Modeling and Design of Voltage Support Control Schemes for Three ...

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 11, NOVEMBER 2014

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Modeling and Design of Voltage Support Control Schemes for Three-Phase Inverters Operating Under Unbalanced Grid Conditions Miguel Castilla, Jaume Miret, Member, IEEE, Antonio Camacho, Luis Garc´ıa de Vicu˜na, and Jos´e Matas

Abstract—Three-phase inverters equipped with voltage support control schemes have been successfully used to alleviate the negative impact of voltage imbalance on electric power systems. With these schemes, dc-link voltage ripple and current harmonics are significantly reduced by processing the positive and negative sequence components separately. However, the design methods for tuning the parameters of these control schemes have one or more of the following limitations: 1) the design is a very time-consuming task, 2) it is conservative, 3) it does not guarantee the specifications for all the considered situations, and 4) the system can be unstable in some abnormal conditions. As an alternative, this paper presents a design method based on the analysis of oscillations in nonlinear systems. The method proceeds by first developing simple and accurate models of the power system, second it reveals the system features through an in deep analysis of the derived models, and third it introduces a systematic design procedure for tuning the parameters of the control schemes. As an example, a voltage support control scheme for a three-phase inverter operating under an unbalanced voltage sag is designed and validated experimentally. Index Terms—Averaging, circuit modeling, control design, power conversion, voltage control.

I. INTRODUCTION RADITIONALLY switching power converters have been predominantly employed in domestic, industrial, and information technology applications. However, due to new advances in power semiconductor and digital signal processor technologies, the application of power converters in electric power systems has recently gain considerably more attention. Thus, power converters are increasingly employed in power systems for power conditioning [1]–[4], compensation [5]–[7], and filtering [8]–[12]. In power systems, the voltage imbalance can cause severe problems to energy sources, rotating equipments, and loads. The power production of energy sources reduces with grid voltage imbalance due to the high ripple current placed in the dc bus [13], [14]. The presence of grid voltage imbalance degrades

T

Manuscript received July 17, 2013; revised September 15, 2013 and November 1, 2013; accepted December 21, 2013. Date of publication January 2, 2014; date of current version July 8, 2014. This work was supported by the Ministerio de Econom´ıa y Competitividad of Spain under Project ENE201237667-C02-02. Recommended for publication by Associate Editor F. Wang. The authors are with the Department of Electronic Engineering, Technical University of Catalonia, 08800 Vilanova i la Geltr´u, Spain (e-mail: [email protected]; [email protected]; antonio.camacho.santiago@ upc.edu; [email protected]; [email protected]). Digital Object Identifier 10.1109/TPEL.2013.2296774

the lifespan of the rotating machinery due to the limited negative sequence capability [15]. Then, power converters equipped with voltage support control schemes are employed to alleviate these adverse effects [16]–[20]. In these schemes, a positive sequence voltage loop is responsible to correct the voltage deviation, by injecting the suitable positive sequence reactive power [16]. By operating in parallel, a negative sequence voltage loop attenuates the voltage imbalance through the provision of the necessary negative sequence reactive power [17], [18]. Therefore, the power quality of the grid voltage is notably improved. The design of these voltage control schemes for unbalanced power systems has been usually carried out by case studies and simulation tools [19], [20]. Therefore, a large number of experiments under different set of conditions were necessary to guarantee a proper operation for all the considered situations, which results in a very time-consuming task. Alternatively approximate models of the system have been derived to design the control schemes. These models are generally based on a power balance equation which does not take into account the complete dynamics of the system [21]–[23]. Although some improvement of the approximate models can be found in the literature [24], the control design is in general conservative. Moreover, in certain abnormal grid conditions, the system performance is poor and it can be even unstable [24]. This paper introduces a mathematical tool to design voltage support control schemes for grid-connected power converters. First, a modeling method based on the analysis of oscillations in nonlinear systems is used to model the power converter [25], [26]. This method extracts the low-frequency dynamics of the power converter by separating the variables in fast and slow time-varying signals and by averaging the resulting models. A similar method was previously presented in [27] to model resonant power converters. In this paper, the method is revised and adapted to the new scenario of grid-connected power converters. This revision includes the classification of the power converter variables and the derivation of the suitable averaged models. Second, an in-depth analysis of the power converter is carried out by making use of the averaged models. The exact features and the real performance of the power converter are revealed by the analysis, and a full and deep understanding of the system behavior is achieved by the identification of previously unknown characteristics. Third, a systematic design procedure for tuning the parameters of the voltage control schemes is presented. The closed-loop characteristics of the power converters are given by fulfilling the typical frequency domain specifications (i.e., control bandwidth and phase margin). Thanks to the

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frequency ω [20]

Fig. 1.

vα+ = V + cos(ωt + ϕ+ )

(1)

vβ+

(2)

=V

+

+

sin(ωt + ϕ )

vα− = V − cos(−ωt − ϕ− )

(3)

vβ−

(4)

−

−

= V sin(−ωt − ϕ ).

