A Lyapunov-Based Nonlinear Power Control Algorithm for Grid-Connected VSCs

—For grid-connected voltage source converters (VSCs), it is commonly required that its power outputs can track the power references given by operators under various grid conditions. To achieve this objective, this article presents a Lyapunov-based nonlinear power control algorithm. The system dynamics is developed in the stationary frame, which facilitates the design of the control algorithm in the absence of phase-locked loops (PLLs) that may cause instability issues in ultra-weak grids. A virtual resistance is then introduced to allow fast power tracking performance and relax the requirement on accurate system parameters. Further, to simplify the control design, the quasi-stationary line (QSL) impedance model is applied, based on which a nonlinear control algorithm that only utilizes the output voltage and current information is developed. Afterward, the stability of the closed-loop system is analyzed via the Lyapunov theory. The theoretical results illustrate that the power regulation goal can be achieved and the VSC can maintain synchronization with the power grid. Finally, experimental results demonstrate the effectiveness of the proposed control algorithm under temporary grid faults and various short circuit ratios (SCRs).


I. INTRODUCTION
I N the past decades, voltage source converters (VSCs) have been widely applied in power grids due to their flexibility in control and operation, e.g., microgrids [1], renewable energy systems [2], flexible ac transmission systems [3], etc.For a grid-connected VSC, a common control objective is to regulate its active and reactive power outputs to track their references [4], [5].
Traditionally, voltage-oriented vector-current control [6], [7] methods are utilized to achieve this objective.A phase-locked loop (PLL) is adopted to synchronize the VSC with the grid [8].However, the PLL-based synchronization strategy may suffer from stability issues, especially when the VSC is connected to a weak grid [9], [10].To address the instability issue induced by the PLLs, one solution is to introduce extra feedforward or feedback terms in the current control loop, such as the output impedance reshaping method [11] and small-signal disturbance compensation control [12].However, these solutions increase the implementation complexity of the current control loop and are dependent on the operating point of the system.An alternative is to modify the PLL based on the concept of complex phase angle vectors to eliminate the frequency-coupling terms induced by traditional PLLs [13].
Recently, an improved parameter tuning method is presented in [14] to alleviate the negative impact of PLLs in weak grids.Although the instability effect of the PLL is mitigated by these strategies, it is still difficult to maintain the stability of VSCs connected to ultra-weak power grids [15].Moreover, the nonlinearities of PLL dynamics, which are usually ignored in the aforementioned literature, may jeopardize the stability of these PLL-based controllers under large grid disturbances [16], [17].
Based on the aforementioned analysis, control algorithms without PLLs are preferred to ensure the stability of gridconnected VSCs.In [18], a PLL-free direct power control (DPC) strategy is developed based on the instantaneous active and reactive power theory.Further, a model predictive DPC strategy is proposed to optimize the switching frequency of the VSC [19].However, since the switching frequency varies with the VSCs' power outputs when using these strategies, undesirable broadband harmonics may be injected into the power grid.Extra efforts on the filter design are therefore required to eliminate these harmonics.To deal with this problem, a DPC strategy using space-vector modulation is proposed with constant switching frequency [20].Nevertheless, the stability analyses in weak grids are not performed in [20].Recently, Gui et.al. [21] developed a voltage-modulated DPC method for weak grid-connected VSCs.Better control performance is achieved by eliminating nonlinear terms in active and reactive power dynamics, yet the stability of the system is analyzed based on the linearized small-signal model.
To allow a wide application of grid-connected VSCs, nonlinear control and stability analysis approaches are needed, e.g., the Lyapunov method [22].In [23], a Lyapunov-based controller is developed for a generalized grid-connected VSC to regulate its power outputs.Further, Altin et.al. [5] designed a sliding-mode controller in the natural frame with a reduced number of voltage and current sensors.However, these algorithms require accurate information of the system, e.g., transmission line parameters, grid frequency and voltage, etc., which is difficult to obtain in practice.To relax this requirement, a robust DPC is designed where the parameter uncertainties are compensated by a robust design [24].But the discontinuous signum function used in the robust design can introduce high-frequency oscillations in the voltage control signals, which increases the power losses of power converters.Moreover, the upper bounds of parameter uncertainties are still required in the robust DPC.Therefore, there is a need to design a nonlinear power controller for grid-connected VSCs without the requirement of system parameters.
To this end, in this study, a Lyapunov-based nonlinear power control algorithm is proposed to regulate the active and reactive power outputs of grid-connected VSCs.To facilitate the design of control algorithms in the absence of PLLs, the dynamic model of the system is developed in the stationary frame.A virtual resistance is introduced to allow fast power tracking performance and relax the requirement on accurate system parameters.The main contributions of this study are summarized as follows: 1) To meet the control requirement of grid-connected VSCs, a nonlinear power control algorithm is presented in this study to regulate the active and reactive power outputs to track their references accurately; 2) The proposed algorithm can work robustly under a wide range of grid conditions, e.g., temporary grid faults, various short circuit ratios (SCRs).The synchronization of the VSC with the grid can be guaranteed in the presence of grid frequency deviations; 3) The stability and robustness of the closed-loop system are analyzed through the Lyapunov synthesis.The system states are proved to converge to their equilibria exponentially.
The remainder of this article is organized as follows.Section II illustrates the QSL model of a grid-connected VSC along with a virtual resistance.In Section III, the nonlinear control algorithm, as well as its stability and robustness analyses, is developed based on the Lyapunov theory.Experimental results are provided in Section IV to show the merits of the designed algorithm.Finally, Section V concludes this study.
Notation: For a complex variable , let  * and  denote its conjugate and modulus, respectively.

