# Multivariable Control of Solar Battery Power by Extremum Seeking: Starting from Linear Analysis

^{*}

## Abstract

**:**

## 1. Some of the Most Important Aspects of the Problem

- The requirements for the quality of transient processes are rarely imposed on closed search engines. The main requirement is the convergence to the minimum/maximum point as fast as possible (studied within the framework of asymptotic stability [28,33]), which is determined by the parameters of the extremal control loops, as well as by the amplitudes and frequencies of the modulating signals.
- The multiparameter system of extremum seeking has the properties of multiply connectedness and requires the use of nonlinear methods of analysis (some of them were mentioned in the annotation, as well as in [28]).
- Introduction to the system of a nonlinear element, which is difficult to reduce to a simple quadratic function with displacement along the axes of the abscissas and ordinates [34], increases the complexity of the exact analytical studies and does not allow them to be made by direct paths.

_{d}(V), diffusion capacitance C

_{d}(V, ω) and transition capacitance C

_{t}(V) were determined by the method of impedance spectroscopy. For simplicity, we took the averaged values from the data available in [44], and a simple calculation of the cutoff frequency showed that it is large enough to consider the solar panel as an inertia-free gain with a variable value of load resistance R

_{reg}, which is the control channel MPPT controllers.

- with the construction of methods for calculating the parameters of the solar panel using elements of special functions (for example, the Lambert function);
- with accurate dynamic models on their basis that allow us to evaluate the dynamics of the reaction of voltage changes to changes in photocurrent when the angular position changes during modulation.

## 2. Solar Battery—Multiparameter Object of Extremum Seeking Control. Parametric Identification of the Voltage-Current and Volt-Watt Characteristics of the Solar Panel in an Analytical Way

_{sh}are connected, as shown in Figure 1 [46] (p. 40). Resistance R

_{s}is the series resistance of each cell.

_{exH}, falling onto the surface of the solar panel (and increases with increasing temperature of the solar cell):

_{exH}—extraterrestrial solar radiation on the surface irradiance, Wt/m

^{2}; G

_{0}—irradiance at the standard test conditions (STC) (G

_{0}= 1000 Wt/m

^{2}); ${\mathrm{k}}_{{\mathrm{I}}_{\mathrm{sc}}}$—current growth factor versus temperature [39]; $\mathsf{\Delta}\mathrm{T}$—difference between ambient temperature and temperature under standard test conditions (STC) = 25 °C.

_{exH}/G

_{0}affects the production of photocurrent ${\mathrm{I}}_{\mathrm{ph}}$, i.e., if G

_{exH}= 0, the solar cell is inactive and produces neither current nor voltage. However, if light hits a solar cell, it generates a photocurrent I

_{ph}.

- Short-current mode (V = 0; I = I
_{sc}); - Maximum power point mode (V = V
_{mp}; I = I_{mp}); - Open-circuit mode (V = V
_{oc}; I = 0);

_{s}, R

_{sh}, I

_{0}, etc., which should be as close as possible to those specified by the manufacturer of the solar panel. This requires finding the exact analytical solution to the system of Equation (3), which has not yet been discovered due to unsolvable recursive relations. Since in Equation (3) there are quantities of different orders, among them are small quantities, after neglecting which the task is simplified so much that its exact solution is possible. In this case, the first step in solving the problem is to accurately solve the simplified problem. It should be noted the peculiarity of this class of models, namely, that any simplifying assumption makes a system with 3–4 free parameters from the I–V characteristics and P–V characteristics. Thus, it becomes possible, due to the selection of these parameters, to ensure that the I–V and I–V characteristics pass near the characteristic modes, but these parameters will be far from those set by the manufacturer (see Appendix A). After analyzing the structure of the system of Equation (3), as well as the data provided in [41] in more detail, we came to the following conclusions:

