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High performance decoupled active and reactive power control for threephase gridtied inverters using model predictive control
Protection and Control of Modern Power Systems volume 6, Article number: 25 (2021)
Abstract
Finite control setmodel predictive control (FCSMPC) is employed in this paper to control the operation of a threephase gridconnected string inverter based on a direct PQ control scheme. The main objective is to achieve highperformance decoupled control of the active and reactive powers injected to the grid from distributed energy resources (DER).
The FCSMPC scheme instantaneously searches for and applies the optimum inverter switching state that can achieve certain goals, such as minimum deviation between reference and actual power; so that both power components (P and Q) are well controlled to their reference values.
In addition, an effective method to attenuate undesired cross coupling between the P and Q control loops, which occurs only during transient operation, is investigated. The proposed method is based on the variation of the weight factors of the terms of the FCSMPC cost function, so a higher weight factor is assigned to the cost function term that is exposed to greater disturbance. Empirical formulae of optimum weight factors as functions of the reference active and reactive power signals are proposed and mathematically derived. The investigated FCSMPC control scheme is incorporated with the LVRT function to support the grid voltage in fulfilling and accomplishing the uptodate grid codes. The LVRT algorithm is based on a modification of the references of active and reactive powers as functions of the instantaneous grid voltage such that suitable values of P and Q are injected to the grid during voltage sag.
The performance of the elaborated FCSMPC PQ scheme is studied under various operating scenarios, including steadystate and transient conditions. Results demonstrate the validity and effectiveness of the proposed scheme with regard to the achievement of highperformance operation and quick response of gridtied inverters during normal and fault modes.
Introduction
Literature review
A gridconnected inverter constitutes an essential part of modern DER (distributed energy resources) grid integration systems [1,2,3,4,5,6,7,8,9,10]. During the last few years, many efforts have been made to ensure reliable operation of the gridtied inverter by taking into consideration modern grid codes and standards [11,12,13,14,15,16,17,18,19,20]. The increasing computation capability of highspeed digital signal processors (DSPs) and the availability of various hardwareinthe loop (HIL) control boards have facilitated the development and implementation of sophisticated control algorithms for achieving reliable gridintegrated systems [21,22,23,24,25,26,27,28,29,30]. Accordingly, some new functions and features (such as LVRT) recommended by updated grid codes and standards can be added to the inverter control algorithm [19, 20, 23, 26,27,28,29,30,31,32,33,34]. Such regulations obligate the gridtied inverters to withstand unintentional grid voltage sag for a specific duration based on some LVRT profiles, which are customised in many countries [19, 20, 31, 35,36,37,38]. A modern control technique that has been utilised in recent years to control the operation of gridtied inverters is the FCSMPC [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55]. Successful utilisation of the FCSMPC algorithm has been reported in many studies [56,57,58,59,60,61,62,63,64,65,66,67,68,69]. Most of the existing FCSMPC schemes for gridconnected inverters are either current controlled [51, 61, 63, 66, 70, 71] or voltage controlled [72,73,74,75].
In currentcontrolled schemes, FCSMPC regulates and controls as per the desired value the current injected to the grid from the DER by applying the adequate inverter switching state. In voltagecontrolled schemes, FCSMPC controls the inverter output voltage (voltage space vector), which indirectly controls the current injected to the grid. In this scheme, the computation of the voltage space vector is similar to the one used in the wellknown SVM technique.
In this paper, FCSMPC is employed to apply a direct PQ control strategy such that the active and reactive powers (P and Q) injected to the grid from the DER are directly controlled to their reference values; this is done through the application of the optimum inverter switching state to ensure a quick response. In recent years, considerable efforts have been made to overcome the major challenges of FCSMPC that negatively affect its overall performance. One of these challenges is choosing the weight factors of the cost function. Thus, the topic of selecting the weight factors of the FCSMPC cost function to optimise the scheme’s performance has gained much attention [41, 44, 46, 55, 76]. Most of the weight factor determination methods are based on iterative rangesweeping techniques or evolutionary search algorithms because of the lack of theoretical design methodologies or analytical approaches for designing and adjusting these parameters. In general, all methods aim to assign a higher weight factor to a given objective term whenever it presents an unaccepted error [46]. To achieve this task, the errors between the desired and actual values of the associated variables are usually used as inputs to the tuning algorithm [41, 55, 76].
