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Dynamic Stability of a Microgrid With an Active Load


IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 11, NOVEMBER 2013

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Dynamic Stability of a Microgrid With an Active Load
Nathaniel Bottrell, Student Member, IEEE, Milan Prodanovic, Member, IEEE, and Timothy C. Green, Senior Member, IEEE

Abstract—Recti?ers and voltage regulators acting as constant power loads form an important part of a microgrid’s total load. In simpli?ed form, they present a negative incremental resistance and beyond that, they have control loop dynamics in a similar frequency range to the inverters that may supply a microgrid. Either of these features may lead to a degradation of small-signal damping. It is known that droop control constants need to be chosen with regard to damping, even with simple impedance loads. Actively controlled recti?ers have been modeled in nonlinear state-space form, linearized around an operating point, and joined to network and inverter models. Participation analysis of the eigenvalues of the combined system identi?ed that the low-frequency modes are associated with the voltage controller of the active recti?er and the droop controllers of the inverters. The analysis also reveals that when the active load dc voltage controller is designed with large gains, the voltage controller of the inverter becomes unstable. This dependence has been veri?ed by observing the response of an experimental microgrid to step changes in power demand. Achieving a well-damped response with a conservative stability margin does not compromise normal active recti?er design, but notice should be taken of the inverter–recti?er interaction identi?ed. Index Terms—Active loads, constant power loads (CPLs), inverters, microgrids (MGs), recti?ers, small-signal stability.

I. INTRODUCTION

T

HE presence of distributed generation (DG) in a distribution network creates the possibility of microgrid (MG) formation [1]. If a MG is formed, whether after a line outage or during planned maintenance, there is a need for the DG to respond to changes in load, and share the load such that the DG operates within their limits. An MG will have to ensure small-signal stability due to small changes to the operating conditions or load perturbations. It is known that, in general, load dynamics interact with generation dynamics and may in?uence the stability of a network [2]. Therefore, when investigating the stability of an MG, both the generation dynamics and the load dynamics must be considered.

Manuscript received February 15, 2012; revised November 26, 2012; accepted January 3, 2013. Date of current version May 3, 2013. This work was supported by a Power Networks Research Academy scholarship. Recommended for publication by Associate Editor Prof. J. M. Guerrero. N. Bottrell and T. C. Green are with the Department of Electrical and Computer Engineering, Imperial College London, London, SW7 2AZ, U.K. (e-mail: nathaniel.bottrell04@imperial.ac.uk; t.green@imperial.ac.uk.). M. Prodanovic is with the Instituto Madrileno de Estudios Avanzados Energa Institute, Madrid 28935, Spain (e-mail: milan.prodanovic@imdea.org). Color versions of one or more of the ?gures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identi?er 10.1109/TPEL.2013.2241455

Typically, electrical power in MGs is generated by rotating machines or by power electronics. Rotating machines include synchronous machines and power electronics include voltage or current source inverters. This study only considers MGs with voltage source inverters. The literature on control approaches that enable the DG to share the load and remain within their operating limits discusses either use of a communications link or use of a droop method. Communication approaches may involve a master–slave link, where the DG outputs are controlled using a dispatch signal [3]. If the master DG unit regulating the grid voltage is not functioning or does not have enough capacity, the MG may not satisfy voltage and frequency limits. The use of a droop control method has the advantages of not requiring a communication link and allows DG to support MGs irrespective of which sources are available. The droop control method has been widely discussed, for example [4]–[7]. However, inverter-interfaced DGs operated with droop controllers have relatively complex and dynamic properties. It is known that in droop-controlled MGs, the low-frequency modes (oscillations that are represented by conjugate eigenvalue pairs) are associated with the droop controllers [7]. The low-frequency modes are most likely to be poorly damped, and at a risk of instability during operating point or parameter changes. The droop controllers give rise to low-frequency modes because of their use of low-pass ?lters to reject harmonic and negative sequence disturbances from the power measurements. The ?ltered power measurements are used to determine the frequency references for the ac voltage controllers of the inverters. Several strategies have been proposed to increase the damping of the low-frequency modes during both steady-state operation and transient behavior. Improvements include adjusting the droop parameters while the MG is functioning by the use of either an energy manager [8] or a grid-impedance estimation strategy [9]. Feedforward terms have been proposed in [10] and using an inverter to imitate a voltage source with a complex output impedance is proposed in [11]. The usage of proportional, integral, and derivative controllers within the droop calculation has been proposed in [12]–[15]. For simplicity, this study will not consider the enhancements of droop controllers to improve stability and will only consider the simple droop controller, as presented in [7]. This is so that the in?uence of load dynamics within the MG can be more readily observed. One possibility to further simplify the MG model is to represent the inverters only by their low-frequency dynamics as in [16] (which was extended to larger system sizes in [17]).