Diagram of the grid-connected power converter.

method, good performance is ensured even during abnormal grid conditions. The rest of the paper is organized as follows. In Section II, the power system is described and the averaging procedure is introduced. In Section III, the model of the complete power system is derived, including the power converter, the voltage control scheme, and the electrical grid. In Section IV, the particular features of the power system are revealed through the analysis of the small-signal model. In Section V, the systematic design procedure for tuning the parameters of the voltage control is presented. In Section VI, the design method is validated by testing a low power experimental prototype during an unbalanced voltage sag. Section VI is the conclusion of the paper. II. SYSTEM DESCRIPTION AND MODELING APPROACH This section describes the power system considered in this study and presents the modeling approach used to analyze it. A. Power System Fig. 1 shows the diagram of the power system including the power converter, the voltage control, and the electrical grid. The power converter is a three-phase three-wire voltage-source inverter. The voltage control senses both the voltage at the point of common coupling (PCC) and the output current of the power converter, and it generates the reference current for the internal control loop of the power converter. The ac network is represented as a series connection of the grid impedance and the ac voltage source. The purpose of the voltage control is to regulate the voltage of the PCC. In normal operation, the amplitude of the voltage v is regulated to the reference voltage Vref . In this study, the voltage control is designed to also cope with unbalanced grid voltage. Thus, the injection of reactive power by positive and negative sequences is necessary to accomplish the task of voltage regulation [19], [20]. The mathematical models developed in this study are based on the stationary reference frame. The control schemes are also implemented in this frame. Fig. 2 shows a diagram of the complete power system model. In this diagram, the voltage control has been divided into three main blocks: the sequence extractor, the voltage control loop, and the reference current generator. The sequence extractor is responsible for estimating the positive and negative sequence components of the PCC voltage. The symmetrical components of the PCC voltage can be written in three-wire systems as sinusoidal signals rotating at the grid

Note that, to simplify the study, the zero sequence voltage components present in some fault conditions (for instance, in single-line-to-ground faults with ac grid grounded) are not considered. The outputs of the sequence extractor are also sinusoidal signals but operating at the estimated grid frequency ωe vα+e = Ve+ cos(ωe t + ϕ+ e )

(5)

vβ+e

(6)

vα−e

=

Ve+

=

Ve−

sin(ωe t +

ϕ+ e ) ϕ− e)

(7)

vβ−e = Ve− sin(−ωe t − ϕ− e ).

(8)

cos(−ωe t −

Note that the amplitudes of the variables defined in (1)–(8) are supposed to be slowly varying functions of time. Precisely the main objective of the voltage loop is to regulate these slow variables to the voltage references. The final purpose is to achieve, in steady state, the following results: + V + = Vref − V − = Vref .

(9) (10)

To this end, the voltage loop provides the reference reactive power necessary for the voltage regulation; see Fig. 2. The reference generator determines the reference current for the internal control loops. The outputs of the power converter and the electrical grid blocks depicted in Fig. 2 are the output currents and the PCC voltages, respectively. It is worth mentioning that only slow time-varying signals are of interest for the analysis and design of the voltage control loop. Below, a modeling approach for extracting the significant low-frequency information of the power system is described. B. Analysis of Oscillations in Nonlinear Systems Several methods can be found in the literature for the study of oscillations in nonlinear systems [25], [27]. In these methods, oscillating signals are identified as fast time-varying variables. Amplitudes of the oscillating signals are considered to be slowly time-varying variables. A step-by-step procedure intended for the derivation of averaged large-signal models of nonlinear systems is presented below [27]. The method is adapted in this study to the modeling of grid-connected power converters. 1) Write the power system model and derive its steady-state solution (i.e., the fast variables). 2) Insert this solution into the system model. If the resulting system is nonlinear, the nonlinear terms should be replaced by their fundamental harmonic components (which is known as harmonic linearization). At this point, the model has been linearized in a large-signal sense, and it

CASTILLA et al.: MODELING AND DESIGN OF VOLTAGE SUPPORT CONTROL SCHEMES FOR THREE-PHASE INVERTERS

Fig. 2.

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Diagram of the complete power system model including the power converter, the electrical grid, and the voltage control.

The reference currents can be separated in terms of active and reactive reference currents as

Fig. 3.

(18)

i∗β = i∗β ,P + i∗β ,Q

(19)

where

Diagram of the averaged model of the power system.

i∗α ,P = can be written in a general form as

vα+e P 2 + 2 3 (vα e ) + (vβ+e )2

(20)

vβ+e P 2 (21) 3 (vα+e )2 + (vβ+e )2   vβ+e Q+ vβ−e Q− 2 ref ref = + − 2 (22) 3 (vα+e )2 + (vβ+e )2 (vα e ) + (vβ−e )2   − − −v Q −vα+e Q+ 2 α e ref ref = + − 2 . (23) 3 (vα+e )2 + (vβ+e )2 (vα e ) + (vβ−e )2

i∗β ,P =

f1c (t) cos(ωt) + f1s (t) sin(ωt) = f2c (t) cos(ωt) + f2s (t) sin(ωt).

i∗α = i∗α ,P + i∗α ,Q

(11)

In (11), the sinusoidal signals are fast variables and the amplitude of these signals are supposed to be slowly varying functions of time. 3) The averaged large-signal model is derived by imposing a harmonic balance. That is, f1c (t) = f2c (t)

(12)

f1s (t) = f2s (t).