- Difference between the photocurrent I
_{ph}and short-circuit current I_{sc}, determined by the ratio through the value (R_{s}+R_{sh})/R_{sh}[41], is 0.007–0.3%, therefore, it is permissible to take I_{ph}= I_{sc}. - From this it follows that from the third expression of the system of Equation (3), we can get the formula for calculating the saturation current of the diode:$${\mathrm{I}}_{0}=\frac{{\mathrm{I}}_{\mathrm{ph}}-\text{}\frac{{\mathrm{V}}_{\mathrm{oc}}}{{\mathrm{R}}_{\mathrm{sh}}}}{{\mathrm{e}}^{\frac{\mathrm{q}\cdot {\mathrm{V}}_{\mathrm{oc}}}{\mathrm{n}\cdot \mathrm{k}\cdot \mathrm{T}}}-1}$$
- Term $\left(\mathrm{V}+\mathrm{I}\cdot {\mathrm{R}}_{\mathrm{s}}\right){/\mathrm{R}}_{\mathrm{sh}}$ can be decomposed into two components ${\mathrm{V}/\mathrm{R}}_{\mathrm{sh}}+\mathrm{I}\cdot {\mathrm{R}}_{\mathrm{s}}{/\mathrm{R}}_{\mathrm{sh}}$. Due to the fact that ${\mathrm{R}}_{\mathrm{s}}{/\mathrm{R}}_{\mathrm{sh}}{=\text{}8\text{}\times \text{}10}^{-6}\dots 0.003$, it is permissible not to take into account the variable component ${\mathrm{I}\text{}\times \text{}\mathrm{R}}_{\mathrm{s}}{/\mathrm{R}}_{\mathrm{sh}}$. Difference between $\left({\mathrm{V}}_{\mathrm{mp}}{\text{}+\text{}\mathrm{I}}_{\mathrm{mp}}\cdot {\mathrm{R}}_{\mathrm{s}}\right){/\mathrm{R}}_{\mathrm{sh}}$ and $\left({\mathrm{V}}_{\mathrm{mp}}\right){/\mathrm{R}}_{\mathrm{sh}}$ = 0.108…8.805%, therefore, at the point of maximum power ${\mathrm{I}}_{\mathrm{mp}}\cdot {\mathrm{R}}_{\mathrm{s}}$ can be ignored.
- At the maximum power point, according to [50], voltage derivative versus current ${\left(\partial \mathrm{I}/\partial \mathrm{V}\right)|}_{\left[{\mathrm{I}}_{\mathrm{mp}}{,\mathrm{V}}_{\mathrm{mp}}\right]}=\u2013{\mathrm{I}}_{\mathrm{mp}}/{\mathrm{V}}_{\mathrm{mp}}$

_{ideal}(this will be shown in more detail in Appendix A).

_{s}> 0 and R

_{sh}> 0. Solving Equations (6) and (7) as inequalities, we can find the range of variation of n at which these conditions are observed:

_{exH}on the surface in the selected geographical location can be calculated as [46]:

^{2};

- Solar cell (nonlinear element with extreme characteristic);
- High-pass filter (HPF
_{i}); - Demodulator multiplier;
- Master integrator I
_{i}with gain k_{i}; - Modulating summator;
- Two high-speed coordinate electric drives W
_{act_i}, tuned to a symmetrical optimum with an additional integral element for converting speed (rad/s) into a rotation angle (rad).

_{hpf}and h are time constant and high pass filter constant.

_{D}, C

_{T}and R

_{d}are unknown. The values of these parameters, obtained using impedance spectroscopy, are given in [44]. As will be shown, if the actual values of the capacitance C

_{D}, C

_{T}s and differential resistance R

_{d}have approximately the same numerical order, then the solar panel can be represented in the form of a proportional link with an adjustable gain factor in the I

_{ph}→V transmission channel.

**A**—reduced incidence matrix of an electrical circuit without a string corresponding to a zero node:

**0**,

**1**—zero and identity matrices; M—source matrix, in which elements on the main diagonal are formed according to the principle “if the element is a current source, then 0, if the element is a voltage source or not a power source at all, then 1”;

**e**,

**v**,

**i**—unknown vectors of nodal potentials, voltages and currents on the elements;

**ui**—vector of known values of voltages or supply currents (V or A).

_{s}_{ph}to voltage V:

^{−7}− 10

^{−6}, which allows us to represent the solar panel as a non-inertia gain from I

_{ph}to voltage V:

_{reg}:

_{i}—gain of master integrator.

