomial equations as well as transfer functions are used to describe linear systems. Nonlinear systems can be described by the above linear equations, but linearization of the nonlinear systems has to be carried out. Nonlinear estimators explain nonlinear behaviors of nonlinear system. Linear and nonlinear graphical tools are used to describe behaviors of systems regarding controllability, stability and so on.

4. Experimental

In this work, we model one type of a commercial grid connected single phase inverters, rating at 5,000 W. The experimental system setup composes of the inverter, a variable DC power supply (representing DC output from a PV array), real and complex loads, a digital power meter, a digital oscilloscope, , a AC power system and a computer, shown