Some methods assign discrete values to the weight factors (usually dual values) [41] or employ evolutionary search algorithms to determine the optimum weight factors involving the operating range [44], while other methods compute the dynamic weighting factor gain as functions of the errors such that the weight factors are tuned online [55, 76]. Nevertheless, inappropriate dynamic weight factors can lead to the overoptimisation of the terms of the cost function. This deteriorates overall performance. Thus, there is a tradeoff between simplicity and accuracy. To address such issues, this paper proposes an empirical weight factortuning method for achieving an optimised transient response of the overall FCSMPC system.
Objective of the paper
The primary objective of the paper is to study and investigate an integrated control scheme for a threephase gridtied string inverter that can guarantee satisfactory steadystate performance as well as quick transient performance by employing decoupled control of the active and reactive powers (PQ) injected into the grid. The control scheme also attempts to consider the LVRT mode during grid voltage sag to satisfy the existing grid codes related to gridconnected inverters. The performance of the investigated scheme is studied under various operating conditions, including the normal and fault modes (grid voltage sag). Qualitative and quantitative analyses of steady state and transient responses are undertaken and addressed. Results demonstrate the validity and effectiveness of the discussed scheme in terms of the achievement of a highperformance threephase grid integration system during both the normal and fault modes of operation. The rest of the paper is organized as follows. Section 2 provides an overview of the currentcontrolled FCSMPC scheme for threephase gridtied inverters, and Section 3 details the different parts of the investigated system, including the computation of grid voltages and currents in (αβ) coordinates, the computation of power components (P and Q), the formulation of the cost function, and the adjustment of weight factors and also describes the LVRT mode. In Section 4, selected simulation results are presented and discussed, and qualitative and quantitative assessments on the results are also provided. Conclusions are drawn in Section 5, and the empirical formulae of variable weight factors and their mathematical derivations are presented in Additional file 1.
Main contribution
The main contributions of the paper are as follows:
(1) It proposes a simple and effective method of optimising the weighting factors of the FCSMPC cost function for reducing undesired cross coupling between the P and Q control loops. Empirical formulae of optimum weight factors (as functions of active and reactive power references) are formulated and mathematically derived.
The proposed method varies the weight factors of the cost function only during detected transient periods and retains equal weight factors during steadystate operation. Thus, during steadystate operation, both terms of the cost function have the same levels of priority and contribution. However, during the transient period, the weight factors adjustment is governed by a simple, logical rule, i.e., granting a higher weight factor to the term that is exposed to higher undesired coupling until the error is restricted to predetermined satisfactory limits.
(2) It applies the concept of decoupled PQ control to incorporate LVRT capability, which is an essential option demanded by recently published grid codes (such as IEEE 1547, VDEARN 4120 and IEC 62477–1). Most existing LVRT schemes are currentcontrol based. They inject a suitable reactive current to the grid; consequently, reactive power is indirectly injected to the grid. However, in this paper, the LVRT is achieved through the direct injection of the optimum values of P and Q during a fault condition using the same FCSMPC scheme; the corresponding reference powers (P_{ref} & Q_{ref}) are instantaneously calculated within the FCSMPC algorithm.
Current controlled FCSMPC scheme for 3Φ grid connected string inverter
Singleline diagram and inverter power circuit
The singleline diagram of a typical threephase PV grid integration system is illustrated in Fig. 1. In this system, all PV arrays (considered as one of the DERs) are connected to a common DC bus of 600 V through the individual MPPT tracking units and suitable DCDC converters incorporated with each PV array. The string inverter injects both active and reactive powers (P and Q) to the grid according to the mode of operation. The circuit diagram of a 3Φ gridconnected inverter is shown in Fig. 2; as shown, the energy produced from the distributed resource is injected to the grid through the inverter. A 3Φ inductor L_{S} having a small equivalent series resistance R_{S} is inserted at PCC between the inverter output and the grid [24]. The voltage space vector \({\overline{\mathrm{U}}}_{\mathrm{S}}\) can be described as a function of the DC link voltage and inverter switching states as follows:
where S_{1}, S_{3} and S_{5} are the switching states of the upper power transistors of the inverter. The voltage space vector \({\overline{\mathrm{U}}}_{\mathrm{S}}\) can be resolved into two orthogonal components (U_{α} and U_{β}) in the (αβ reference frame, where their equivalent values are computed by:
The resultant components of inverter voltage space vector \({\overline{\mathrm{U}}}_{\mathrm{S}}\) for all switching states are presented in Table 1. The six active inverter switching states and two nil switching states are used to control the operation of the gridtied inverter in different configurations such as currentcontrolled mode or SPWM or SVM techniques.
In FCSMPC, the optimum inverter switching state is instantaneously selected and applied to the inverter to minimise a specific cost function; this is explained in the following sections.
Principle of current controlled FCSMPC approach
In recent years, FCSMPC has been adopted to control the operation of switching power converters such as gridtied inverters [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70, 76, 77]. In the FCSMPC technique, the future behaviour of the system is predicted for a finite time frame [40]. Accordingly, the optimum future control action is applied to the system to satisfy a customised goal function [57,58,59,60,61,62,63,64,65,66], and the FCSMPC algorithm is repeated at every sampling period [67,68,69,70, 76, 77]. Generally, it is characterised by fast transient response and the ability to consider the nonlinearities and constraints in the control law [41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60].
Applying KVL to the circuit in Fig. 2 yields:
The rates of change of grid currents \(\frac{di_a}{dt}\), \(\frac{di_b}{dt}\) and \(\frac{di_c}{dt}\) are rearranged:
Accordingly, (4.a) is employed to derive the instantaneous value of phase current i_{a} at the (k + 1)^{th} sample as:
where T_{S} is the sampling time, L_{S} is the perphase inductance of the inductor and R_{S} is the equivalent series resistance of the inductor L_{S}.
Similarly, the instantaneous grid currents of other phases, i_{b} and i_{c}, at the (k + 1)^{th} sample are predicted as:
From (6.a), (6.b) and (6.c), the grid currents at the (k + 1)^{th} sample can be predicted through online measurement of the grid voltage (e_{an}, e_{bn}, e_{cn}) and grid currents i_{a}, i_{b} and i_{c} at the current k^{th} sample.
The inverter output voltages (v_{an}, v_{bn}, v_{cn}) can be measured directly, while their (αβ) components in the stationary reference frame can be calculated based on (2.a) and (2.b). Unlike the hysteresis controllers or PI controllers driving PWM units, the FCSMPC scheme instantaneously selects the optimum switching state every sampling period and achieves a specific goal function without requiring a PWM unit or hysteresis current controllers [40].
A simplified block diagram of a typical FCSMPC scheme is presented in Fig. 3, in which the predictive model block computes and predicts the grid currents at the (k + 1)^{th} sample for the eight inverter switching states. Inside the cost function computation block, a cost function is computed for all switching states.
The switching state that results in the minimum value of the cost function is considered as the optimum state to be applied. This task is performed inside the optimum switching state selection block shown in Fig. 3. Consequently, if the reference grid currents \({i}_a^{\ast }\), \({i}_b^{\ast }\) and \({i}_c^{\ast }\) are given, the chosen cost function J, described by (7), is repeatedly evaluated during every sampling period for the eight possible switching states. One of them is the optimum switching state that results in the minimum value of the cost function at the (k + 1)^{th} sample. Accordingly, the actual grid current tracks the reference value with accepted error (deviation).
Excellent waveform tracking under FCSMPC approach requires high sampling rates, which can be provided by highspeed data acquisition cards.
Description of the investigated decoupled PQ control system using FCSMPC approach
The block diagram of the investigated FCSMPC PQ system for a 3Φ gridconnected string inverter is illustrated in Fig. 4. The system is composed of two main blocks: (1) LVRT, which determines the mode of operation and computes the suitable reference signals of active and reactive powers; (2) The FCSMPC system, which includes several blocks, such as prediction of grid currents and active and reactive powers, computation of cost function and selection of optimum switching state. All tasks are explained below in Subsections 3.1, 3.2 and 3.3.
Computation of grid voltages and currents in (αβ) stationary reference frame
The (αβ) components of grid currents i_{α} and i_{β} are computed using (8.a) and (8.b), respectively. Similarly, the (αβ) components of grid voltages e_{α} and e_{β} are calculated using (9.a) and (9.b), respectively, based on the concept of the space vector as described by (1). Those components are essential in active and reactive power computations, as explained in Section 3.2.
Prediction of active and reactive power
The instantaneous active and reactive powers (P and Q) injected to the grid are computed as follows [21, 22, 78]:
Resolution of (6.a), (6.b) and (6.c) into the equivalent (αβ) components leads to the predicted i_{α} and i_{β} at the (k + 1)^{th} sample period as follows:
where u_{α} and u_{β} were previously computed using (2) and tabulated in Table 1. e_{α} and e_{β} were previously computed using (10).
From (8) to (11), the instantaneous active and reactive powers are predicted at the (k + 1)^{th} sample as follows [17]:
As the grid voltage has low variation compared to the sampling and switching frequencies, the grid voltage components e_{α} and e_{β} can be considered constant during the sampling period, i.e., (\({e}_{\alpha}^{k+1}\) = \({e}_{\alpha}^k\)) and (\({e}_{\beta}^{k+1}\) = \({e}_{\beta}^k\)).
Formulation of cost function