- Having data ${\mathrm{V}}_{\mathrm{oc}}$;$\text{}{\mathrm{I}}_{\mathrm{sc}}$;${\text{}\mathrm{V}}_{\mathrm{mp}}$;$\text{}{\mathrm{I}}_{\mathrm{mp}}$ calculate ${\mathrm{I}}_{0}$ and resistances ${\mathrm{R}}_{\mathrm{s}}$ and ${\mathrm{R}}_{\mathrm{sh}}$ by the formulas (4), (6) and (7). If necessary, adjust the ideality factor n in accordance with formulas (8).
- Calculate the amplitude incidence coefficient ξ according to formulas (26)–(31).
- Calculate the amplitude of the photocurrent ΔI
_{ph}by the formula (32). - Calculate the voltage amplitude ΔV by formulas (38) and (39).
- Calculate the amplitude of the power ΔP by the formulas (40)–(42).
- Given the parameters of the circuits T
_{hpf}, h, k_{i}, select the transfer function of the compensators in accordance with formula (19) and taking into account the reduction of a high-order object to a low-order object with formulas (25) - Selecting the output amplitude of the oscillation signal of the master integrator ${\mathrm{I}}_{\mathrm{int}}$ for each of the extremum seeking loops, calculate the required frequency of the input modulating effects using the formula (47), which will provide the required amplitude of oscillations ${\mathrm{I}}_{\mathrm{int}}$.
- Knowing the permissible amplitude of oscillations of angular electric drives, by the formula (52) calculate the necessary amplitudes of the main input modulating signals A
_{in}. - In the absence of satisfactory results on convergence at a given point, change the tuning parameters of T
_{hpf}, h, k_{i}contours and repeat steps 7–10. You should strive to make them slightly different from each other in order to ensure that the modulating frequencies differ in accordance with the recommendations [28].

## 3. Results

_{0}. Then we adjusted the value of the scale factor n (formula (8)).

_{max}(Under P

_{max}should understand the maximum power from manufacturer datasheet) volt-ampere and volt-watt curves, calculated by Equation (8), do not differ from datasheet, which is an “almost accurate” solution to this problem. It should be said that if we refuse to search for the exact solution of the system of Equation (3), then it becomes possible to have a whole class of solutions ${\mathrm{R}}_{\mathrm{s}}$,$\text{}{\mathrm{R}}_{\mathrm{sh}}$ and n, leading to the exact passage of the curves through the characteristic modes of operation. The change in current at such values of the parameters with a high of accuracy will still coincide with Equation (2). Table 4 shows the calculation results for a sample of solar panels borrowed from [41].

_{sin}_

_{1}= A

_{sin_2}= 1°. Depending on these values, we will further determine the required search frequencies of the modulating signals.

- The original position is considered aligned. For this case, the reference signals and the angles of the Sun are the same.
- At the beginning of the motion, the angles of the Sun are changed, but the state of the system has not yet been changed. For this case, the angles of the Sun are equal to the final position, and the angles are set to the previous state.
- At the end of the transition to a new state, the angles of the Sun and the angles of installation again coincide.

