MPC Algorithm With Reduced Computational Burden and Fixed Switching Spectrum for a Multilevel Inverter in a Photovoltaic System

Renewable energy has gained significant attention of researchers in the last years, mainly due to the importance of using unlimited energy sources to supply homes, industries, cities and countries. In this context, this document focuses on the solar injection by employing a neutral point clamped (NPC) topology together with utilization of a maximum power point tracking (MPPT) and space vector modulation (SVM) techniques. Model predictive control (MPC) is employed to manage the currents and track their references. The proposed algorithms do not employ a cost function to decide which voltage to apply resulting in a spread frequency spectrum, and instead, a concentrated SVM spectrum is imposed. Notwithstanding, the DC link capacitors voltage balance is ensured and the computational burden is notably reduced as compared to traditional Finite Set Model Predictive Control (FS-MPC). Nevertheless, the consistent results are a consequence of the critical analysis that shows the feasibility of the proposal and guarantees the good performance of the entire system in simulated and experimental platforms.


I. INTRODUCTION
Governments around the globe have been promoting the use of renewable and clean energies, being the most popular sources the solar and wind energy, and also promoting electric mobility as a part of the carbon dioxide emission reduction, [1]. Therefore, investigations that help to improve and make clean energies more efficient are of particular interest. The sun irradiance is a lifetime unlimited source that may be employed to supply the growing number of automatic devices used in this electric-dependent world. On this line, many researchers have been working towards improved renewable energy injection systems. The improvements refer to the The associate editor coordinating the review of this manuscript and approving it for publication was Mauro Gaggero . control algorithms, the power topology and the whole system efficiency [2]. Solar irradiation can be harvested in all around the globe, although the intensity and temperature are different in every single place and also change during the day. Those two parameters -the irradiance and temperature-are critical in solar photovoltaic (PV) systems, because the harvested amount of energy depends directly of these two variables.
To increase the efficiency of the photovoltaic systems, several MPPT algorithms have been proposed and the most used ones are perturb and observe (P&O) [3], incremental conductance (InC) [4] and measuring cells based (MCB) [5]. These MPPT algorithms have been implemented in (i) DC/DC converters [6], where the inverter DC link voltage is controlled and independent of the PV MPPT voltage, or (ii) directly on the inverters control strategy [7], where the DC link voltage is imposed by the MPPT, but the additional DC/DC stage is removed and therefore the total efficiency increases.
Nowadays, monofacial solar panels dominate the market [8]; however bifacial photovoltaic (BPV) technology has been shown to significantly increase the performance of photovoltaic modules [9] and is increasingly being considered in photovoltaic systems, due to their enhanced generation capability in terms of watts per area [10].
To be able to inject the power coming from the array of BPV cells, topologies such as the cascaded H-bridge (CHB) [11], active front end (AFE) [12], and neutral point clamped (NPC) [13] have been proposed, among others. The control algorithm is to be adapted for every topology, where the most common are: (i) linear control, such as proportional-resonant control, repetitive control and classic proportional-integral (PI) controllers, [14], (ii) nonlinear based control and nonlinear feedback linearization [15] and (iii) model predictive control, as finite-set MPC, and deadbeat control (DB), [12], [16], being MPC of particular interest in this work.
This work presents a control of a solar plant directly connected to the power grid, which uses a NPC multi-level inverter, where the proposed control strategy is a combined linear control and predictive control scheme with extended horizon and fixed switching frequency to obtain a low fluctuation of the neutral point (imbalance of the DC capacitors) and a low distortion in the AC currents. To operate the photovoltaic array at the maximum power point (M PP ), a linear discrete PI controller is used to regulate the DC voltage whose reference is generated by the MPPT algorithm. On the NPC AC side, to control the current i abc (t), a variation of the FS-MPC is used, named fast-finite-set MPC (FFS-MPC), whose reference is generated by the DC link voltage linear controller, including a unitary power factor (PF) reference. The FFS-MPC is easy to implement, since it becomes a modified version of the standard FS-MPC, and the computational burden is reduced, requiring no specific hardware.
In conventional MPC, a restricted model is minimized for a finite time horizon, whose minimization delivers the control output. Due to the rapid dynamics of electrical systems, a relatively high sampling frequency is generally required, which leaves little time available for the acquisition of the variables and the calculation of the optimization problem in real time. An alternative to the conventional MPC proposed in [17], which only has one cost function, is the finite set model predictive control that separates the cost functional in order to reduce the computing time [18]. In this algorithm, control variables are predicted for each valid switching state; then, a cost function is evaluated and, finally, the state that minimizes the cost function is applied throughout the sampling period. Thus, FS-MPC has become an interesting approach, thanks to its advantages, such as rapid dynamic response, simple inclusion of nonlinearities, restrictions, implementation in multi-objective control systems, etc.
In this work, the mathematical model associated with the various elements of the proposed scheme is obtained in the form of state variable equations. Next, a discrete approximation of the state equations in the time domain of the system 77406 VOLUME 8,2020 is deduced in order to develop and propose a discrete mixed control strategy. Then, the proposed control strategy is simulated and applied in a photovoltaic system in PSim. Finally, several experimental tests are carried out to validate the model and the proposed control algorithm and the key design guides are also provided. The most relevant conclusions of the work are also included.

