Bond graph based control of a solar array

Abstract

In this paper, the control of a solar array is proposed. The bond graph methodology is applied in order to get the structure control law for a solar array connected to the electrical network. For this, the different models of the complete system are presented. The solar array is built up by considering the individual solar panels. The inverter bond graph model is used for the development of the proposed control law. This model is inverted graphically to get the control structure. Also, the Park transformation concept is adapted to handle the control structure. This allows us to get a different control structure. Two different controls are considered in the proposed control structure, with their robustness tested for different power factors. The control law is tested using a real-time simulator. For this test, the rapid control prototyping concept is used. The obtained results demonstrate the viability of the proposed control law.

Keywords

Bicausality bond graph inverter solar cell

1. Introduction

Photovoltaic energy is one of the most widely used renewable energy sources around the world; it is most commonly used to generate power for domestic use. Photovoltaic arrays are installed on many houses, and also in rural communities near big cities.

There are basically two different ways of connecting the photovoltaic array: stand-alone and network-connected. The configuration depends on the application itself. This means that if the photovoltaic array is used in a rural place, it will be installed in a stand-alone configuration, and batteries are needed. On the other hand, if the photovoltaic array is installed in a city or near to the power network, this can be configured as a network-connected array. The network-connected configuration accounts for over 99% of the installed photovoltaic capacity.¹ This is the reason why this paper focuses on the network-connected configuration.

The bond graph methodology consists of subsystems linked together by half arrows, representing power bonds.^2–5 These exchange instantaneous power at places called ports. The power variables are forced to be identical when two ports are connected; the power variables are assumed to be functions of time. The different power variables are classified using a universal scheme, called either effort e(t) or flow f(t). Their product P(t) = e(t) ×f(t) is the instantaneous power flowing between the ports. The main advantages of the bond graph tool for modeling purposes are: It represents a unified graphical language with a physical insight to energy dissipation and storage in the dynamic elements, the visualization of the causality allows obtaining the mathematical expression of the model and others.⁶

Different models for the solar cell in the bond graph have been developed.^7–10 In the first of these three references, the model is basically the same. It is developed from the equivalent electrical circuit of a solar cell, and in each of these cases the causality problem is solved from a different perspective. In Tapia Sánchez and colleagues’ paper,¹⁰ a more complete bond graph model is presented, and the physical structure of the solar cell is developed, including the thermal part.

In this paper, the model reported by Andoulsi and colleagues⁸ and Roboam and colleagues⁹ is considered. This is because it is not required to have a very detailed model of the solar array, since this contribution is focused in the control part of the completed system. However, a more detailed model can be considered, as the presented by Tapia Sánchez and colleagues,¹⁰ which can be a good alternative when the shadow effect is considered.¹¹

On the other hand, the bond graph model of a three-phase inverter has been addressed before.^12,13 Each model has been used for different applications. They have the particularity of being established with an electronic switch (MTF) for modeling the three commutation cells of the power converter, taking into account phase-to-neutral voltages. In this paper, the three-phase electrical inverter, which is modeled with only two electronic switches¹⁴ and takes phase–phase voltages, is used. This model is general since it is valid for unbalanced or balanced operation conditions.

The use of a graphical methodology instead a traditional one allows us to apply different graphical techniques that facilitate the analysis, modeling, or control of different systems,^15,16 such as the control system, addressed in this contribution.

The concept of the graphical inversion (using the bicausality bond graph) to obtain the control law structure has been applied before.¹⁴ Nevertheless, in this paper the combination of the Parks transformation and the inverse bond graph model allows us to obtain a different control law structure. The control robustness is proved by simulations and an experimental essay.

The outline of the paper is as follows. In Section 2, the system is described. The bond graph models are presented in Section 3. In Section 4 the proposed control law is developed and its robustness tested. Section 5 shows the simulation and experimental results. Discussion is given in Section 6. Finally, in Section 7 the conclusion and suggestions for future work are given.

2. Solar array connection

Figure 1 shows the schema of a solar array connected to the electrical network.

Figure 1.

Schematic system.

The solar array delivers the primary power through the solar array filter. This power is sent to a converter (known as a chopper converter) in order to maintain the available power generated by the solar array. Another filter placed between the converter and the three-phase inverter is called the DC bus (direct current bus). In the DC bus, the objective is to maintain a constant voltage to be transformed into AC power (alternate current) via the power three-phase inverter. The introduction of a capacitor on the DC bus is necessary to allow decoupling between the harmonics generated by both converters.

The electronic power inverter can be considered as an amplifier. It is composed of three pairs of electronic switches (IGBTs). Each pair of switches is called a commutation cell.

As the output inverter contains harmonics, the use of a three-phase filter is required. There are different filter configurations, the basic one being known as an LR filter. This filter consists only of an inductance (L_f1) in series with a resistance (R_f1); it can be taken as a base for more complex filters, simply by adding components. Each filter configuration has its own characteristics and is used for specific requirements or analysis from the point of view of stability and harmonics analysis.