II. MODELING OF GRID-CONNECTED VSCS
Fig. 1 illustrates the overall control diagram of a typical three-phase VSC connected to the grid through an LC-filter Simplified VSC Grid Since the timescale of the inner dual-loop PR control is designed much faster than that of the outer nonlinear power control, the VSC and its output LC-filter with the inner dualloop control can be assumed as a voltage source with ideal reference tracking [25], i.e.,  αβ =  (   )αβ .The simplified topology of the grid-connected VSC is given in Fig. 2.

A. System Model With Virtual Resistance
The dynamics of the simplified system given in Fig. 2 can be expressed as where   > 0,   > 0 are the unknown inductance and resistance of the transmission line,  a ,  b ,  c denote the output currents,  a ,  b ,  c denote the grid voltages, and  a ,  b ,  c denote the output voltages which are treated as the control inputs of the simplified system.Next, according to the Clark transformation, the line dynamic model in the stationary frame (- frame) is given as where  α ,  β denote the transformed output currents,  α ,  β denote the transformed grid voltages, and  α ,  β denote the transformed voltage control inputs.
Based on the linear system theory [26], for given voltage control inputs  α and  β , it can be proved that the line currents  α ,  β will exponentially converge to their steady-state trajectories with a convergence rate of   /  .Therefore, to allow fast power tracking performance, it requires that the output currents converge to their steady-state trajectories fast enough.Hence, a virtual resistance is applied in this study to obtain a larger output current convergence rate, which is expressed where   denotes the virtual resistance,  α and  β denotes the virtual voltage control inputs behind the virtual resistance.Substituting (3) into (2) yields the system dynamics with the virtual resistance as Hence, for given virtual voltage control inputs  α ,  β , the line currents  α ,  β will exponentially converge to their steadystate trajectories with a convergence rate of (  +   )/  , which is larger than the original one   /  with a positive virtual resistance   .Moreover, a larger   allows the designs of power controllers with faster power tracking performance.
Thereafter, as in [27], [28], the QSL approximation is applied for system (2) to allow simple control designs.In the - frame, the QSL impedance model of ( 4) can be expressed as where   is the angular frequency of the grid voltage  abc .For ease of expression, define  αβ =  α + j β ,  αβ =  α + j β ,  αβ =  α + j β , and  αβ =  α + j β with j being the imaginary unit.The relationship between  αβ and  αβ can be then expressed as and the system model ( 5) can be rewritten as where   = 1/(  +   + j    ) represents the virtual line admittance.

B. Control Objective
The instantaneous active and reactive power outputs ( and ) of the VSC can be expressed as In this study, the main control objective is to regulate the active and reactive power outputs of the VSC to track given references (    and     ).Based on (8), the complex power regulation error is defined as where  =  + j denotes the complex apparent power output,     =     + j    denotes the complex apparent power reference.Hence, the control objective is achieved if   = 0 holds in the steady-state.