II. MATHEMATICAL MODEL
A photovoltaic system connected directly to the network is presented in FIGURE 1. The scheme has a large array of bifacial photovoltaic modules (BPVs) and 2 capacitors (C p , C n ) on the DC side of the multilevel inverter and a first order filter (R-L) connected to the simplified AC network (v s ). In the following, the mathematical model on the AC side, DC side of the converter, the three levels NPC converter, and the BPV array model are developed.

A. MODEL AT THE AC SIDE OF THE NPC
Applying the voltage Kirchhoff law on the AC side FIGURE 1, the following relationship is obtained: where v abc is the output voltage of the converter, i abc is the injected current, and v abc s is the grid voltage. Using the Clarke power invariant transformation and rewording, the model in αβ axis is: where the power invariant transformation is:

B. MODEL AT THE AC SIDE OF THE NPC
The NPC inverter has 3 legs (one leg per phase, k = {a,b,c}) and each leg has 4 switches (j = {1, 2, 3, 4}), the combinations of these switches are limited (so as not to damage the equipment) and can only generate 3 possible states per phase. The states (γ k ) of each leg k The 19 possible normalized output voltages (v j , j = {0, 1, . . . , 18}), where r 3 = √ 2/3, which depend on the switching state of each section (γ k ) and therefore on the 27 possible states (ST). The voltage 0 can be generated by 3 possible states (-1, 0, 1), the short voltages (1, 2,. . . , 6) can be generated by 2 combinations each one and the medium (7,. . . , 12) and long voltages (13,. . . , 18) are generated only by one state.

C. MODEL AT THE DC SIDE OF THE NPC
The inverter has two capacitors (C p , and C n ) at the DC side, but to simplify the model both capacitors will have the same value (C). Using the current Kirchhoff law, the currents i cp VOLUME 8, 2020 and i cn on each capacitor are given by: The top and bottom currents (i p and i n ) on the DC side of the NPC depend upon the state of the switches of each leg upon the output currents (i a , i b , & i c ), as: Furthermore, applying the voltage Kirchhoff law in the BPV array:

D. BPV ARRAY MODEL
The BPV array in FIGURE 1 is based on N s series and N p parallel strings connection of BPV cells. Each cell is modeled using the Single Diode Model [19], whose equivalent electric circuit is shown in FIGURE 3. The arrangement in this work is capable of generating 600kW operating under Standard Test Conditions, i.e., irradiance S 0 = 1000 W/m 2 , temperature T 0 = 25 • C and the Air Mass AM = 1.5. Assuming equal cells operating in equal environmental conditions, the power generated by the entire array, at a certain given time, can be calculated with the following equations: where, The cell parameters are the rated short-circuit current (I sco ), the reverse saturation current of the diode (I o ), series resistance (R s ), parallel resistance (R sh ), ideality factor (n) and the electron charge and Planck's constant, q and k respectively. Table 1 summarizes the parameters of a typical cell. Clearly, i pv depends directly on the irradiance (S) and on the temperature (T ) to which the cells operate.

III. DC LINK CONTROL
The BPV solar array is connected directly to the NPC inverter, and to follow the maximum power point, the MPPT algorithm of [7] is implemented. This algorithm is based on measuring cells that together with the cell model estimate the M PP voltage which is set to be the DC-link voltage reference. A PI controller is designed for the DC-link voltage, where the dynamic of the capacitors can be defined as: by the procedure shown in [12]. Considering the model in (12), and a PI controller, h dc c (s) = k c (1 + 1/(T i s)), the closed loop transfer function becomes: Now, by matching the desired characteristically polynomial p c (s) = s 2 + 2ξ ω n s + ω 2 n , with the characteristic polynomial in (13), the parameters results to be: The sample time (T s ), proportional gain (k c ) and integrative time (T i ) parameters of the PI controller are shown in Table 2.
The i q reference is set to zero to obtain a unitary displacement power factor at the grid, the i d reference is set by the control of v dc (see FIGURE 1). The αβ current references based on the dq components are obtained using the transformation: The complete control scheme is shown in FIGURE 1.