3. Individual models

In this section, the individual bond graph models for the three-phase inverter, the solar array, and the converter with its filter are presented.

3.1. Inverter with filter model

For the power converter, the model is taken from the work of Sánchez and colleagues.^14,17 The model presented in these references corresponds to a three-phase converter using an LC filter connection. An RL filter connection is considered. The difference in the chosen filter is due to an injector current behavior given to the power converter, unlike the voltage source behavior. Figure 2 shows the inverter with the three-phase filter.

Figure 2.

Inverter with filter bond graph model.

As with the model presented by Sánchez and colleagues,^{14, 17} this model shows only two phases (two MTF elements) due to the fact that phase-to-phase voltages are considered. The average behavior of the inverter is considered, as only the meantime impact of the inverter on the filter is of concern. The third phase of the filter (L_f3, R_f3) is considered as the voltage reference, making the voltages u_m1 and u_m2 appear in the bond graph model in the 0-junction between the inverter and filter. The equations describing the filter in the model are given in Appendix A.

3.2. Solar panel model

Two models are commonly used to describe the equivalent electric circuit of a solar panel: the one-diode model (Figure 3)¹⁸ and the two-diode model.^19–22 The most commonly used configuration that represents the electrical behavior of the P–N junction is the one-diode model. This is because such a configuration has a convenient balance between accuracy and simplicity.

Figure 3.

Electric circuit of a solar panel.

In terms of bond graph methodology,^3–6 the ideal solar cell model is shown in Figure 4.

Figure 4.

Ideal solar panel bond graph electric model.

The diode is represented by a nonlinear resistive element (RD). Then, the current–voltage relation becomes nonlinear. This model has been presented previously.^7–9

The two resistances R_s and R_sh (see Figure 3) have a direct impact on the solar cell characteristic responses. Actually, these resistances cause problems of resolution and computing time due to the dual choice of causality and algebraic loops between the resistances. This problem has been presented in detail elsewhere.^8,9 For Andoulsi and colleagues,⁸ the problem was solved with the addition of capacitance in parallel to the diode (Figure 5(b)). This capacitance represents the depletion region and diffusion of carriers, depending on the light and the temperature at the cell. For Madansure and colleagues⁷ and Roboam and colleagues,⁹ the series resistance was neglected (Figure 5(a)).

Figure 5.

Solar panel: (a) with resistance R_sh; (b) with capacitance.

The equations that describe the elements in the bond graph²³ of Figure 5(a) are as follows:

i_{pv} = i_{ph} - i_{D} - i_{R_{sh}}

(1)

\begin{matrix} i_{ph} = i_{cc} + Δ i \\ Δ i = i_{cc} ((Δ i_{sc} \frac{G}{1000} (T - 298)) + (\frac{G}{1000} - 1)) \end{matrix}

(2)

\begin{matrix} i_{D} = i_{cc} C_{1} \exp (\frac{v_{cell} - Δ v_{cell}}{C_{2} v_{co}} - 1) \\ Δ v = v_{co} Δ v_{co} (T - 298) \end{matrix}

(3)

where i_cc is the short-circuit current, v_co is the open-circuit voltage, i_D is the diode current, and Δ represents an increment. The constant C₁ and C₂ are defined by Equation (4):

C_{1} = (1 - \frac{i_{mp}}{i_{cc}}) \exp (\frac{- v_{mp}}{C_{2} v_{co}}); C_{2} = \frac{\frac{v_{mp}}{v_{co}} - 1}{\ln (1 - \frac{i_{mp}}{i_{cc}})}

(4)

where i_mp is the current at the rated power (P_max) and v_mp is the voltage at the rated power.

The generalized and ideal model of the solar cell is affected by the temperature (T) and the solar irradiance (G), which are the most important parameters in the behavior of the voltage (v_cell) and current (i_cell). Thus, the bond graph model of Figure 5(a) is modified to consider the temperature and irradiance condition by replacing the Sf source with an MSf source, as shown in Figure 6.

Figure 6.

Solar panel.

It is important to note that the MSf source is modulated by two signals, coming from the temperature and irradiance conditions. Also, the nonlinear RD element is modulated only by the temperature signal.

Depending on the structure of the solar panel, this provides different powers, which are between 150 W and 300 W for a conventional panel. In order to increase the power, a few solar panels need to be connected in series, as shown in Figure 7.

Figure 7.

Solar panel array.

The solar panel of Figure 6 is encapsulated in a submodel, represented by the block panels of Figure 7. The array shows the parallel connection of two arrays of nine panels connected in series. Each panel is joined to a common connection point by considering a capacitive effect (C-element) in their interconnections. It is important to notice that there are two C-elements in a derivate causality (C_p10 and C_p8). This is because the solar array imposes the current (flow) as output in the two parallel arrays. This allows increasing the current provided by the solar array. If more power is needed, more panels need to be connected to the array.

In order to verify the behavior of this model, a simulation is conducted by considering a temperature of 300 K and an irradiance of 1000 W/m². Table 1 shows the numerical parameters for a BP3160 solar panel.

Table 1.