## 4. Discussion. Interpretation of the Results. Limitations of the Study in Question

- In almost all experiments, the required parameters of the power fluctuation of the solar panel approximately correspond to the set. We cannot provide an arbitrary amplitude of oscillations of the coordinate electric drive, therefore, we chose a limitation, which, judging by the calculations, is satisfied and corresponds to the given = 1°. The influence of integrators on the input modeling signal is minimized so as to ensure convergence. In experiments 2 and 3, which turned out to be the least successful, we explain the main errors by the unsuccessful choice of the parameters of the optimization loops, amplitudes and frequencies of the modulating signals. The effect of convergence success depending on the power of the solar panel has so far been rejected, but requires additional verification, so the number of experiments needs to be increased from 6 to 20–25 and the power range of the solar panels expanded. It is required to find out how to choose the parameters of the high-pass filter and the integrator. So far, the problem is being solved for the already known or given parameters of the HPF and integrator.
- The effect of inconstancy in the amplitude of the fluctuation of the photocurrent and power, depending on the angular position of the solar panel and the angular position of the Sun in the chosen geographical location, shows that, provided the system is opened, multiplicity can be ignored. In other words, the adjustment of extremum seekeng loops can be done by “removing” nonlinearity from the system, leaving only the change in the effect itself.
- A new problem is highlighted from the previous paragraph. Our calculation shows that the effect varies over time. Consequently, adaptation of amplitudes and frequencies will be required. It is possible that the algorithm can be built on the basis of the presented calculations.
- Particular attention should be paid to restrictions. The number of parameters is limited to 3 (two angular positions and load resistance). Despite the fact that these are the main properties of the control object, additional non-linearities (for example, when expanding the number of adjustable coordinates) can complicate the calculation, because additional nonlinear interconnections appear in an open circuit.
- In general, the presented method can be extended to other nonlinearities, as well as the number of optimized parameters. You only need to learn how to correctly determine the amplitude of the fluctuations in the quantity coming to the extremum seekeng loops and to “remove” nonlinearity from the system (see point 3 of the Discussion section). We assume this is relevant for position-tracking systems of other classes.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

- On the interval 0 ≤ V ≤ V
_{mp}, the term ${\mathrm{I}}_{0}\cdot \left({\mathrm{e}}^{\frac{\mathrm{q}\cdot \left(\mathrm{V}+\mathrm{I}\cdot {\mathrm{R}}_{\mathrm{s}}\right)}{\mathrm{n}\cdot \mathrm{k}\cdot \mathrm{T}}}-1\right)$ in first equation of the system of Equation (A3) practically does not affect the calculations, so it is assumed to be zero. To calculate the resistances ${\mathrm{R}}_{s}$ and ${\mathrm{R}}_{\mathrm{sh}}$, it is advisable to enter an additional point in the middle of ${\mathrm{I}}_{\mathrm{sc}}$ and ${\mathrm{I}}_{\mathrm{mp}}$, namely (a graphic explanation is given in Figure A1 and Figure A2:$${\mathrm{I}}_{\mathrm{ph}}-\frac{\frac{{\mathrm{V}}_{\mathrm{mp}}}{2}+\frac{{\mathrm{I}}_{\mathrm{sc}}{+\mathrm{I}}_{\mathrm{mp}}}{2}\cdot {\mathrm{R}}_{\mathrm{s}}}{{\mathrm{R}}_{\mathrm{sh}}}=\frac{{\mathrm{I}}_{\mathrm{sc}}{+\mathrm{I}}_{\mathrm{mp}}}{2}$$ - In a sufficiently small-scale neighborhood of the peak power at the voltage of the second term V≈V
_{mp}effect begins to show enough that it cannot be disregarded. This is important, i.e., leads to a specific law of change, on the basis of which the following assumption is made. - On the section V
_{mp}≤ V ≤ V_{oc}, the volt-watt curve is close to a quadratic law:$$-\left({\mathrm{K}}_{\mathrm{c}}\cdot {\left({\mathrm{V}-\mathrm{V}}_{\mathrm{mp}}\right)}^{2}-{\mathrm{I}}_{\mathrm{mp}}\right)=\mathrm{I}$$_{c}is determined from the condition of equality of the 0 curve by Equation (A5) at the idle point ${\mathrm{V}}_{\mathrm{oc}}$:$${\mathrm{K}}_{\mathrm{c}}=\text{}\frac{{\mathrm{I}}_{\mathrm{mp}}}{{\left({\mathrm{V}}_{\mathrm{oc}}-{\mathrm{V}}_{\mathrm{mp}}\right)}^{2}}$$

_{s}and R

_{sh}and, we get:

**Table A1.**1-Diode/2-resistors circuit model parameter values from different solar cells (with data from [41] and by alternative approach).