A. Lyapunov-Based Control Design and Stability Analysis
In this section, a nonlinear power controller based on the Lyapunov theory is presented to achieve the control objective defined in Section II-B.
According to the Lyapunov theory, for a function  that is positive definite with respect to   , if  ≤ − with  being a positive constant, then   is bounded for all time and will converge to zero exponentially.Based on this principle, consider the Lyapunov function candidate  =   2 , whose time derivative is The time derivative of   can be given as The detailed mathematical deduction procedure for   can be found in Appendix A.
Next, based on  and the power regulation error dynamics in (11), a nonlinear power control law is designed as where  =   + j  is a user-defined control gain with   and   being two non-negative constants whose selections will be discussed later.Notice that the first part in ( 12) is designed to generate a negative definite term in .The second part is to compensate the dynamic term j   αβ in (11).Now, substituting (12) into the time derivative of   yields the closed-loop system dynamics as Then  becomes Therefore, the stability of the closed-loop system can be guaranteed if  is negative definite.By exploring the conditions for  ≤ 0, the stability results of the proposed controller (12) are stated in the following theorem.
Theorem 1: Considering a grid-connected VSC modeled by (7), if the power controller is designed as (12), then   will converge to zero exponentially if where  is a constant dependent on the control parameter  and the virtual line admittance   , expressed as Proof: Please see Appendix B. Hence, according to Theorem 1, the complex apparent power output  of the VSC can track its reference     accurately with the proposed nonlinear power controller (12) in the steady-state.The control objective is thus achieved.Moreover, notice that ( 15) and ( 16) illustrate the impact of the control parameter  and the virtual line admittance   on the stable operating range of the output voltage  αβ .By recalling the definitions of   and , the square of ( 16) can be rewritten as With the help of the Cauchy-Schwarz inequality, one can obtain that  ≤ 1 holds, and Hence, the largest stable operating range of the output voltage  αβ can be obtained if the control gain  is properly selected according to (18).However, in practice, it is hard to directly select the control parameter  to satisfy (18) due to the the unknown line parameters   and   .To deal with the line parameter uncertainties, one can select a large enough virtual resistance, e.g.,   ≥ 2.5/SCR in per unit value.Then the control gain can be simply selected as   > 0,   = 0, which is sufficient to ensure an acceptable stable operating range of  αβ as illustrated in Section IV.
In Fig. 3, the diagram of the proposed nonlinear power control algorithm is demonstrated.With the information of the VSC's active and reactive power outputs and the given power references, the power regulation error   is calculated.Further, with the constant   and the user-defined control gain , the virtual voltage behind the virtual resistance can be obtained by (12).Finally, the output voltage reference can be calculated according to (6) with  αβ =  (   )αβ , which is further realized by the inner dual-loop PR controller as shown in Fig. 1.
Remark 1: Notice that based on ( 6), (7), and  αβ =  exp(j), the controller in (12) can be rewritten as where ℓ  = Re{ (1 −     )} and ℓ  = Im{ (1 −     )} are two constants.Hence, the proposed nonlinear power control algorithm becomes an integral controller of the coupled power regulation errors with voltage-dependent control gains.However, the integral controller (19) will need a longer time to adjust   to zero compared to the proposed method since it eliminates the positive impact of   on the convergence speed of the line current.Besides, when the grid angular frequency   is not equal to the nominal one, the proposed controller cannot be simply expressed as an integral one given in (19).Remark 2: It should be noted that for VSCs with Lfilters, the dual-loop PR controller is not required.The voltage reference  (   )αβ will be directly used for the modulation of the VSC.The proposed control algorithm becomes a DPC one [18].