Solar panel parameters.

i_cc	4.8 A	C ₁	1.29e-5
v_co	44.2 V	C ₂	82e-3
R_sh	8.7 Ω	Δi_sc	6e-4 A
Δv_co	3.6e-3 V	C_p	5e-5 F

Figure 8(a) and (b) show the current and the power curves, respectively. The curves are plotted with respect to the array output voltage, as usually made when a solar panel is considered. As the array is composed of 18 panels, and an individual panel has a power rating of 160 W (provided by the manufacturer at nominal conditions), the total power delivered by the array is 2880 W. However, the total power obtained in the simulation is 2750 W, at their maximum point. This difference remains within the power tolerance given by the manufacturer (power tolerance ±5%).

Figure 8.

Responses of the solar array.

The previous simulation shows the correct behavior of the solar panels array.

3.3. Solar array+filter+converter

The characteristic curve of the solar array makes it necessary to introduce a filter and a chopper converter in order to reduce the harmonics and to maintain the maximum power in the array, respectively. Then, the solar panel array shown in Figure 7 is encapsulated in a submodel. The filter and the chopper converter are added to obtain the model of the solar array as shown the Figure 9.

Figure 9.

Solar array + filter + chopper bond graph model.

4. Inverter control

The control of a three-phase inverter has been the purpose of study of many research papers. In this section the control structure is presented, which is obtained by using the inverse bond graph model concept²⁴ and the bicausality concept.^25,26

A current control is then developed for the power inverter. By considering the model presented in Figure 2, the inversion of the model is made in two stages.

Inversion for m₁ control signal. The causality is inverted from detector Df: i_f₁ to the MTF₁ using the bicausality inversion. This is made as follows: first, the detector Df: i_f₁ is replaced by a source SSf which imposes zero effort because an ideal sensor is considered. Then, the bicausality bonds are propagated to the MTF₁ element in which the modulated signal is inverted, which becomes an output signal in the inverse bond graph model. Figure 10(a) shows the bicausality applied to the model.

Inversion for m₂ control signal. As in the first stage, the causality is inverted from detector Df: i_f₂ to the MTF₂ using the bicausality inversion. Then, the same bicausality process is applied. Figure 10(b) shows the bicausality applied in this stage.

Figure 10.

Power converter inverse bond graph: (a) phase 1; (b) phase 2.

It is important to mention that the two stages can be applied simultaneously, and the inverse model can be obtained.¹⁴ However, the inversion is made here in two stages in order to show a simpler manner of implementing the technique. Then, the mathematical expressions are obtained for both inverse models.

In the 1₁-junction of Figure 10(a), the mathematical expression is given by Equation (5):

e_{5} - e_{9} - e_{15} - e_{18} - e_{12} - e_{21} = 0

(5)

with e₁₂= e₁₄= –(e₁₀+ e₁₇+ e₂₀). In Equation (5), the effort in the SSf:i_f₁ source is zero (e₂₁= 0), since this source does not impose any effort to the inverse system. The MTF element has the next relation: m₁= e₅/e₃. Then, in terms of the electrical variables, Equation (5) is represented by Equation (6):

m_{1} = \frac{1}{u_{s}} ({\hat{R}}_{f 1} i_{f 1} + {\hat{L}}_{f 1} \frac{d i_{f 1}}{dt} + v_{1} - {\hat{R}}_{f 3} i_{f 3} - {\hat{L}}_{f 3} \frac{d i_{f 3}}{dt} - v_{3})

(6)

where ${\hat{R}}_{fi}$ and ${\hat{L}}_{fi}$ are estimated parameters of the original system parameters. The second expression in the 1₂-junction of Figure 10(b), is given by Equation (7):

e_{8} - e_{16} - e_{19} - e_{10} - e_{13} - e_{22} = 0

(7)

with e₁₃= e₁₄= –(e₁₇+ e₂₀+ e₁₀), e₂₂ = 0, and considering m₂=e₆/e₄, the mathematical expression (represented in terms of the electrical variables) for the second control signal is given by Equation (8):

m_{2} = \frac{1}{u_{s}} ({\hat{R}}_{f 2} i_{f 2} + {\hat{L}}_{f 2} \frac{d i_{f 2}}{dt} + v_{2} - {\hat{R}}_{f 3} i_{f 3} - {\hat{L}}_{f 3} \frac{d i_{f 3}}{dt} - v_{3})

(8)

Equations (6) and (8) represent the control law in the open loop. So, in order to establish a closed loop control law, it is necessary to impose the dynamic errors for each phase (j) $ε_{ifj} = i_{fjref} - i_{fj}$ , as ${\overset{\cdot}{ε}}_{ifj} + k_{j} ε_{ifj} = 0$ where j = 1, 2, 3.