The cost function is formulated to account for the active and reactive powers as follows:
where P_{ref} and Q_{ref} are the desired reference values of the active and reactive powers to be injected to the grid from the DER through the 3Φ inverter. In (14), the first term of the objective function aims to minimise the active power deviation (ripple), while the second term aims to minimise the reactive power ripple, and both terms have the same degree of importance and make equal contribution. Accordingly, the instantaneous value of the cost function J in (14) is computed for all inverter switching states. The resultant values of the cost function are plotted in Fig. 5 (a) for a small time frame of 70 μs (seven samples, each with a sampling period of 10 μs). From the results, applying inverter vectors U_{4} and U_{5} alternatively produces the minimum values of the cost function (J_{4} and J_{5}) during the selected time frame (in six samples out of seven) in conjunction with the null vectors U_{0} or U_{7} (in one sample out of seven). Thus, switching states 4 and 5 is the optimum selection (also see Fig. 5 (b)). In addition, the null vector is considered as the optimum selected vector in one of the seven samples (at t = 0.02005 s; J_{0} is the minimum). Fig. 5 (a) also shows that applying vectors U_{2} and U_{3} will yield the worst values of cost function (J_{2} and J_{3}) during the selected time frame. Similarly, applying vectors U_{1} and U_{6} will not result in an optimised cost function. Thus, vectors U_{2}, U_{3}, U_{1} and U_{6} are not used during the investigated time period. In Fig. 5 (b), the overall minimum possible cost function J_{min} (dotted line) is plotted together with J_{4} and J_{5}, which correspond to vectors U_{4} and U_{5}, respectively. In most FCSMPC systems, weight factors are included in the terms of the cost function. Involving such weight factors allows the FCSMPC system to assign a priority to the controlled variables on the design criteria. Thus, (14) is rewritten to involve weight factors in both terms of the cost function:
where w_{p} and w_{q} are the weight factors of active and reactive power terms, respectively.
Flowchart of the investigated FCSMPC PQ control system
The flowchart of the FCSMPC algorithm is shown in Fig. 6 (a); (2) and (11)–(15) are computed at each sample for all possible inverter switching states such that an optimum switching state is determined and applied during the next sample.
Initially, the system is investigated with constant weight factors (w_{p} = w_{q} = 1) during the whole operation of the FCSMPC system.
Then, those weight factors are varied during the transient periods, as explained in the flowchart of Fig. 6 (b); this is described in detail in Additional file 1.
Adjustment of weight factors (Wp &Wq) of cost function
As shown in the flowchart presented in Fig. 6 (b), the weight factors in (15) are adjusted based on a simple rule that imposes a penalty to the term causing cross coupling during the transient operation (by reducing its weight factor).
The same result can be obtained by granting higher priority or rewarding the negatively affected term of the cost function (by increasing its weight factor) during the transient periods. During steadystate operation, since no change occurs in the reference power, the weight factors are assigned to their initial values of unity. Once the algorithm detects a transient change in any of the reference signals (P_{ref} or Q_{ref}), the weight factors are adjusted accordingly based on the rule summarised in Table 2 and previously illustrated in the flowchart of Fig. 6 (b).
As shown in Table 2, when the algorithm detects an abrupt change in P_{ref} while Q_{ref} has no change (Sp > Sq), the weight factor w_{p} is reduced to alleviate the disturbance that occurred on the Q loop from the P loop, while the weight factor w_{q} is kept constant at unity.
Similarly, when the algorithm detects an abrupt change in the Q_{ref} signal while P_{ref} has no change (Sq > Sp), the weight factor w_{p} is kept constant at unity, while w_{q} is reduced to alleviate the disturbance that occurred on the P loop from the Q loop.
This paper proposes a simple and effective method of tuning the weight factors (w_{p} and w_{q}). The details of the proposed method, including the mathematical derivation of the proposed formulae of w_{p} and w_{q}, are presented in Additional file 1. The optimum values of w_{p} and w_{q} are plotted in Fig. 7 (a) and (b), respectively.
LVRT mode and modification of reference signals
The inverter control scheme is incorporated with an LVRT option to make it compatible with modern grid integration regulations and standards such as VDEARN 4120, IEC 62477–1 and IEEE 1547 [19, 20, 26, 27, 31, 35,36,37].
In cases of fault and occurrence of voltage sag, the LVRT mode is enabled. Hence, the LVRT subroutine determines the level of voltage sag, and the reference power signals (P_{ref} and Q_{ref}) are then computed such that the inverter injects a combination of active and reactive powers (or only reactive power) to the grid based on the actual value of the grid voltage sag (V_{sag}), as shown in Fig. 6 (c). The flowchart of the LVRT subroutine illustrated in Fig. 6 (c) demonstrates that the numerical value of the active and reactive powers to be injected to the grid is considered as a function of (V_{sag}). This relation is also presented as follows:
In (16), the voltage sag is discriminated in three zones:

In the first zone, V_{sag} is between 0.9 and 1, and the inverter injects only active power to the grid.

In the second zone, V_{sag} is between 0.5 and 0.9, and the inverter injects both active and reactive powers with fixed apparent power.

In the third zone, V_{sag} is below 0.5, and the inverter injects only reactive power to the grid with nil active power.
Most developed countries have their own LVRT profiles, which the operation of modern gridtied inverters should guarantee [23, 26, 27, 29, 38].
Simulation results
The overall FCSMPC system for a threephase gridtied string inverter was modelled and investigated in PSIM software®. The simulation parameters are summarised in Table 3. The simulation platform is PSIM and the sampling time is 20 μs (sampling rate is 50 kHz). In this section, the simulation results are categorised in three groups:
(1) Steadystate performance of the elaborated FCSMPC PQ system at normal operating conditions with quantitative assessment;
(2) Transient response of the FCSMPC PQ system at normal operating conditions with quantitative assessment;
(3) Transient performance of LVRT under grid voltage sag.
Steady state performance

a.
Unity PF Operation (P_{ref} = 10 kW, Q_{ref} = 0 VAR)
The steady state performance of the investigated 3Φ gridtied inverter system was studied for unity PF operation. The reference active power was set to 10 kW, and the reference reactive power was zero. The instantaneous values of the resultant P and Q and the current injected to the grid are plotted in Figs. 8 (a), (b) and (c), respectively. As the reference reactive power was zero, the current injected to the grid was inphase with the grid voltage. The results shown in Figs. 8 (a) and (b) indicate that both P and Q were well controlled to their desired values. The corresponding harmonic spectra of P, Q and the grid current are illustrated in Figs. 9 (a), (b) and (c), respectively.
Quantitative analysis of steady state performance at unity PF
The quantitative analysis of steady state performance was conducted, and the results are presented in Table 4. The active power injected to the grid was 9.99 kW (desired reference value was 10 kW), while the reactive power was 68 VAR (desired reference value was zero VAR).
However, it had negligible effect on the resultant PF (in this case, the actual PF was 0.999). Moreover, the computed THD of the grid current was 3.35%.