IV. PREDICTIVE CURRENT CONTROL
To track the current references in (15), a MPC is used. The MPC is one of the most popular approach to control power converters. This control technique is based on the inverter model, defined, in this case, in (1) to (8). Those equations can be discretized by the Euler-forward approximation, where dx/dt = (x(k+ 1)x(k))/T s . The active and reactive power control is performed through the currents, therefore, the currents need to follow the references to achieve what is desired. In the case of the current prediction, based on (2), it is defined as: whereî αβ (k + 1) represents the estimation of i αβ (k+ 1). This method is used for both the traditional FS-MPC and the proposed FFS-MPC.
In the traditional FS-MPC, all the possible states must be tested, using the model, and then, applying the state which minimizes a cost function. Furthermore, due to the computation delay, (16) must be forwarded one step ahead, [12].
Despite the NPC has 19 voltages, the short and zero voltages have more than one state, FIGURE 2 (a), resulting in 27 states. Those additional states are used to reduce the imbalance between the voltage v p and v n . The equations (6) to (8) are used together with the Euler approximation, with the following results: The voltage deviation between v p and v n , assuming that C p = C n = C, becomes: As the calculated states are to be applied in the time k+ 1, the voltage difference in (18) need to be one step forwarded: The simplest FS-MPC algorithm defines a cost function as: VOLUME 8, 2020 which guarantees the tracking of the current reference and, at the same time, reduces the voltage difference between v p and v n . Thus, as there are 27 valid states, the algorithm needs to compute the model 27 times to choose the best one to minimize g 3 . One improvement on this algorithm is to choose the best voltage (instead of the best state), among the 19 possible voltages, and go to a second loop only if there is chosen a short or zero voltage (which have more than one state), leading to include at most 22 iterations to find the best state. Thus, two cost functions are defined: for the first loop (voltage selection) and for the second loop (state selection). The entire FS-MPC algorithm is shown in FIGURE 4 (b).

B. PROPOSED FFS-MPC
The proposed FFS-MPC also uses (16), but instead of trying all the valid states, it defines directly the voltage to be applied in the next step as: To impose the voltage v αβ (k+ 1), it is synthetized by SVM modulation technique, which guarantees a fixed switching spectrum. In other words, the voltage in (23) is obtained by using a combination in time of the three nearest voltages, see   Now, if, at least, one or more of the three nearest voltage are the short and/or zero voltage, a second loop is implemented to minimize the voltage difference, applying the same g 2 as in (22). The proposed FFS-MPC algorithm is illustrated in FIGURE 4 (a).

V. COMPUTATIONAL EFFORT AND COMPARISON WITH FS-MPC
The FFS-MPC obtains the converter voltage to be applied directly, where this voltage is synthetized by SVM technique. Once the converter voltage is decided, if and only if there exist redundancy, the DC link voltage balancing subroutine takes place by choosing the one that minimize this difference. If the FFS-MPC algorithm is compared with the traditional FS-MPC, shown in FIGURE 4 (b), the computational cost is notably reduced, because FS-MPC tries all the possible states choosing the one that minimizes the cost functional and therefore it needs to evaluate the complete model several  times, more details of this algorithm can be found in [12], [17], [18]. In contrast, the proposed algorithm, shown in FIGURE 4 (a), sets the voltage to be applied directly. In fact, VI-A shows the total amount of operations that requires both algorithms, giving a fair comparison to evaluate the computation requirements. The VI-A shows the operations needed to implement the FS-MPC and the FFS-MPC, where the acquisition, the PLL, the Clark Transformation and the states writing times, are not considered as these stages are the same for both algorithms. Just to give an example, the fourth column of VI-A depicts the computing time for every operation considering the TMS320F28335 DSP based board [20] to highlight the difference in the total computing time. As it can be seen, the FS-MPC takes about 44 µs to execute the code, and the FFS-MPC takes around 7 µs, being the FS-MPC at least 6 times heavier than the FFS-MPC. This allows to implement the proposal algorithm in a greater variety of digital boards, reducing the cost related to the controller.
Additionally, the FFS-MPC control has a better frequency spectrum. FS-MPC has a typical spread out spectrum, as can be seen in FIGURE 5. However, the proposed FFS-MPC shows a lower THD and a more concentrated spectrum that also allows an easier filter design, FIGURE 7.  To compare the time response, two simulations are performed to notice the current control bandwidth, which can be seen in FIGURE 6. This figure shows a similar response, the proposed FFS-MPC is even a quietly faster from this comparison, and therefore the computational reduction does not affect in the dynamic performance.