The k_j gains are selected in order to compensate the poles of the system in the open loop. By replacing i_f₁ for i_f_1ref–ε_if₁, i_f₂ for i_f_2ref–ε_if₂ and i_f₃ for i_f3ref–ε_if3 in Equations (6) and (8), and using the dynamics errors, the control law is given by Equations (9) and (10):

\begin{array}{l} m_{1} = \frac{1}{u_{s}} (({\hat{R}}_{f 1} - {\hat{L}}_{f 1} k_{1}) i_{f 1} + {\hat{L}}_{f 1} (\frac{d i_{f 1 r e f}}{d t} + k_{1} i_{f 1 r e f}) \\ + v_{1} - ({\hat{R}}_{f 3} + {\hat{L}}_{f 3} k_{3}) i_{f 3} - {\hat{L}}_{f 3} (\frac{d i_{f 3 r e f}}{d t} + k_{3} i_{f 3 r e f}) - v_{3}) \end{array}

(9)

\begin{array}{l} m_{2} = \frac{1}{u_{s}} (({\hat{R}}_{f 2} - {\hat{L}}_{f 2} k_{2}) i_{f 2} + {\hat{L}}_{f 2} (\frac{d i_{f 2 r e f}}{d t} + k_{2} i_{f 2 r e f}) \\ + v_{2} - ({\hat{R}}_{f 3} + {\hat{L}}_{f 3} k_{3}) i_{f 3} - {\hat{L}}_{f 3} (\frac{d i_{f 3 r e f}}{d t} + k_{3} i_{f 3 r e f}) - v_{3}) \end{array}

(10)

In Equations (9) and (10), the reference currents (i_f_1ref, i_f_2ref, i_f_3ref) and their respective derivatives are present to obtain the controller signals m₁ and m₂. In order to calculate the i_f_1ref, i_f_2ref, and i_f_3ref, the reference i_dref and i_qref using the Park transformation²⁷ are used as:

[\begin{matrix} i_{f 1 ref} \\ i_{f 2 ref} \end{matrix}] = \sqrt{\frac{2}{3}} [\begin{matrix} \cos (θ) & - \sin (θ) \\ \cos (θ - \frac{2 π}{3}) & - \sin (θ - \frac{2 π}{3}) \end{matrix}] [\begin{matrix} i_{dref} \\ i_{qref} \end{matrix}]

(11)

The i_f_3ref will be formulated by the equilibrium consideration of a three-phase system. This mean that i_f_3ref = –i_f_1ref–i_f_2ref. Then, from Equation (11) the i_f_1ref will be given by Equation (12):

i_{f 1 ref} = \sqrt{\frac{2}{3}} (\cos (θ) i_{dref} - \sin (θ) i_{qref})

(12)

Meanwhile its derivative is as follows:

\frac{d i_{f 1 ref}}{dt} = \sqrt{\frac{2}{3}} (- \sin (θ) i_{dref} - \cos (θ) i_{qref}) \overset{•}{θ}

(13)

And for the i_f2ref it is as follows:

i_{f 2 ref} = \sqrt{\frac{2}{3}} (\cos (θ - \frac{2 π}{3}) i_{dref} - \sin (θ - \frac{2 π}{3}) i_{qref})

(14)

\frac{d i_{f 2 ref}}{dt} = \sqrt{\frac{2}{3}} (- \sin (θ - \frac{2 π}{3}) i_{dref} - \cos (θ - \frac{2 π}{3}) i_{qref}) \overset{•}{θ}

(15)

Figure 11 shows the block diagram of the proposed control law.

Figure 11.

Control law block diagram.

In Figure 11, Equations (11) to (15) are implemented into the “calculation i_f_1ref” and “calculation i_f_2ref” blocks, respectively. The current references (i_dref, i_qref) are the inputs in the control and set the exchanger current to the load or network. In the block diagram, one can see that references for the third phase (i_f_3ref, di_f_3ref) are formulated from the other two phases. Nevertheless, three current sensors (i_f₁, i_f₂, i_f₃) are used.

Note: It is important to mention that the development of the control law presented in this section is based on proportional gains for the controller (k₁, k₂, k₃). Nevertheless, these gains can be replaced by any different control, as for example a proportional–integral (PI) or proportional–integral–derivative control (PID). The problem with using a more complete or complex control is that a more complex calculus of the controller parameters is needed. However, the whole system (model and control) will be more robust. At this point, a discussion could be made about the complexity and functionality of the system. This discussion will be presented in Section 6. It is important to mention that in the block diagram of Figure 11, the $\hat{R}$ and $\hat{L}$ elements correspond to the ${\hat{R}}_{fi}$ and ${\hat{L}}_{fi}$ , respectively, from Equations (9) and (10), and they are considered to have the same numerical value.