Reference | ${\mathbf{R}}_{\mathbf{s}}$ | ${\mathbf{R}}_{\mathbf{sh}}$ | ${\mathbf{I}}_{0}$ | ${\mathbf{P}}_{\mathbf{max}}$ | $\mathbf{n}$ |
---|---|---|---|---|---|

Kennerud, 1969 | 2.06 × 10^{−6} | 2.98 | 5.25 × 10^{−6} | 0.208 | 1.37 |

Charles, 1981 | 1.37 × 10^{−8} | 26.98 | 3.4 × 10^{−9} | 0.0385 | 1.2 |

Charles, 1981 | 5.12 × 10^{−8} | 3.58 | 7.74 × 10^{−8} | 0.186 | 1.25 |

Lo Brano, 2010 | 2.67 × 10^{−11} | 1.045 | 3.99 × 10^{−10} | 3.52 | 1 |

Cubas | 2.91 × 10^{−6} | 43.83 | 9.1 × 10^{−7} | 192.5 | 80 |

PSM-150 | 1.55 × 10^{−5} | 116.7 | 2.13 × 10^{−6} | 147 | 115 |

**Figure A1.**Volt-ampere curves and their quadratic equivalent in range ${\mathrm{V}}_{\mathrm{mp}}\le \mathrm{V}\le {\mathrm{V}}_{\mathrm{oc}}$ for first three experiments as an example of an alternative procedure. (

**a**) Experiment No.1, (

**b**) Experiment No.2, (

**c**) Experiment No.3

**Figure A2.**(

**a**) Plot of the volt-watt characteristic on the interval 0 ≤ V ≤ V

_{mp}, (

**b**) adjustment of the ideality factor.

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**Figure 2.**General extremum seeking control structure for Sun tracking system. Red line is the place of “open” of a closed system. Blue line is the outline of the MPPT controller.

**Figure 3.**Different impulse characteristics of ${\mathrm{W}}_{\mathrm{act}}=1/{\left({\mathrm{T}}_{\mathrm{hpf}}\cdot \mathrm{s}+\mathrm{h}\right)}^{\mathrm{n}}$ (

**a**), and impulse characteristics of n = 2 and n = 4 objects after corrections in time domain (

**b**).

**Figure 5.**Equivalent dynamic electrical circuit for photovoltaic (PV) module ([44]).

**Figure 6.**Volt-ampere curves in range $0\text{}\le \text{}\mathrm{V}\text{}\le {\text{}\mathrm{V}}_{\mathrm{oc}},$ (

**a**) Experiment 1, (

**b**) Experiment 2, (

**c**) Experiment 3, (

**d**) Experiment 4, (

**e**) Experiment 5, (

**f**) Experiment 6.

**Figure 7.**Volt-watt curves in range $0\text{}\le \text{}\mathrm{V}\text{}\le {\text{}\mathrm{V}}_{\mathrm{oc}}.$ (

**a**) Experiment 1, (

**b**) Experiment 2, (

**c**) Experiment 3, (

**d**) Experiment 4, (

**e**) Experiment 5, (

**f**) Experiment 6.

**Figure 9.**General incidence coefficient amplitude deformation in the process of daily movement of the Sun.

**Figure 16.**Comparison of photocurrent amplitudes and incidence coefficient ξ for various experiments (No.1–6). (

**a**) Experiment No.1, (

**b**) Experiment No.2, (

**c**) Experiment No.3, (

**d**) Experiment No.4, (

**e**) Experiment No.5, (

**f**) Experiment No.6.

**Figure 17.**Comparison of the amplitudes oscillations of the angles for various. (

**a**) Experiment No.1, (

**b**) Experiment No.2, (

**c**) Experiment No.3, (

**d**) Experiment No.4, (

**e**) Experiment No.5, (

**f**) Experiment No.6.

**Table 1.**The conditions by which the lower and upper boundaries of oscillations of the second term of Equation (16) are calculated.