B. Robustness Analysis
As illustrated in (15), the stable operating range of the proposed power controller is given.Hence, the proposed controller is robust against grid contingencies if ( 15) is not violated, and thus allows a wide range of grid conditions, such as temporary grid voltage drops, various SCRs, etc. Besides, the accurate power tracking performance can still be achieved in the steady-state if the equilibrium points exist.
Notice that the theoretical results obtained in Theorem 1 is based on the condition that the angular frequency in ( 12) is exactly the one of the grid.However, in practice, the grid angular frequency may deviate from its nominal value and the proposed power controller should be able to synchronize with the grid in the presence of PLLs.
For ease of exposition, ( 12) is rewritten as with the constant  0 being the nominal grid angular frequency, which can be further expressed as (21) where ω =   −  0 is the error between the grid angular frequency and its nominal one, and Notice that according to (21), the VSC can synchronize with the grid if  = 0 in the steady-state.
Based on the Lyapunov theory, the following theorem can be given regarding the control law (20).
Theorem 2: Considering a grid-connected VSC modeled by (7), if the power controller is designed as (20), then  will converge to zero exponentially if The VSC can maintain synchronization with the grid in the steady-state.
Proof: Please see Appendix C. Therefore, compared with (15), one can notice that the stable operating region of  αβ in ( 23) is reduced due to the deviation between  0 and   .According to Theorem 2,  will converge to zero exponentially instead of   .Similar to the idea of the power synchronization control (PSC) [29], according to (22), the power regulation error   will converge gradually to frequency deviations between the VSC and the grid to keep the VSC synchronized with the grid.Thus, the  proposed power control algorithm ( 20) is robust against grid frequency deviations.
In the case that  0 ≠   , one can notice that the accurate power regulation cannot be ensured according to Theorem 2. Fortunately,   can be decreased by increasing  for better power tracking performance.Moreover, as demonstrated in (23), a larger  can enlarger the stable operating region of  αβ .The impact of the grid frequency deviations can be alleviated by selecting a proper control gain , e.g.,   ≥ 40  and   = 0 in per unit value as in Section IV.
Remark 3: In Section II, a precondition used in this study is that the timescale of the inner dual-loop PR control is much faster than that of the nonlinear power control one.If the control gain   is too big, then this precondition will not be met and thus the stability of the controller may not be ensured.Hence,   must be upper-bounded and cannot be selected openly in practice.

IV. VERIFICATION STUDIES
The effectiveness of the proposed nonlinear power control algorithm is tested on a prototype experimental setup as shown in Fig. 4. A three-phase VSC with an LC-filter is connected to a power grid emulated by the Grid Simulator Chroma 61845 through a transmission line.The sampling and switching frequencies are set as 10 kHz.The proposed control algorithm is implemented in the DS1007 PPC processor board with an NXP QorlQ P5020 dual-core real-time processor (64bit, 2 GHz) and a 32-bit I/O bus.The voltages and currents are measured through the DS2004 high-speed A/D board with a 16-bit resolution and 800 ns conversion time.The switching pulses are generated via the DS5101 digital waveform output board with a 25 ns resolution.The system parameters for different cases are listed in Table I.The control parameters for the dual-loop PR controller and the proposed power controller are given in Table II.To show the advantages of the proposed method, a PSC-based VSC with its reactive power regulated by the PI controller [30] is used for comparison, whose control parameters are shown in Table III.

A. Case I: Power Tracking Performance
Firstly, the system parameters in Set 1 are used.The SCR is approximate 28.53.The corresponding results are illustrated in  Figs.5-6.From Fig. 5, one can notice that both the active and reactive power outputs can track their references accurately in the steady-state with the proposed method.The settling time is around 20 ms due to the fast regulation of the output voltage and current.However, as shown in Fig. 6, the PSCbased control strategy cannot synchronize the VSC with the grid when the active power reference increases.
Next, the power tracking performance is tested with system parameters in Set 2, where the SCR is approximate 5.5.The experimental results are given in Figs.7-8.Again, in Fig. 7, one can see that the proposed controller can achieve accurate active and reactive power tracking performance with a settling time of around 30 ms.Besides, from Fig. 8, with the SCR decreases, the PSC-based control strategy can also achieve the active and reactive power tracking objective.However, the settling time is much longer than the proposed method, which is around 100 ms.
Finally, the performance of the proposed and the PSC-based controllers is tested with system parameters in Set 3 with the SCR≈0.95,i.e., the grid is ultra-weak.Notice that in this case, the reactive power reference is selected as 2 kvar to ensure the existence of equilibria.The corresponding results are shown in Figs.9-10.Again, the power outputs can track their references accurately in steady-state by both controllers.The settling time of the proposed controller and the PSC-based one is  around 50 ms and 200 ms, respectively, which is longer than that of a grid with SCR≈5.5.for both controllers.

B. Case II: Voltage Drop
In this case, the performance of both controllers under the condition that the grid voltage drops from 1 p.u. to 0.6 p.u. for 200 ms is tested.The system parameters in Set 2 are used.The corresponding results are delivered in Figs.11-12.As shown in Fig. 11, during the grid voltage drop, with the proposed control strategy, the active power output is slightly affected, while the reactive power output deviates from its reference during the transient-state and converges to zero in about 40 ms.The control objective is achieved.In comparison, from Fig. 12, one can see that when the grid voltage drops, the active and reactive power outputs with the PSC-based controller deviate from their set points with the errors around 2500 W and 2500 var, respectively, which are much larger than those of the proposed method.