If the proportional controller is changed by a proportional–integral one, Equations (9) and (10) are reformulated as shown in Equations (16) and (17), respectively:

m_{1} = \frac{1}{u_{s}} (({\hat{R}}_{f 1} - {\hat{L}}_{f 1} (P I_{1})) i_{f 1} + {\hat{L}}_{f 1} (\frac{d i_{f 1 ref}}{dt} + (P I_{1}) i_{f 1 ref}) + v_{1} - ({\hat{R}}_{f 3} + {\hat{L}}_{f 3} (P I_{3})) i_{f 3} - {\hat{L}}_{f 3} (\frac{d i_{f 3 ref}}{dt} + (P I_{3}) i_{f 3 ref}) - v_{3})

(16)

m_{2} = \frac{1}{u_{s}} (({\hat{R}}_{f 2} - {\hat{L}}_{f 2} (P I_{2})) i_{f 2} + {\hat{L}}_{f 2} (\frac{d i_{f 2 ref}}{dt} + (P I_{2}) i_{f 2 ref}) + v_{2} - ({\hat{R}}_{f 3} + {\hat{L}}_{f 3} (P I_{3})) i_{f 3} - {\hat{L}}_{f 3} (\frac{d i_{f 3 ref}}{dt} + (P I_{3}) i_{f 3 ref}) - v_{3})

(17)

where PI₁, PI₂ and PI₃ are represented by $P I_{i} = k_{pi} (1 + \frac{1}{T_{i} s})$ . The simulation control schema of Equations (16) and (17) is presented in Figure 12.

Figure 12.

Control law block diagram considering a proportional–integral controller.

In Figure 12, the “calculation i_f_1ref” and “calculation i_f_2ref” blocks are the same as in Figure 11. Also, the k_pi and T_i gains for the proportional–integral controller for each branch are shown in Figure 12.

4.1. Inverter control

A simulation is carried out in order to validate the proposed control law. Figure 13 shows the simulation scheme.

Figure 13.

Simulation scheme for the current control.

The inverter is connected to the network (represented by the three MSe sources). In the DC bus, a source of current (Sf) and a parallel capacitor (C) are considered. The block placed between the filter and network allows for calculating the active (P) and reactive (Q) power sent by the inverter to the network. The traditional expressions for the power calculation are used in this block. Two delay blocks are added in the control signal in order to consider a practical delay time.

The references i_qref and i_dref are set to 10 A at t = 0.3 s, and 15 A at t = 0.5 s, respectively. The values of the filter in the three phases are the same, i.e. R₁= R₂= R₃= 0.002 Ω and L₁= L₂= L₃= 1 mH. In the simulation, the estimated parameters of the implicit values (in the control) matched exactly with the system parameters. This means that ${\hat{R}}_{f}$ ₁ = ${\hat{R}}_{f}$ ₂ = ${\hat{R}}_{f}$ ₃ = 0.002 Ω and ${\hat{L}}_{f 1} = {\hat{L}}_{f 2} = {\hat{L}}_{f 3} = 1 mH$ . Figure 14 shows the results.

Figure 14.

Proportional control responses: (a) three-phase voltages; (b) three-phase currents; (c) active (P) and reactive (Q) power.

In Figure 14(a) the output voltage of the converter is shown. There is no output current (Figure 14(b)) from t = 0 s to t = 0.2 s, and thereafter from t = 0.2 s to t = 0.4 s the current increases when i_qref increases. The same behavior takes place when the i_dref increases at t = 0.4 s. Besides the increase of the current, it can be observed from Figure 14(c) that i_q and i_d references control the active and reactive power exchanged to the network, respectively. For the simulation, the values of k₁ = k₂ = k₃ = 1200 are considered.

A second simulation is carried out in order to compare the control law using the proportional and the proportional–integral controller. For the proportional, the same gains as previously have been considered; for the proportional–integral, k_p₁= k_p₂= k_p₃= 4200 and T₁= T₂= T₃= 4.66e-3 have been used. The gains in both controllers have been calculated in order to have the same time response.

Figure 15 shows the current i_f₁ in both models.

Figure 15.

Comparison current response.

In Figure 15, the current i_f₁ matches in both responses, and only a small difference is shown in the transient change when the i_dref is activated. It can be observed that the proportional–integral (PI) response presents a greater value in this transient change.

Despite this difference, the result in both controllers is the same.

4.2. Voltage control

The inverter control is not complete until a regulation in the DC bus has been made. This means that the voltage in the DC bus needs to be maintained at a constant value while the current input changes (current coming from the solar array).

The voltage control loop is made taking the i_qref as the correction gain (k_c) multiplied to the error, as shown in Equation (18):

i_{qref} = k_{c} (u_{sref} - u_{s})

(18)

As the derivative current is needed as a reference, this means that di_qref is required. Then, the derivative (Equation (18)) is given as:

\frac{d i_{qref}}{dt} = - \frac{k_{c}}{C_{bus}} (I_{hac} - i_{m})

(19)

In Equation (19) the I_hac is the output current chopper and i_m is the converter current input.

The reference and its derivative are applied directly into the references di_f_2ref/dt and di_f_1ref/dt as:

\frac{d i_{f 1 ref}}{dt} = \sqrt{\frac{2}{3}} [(- \sin (θ) i_{dref} - \cos (θ) i_{qref}) \overset{•}{θ} + \cos (θ) \frac{d i_{dref}}{dt} - \sin (θ) \frac{d i_{qref}}{dt}]

(20)

\frac{d i_{f 2 ref}}{dt} = \sqrt{\frac{2}{3}} [(- \sin (θ - \frac{2 π}{3}) i_{dref} - \cos (θ - \frac{2 π}{3}) i_{qref}) \overset{•}{θ} + \cos (θ - \frac{2 π}{3}) \frac{d i_{dref}}{dt} - \sin (θ - \frac{2 π}{3}) \frac{d i_{qref}}{dt}]

(21)

Figure 16 shows the control law modification.