Condition | $\mathbf{Minimum}\text{}{\mathsf{\Delta}\mathsf{\xi}}_{2\mathbf{min}}$ | $\mathbf{Maximum}\text{}{\mathsf{\Delta}\mathsf{\xi}}_{2\mathbf{max}}$ |
---|---|---|

0 < θ < 90° ∧ 0< γ < 90° | $\mathrm{sin}\left(\mathsf{\theta}+\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}-\mathrm{A}\right)\right)$ | $\mathrm{sin}\left(\mathsf{\theta}-\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}+\mathrm{A}\right)\right)$ |

0 < θ < 90° ∧ γ = 90° | $\mathrm{sin}\left(\mathsf{\theta}+\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}-\mathrm{A}\right)\right)$ | $\mathrm{sin}\left(\mathsf{\theta}+\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}+\mathrm{A}\right)\right)$ |

θ = 90° ∧ 0 < γ < 90° | $\mathrm{sin}\left(\mathsf{\theta}+\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}-\mathrm{A}\right)\right)$ | $\mathrm{sin}\left(\mathsf{\theta}-\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}+\mathrm{A}\right)\right)$ |

θ = 90° ∧ γ = 90° | $\mathrm{sin}\left(\mathsf{\theta}+\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}+\mathrm{A}\right)\right)$ | |

0 < θ < 90° ∧ 90° < γ < 180° | $\mathrm{sin}\left(\mathsf{\theta}-\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}-\mathrm{A}\right)\right)$ | $\mathrm{sin}\left(\mathsf{\theta}+\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}+\mathrm{A}\right)\right)$ |

θ = 90° ∧ 90° < γ < 180° | $\mathrm{sin}\left(\mathsf{\theta}-\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}+\mathrm{A}\right)\right)$ | |

0 < θ < 90° ∧ γ = 180° | $\mathrm{sin}\left(\mathsf{\theta}-\mathrm{A}\right)\mathrm{sin}\left(\mathsf{\Theta}\right)\mathrm{cos}\left(\mathsf{\Gamma}-\left(\mathsf{\gamma}-\mathrm{A}\right)\right)$ |

**Table 2.**Netlist for scheme on Figure 4 from Microcap.

Element | Exit Node | Entry Node |
---|---|---|

I_{ph} | 1 | 2 |

R_{sh} | 1 | 2 |

R_{d} | 2 | 1 |

R_{s} | 2 | 1 |

C_{D} | 2 | 1 |

C_{T} | 2 | 1 |

**Table 3.**I–V curve data (short circuit V = 0, I = I

_{sc}, open circuit V = V

_{oc}, I = 0, and the maximum power V = V

_{mp}, I = I

_{mp}points; the slopes of the I–V curve at the open circuit and short circuit points) of several solar cells [41].

Reference | ${\mathbf{V}}_{\mathbf{oc}}$ | ${\mathbf{I}}_{\mathbf{sc}}$ | ${\mathbf{V}}_{\mathbf{mp}}$ | ${\mathbf{I}}_{\mathbf{mp}}$ | T |
---|---|---|---|---|---|

Kennerud, 1969 | 0.420 | 0.804 | 0.316 | 0.698 | 330 |

Charles, 1981 | 0.536 | 0.1023 | 0.437 | 0.0925 | 300 |

Charles, 1981 | 0.524 | 0.561 | 0.390 | 0.481 | 307 |

Lo Brano, 2010 | 0.608 | 7.665 | 0.513 | 7.174 | 298 |

Cubas | 32.9 | 8.21 | 26.3 | 7.61 | 298 |

PSM-150 | 43.2 | 4.8 | 35 | 4.5 | 298 |

**Table 4.**1-Diode/2-resistors circuit model parameter values from different solar cells (with data from [41]).

No. of Experiment | Reference | ${\mathbf{R}}_{\mathbf{s}}$ | ${\mathbf{R}}_{\mathbf{sh}}$ | ${\mathbf{I}}_{0}$ | ${\mathbf{P}}_{\mathbf{max}}$ | $\mathbf{n}$ |
---|---|---|---|---|---|---|