(13)

The result is a model that describes the behavior of the slow variables.

i∗α ,Q i∗β ,Q

The active power P is transferred to the grid by the use of (20) and (21) [28]. The reactive power is injected via both positive and negative sequences by making use of (22) and (23) [29]. The averaged model of the electrical grid and the power converter is derived using the modeling procedure described in the previous section; see details on this derivation on Appendix I. The obtained model can be written as

III. AVERAGED MODEL OF THE POWER SYSTEM

V + = Vg+ +

2 ωe Lg Q+ ref 3 Ve+

(24)

This section is devoted to the derivation of the averaged model of the power system. A diagram of this model is shown in Fig. 3.

V − = Vg− −

2 ωe Lg Q− ref . 3 Ve−

(25)

A. Averaged Model of the Grid and the Power Converter From Fig. 1, the model of the electrical grid in stationary reference frame can be written as diα dt diβ . + Lg dt

vα = vg α + Lg

(14)

vβ = vg β

(15)

For the design of the external control loop, the internal current control is assumed to be a perfect tracking loop and therefore the output current is normally expressed as [17] iα = i∗α

(16)

iβ = i∗β .

(17)

The support mechanism to the PCC voltage can be identified from (24) and (25). As usual in inductive networks, the effect of the active power is negligible and then P is not present in (24) and (25). The injection of reactive power via positive sequence increases the positive sequence voltage at the PCC. Besides the injection of reactive power via negative sequence decreases the negative sequence voltage at the PCC. The ability to reduce the voltage deviations depends largely on the grid impedance and the converter power rating (which, in fact, limits the maximum reactive power that can be injected into the electrical grid). B. Averaged Model of the Sequence Extractor Fig. 4 shows the diagram of the voltage sequence extractor considered in this study [30], [32]. It consists of three main blocks: a direct/quadrature voltage extractor, a frequency

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Fig. 4.

Fig. 5.

Diagram of the voltage sequence extractor.

Diagram of the voltage support control loop.

model can be expressed as estimator, and a sequence voltage calculator. The first block is used to extract the direct and quadrature components of the input signals. To this end, the extractor behaves as a band-pass filter for the direct signals and as a low-pass filter for the quadrature signals. The dynamic model of the extractor can be derived from [28] and it can be written as dvα de = ωe [2ξ(vα − vα de ) − vα q e ] dt dvα q e vα q e dωe = ωe vα de + dt ωe dt dvβ de = ωe [2ξ(vβ − vβ de ) − vβ q e ] dt vβ q e dωe dvβ q e = ωe vβ de + dt ωe dt

(26) (27) (28) (29)

being ξ the damping factor of the filters. In Fig. 4, the estimation of the grid frequency is employed to online tune the direct and quadrature filters. Although several frequency estimators can be found in the literature, the implementation presented in [31] is chosen here due to its high robustness to amplitude variations in the grid voltage. The dynamic model of the frequency estimator is (vα − vα de )vα q e + (vβ − vβ de )vβ q e dωe = −γωe + 2 dt (vα e ) + (vβ+e )2 + (vα−e )2 + (vβ−e )2

(30)

where the parameter γ determines the speed of convergence. The sequence calculator block provides the positive and negative sequence components of the grid voltages [31] vα de − vβ q e 2 vα de + vβ q e = 2 vβ de + vα q e = 2 vβ de − vα q e . = 2

vα+e =

(31)

vα−e

(32)

vβ+e vβ−e

(33) (34)

The averaged model of the sequence extractor is obtained by applying the averaging procedure described in Section II. The details on the model derivation can be found in Appendix I. The

γ Ve+ (V + Ve+ + V − Ve− ) ω − ωe dVe+ = ξωe (V + − Ve+ ) + dt 2 ωe (Ve+ )2 + (Ve− )2

(35) −

γ dVe = ξωe (V − − Ve− ) + dt 2

−

Ve (V + Ve+ + V − Ve− ) (Ve+ )2 + (Ve− )2

ω − ωe (36) ωe

V + Ve+ + V − Ve− dωe =γ (ω − ωe ). dt (Ve+ )2 + (Ve− )2

(37)

Note that the sequence extractor behaves as a third-order nonlinear system. The complexity of the system is given by its large number of nonlinear terms and the coupling between the equations. C. Averaged Model of the Voltage Control Loop Fig. 5 shows the diagram of the voltage control loop analyzed in this study [32]. It is formed by three blocks: a grid voltage and impedance estimator, a reference reactive power calculator, and a low-pass filter. The first block estimates the grid impedance by using the method reported in [33]. Then, the grid voltage is calculated as  ωe Lg c Q+ 2 ref (38) Vg+e = (vα+e )2 + (vβ+e )2 −  3 (v + )2 + (v + )2 αe βe Vg−e =