b.
Zero PF Operation (P_{ref} = 0 W, Q_{ref} = 10 kVAR)
The steadystate performance of the FCSMPSMPC was investigated for a different operating scenario: the reference active power was zero, while the reference reactive power was − 10 kVAR. The instantaneous values of the corresponding P, Q and the current injected to the grid are depicted in Fig. 10 (a), (b) and (c), respectively. The results indicated that P and Q were controlled to their desired values. As the reference active power was zero (Zero PF), the grid current i_{a} lagged the grid voltage by 90°, as seen in Fig. 10 (c).
The corresponding harmonic spectra of P, Q and the grid current are depicted in Figs. 11 (a), (b) and (c), respectively.
Quantitative analysis of steady state performance at zero PF
Quantitative analysis of the steady state performance was carried out, and the results are presented in Table 5. The reactive power injected to the grid was 10.005 kVAR, while the active power was − 100.6 W. Again, this only had a small effect on the resultant PF (in this case, the actual PF was 0.01). The computed THD of the grid current was 3.12%.
Transient performance

a.
Step change of P_{ref} (0 → 10 kW) with Q_{ref} = 0 VAR
The transient performance of the FCSMPC system was investigated for different operating scenarios. In this subsection, the reference active power P_{ref} has a step change from 0 to 10 kW, while the reactive power reference Q_{ref} is zero.
The transient responses were studied with fixed as well as adjustable weight factors, and the results are illustrated in Figs. 12 and 13, respectively.
Quantitative analysis of step response of active power P_{ref} (0 → 10 kW) with Q_{ref} = 0 VAR
A quantitative analysis was conducted of the transient response presented above, and the results are summarised in Table 6 for both FCSMPC systems.
As shown in Fig. 12(a), the active power with fixed weight factors had a fast transient response with a settling time of less than 2 ms (1.77 ms). Similarly, the observed settling time in the case of variable weight factors was also less than 2 ms (1.82 ms), as illustrated in Fig. 13(a).
In addition, both FCSMPC schemes did not produce overshoot for the active power.
However, the reactive power component Q was negatively affected by the active power step change. The results indicated that the Q component was subjected to undesired cross coupling during the transient period in the case of the FCSMPC scheme with fixed weight factors. This cross coupling on Q was approximately 1.6 kVAR and the time elapsed was 2 ms, as shown in Fig. 12(b).
The results demonstrated that the proposed FCSMPC scheme with variable weight factors effectively eliminates the cross coupling and successfully attenuates the disturbance to a negligible level during the transient period, as shown in Fig. 13(a) and (b).
Figure 12(c) indicates that the weight factors were fixed to unity in the case of the fixed weight factors, while with variable weight factors, the weight factor w_{p} (as depicted in.
Figure 13(c)) was changed according to the adjustment rule previously addressed in both Table 2 and Fig. 7 (a). The grid currents sketched in Figs. 12(d) and 13(d) indicate the successful operation of both schemes at unity PF during the transients. In addition, once the transient period had passed, both the fixed and variable weight factor schemes resulted in similar PF and THD of the current injected to the grid, as seen in Table 6.