1 | Kennerud, 1969 | 0.31606 × 10^{−1} | 9.9227 | 5.25 × 10^{−6} | 0.208 | 1.37 |

2 | Charles, 1981 | 0.92837 × 10^{−2} | 258.74 | 3.4 × 10^{−9} | 0.0385 | 1.51 |

3 | Charles, 1981 | 0.61621 × 10^{−1} | 14.423 | 7.74 × 10^{−8} | 0.186 | 1.72 |

4 | Lo Brano, 2010 | 0.31318 × 10^{−3} | 16.570 | 3.99 × 10^{−10} | 3.52 | 1.2867 |

5 | Cubas, 2014 | 0.83777 | 44.493 | 9.1 × 10^{−7} | 192.5 | 70.2 |

6 | PSM-150 | 0.59484 | 123.76 | 2.13 × 10^{−6} | 147 | 70.2 |

**Table 5.**Topocentric elevation angle (TEA) (corrected) and Azimuth angle (AA) (eastward from N) in Chelyabinsk on July 15, 2019 from 8:00 AM to 8:00 PM according to NREL MIDC SPA.

Date | Time | TEA (corrected) [°] | AA (eastward from N) |
---|---|---|---|

7/15/2019 | 8:00:00 | 26.062834 | −90.849174 |

7/15/2019 | 9:00:00 | 34.561596 | −78.02713 |

7/15/2019 | 10:00:00 | 42.627950 | −63.434196 |

7/15/2019 | 11:00:00 | 49.610008 | −45.947623 |

7/15/2019 | 12:00:00 | 54.564102 | −24.62469 |

7/15/2019 | 13:00:00 | 56.401837 | −0.100691 |

7/15/2019 | 14:00:00 | 54.579909 | 24.433072 |

7/15/2019 | 15:00:00 | 49.635714 | 45.775489 |

7/15/2019 | 16:00:00 | 42.656412 | 63.278413 |

7/15/2019 | 17:00:00 | 34.586795 | 77.880929 |

7/15/2019 | 18:00:00 | 26.079756 | 90.706762 |

7/15/2019 | 19:00:00 | 17.596336 | 102.636741 |

7/15/2019 | 20:00:00 | 9.518739 | 114.313934 |

**Table 6.**Allocated time intervals for modeling extremum seeking control systems in Chelyabinsk on July 15, 2019 from 8:00 AM to 8:00 PM according to NREL MIDC SPA.

Experiment | Reference | Time | TEA (corrected) [°] | AA (eastward from N) |
---|---|---|---|---|

1 | Kennerud, 1969 | 8:00:00 | 26.062834 | −90.849174 |

9:00:00 | 34.561596 | −78.02713 | ||

2 | Charles, 1981 | 11:00:00 | 49.610008 | −45.947623 |

12:00:00 | 54.564102 | −24.62469 | ||

3 | Charles, 1981 | 14:00:00 | 54.579909 | 24.433072 |

15:00:00 | 49.635714 | 45.775489 | ||

4 | Lo Brano, 2010 | 17:00:00 | 34.586795 | 77.880929 |

18:00:00 | 26.079756 | 90.706762 | ||

5 | Cubas | 9:00:00 | 34.561596 | −78.02713 |

10:00:00 | 42.627950 | −63.434196 | ||

6 | PSM-150 | 19:00:00 | 17.596336 | 102.636741 |

20:00:00 | 9.518739 | 114.313934 |

Experiment | ξ_{min}/ξ_{max}/Δξ/ | I_{ph_min}/I_{ph_max}/ΔI _{ph}/ | V_{min}/V_{max}/ΔV/ | P_{min}/P_{max}/ΔP/ |
---|---|---|---|---|