(vα−e )2 + (vβ−e )2 +

ωe Lg c Q− 2 ref  3 (v − )2 + (v − )2 αe βe

(39)

where Lg c is a control parameter that in [32] was updated by the online measure of the grid impedance. Note that (38) and (39) can be easily derived from the averaged model expressed in (24) and (25). The second block calculates the reactive power under the assumption that the voltage at the PCC tracks accurately the reference voltage. In this case, the reactive power is written, by making use again of (24) and (25), as Q+ =

+ + 3 Vref (Vref − Vg+e ) 2 ωe Lg c

(40)

Q− =

− − (Vg−e − Vref ) 3 Vref . 2 ωe Lg c

(41)

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The third block is a first-order low-pass filter that governs the transient response of the reference reactive power dQ+ ref = ωc (Q+ − Q+ ref ) dt

(42)

dQ− ref = ωc (Q− − Q− ref ) dt

(43)

where ωc is the filter cut-off frequency. This filter provides the required time delay to ensure that the measures and previous estimations have reached the final values before they are used to control the system. The averaged model of the voltage control loop can be derived by inserting (5)–(8) in (38) and (39), and then rearranging (40)– (43). The resulting model is written as  + +  Qref dQ+ 3 Vref + ref = ωc + − Ve+ ) (44) (Vref dt 2 ωe Lg c Ve+  − −  dQ− Qref 3 Vref − ref = ωc − − Ve− ). (45) (Vref dt Ve− 2 ωe Lg c D. Discussion on the Derived Averaged Models As far as the authors know, there are no available models in the literature that predict accurately the low-frequency dynamics of the sequence extractor introduced in [31] and the voltage control loop recently presented in [32]. Thus, one of the contributions of this paper is the averaged models expressed in (35)–(37) for the sequence extractor and in (44)–(45) for the voltage control loop. Note that these models are essential to predict the features of the complete power system and to design the control parameters (i.e., ξ, γ, ωc , and Lg c ).

Fig. 6. Diagram of the small-signal model of the power system. (a) Positivesequence model, (b) negative-sequence model.

The damping factor ξ governs the dynamics of the voltage sequence extractor and the parameter γ controls the dynamics of the frequency estimator. From (46)–(48), the voltage and frequency settling times can be expressed as ts,v =

5 ξωo

(49)

ts,ω =

5 . γ

(50)

The transfer functions of the power system are derived from (24) and (25) and (46) and (47). They can be written as G+ ps (s) =

s + ξωo 2 ωo Lg   3 Vo+ s + 1 + 2 ω o L g Q +o ξω + o 3 (V ) 2

(51)

o

IV. SMALL-SIGNAL MODEL OF THE POWER SYSTEM This section reveals the particular features of the voltage control scheme considered in this study through the analysis of the small-signal model characteristics. A. Small-Signal Model The derivation of small-signal models is well documented in the literature and then details on the model derivation will be omitted here. The definition of the involved variables is introduced in Appendix II. Fig. 6 shows the diagram of the smallsignal models of the power system. Note that the positive- and negative-sequence models are decoupled, which facilitates the analysis of the system. The transfer functions of the sequence extractor are derived from (35)–(37), and they can be expressed as ξωo s + ξωo

(46)

ξωo = s + ξωo

(47)

G+ se (s) = G− se (s)

ω ˆe γ (s) = . ω ˆ s+γ

(48)

s + ξωo 2 ωo Lg   G− . ps (s) = − − 3 Vo s + 1 − 2 ω o L−g Q −o ξω o 3 (V o ) 2

(52)

Note that the poles of the transfer functions depend on the reactive power injection. As the injection increases, then the frequency of the pole in (51) increases and in (52) decreases. With no injection, a zero-pole cancellation takes place and the power plant is modeled as a variable gain. In fact, this gain relies on the grid frequency, the grid impedance, and the amplitude of the grid voltage. Probably the most interesting result of this study is the transfer functions of the voltage control loop. From (44) and (45), these transfer functions can be derived   ωc Q+ 3 Vo+ k+ o (53) (s) = + = i G+ vl + s Vo 2 ωo Lg c s   ωc Q− 3 Vo− ki− o G− . (54) (s) = − = vl s Vo− 2 ωo Lg c s Note that (53) and (54) are adaptive integral regulators. The integral gains change with the injection of reactive power and the characteristics of the electrical grid (both amplitude and frequency). Besides two control design parameters ωc and Lg c are available to establish the preferred features for the closed-loop

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system. What is even more important is that the integral regulators will eliminate the steady-state error in the sequence voltages (see the voltage loop blocks in Fig. 6) and thus the online estimation of the grid impedance is no further required. In other words, the restriction Lg c =Lg is not really necessary for the proper operation of the control system. The main consequence is a simplification of the voltage control scheme presented in [32].