Figure 16.

Modification of the control law.

Equations (20) and (21) are implemented in two blocks of calculations. Equations (18) and (19) are implemented outwardly, as shown in the Figure 16.

4.3. Control robustness

Since in the control law the parameters $\hat{R}$ and $\hat{L}$ are explicit, these parameters are taken as the approach for the original ones (R_fi and L_fi). Then, a robustness study is made in order to verify the effectiveness of this control.

The inverter is connected to a three-phase load composed of a parallel inductance and resistance (RL), as shown in Figure 17.

Figure 17.

Simulation scheme for the poles.

The inverter needs to deliver the available power from the solar array, modeled in this case as an ideal current source, to the load. In this context, the load varies from a power factor of 0.5 to 1 in order to show the control robustness. Besides, as the current solar array varies depending on the available irradiation, the current solar array (Sf:I_hac) is varied from 1 A to 10 A, while the voltage of the DC bus is set to 1000 V. Then, the pole plot is taken directly from the simulator (Figure 18).

Figure 18.

Three-dimensional pole plot for different values of the load and the current variation.

When the load has a unit power factor, the poles of the system and controller are closely located among them (red color in Figure 18), and the current variation does not have a significant impact on the pole place. As the power factor decreases, the poles start to move away in agreement with the current variation. The effect of the current increase has a direct impact in the imaginary axis position of some poles, presenting a greater separation of these poles with respect to the real axis. In other words, with higher current, higher damping will be present in the system. This means that the transient oscillation will be more significant for a higher current. Even though there are damping responses (in the transient state) for a higher current, the poles positions show that this damping variation is not very significant since the poles keep closer to their original position.

These results verified the robustness of the proposed control law.

4.4. Solar arrangement control

The characteristic curve of the solar array makes it necessary to introduce a filter and a chopper converter. In order to maintain the available power delivered by the variant characteristic of the solar array, different techniques can be applied to the model, i.e. MPPT (maximum power point tracking) using an adaptive fuzzy logic control,²⁸ genetics algorithms²⁹ or advanced techniques,³⁰ and others. A simple proportional–integral controller will be used to maintain as constant the voltage delivered by the solar array. The simplicity of the control is because an output filter (C_a and L_pv elements) is used in the solar array, allowing not only filtering the solar cell array, but also to have a dynamical element that allows controlling the behavior of the system.

The solar array, the filter, and the converter are shown in Figure 19. It is important to mention that the capacitance C_a does not represent the capacitive effect of the complete solar array. This capacitance corresponds to the output filter and inside the so-called configuration block, each solar panel has its own capacitive effect, as shown in Figure 7.

Figure 19.

Solar array, filter, and chopper converter connection.

The filter and control numerical parameters are shown in Table 2. The k_pv and k_iv gains have been calculated in order to have a 0.5 s response in the control. The known Ziegler and Nichols method³¹ has been used in order to have the desired time response.

Table 2.

Solar array control parameters.

k_pv	3.64e-3	k_iv	53.8e-3
L_pv	0.01 mH	C_a	10e-6 F

In the next section we present the simulation and practical results of the complete system.

5. Simulation and experimental results

In this section the simulation and practical results are presented.

5.1. Simulation of the complete system

For the simulation, the single models are shown before they are connected, as in Figure 1, in order to simulate the complete system. Figure 20 shows the complete bond graph model.

Figure 20.

Complete simulation schema.

The solar array is connected to the network to deliver the total power available due to the irradiance and temperature influence. The same characteristics presented in Section 3.3 for the solar array are considered. The control law with the proportional controller is considered in this simulation.

The conditions taken into account for this simulation are as follows:

Initial condition is considered only for the DC bus capacitance;

The DC bus voltage reference is set to 1000 V; and

The average irradiance is considered as (1000 W/m²) when the simulation starts (at 2 s), and when the simulation finishes (after 2 s). A decrease of the irradiance from 1000 W/m² to 600 W/m² is made from 2 s to 4s.

The voltage bus is shown in Figure 21(a). The transient changes are presented when the irradiance reference (Figure 21(d)) decreases. The same effect is visualized in the solar array current I_pv (Figure 21(b)) and the i_f₁ current is supplied by the inverter to the network. These two currents decrease during the irradiation depletion, opposite to the voltage which is maintained in the fixed reference.

Figure 21.

System simulation: (a) DC bus; (b) array current; (c) current phase 1; (d) irradiance.

The total power available in the solar array needs to be supplied to the network, while the reactive power needs to be reduced. Figure 22 shows that the active power (P) gets closer to the available solar array power (P_pan), and the reactive power is set to zero.

Figure 22.

System simulation power curves.

These results confirm the performance of the control in the complete solar array system connected to the network.

In the next subsection, the obtained practical results are presented in order to validate the proposed control law.