1 | 0.979/ | 0.783/ | 0.175/ | 0.05/ |

0.984/ | 0.787/ | 0.272/ | 0.12/ | |

0.005 | 0.004/ | 0.097 | 0.07 | |

2 | 0.949/ | 0.097/ | 0.239/ | 0.004/ |

0.958/ | 0.098/ | 0.333/ | 0.012 | |

0.01 | 9.847 × 10^{−4}/ | 0.094/ | 0.008/ | |

3 | 0.941/ | 0.528/ | 0.27/ | 0.055/ |

0.966/ | 0.542/ | 0.478/ | 0.128/ | |

0.025 | 0.014/ | 0.218/ | 0.073/ | |

4 | 0.968/ | 7.423/ | 0.418/ | 6.017/ |

0.997/ | 7.639/ | 0.588/ | 6.873/ | |

0.028 | 0.216 | 0.171/ | 0.9/ | |

5 | 0.974/ | 7.994/ | 24.061/ | 204.182/ |

0.981/ | 8.057/ | 35.101/ | 742.08/ | |

0.008 | 0.063/ | 11.04/ | 40/ | |

6 | 0.987/ | 4.736/ | 4.067/ | 278.727/ |

0.991/ | 4.758/ | 4.257/ | 288.441/ | |

0.005 | 0.022/ | 0.191/ | 32/ |

Experiment | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

k_{1} | 50 | 2 | 5 | 5 | 1 | 0.5 |

k_{2} | 100 | 2.5 | 6 | 6 | 2 | 0.6 |

k_{3} | 3 | 1 | 1 | 1 | 1 | 1 |

A_{sin_1}, deg. | 1 | 1 | 1 | 1 | 1 | 1 |

A_{sin_2}, deg. | 1 | 1 | 1 | 1 | 1 | 1 |

A_{sin_3} = A_{reg}, Ohm | 0.13 | 0.968 | 0.239 | 0.012 | 0.804 | 1.708 |

T_{hpf_1} | 10 | 1 | 1 | 1 | 10 | 10 |

T_{hpf_2} | 10 | 1 | 1 | 1 | 10 | 10 |

T_{hpf_3} | 1 | 1 | 1 | 1 | 1 | 1 |

h_{1} | 5 | 1.5 | 4 | 1 | 10 | 10 |

h_{2} | 10 | 2 | 3 | 1 | 10 | 10 |

h_{3} | 1 | 1 | 1 | 1 | 1 | 1 |

P_{in} = ΔP, W | 0.01 | 0.007 | 0.08 | 2 | 65 | 30 |

i_{out} | 0.065 | 0.002 | 0.05 | 1.75 | 3 | 0.5 |

ω_{1}, rad/s | 4.975 | 1.323 | 3 | 4.899 | 3.873 | 1.249 |

ω_{2}, rad/s | 9.95 | 1.5 | 5.196 | 5.916 | 7.937 | 1.639 |

ω_{3}, rad/s | 30 | 10 | 80 | 10 | 10 | 10 |

A_{in_1}, deg. | 80.542 | 1.176 | 21.564 | 13.683 | 8.757 | 1.401 |

A_{in_2}, deg. | 54.734 | 1.58 | 13.282 | 19.704 | 35.03 | 2.018 |

A_{in_3} = A_{sin_3}, Ohm | 0.13 | 0.968 | 0.239 | 0.012 | 0.804 | 1.708 |

K_{4}_{→2_1} | 1.242 | 1.512 | 0.072 | 1.827 | 1.827 | 1.827 |

K_{4}_{→2_2} | 1.827 | 0.989 | 0.301 | 1.827 | 1.827 | 1.827 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kirpichnikova, I.M.; Sologubov, A.Y.
Multivariable Control of Solar Battery Power by Extremum Seeking: Starting from Linear Analysis. *Machines* **2019**, *7*, 64.
https://doi.org/10.3390/machines7040064

**AMA Style**

Kirpichnikova IM, Sologubov AY.
Multivariable Control of Solar Battery Power by Extremum Seeking: Starting from Linear Analysis. *Machines*. 2019; 7(4):64.
https://doi.org/10.3390/machines7040064

**Chicago/Turabian Style**

Kirpichnikova, I. M., and A. Yu. Sologubov.
2019. "Multivariable Control of Solar Battery Power by Extremum Seeking: Starting from Linear Analysis" *Machines* 7, no. 4: 64.
https://doi.org/10.3390/machines7040064