TABLE I PARAMETERS OF THE EXPERIMENTAL SETUP AND DESIGN SPECIFICATIONS

B. Closed-Loop Characteristics The closed-loop characteristics, including system stability and transient response, can be evaluated by analyzing both the loop gain and the closed-loop transfer functions. From Fig. 6, these transfer functions can be written as + G+ vˆ+ ps (s)Gv l (s) (s) = 1 + T + (s) vˆr+ef

(55)

− G− vˆ− ps (s)Gv l (s) − (s) = 1 + T − (s) vˆr ef

(56)

+ + T + (s) = G+ ps (s)Gse (s)Gv l (s)

(57)

where −

T (s) =

− − G− ps (s)Gse (s)Gv l (s).

(58)

C. Discussion on the Derived Small-Signal Models Another contribution of this paper is the small-signal models expressed in (51) and (52) for the power converter and the electrical grid and in (53) and (54) for the voltage control loop. As far as the authors know, there are no available models in the literature that predict so accurately the small-signal dynamics of these two subsystems. Note that these models provide insight about the expected behavior of the complete system. In addition, they are essential to design the control parameters, as shown in next section. V. DESIGN OF THE VOLTAGE CONTROL LOOP This section presents a step-by-step procedure to calculate the four design parameters of the voltage control scheme: ξ, γ, ωc , and Lg c . An example illustrates the application of the design procedure. A. Design Procedure The settling time of a sequence extractor is usually between one and two periods of the grid voltage [30], [31]. By fixing the settling time at one period of the grid voltage in (49) and (50), the control parameters of the sequence extractor can be calculated as 5 (59) ξ= 2π 5ωo γ= . (60) 2π The control parameters of the voltage loop (ωc and Lg c ) determine the values of the regulator integral gains, as shown in (53) and (54). Thus, ωc and Lg c will have a strong influence on

the magnitude of the loop gain transfer functions; see (57) and (58). However, the effect on the phase of these transfer functions is negligible. Note that the poles and zeroes of the loop gain transfer functions do not rely on the control parameters ωc and Lg c . Taking in mind this fact, Lg c can be fixed to a known value and then ωc could be chosen to accomplish the frequency domain specifications for the control bandwidth and the phase margin. It is worth mentioning that the value of Lg c is established in the design process and it is not necessary to online measure the grid impedance to update the value of this parameter. In this study, Lg c is set to the value of the grid impedance in the worst-case scenario (i.e., weak grid condition) Lg c = Lg ,w .

(61)

With this choice, the accomplishment of the frequency domain specifications is guaranteed for all the range of the grid impedance, as shown below. B. Design Example A low-power experimental setup is used in the next section to validate the proposed design procedure. Here, the voltage control scheme of this setup is designed. Table I lists the parameters of the power system and the design specifications. A type C voltage sag is used to evaluate the features of the system. The grid voltage components can be written as a function of the voltage unbalance factor (VUF) as [34] 1 Vg (62) 1 + VUF VUF Vg− = Vg (63) 1 + VUF where VUF = 0.3 is chosen in the experiments. The reference voltages are selected in order to improve the PCC voltages during the sag. In particular, the positive-sequence voltage is increased 6%, and the negative-sequence voltage is decreased Vg+ =

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TABLE II SOLUTION TO THE DESIGN EXAMPLE

Fig. 7. Bode diagram of the loop gain T + (s) for the maximum and minimum grid impedances. TABLE III CONTROL BANDWIDTH AND PHASE MARGIN AS A FUNCTION OF L g

Fig. 8. PCC voltages and inverter currents for (a) L g = 0.12 p.u. and (b) L g = 0.06 p.u.

6% + Vref = Vo+ = 1.06Vg+

(64)

− Vref = Vo− = 0.94Vg− .

(65)

This is an arbitrary selection and other percentages are possible by taking into account the limitations derived in Appendix III. The design is performed for weak grid conditions and the corresponding reactive power symmetrical components for this worst-case scenario are listed in Table I. The solution to the design example is summarized in Table II. The design parameters ξ, γ, and Lg c are calculated from (59)– (61), respectively. The parameter ωc is determined by guaranteeing a BW of 100 rad/s in the positive-sequence loop gain for Lg =Lg ,w . The reaching of this specification is verified by the Bode diagram shown in Fig. 7. In addition, Table III lists the measured control bandwidth and the phase margin for different

values of the grid impedance. In view of these results, it is possible to conclude that the frequency domain specifications are met for all the considered grid impedance values. In particular, the following observations can be made: 1) The system is stable (the phase margin is positive). 2) The transient response becomes slower (i.e., the control bandwidth reduces) as the value of the grid impedance decreases. 3) The measures of both loop gains are very similar. Thus, the design can be carried out with any of the two loop gain transfer functions. VI. EXPERIMENTAL RESULTS This section validates the proposed design procedure with selected laboratory tests on the low-power experimental setup. A. Experimental Setup The operation of the power converter shown in Fig. 1 was reproduced in the laboratory by a Semikron three-phase IGBT inverter. The electrical grid was emulated by a programmable Pacific Power ac power supply and a series connected threephase inductor. A TMS320F28335 floating point DSP from Texas Instruments was employed as digital control platform. The experiments were designed with the following sequence of events: 1) initially, from 0 to 0.1 s, the ac power supply

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Fig. 9. Positive and negative sequence components of the estimated voltage (in black) and reference voltage (in grey) for (a) L g = 0.12 p.u. and (b) L g = 0.06 p.u.