5.2. Experimental results

As this article is centered on the development of the control law, for the practical results a different system configuration is used.

The rapid control prototyping (RCP) concept using a real-time simulation (Opal OP5600) is used in order to implement the control law. An actual inverter (LabVolt 8857-12, Edutelsa, tecnología para la educación. Saltillo, Coahuila, México)) is connected to a three-phase source (LabVolt 8525-22), which represents the electrical network, through a three-phase inductor module (LabVolt 8374-A2). A DC voltage source connected in series with a resistance is used in order to represent the current supplied by the solar array. Figure 23 shows this practical implementation.

Figure 23.

Practical implementation.

In Figure 23, the connection between the real-time simulator (RTS) and the actual components is made through the data acquisition OP8660. This module allows sensing the three-phase currents and voltages between the network and the filter for the active and reactive power calculation. The DC bus current and voltage between the series resistance and the inverter (which are considered as the solar array) are also sensed by the data acquisition module.

The characteristics for each module are as follows:

DC voltage source: variable voltage from 0 V to 120 V, 25 A;

Series resistances: parallel resistance 240 Ω/0.5 A, 120 Ω/1 A, 60 Ω/2 A, 30 Ω/4 A. 1140 W – 120 V CA/CD;

Inverter: DC bus 420 V, 10 A; switching commutation 0/5 V, 0–20 kHz;

Filter: 14.5 mH, 7 A, 50/60 Hz; and

Three-phase source: 0–120/208 V, 15 A.

For the practical implementation, the conditions of the test are as follows:

The inverter is always connected to the three-phase voltages;

The voltage in the three-phase voltage has a magnitude of 80 V peak-to-peak;

When the test starts, the DC bus current inverter is near zero (around 0.5 A);

The DC voltage source is varied in order to change the inverter input current. This is made approximately at t = 7.5 s, t = 19 s, and t = 28 s, and during 6 s, 5 s, and 4 s, respectively; and

The DC bus voltage in the inverter is set to 110 V during the test, and it does not change.

Figure 24 shows the inverter input current (DC bus). The current curve response has considerable noise. This is because the DC source is not an ideal source, and also due to the selected values for the current and the resistance, respectively. It can be observed that current changes (in the three intervals) close to 1.8 A, if their average value is considered. The current change is due to the voltage magnitude variation in the DC source and not due to resistance change.

Figure 24.

Practical I_pv current.

The power delivered by the inverter to the network (three-phase voltage) is shown in Figure 25. The total active power is delivered to the three-phase voltages and the profile of this power varies in the same ratio as the currents I_pv (see Figure 25(a)).

Figure 25.

Practical power delivered by the inverter: (a) active power; (b) reactive power.

Otherwise, the reactive power (Figure 25(b)) is set to zero, because no reactive power is delivered by the inverter. The reactive power is not completely zero; however, it maintains a constant profile during the transitions of I_pv. The two powers in Figure 25 show considerable noise due to the DC bus and the harmonics of the three-phase voltage source.

The DC bus voltage needs to be maintained at the set reference V_busref = 110 V. Figure 26 shows the actual V_bus; its average value is 110 V.

Figure 26.

Practical DC bus voltage.

The experimental results shown in this subsection, in terms of conditions and configuration, do not match exactly with the simulation results presented in the last subsection. This is due to the fact that the actual sources and elements needed are not available in the laboratory. As the objective is to validate experimentally the control law, this different system condition (as compared to the simulation) allowed us to do so. The results shown in both systems (simulation and experimentation) are similar in terms of conditions (power changes), instead of magnitudes.

6. Discussion

This research work deals with the development of a control law applied to a power electronic converter. The development of the control law considers the model, creating a graphical inversion to get the control law. The Park transformation, together with its derivative, allowed us to develop a different control law that has not been reported in the literature before.

The proportional and proportional–integral controls used in the simulation results show that different controllers can be applied to the proposed control structure. Then, as noted before, the problem is that using a more complete or complex control results in a more complex calculus of the controller parameters. It is not necessary to show that it is easier to implement the control law using a proportional controller instead of a proportional–integral–derivative, since their structures (when compared) required different numbers of elements. Therefore, it will be easier to implement the control in a microprocessor or develop a target with fewer components. The aforementioned considerations need to be evaluated, as in many applications simplicity or practicality are more important than robustness, or maybe the opposite holds. The results presented in this research show that the proposed control achieves both simplicity and robustness.

The proposed control laws still need to be tested in different scenarios, such as unbalanced, fault conditions, and harmonic distortion. The investigation of the proposed control is already being progressed in this direction.

7. Conclusions and future work

A control law for a solar array system has been proposed. The individual components of the whole system have been developed using the bond graph methodology. Specifically, the solar array model was formulated through the combination of solar panels. This allows increasing the total power delivered by the array, and it can also be useful for a specific analysis (e.g., fault, non-uniform solar irradiance distribution).