b.
Step change of Q_{ref} (0 → 10 kVAR) with P_{ref} = 0 W
In this subsection, the transient performance of the FCSMPC system is investigated in another operating scenario. The reference active power P_{ref} was set to zero, while the reactive power reference Q_{ref} was step changed from 0 to 10 kVAR. The transient responses were investigated with fixed as well as adjustable weight factors, and the results are illustrated in Figs. 14 and 15, respectively.
Quantitative analysis of step response of reactive power Q_{ref} (0 → 10 kVAR) with P_{ref} = 0 W
As shown in Fig. 14(a), the reactive power with the FCSMPC system of fixed weight factors had a fast transient response with a settling time of less than 2 ms (1.57 ms).
In the case of the FCSMPC system with variable weight factors, the observed settling time was 0.42 ms (much less than the fixed weight factors case), as illustrated in Fig. 15(a).
Both FCSMPC schemes did not produce overshoot for the reactive power (as can be observed from Figs. 14(a) and 15(a)). However, the FCSMPC system with fixed weight factors slightly undershot before the reactive power settled down around the reference value, as illustrated in Fig. 14(a).
In addition, the active power component P was negatively affected by the step change in the Q component.
The results indicated that the P component was subjected to severe undesired cross coupling during the transient period in the case of the fixed weight factors. The active power drop was approximately 2.7 kW, and it elapsed in 1.8 ms, as shown in Fig. 14(b).
The results demonstrated that the proposed FCSMPC scheme with variable weight factors effectively alleviates the cross coupling and successfully attenuates the disturbance to a negligible level during the transient period, as shown in Figs. 15(a) and (b).
Figure 14(c) shows the unity weight factors for the fixed weight factors case, while in the case of variable weight factors, the weight factor w_{q} was changed, as depicted in Fig. 15(c), according to the adjustment rule previously addressed in Table 2 and Fig. 7 (b).
The waveforms of the grid current sketched in Figs. 14(d) and 15(d) indicate the successful operation of both schemes at zero PF during the transients.
Once the transient period had passed, both schemes (fixed and variable weight factors) resulted in similar PF and THD of the current injected to the grid, as seen in Table 7.
LVRT mode of the FCSMPC system under grid voltage sag
The proposed FCSMPC PQ control scheme is incorporated with an LVRT mode so that it withstands grid voltage sag for a short duration (determined by standard LVRT profiles) to be consistent with the uptodate grid code standards [27].
The operational scenario depends on the level of voltage sag, as explained in Section 3.6 and shown in the flowchart of Fig. 6 (c). The first part of Section 4.3 presents an emulation of grid fault and voltage sag based on a standard LVRT profile. The LVRT profile is presented in Fig. 16(a), and the corresponding grid voltage sag is plotted in Fig. 16(b).
Consequently, the FCSMPC PQ control unit generates the suitable reference active and reactive powers P_{ref} and Q_{ref} based on (16), as illustrated in Figs. 16(c) and (d), respectively. The second part of Section 4.3 illustrates the transient response of the FCSMPC schemes under grid voltage sag. Figures 17 and 18 present the LVRT performance of the FCSMPC PQ scheme with fixed and variable weight factors, respectively. Both groups of results prove that the FCSMPC PQ scheme provides quick response and is able to enhance the LVRT capability of a gridtied string inverter such that the LVRT option incorporated with the inverter control scheme is consistent with grid codes and standards.
Conclusion
In this paper, an FCSMPC approach was used to control the operation of a threephase gridtied string inverter based on the concept of direct PQ control, providing a decoupled control of both active and reactive powers injected to the grid from the DER.
The crosscoupling problem associated with PQ control loops during transient operation was addressed and investigated, and an efficient method to minimise the undesired cross coupling between the P and Q control loops was proposed. The attenuation of cross coupling was achieved by varying the weight factors of the FCSMPC cost function during the transient period.
The proposed method can be considered as a feedforwardtuning scheme for weight factors. In addition, the paper deduced empirical formulae to compute and tune the optimum weight factors as functions of reference active and reactive power signals. The relevant mathematical derivations were also presented. Results prove the validity and effectiveness of the proposed method with regard to the attenuation and alleviation of undesired cross coupling between the P and Q components.
The study involved both steady state and transient performances of the FCSMPC scheme under variable operating scenarios, such as unity PF and zero PF operations. In addition, qualitative and quantitative analyses of the obtained results were carried out.
The elaborated FCSMPC scheme inherently provided the LVRT mode, as requested by the updated grid codes and standards. The concept of a decoupled PQ control approach was applied to inject the necessary active and reactive powers to the grid during grid faults by modifying the instantaneous reference active and reactive powers based on the instantaneous grid voltage.
The results indicated that the adopted method was successful and able to support the LVRT capability of a gridtied string inverter.
The results demonstrate the capability of the FCSMPC approach in achieving a highperformance DER grid integration system that can be operated with different operating conditions, including fault mode
Availability of data and materials
Not applicable.
Abbreviations
 ESR:

Equivalent series resistance
 DER:

Distributed energy resources
 DSP:

Digital signal processor
 FCSMPC:

Finite control set model predictive control
 HIL:

Hardware in the loop
 KVL:

Kirchhoff voltage law
 MPC:

Model predictive control
 LVRT:

Low voltage ride through
 kW:

Kilo Watt
 PCC:

Point of common coupling
 PF:

Power Factor
 PI:

Proportional integral controller
 PLL:

Phase locked loop
 SVM:

Space vector modulation
 PV:

Photovoltaic
 PU:

Per unit
 V_{sag} :

Per unit value of grid voltage sag
 THD:

Total harmonic distortion
 VSI:

Voltage source inverter
 VAR:

Voltampere reactive
 3Φ:

ThreePhase
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Acknowledgements
The author is grateful to Mr. Bahaaeldin M. A. Moustafa, from Nile University (NUEgypt) for his effort to download the needed references of the paper. The author is also grateful to Dr. Ali AbouSena, from Karlsruhe Institute of Technology (KITGermany) for his effort in revising the English language of this paper.
List of Symbols
V_{DC} DC bus voltage
U_{S} Inverter voltage space vector
e_{an} Grid voltage of phase a
e_{bn} Grid voltage of phase b
e_{cn} Grid voltage of phase c
e_{α} Grid voltage component of αaxis of (α−β) stationary reference frame
e_{β} Grid voltage component of βaxis of (α−β) stationary reference frame
V_{an} Inverter o/p voltage of phase a
V_{bn} Inverter o/p voltage of phase b
V_{cn} Inverter o/p voltage of phase c
U_{α} Inverter o/p voltage component of αaxis of (α−β) stationary reference frame
U_{β} Inverter o/p voltage component of βaxis of (α−β) stationary reference frame
L_{S} Perphase grid filter inductor
R_{S} Perphase ESR of inductor L_{S}
i_{a} grid current of phase a
i_{b} grid current of phase b
i_{c} grid current of phase c
i_{α} Grid current component of αaxis of (α−β) stationary reference frame
i_{β} Grid current component of βaxis of (α−β) stationary reference frame
\( {i}_a^{k+1} \) Predicted current of phase a at sample (k+1)
\( {i}_b^{k+1} \) Predicted current of phase b at sample (k+1)
\( {i}_c^{k+1} \) Predicted current of phase c at sample (k+1)
\( {i}_{\alpha}^{k+1} \) Predicted current component of αaxis at sample (k+1)
\( {i}_{\beta}^{k+1} \) Predicted current component of βaxis at sample (k+1)
T_{S} Sampling period of MPC algorithm
t_{d} Time delay of reference signal
k Sample number k
i Switching state number (0 → 6)
P Inst. value of active power injected to the grid
Q Inst. value of reactive power injected to the grid
P^{k + 1} Predicted value of active power at sample (k+1)
P_{ref} Reference value of active power to be injected to grid
Q^{k + 1} Predicted value of reactive power at sample (k+1)
Q_{ref} Reference value of reactive power to be injected to grid
J Cost function of FCSMPC
J_{i} Cost function corresponding to switching state i
J_{i} (k+1) Cost function at the sample (k+1) corresponding to inverter switching state i
w_{p}, w_{q} Weight factors of MPC cost function
δ_{p} Absolute error between reference and delayed reference active power signals
δ_{q} Absolute error between reference and delayed reference reactive power signals
V_{sag} Per unit value of grid voltage sag
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The article has been prepared and edited by a single author “Mohamed Azab” who carried out all tasks of the article. English language revision has been carried out by a colleague from Karlsruhe Institute of Technology (KIT) and the final draft has been revised by a professional proofreading office “Paper True”. The author(s) read and approved the final manuscript.
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Azab, M. High performance decoupled active and reactive power control for threephase gridtied inverters using model predictive control. Prot Control Mod Power Syst 6, 25 (2021). https://doi.org/10.1186/s4160102100204z
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Keywords
 Model predictive control
 Finite control set
 Gridconnected inverter
 Active power
 Reactive power
 Distributed generation
 Low voltage ride through
 PQ theory
 FCSMPC
 LVRT