Fig. 10. Positive and negative sequence components of the reference reactive power for (a) L g = 0.12 p.u. and (b) L g = 0.06 p.u. TABLE IV CONTROL BANDWIDTH AND PHASE MARGIN AS A FUNCTION OF ω c

emulates a normal grid condition; 2) from 0.1 to 0.6 s, a type C voltage sag with VUF = 0.3 was programmed in the ac power supply; 3) the voltage control was intentionally inactive from 0.1 to 0.2 s, in order to clearly show the PCC voltages during the abnormal grid condition; 4) from 0.2 to 0.6 s, the voltage control was active and therefore the reactive power injection and the PCC voltage regulation were put into operation. The transient response of the power converter was just evaluated in this interval, once the voltage control scheme was activated. The power production of the system was maintained during the voltage sag at P = 750 W; see Table I. B. Tests for Weak Grid Conditions Figs. 8–10 show the main waveforms of the power system during the sequence of events described above. In particular, the results of the tests for weak grid conditions are shown in part (a) of these figures, while part (b) shows the results for a lower grid impedance. Note that the PCC voltages become unbalanced and their amplitudes are reduced during the voltage sag; see Fig. 8(a). This loss of voltage quality can also be observed in the reduction of positive sequence voltage and the increase of negative sequence voltage; see Fig. 9(a). The reactive power injection activates the voltage regulation mechanism and places the symmetrical components of the PCC voltage at their reference values with no steady-state error. The inverter

current can be seen in Fig. 8(a). In normal condition, only active current is injected to the grid. During the sag, a current increase is noticed, especially when the voltage control is activated. Note that, in this case, the current is unbalanced due to the presence of positive and negative sequence components. Thanks to this unbalanced current both positive and negative sequence voltages can be regulated to their reference voltages. With balanced current, only one of the voltage sequences could be controlled and then the control objectives expressed in (9) and (10) cannot be meet simultaneously. Probably the best waveforms to evaluate the transient response are the reference reactive power

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Fig. 11. Reference reactive power Q + for several values of the control bandwidth. (a) 300 rad/s, (b) 200 rad/s, (c) 150 rad/s, (d) 100 rad/s, (e) 75 rad/s, re f (f) 50 rad/s, (g) 30 rad/s, (h) 20 rad/s, and (i) 15 rad/s.

components. Fig. 10(a) shows a fast transient response with a settling time of 22.5 ms. This value is in good agreement with the theoretical result (28.3 ms) derived by the analysis of (55).

smaller values of Lg (see Table III). Thus, the filtering action of the closed-loop system improves for lower values of the grid impedance.

C. Tests Using a Different Grid Impedance

D. Tests Using Different Values of ωc

The next experiment was performed to evaluate the effect of using a different impedance in the experimental setup. In particular, a 2.5 mH (0.06 p.u.) inductor was used to emulate the grid impedance. The parameters of the voltage control listed in Table II were maintained during the test (i.e., the control design was not updated for the new impedance). Also the reference voltages listed in Table I were maintained to facilitate the comparison between the two experiments. Figs. 8(a)–10(b) show the results. Note that a higher current and reactive power are necessary in this case to achieve the same reference voltages. What is even more interesting is that no steady-state error is noticed in the symmetrical components of the PCC voltage, see Fig. 9(b). This fact confirms that 1) the voltage control scheme behaves as an integral regulator, and 2) the online measurement of the grid impedance is not really necessary to regulate the PCC voltage. The transient response of the power system is slower in this case in comparison with the results shown in Fig. 8(a)–10(a). The measured settling time of the reference reactive power components is 80.9 ms; see Fig. 10(b). This result is in good agreement with the theoretical value (82.1 ms) predicted by (55). The measured results show a small ripple caused mainly by the practical errors associated with the digital implementation of the control scheme (i.e., coefficient mismatch, quantization, delay). The magnitude of this ripple is reduced by decreasing the value of Lg , as shown in Figs. 9 and 10. This valuable effect is due to the reduction of the control system bandwidth for

The last experiment evaluates the effect of using different values for ωc . Table IV lists the control bandwidth and the phase margin for a broad range of ωc values. Fig. 11 shows the transient response of the reference reactive power for the considered values of ωc . Note that for high values, the transient response is very fast. However, the capacity to reject low-frequency harmonics is low and the start-up overshoot is high. These harmonics cause unwanted ripple in the reference signals, as shown in Fig. 11(a). Quite the opposite, a high harmonic rejection is reached and no overshoot is noticed for low values of ωc , as shown in Fig. 11(i). In this case, the main limitation is a slow transient response. In view of the results, it seems obvious that a good solution for the design tradeoff is to set the control bandwidth BW at 1/3 of the bandwidth of the sequence extractor ξωo , as proposed in Section V. VII. CONCLUSION A theoretical methodology formulated for the analysis of the external control loops of three-phase grid-connected converters has been presented in this paper. The method proceeds as follows. First, the large-signal averaged models of the power system are developed by using a simple method to analyze oscillations in nonlinear systems. Second, the system features are derived by making use of the linear analysis techniques based on small-signal models. Third, a control design method can be established with the system knowledge gained by the small-signal