C. Case III: Frequency Variation
In this case, the control performance when grid frequency decreases 49.   and 15, one can notice that the active power outputs are almost unaffected, while the reactive power output in the steady-state is no longer zero with a steady-state tracking error around 100 var.According to Theorem 2, when the grid frequency changes, both the active and reactive power outputs cannot accurately track their references if ω ≠ 0 since the power regulation errors are used to keep the VSC synchronized with the grid.From Figs. 14 and 16, for the PSC-based control methods, the reactive power output is not affected, while the steady-state active power tracking error will be larger than 500 W.Although this steady-state tracking error can be decreased by increasing the active power control gain, the system may suffer from instability issues.

D. Case IV: Voltage Harmonics
In this case, the control performance when a 10% 5th harmonic appears in the grid voltage for 200 ms with the system parameters in Set 2 used.The experimental results are illustrated in Figs.17-18.One can notice that the active and reactive power tracking performance of both controllers is similar.Compared with the output voltage of the PSC-based controller, which is almost sinusoid, the proposed controller is more sensitive to the harmonics.A possible reason is  that the proposed method is more sensitive to the power regulation error to achieve a fast and accurate power tracking performance.

V. CONCLUSION
In this study, a Lyapunov-based nonlinear control algorithm is presented to achieve the regulation of the active and reactive outputs of a grid-connected VSC.A virtual resistance is introduced to allow fast power tracking performance and relax the requirement on accurate system parameters.Based on the QSL impedance model, a nonlinear power control algorithm that only utilizes the output voltage and current information is designed.The closed-loop system stability is analyzed via the Lyapunov theory.Theoretical results show that the VSC can keep synchronization with the grid in the presence of grid frequency deviations.Finally, experimental tests under temporary grid faults and various SCRs illustrate the performance and robustness of the proposed control algorithm.
In the future, how to achieve accurate power regulation when grid frequency deviates from its nominal value will be investigated.The explicit relationship between the convergence of the power regulation error and the satisfaction of the requirement on the output voltage will be studied.Besides, the impacts of the inner control loops on the power control performance will be explored.This is the author's version of an article that has been published in this journal.Changes were made to this version by the publisher prior to publication.
The final version of record is available at http://dx.doi.org/10.1109/TIE.2021.3065614 Copyright (c) 2021 IEEE.Personal use is permitted.For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.APPENDIX A POWER REGULATION ERROR DYNAMICS Substituting ( 6) and ( 7) into the time derivative of   defined in (9) yields According to (7), the time derivative of  αβ becomes Then, substituting (25) into (24) yields  Further, by recalling (6), the power regulation error dynamics in (11) can be derived.

APPENDIX B PROOF OF THEOREM 1
By revoking (15), there exists a constant 0 <  < 1, such that According to ( 6) and ( 7), the following implication based on ( 27) can be derived as Therefore,  in ( 14) further gives Again, according to (15), the following implication can be derived as Substituting ( 30) into (29) yields where where  0 and  0 are the initial values of  and   at time  0 , respectively.Hence,   will converge zero exponentially.The proof is thus completed.

APPENDIX C PROOF OF THEOREM 2
Firstly, combining (11) and (21) gives the closed-loop dynamics of the power regulation error as To facilitate the stability analysis, substituting (33) into the time derivative of  yields Combining ( 6), (7), and the first equality in (25) yields

Fig. 3 .
Fig. 3. Diagram of the proposed nonlinear power control algorithm.

7
Hz and increases to 50.3 Hz for 200 ms is tested.Again, the system parameters in Set 2 are used.The corresponding results are shown in Figs.13-16.From Figs. 13

Fig. 11 .
Fig. 11.Control performance of the proposed controller when grid voltage drops to 0.6 p.u.

Fig. 12 .
Fig. 12.Control performance of the PSC-based controller when grid voltage drops to 0.6 p.u.

Fig. 13 .
Fig. 13.Control performance of the proposed controller when grid frequency decreases to 49.7 Hz.

Fig. 14 .
Fig. 14.Control performance of the PSC-based controller when grid frequency decreases to 49.7 Hz.

Fig. 15 .Fig. 16 .
Fig. 15.Control performance of the proposed controller when grid frequency increases to 50.3 Hz.