A different control structure has been proposed in this paper. The bicausality concept applied to the inverter model allows us to graphically invert the model. As the reference currents in the proposed control are in the d–q reference frame, the use of the Park transformation was necessary to combine the inverse control structure and the active and reactive power regulation. This was made by means of a derivation of the Park transformation. Two different controls (proportional and proportional–integral) have been compared. It was demonstrated that the two structures are similar and only depend of the selected application. This was mentioned in the discussion section. Further, the robustness of the control was verified, since the control structure strongly depends on the system parameters, which are implicitly found in their structure.

The complete system was simulated by considering the elements and the inverter control. The expected results have been obtained from the conducted simulations. A practical test through real-time simulation allowed us to validate the proposed control law. The rapid control prototyping concept was used; this means that the control law has been implemented in the simulator, with an actual inverter, filter, and sources. The practical results obtained allowed successful validation of the proposed control law.

As a future work, the control laws will be tested in different scenarios (i.e., unbalanced conditions, fault conditions, and harmonic distortion). A more complete filter (LC) will be used in order to eliminate the harmonic distortion. Also, the shadow effect can be included using a detailed model in order to test this control law or develop another one.

Footnotes

Appendix A

The mathematical model that describes the inverter with the three-phase filter of Figure 2 is as follows. In i_f₁ 1-junction (bond 21):

(A.1)

e_{7} - e_{9} - e_{15} - e_{18} - e_{12} = 0

with $e_{12} = e_{14} = - (e_{10} + e_{17} + e_{20})$ .

In terms of the electrical variables, Equation (A.1) is represented by the following:

(A.2)

u_{m 1} - (v_{9} - v_{10}) - R_{f 1} i_{f 1} - L_{f 1} \frac{d i_{f 1}}{dt} + R_{f 3} i_{f 3} + L_{f 3} \frac{d i_{f 3}}{dt} = 0

where v_i with i = 9, 10 corresponds to the voltage in bond i, and u_m₁ corresponds to the voltage in bond 7.

The i_f₂ 1-junction (bond 22) is given by Equation (A.3):

(A.3)

e_{8} - e_{16} - e_{19} - e_{11} - e_{13} = 0

with $e_{13} = e_{14} = - (e_{10} + e_{17} + e_{20})$ .

Then, in terms of the electrical variables, Equation (A.3) is represented by Equation (A.4):

(A.4)

u_{m 2} - (v_{11} - v_{10}) - R_{f 2} i_{f 2} - L_{f 2} \frac{d i_{f 2}}{dt} + R_{f 3} i_{f 3} + L_{f 3} \frac{d i_{f 3}}{dt} = 0

where v_i with i = 10, 11 corresponds to the voltage in bond i, and u_m₂ corresponds to the voltage in bond 8.

Equations (A.2) and (A.4) do not consider the detectors: Df:i_f₁, Df:i_f₂ and Df:i_f₃.

In the last expressions, u_m₁ and u_m₂ correspond to the phase-to-phase voltage. Additionally, if the numerical values of the three inductances and resistances in the filter are assumed to be equal, Equations (A.2) and (A.4) can be expressed in the matrix form:

(A.5)

[\begin{matrix} u_{m 1} \\ u_{m 2} \end{matrix}] = [\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} L_{f} \frac{d i_{f 1}}{dt} + R_{f} i_{f 1} \\ L_{f} \frac{d i_{f 2}}{dt} + R_{f} i_{f 2} \end{matrix}] + [\begin{matrix} v_{9} - v_{10} \\ v_{11} - v_{10} \end{matrix}]

It is important to note that Equation (A.5) is obtained because i_f₃ = –i_f₁– i_f₂, from the 0-junction between bonds 12, 13, and 14. Also, the inductances (L_f₁, L_f₂, L_f₃) and resistance (R_f₁, R_f₂, R_f₃) are considered with an equal numerical value; then they are replaced only by the L_f and R_f notation, respectively, in Equation (A.5).

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Author biographies

R. Tapia-Sánchez received his PhD from Ecole Centrale de Lille, Lille, France, in 2011. He joined the Faculty of Electrical Engineering at the University of Michoacan, Morelia, Mexico, in 2011, where he is currently professor in the Division of Graduate Studies. His research interest focuses on the design and control of renewable energy systems.

A. Medina-Rios obtained his PhD from the University of Canterbury, Christchurch, New Zealand, in 1992. He has worked as a Post-Doctoral fellow at the Universities of Canterbury, New Zealand (1 year) and Toronto, Canada (2 years). At present he is a staff member of the Facultad de Ingenieía Eléctrica, UMSNH, Morelia, Mexico, where he is a professor in the Division for Postgraduate Studies in Electrical Engineering. His research interests are in the dynamic and steady-state analysis of power systems.

A. Ramos-Paz was born in Morelia, Michoacan, Mexico, in 1975. He received his B.Sc. degree in Electrical Engineering and the M. Sc. and Ph. D. degrees in 2001, 2002 and 2007, respectively, all from the UMSNH. Currently, he is an Associate Professor at the División de Estudios de Posgrado, FIE, UMSNH, Morelia, México. His research interests are on power systems analysis and modeling parallel processing.

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