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analysis. In addition to the proposed design methodology, other major contributions of this study are as follows. 1) The averaged models for both the sequence extractor introduced in [31] and the voltage support control scheme presented in [32]. 2) The small-signal models for the power system (including the power converter and the electrical grid) and the voltage support control scheme presented in [32]. With these models, the exact features and the real performance of the analyzed power system were revealed. The features can be summarized as follows. 1) The power system is stable for all the considered range of operation. 2) The transient response strongly depends on the value of the grid impedance. 3) The voltage control scheme behaves as a nonlinear integral regulator. 4) The online measurement of the grid impedance is not really necessary for the correct operation of the system. The application of the proposed methodology can be easily extended to other power converters connected to the utility grid. It could be also employed to devise new control schemes and strategies with improved performance. In particular, several control strategies to determine the reference voltage values in accordance with the grid characteristics (impedance, VUF, etc.) could be developed with the proposed averaged models. These ideas are actually open for further research. APPENDIX I This appendix gives the detailed derivation of the averaged models of the power system and the voltage sequence extractor. A. Averaged Model of the Power System The steady-state voltages of the power system are defined by (1)–(8) and vg+α = Vg+ cos(ωt + ϕ+ )

(A1)

vg+β = Vg+ sin(ωt + ϕ+ )

(A2)

vg−α

cos(−ωt − ϕ )

(A3)

vg−β = Vg− sin(−ωt − ϕ− ).

(A4)

=

Vg−

−

The method proceeds next by inserting this solution in the model (14) and (15) with the help of (16) and (23). The resulting model has the general form written in (11). By applying the harmonic balance (12) and (13) to this model, the averaged model expressed in (24) and (25) is obtained after some simple manipulations. B. Averaged Model of the Sequence Extractor The steady-state solution (1)–(8) is first inserted in the model (26)–(34). The resulting model has the form expressed in (11). By using the harmonic balance (12) and (13), the averaged model (35)–(37) is derived.

APPENDIX II This section defines the small-signal variables used in this study. To this end, the averaged variables are decomposed in steady-state quantities and small signal variables V + = Vo+ + vˆ+

(A5)

Ve+

(A6)

=

Vo+

+

vˆe+

+ + Vref = Vo+ + vˆref

V

−

=

Vo−

−

+ vˆ

Ve− = Vo− + vˆe− − Vref

=

Vo−

+

− vˆref

(A7) (A8) (A9) (A10)

ω = ωo + ω ˆ

(A11)

ωe = ωo + ω ˆe

(A12)

Q+ ref Q− ref

=

Q+ o

=

Q− o

+

+ qˆref

(A13)

+

− qˆref

(A14)

where the subscript o denotes steady-state quantities and the superscript ˆ identifies the small-signal variables. Note that the steady-state quantities of all the positive-sequence voltages have the same value. This can be deduced by calculating the steadystate solution of (35) and (44). The same is true for all the negative-sequence voltages in steady state. This fact can be demonstrated by solving (36) and (45) in steady state. Also the steady-state values of the grid frequency and the estimated grid frequency coincide, as it can be confirmed from (37). APPENDIX III This appendix derives the theoretical limits to the voltage support control analyzed in this paper. The steady-state values of the positive and negative sequence voltages at the PCC can be calculated by inserting the steady-state quantities from (A5), (A6), (A8), (A9) in (24) and (25)

 + 2 8 1 + + + Vg Vg + (A15) Vo = + ωo Lg Qo 2 3

 2 8 1 − − − − Vo = Vg + Vg − ωo Lg Qo . (A16) 2 3 The maximum capability to compensate the negative sequence voltage by the injection of negative sequence reactive power is derived, from (A16), as Vg− 2  2 3 Vg− = . 8 ωo Lg

Vo−m in = Q− o m ax

(A17) (A18)

The limits for the positive sequence reactive power are only dictated by the converter power rating given that no additional limits are introduced from (A15). The theoretical limits have been verified by simulation results, as shown in Fig. 12. The sequence of events used in the

CASTILLA et al.: MODELING AND DESIGN OF VOLTAGE SUPPORT CONTROL SCHEMES FOR THREE-PHASE INVERTERS

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− Fig. 12. Normalized negative sequence voltage V e− /V g− for several values of the negative sequence reactive power ratio Q − o /Q o m a x . (a) 0.2, (b) 0.4, (c) 0.6, (d) 0.8, (e) 1.0, and (f) 1.01.

experiments described in Section VI is repeated here to validate the expressions (A17) and (A18). The voltage control is activated at 0.2 s and, after a settling time, the negative sequence voltage Ve− reaches the steady-state value Vo− . Note that the negative sequence voltage Vo− reduces as the reactive power Q− o increases. In particular, the minimum value for this voltage is obtained when the reactive power is maximum, as shown in Fig. 12(e). For higher values of the reactive power, (A16) has no real solution and the system becomes unstable, as shown in − Fig. 12(f) for the case Q− o = 1.01·Qo m ax .