A. SIMULATED RESULTS
The parameters of the simulations are shown in Table 2 and were performed in PSim. To validate the proposed FFS-  MPC current control, the key waveforms of the simulation of the photovoltaic system are presented in FIGURE 7. At the instant of time t = 98ms a change is made in the current reference, which can be seen in FIGURE 7 (a) and FIGURE 7 (c). At t = 142ms, the DC voltage source increases by 20%, which can be seen in FIGURE 7 (b) and FIGURE 7 (d), it can also be seen that the proposed control strategy is capable of maintaining the voltages balanced on the capacitors. In FIGURE 7 (c) it is seen that the system operates in perfect synchrony with the network voltage and using the changes imposed, it operates with unitary power factor. Finally, FIG-URE 7 (e) and FIGURE 7 (f) show the harmonic content of the current i a and voltage v ab of the converter respectively, operating in steady state, it can be seen that the switching frequency is fixed and around 3 kHz.

B. EXPERIMENTAL RESULTS
The experimental results are obtained using a RL filter, a PV electronic emulator and a programmable three-phase source. The used parameters are given in Table 4 and the topology of FIGURE 1.

1) CURRENT CONTROL TEST: STEP CHANGES
In FIGURE 8 the key waveforms due to a step change in the current references keeping constant the DC voltage are shown. The AC voltage gets adapted in less than 2 ms in order to track the AC current references. The DC link 2) CURRENT CONTROL TEST: FREQUENCY CHANGES FIGURE 12 shows a frequency step change from 45 Hz to 60 Hz, where the current control maintains the power factor after this step, and the DC voltage is not affected, as it can be seen in the v ab injected voltage. The current dynamic is fast, taking less than two cycles to return to unitary power factor.

3) COMPLETE LOOP TEST: TEMPERATURE AND RADIATION CHANGES
The DC voltage is provided by a PV electronic emulator (Magna Power SL600-4.3) that is programmed with 6 dif- ferent BPV profiles (P 1 PV , P 2 PV ,. . . , P 6 PV ), using the PPPE software, FIGURE 12. These different profiles emulate different environmental conditions i.e. changes of irradiation and temperature. A loop is programmed so that every 25 s the P PV profiles change sequentially in an infinite loop, FIGURE 11. The inverter uses an MPPT algorithm to establish the DC voltage reference to guarantee maximum power generation, which is tracked, FIGURE 11 (c), and it can be seen how the system is able to reach the maximum power points, FIGURE 11 (a) and (b).
Key waveforms are shown in FIGURE 11. The DC link voltage, FIGURE 11 (c), changes every 10 s in order to cover sequentially the six profiles and FIGURE 11 (a) shows the correct DC link voltage setting as the maximum DC power generation is achieved in each profile.

VII. CONCLUSION
The proposed FFS-MPC has the ability to inject power from a photovoltaic source, with good performance and, specially, with reduced computational effort and concentrated spectrum. In fact, it does not have a main cost function unlike the typical predictive controls that evaluates a complex cost function up to 27 times, also it does not calculate the imbalance on the future voltage of the capacitors in each possible state, indeed, it only does so in the low voltages to apply up to three times, reducing in 84% the computational effort. On the other hand, the harmonic content of the switching is less dispersed than in the typical predictive control. A laboratory scale plant was built, and the results are in whole agreement with the simulations. In fact, the maximum power point in BPV plant is reached, and the voltage imbalance in the capacitors is zero for all practical purposes. One limitation of this approach is the need of PWM modules to synthetize the SVM technique, because the SVM is timing sensitive. On the other hand, this approach can be extended to micro-grids, where the grid frequency and voltage amplitude may suffer disturbances in the values. Additionally, the aforementioned MPC method can be extended to the voltage control as well, since it can be modeled and therefore a similar technique is possible to be applied.