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[25] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Delhi, India: Hindustan Publishing, 1961. [26] A. Gelb and W. E. Van der Velde, Multiple-Input Describing Functions and Nonlinear System Design. New York, NY, USA: McGraw-Hill, 1968. [27] S. R. Sanders, J. M. Noworolski, X. Z. Liu, and G. C. Verghese, “Generalized averaging method for power conversion circuits,” IEEE Trans. Power Electron., vol. 6, no. 2, pp. 251–259, Apr. 1991. [28] A. Yazdani and R. Iravani, Voltage-Sourced Converters in Power Systems. New York, NY, USA: Wiley, 2010. [29] A. Camacho, M. Castilla, J. Miret, J. Matas, E. Alarcon-Gallo, L. Garcia de Vicuna, and P. Marti, “Reactive power control for voltage support during type C voltage sags,” in Proc. IEEE 38th Annu. Conf. Ind. Electron. Soc., Oct. 2012, pp. 3462–3467. [30] P. Rodriguez, A. Luna, M. Ciobotaru, R. Teodorescu, and F. Blaabjerg, “Advanced grid synchronization system for power converters under unbalanced and distorted operating conditions,” in Proc. IEEE 32nd Annu. Conf. Ind. Electron., 2006, pp. 5173–5178. [31] P. Rodriguez, A. Luna, I. Candela, R. Mujal, R. Teodorescu, and F. Blaabjerg, “Multiresonant frequency-locked loop for grid synchronization of power converters under distorted grid conditions,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 127–138, Jan. 2011. [32] J. Miret, A. Camacho, M. Castilla, L. Garc´ıa de Vicu˜na, and J. Matas, “Control scheme with voltage support capability for distributed generation inverters under voltage sags,” IEEE Trans. Power Electron., vol. 28, no. 11, pp. 5252–5262, Nov. 2013. [33] L. Asiminoaei, R. Teodorescu, F. Blaabjerg, and U. Borup, “Implementation and test of an online embedded grid impedance estimation technique for PV inverters,” IEEE Trans. Ind. Electron., vol. 52, no. 4, pp. 1136– 1144, Aug. 2005. [34] L. Zhan and M. H. J. Bollen, “Characteristic of voltage dips (sags) in power systems,” IEEE Trans. Power Del., vol. 15, no. 2, pp. 827–832, Apr. 2000.

Miguel Castilla received the B.S., M.S., and Ph.D. degrees in telecommunication engineering from the Technical University of Catalonia, Barcelona, Spain, in 1988, 1995, and 1998, respectively. Since 2002, he has been an Associate Professor in the Department of Electronic Engineering, Technical University of Catalonia, where he teaches courses on analog circuits and power electronics. His research interests include the areas of power electronics, nonlinear control, and renewable energy systems.

Jaume Miret (M’98) received the B.S. degree in telecommunications, the M.S. degree in electronics, and the Ph.D. degree in electronics from the Technical University of Catalonia, Barcelona, Spain, in 1992, 1999, and 2005, respectively. Since 1993, he has been an Assistant Professor in the Department of Electronic Engineering, Technical University of Catalonia, Spain, where he teaches courses on digital design and circuit theory. His research interests include dc-to-ac converters, active power filters, and digital control.

Antonio Camacho received the B.S. degree in chemical engineering and the M.S. degree in automation and industrial electronics from the Technical University of Catalonia, Barcelona, Spain, in 2000 and 2009, respectively, where, he is currently working toward the Ph.D. degree in electronic engineering. His research interests include networked and embedded control systems, industrial informatics, and power electronics.

˜ received the Ingeniero de Luis Garc´ıa de Vicuna Telecomunicaci´on and Dr.Ing. degrees from the Technical University of Catalonia, Barcelona, Spain, in 1980 and 1990, respectively, and the Dr.Sci. degree from the Universit´e Paul Sabatier, Toulouse, France, in 1992. From 1980 to 1982, he was an Engineer with Control Applications Company. He is currently a Full Professor in the Department of Electronic Engineering, Technical University of Catalonia, where he teaches courses on power electronics. His research interests include power electronics modeling, simulation and control, active power filtering, and high-power-factor ac/dc conversion.

Jos´e Matas received the B.S., M.S., and Ph.D. degrees in telecommunications engineering from the Technical University of Catalonia, Barcelona, Spain, in 1988, 1996, and 2003, respectively. From 1988 to 1990, he was an Engineer of a consumer electronics company. Since 1990, he has been an Associate Professor in the Department of Electronic Engineering, Technical University of Catalonia. His research interests include power-factorcorrection circuits, active power filters, uninterruptible power systems, distributed power systems, and nonlinear control.