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This modeling technique is valid if the system is not sensitive to the mid-frequency or high-frequency dynamics. Loads within an electrical network are either passive or active loads. Passive loads include devices such as incandescent lighting or resistive space heaters and are typically modeled by a resistor or an inductor–resistor network. Active loads include devices such as machine drives, back-to-back converter con?gurations, and consumer electronics with unity power factor correction. Over a small time period, these devices may be modeled as constant power loads (CPLs). This study considers inverter-based generation with active and passive loads. This study is important since published literature has mostly considered MGs with passive (i.e., impedance) loads, whereas MG implementations are likely to include signi?cant proportions of loads with active front ends. Active front ends are used for providing regulated voltage buses to supply the ?nal use equipment. By regulating the voltage bus, an active load may become a CPL. Constant impedance loads generally increase damping, whereas CPLs tend to decrease damping [18]. To understand how a network responds to different generation technologies, in this case DG, a range of load dynamics must be studied. The results produced from using only passive loads may be misleading, since they might not represent the type of loads connected to a network [19]. CPLs typically destabilize dc MG networks [20]–[22] and were shown to destabilize an ac MG in [23], which went on to conclude that CPLs are only stable if paralleled with constant impedance loads. The study used the small-signal representation of an ideal CPL, which exhibits a negative incremental resistance Δi = ? VP2 Δv but did not consider the dynamics of the bus regulator. Many solutions have been proposed to overcome the problems of CPLs, including the solution discussed in [24] and [25]. Large-signal stability of an MG, with various load types, was investigated in [26]. The conclusion from this was that constant PQ loads and impedance loads have no effect on stability but motor loads do. Although this study did not consider smallsignal stability, it is important because it demonstrates that CPLs have the possibility of being stable without the need to be paralleled with constant impedance loads (contrary to the assertion in [23]). Several approaches for determining network stability exist. Power electronic networks can be analyzed using impedance methods [27]. Impedance methods plot the source and load impedance as a function of frequency. If the source impedance as a function of frequency has a magnitude that is greater than that of the load impedance as a function of frequency, then the system is deemed to be unstable. This method has been used for dc networks and examples in the literature are [28] and [29]. Small-signal stability of power systems is often analyzed using eigenvalues of which some examples from literature are [30]–[32]. The power system is represented as a state-space equation and the “A” matrix is used to determine the eigenvalues. One advantage with eigenvalue analysis is the ability to investigate interactions between states.

The approach to be taken follows that of [7] and [33] by using full dynamic models of all elements. This is justi?ed, since the separation of modes into distinct frequency groups (based on controller bandwidths) is still a matter of choice by the equipment designer. Each inverter and load is modeled on a local rotating (dq) reference frame, and then, the subsystems are combined onto a common reference frame by the use of rotation functions. The models of inverters will be taken from [7] and models of the active load will be taken from [34]. A laboratory MG with three 10-kVA inverters is used for experimental veri?cation of the analytical results. Eigenvalue analysis will be used to assess the stability of the system and the sensitivity and damping to change in the gain of the dc voltage controller of the active load. To determine interactions between the eigenvalues, participation analysis will be performed [35]. Participation analysis allows the investigation of the sensitivity of eigenvalues to the states of the MG and indicates interactions between the dynamics of the inverters and the active load. A full state-space model (rather than a transfer function) allows participation analysis and eigenvalue trace analysis. This allows the in?uences on poorly damped modes to be explored. The objective for the study reported here is not to analyze the in?uence of high controller gains, but to examine whether, ?rst, the negative resistance characteristic of active front-end loads have a signi?cant destabilizing effect on MGs and, second, whether there are signi?cant interactions between the dynamics of the inverters and the active loads such that their controllers need coordination or codesign. A comprehensive ?gure showing all the DGs and loads with corresponding control is shown in Fig. 1. II. MODELING THE ACTIVE LOAD IN THE MG A. CPL Model A theoretical CPL, as shown in Fig. 2, has the characteristic where V · I = P is modeled. For a CPL, the instantaneous value of impedance is always positive and the incremental impedance is always negative. [24] The negative incremental impedance causes the current to increase when the voltage decreases, and the current to decrease when the voltage increases. It is well known that the negative incremental impedance may cause instabilities [20], [22], [25], [36]. The CPL considered is modeled in series with an inductor and is shown in Fig. 3. The large signal equation is described in (1). Equation (1) is linearized around an operating point and converted into the DQ reference plane which enables the smallsignal equation in (3) and (4) to be formed. The small-signal equation is written as a state-space model in (6). The large-signal equation of the CPL is ?Rcpl 1 dicpl = icpl + vg dt Lcpl Lcpl where Rcpl =
2 vg . Pcpl

(1)

(2)

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Fig. 1.

Comprehensive ?gure showing all the DGs and loads with corresponding control.

?Rcpl dΔicplQ = ΔicplQ ? ω0 icplD dt Lcpl + where Rcpl = ?
Fig. 2. V–I characteristic of a CPL.

1 Δvg Q + IcplD Δωcom m Lcpl

(4)

Vg2D . Pcpl

(5)

The state-space model of the CPL is ˙ ] = Acpl [ ΔicplDQ ] + Bcplv [ Δvg D Q ] [ ΔicplDQ + Bcplw [ Δωcom m ] where (6)

Fig. 3.

Circuit of the CPL.

The small-signal equation of the CPL is ?Rcpl dΔicplD = ΔicplD + ω0 icplQ dt Lcpl + 1 Δvg D + IcplQ Δωcom m Lcpl (3)

Acpl

?Rcpl ? Lcpl ? =? ? ?ω0 IcplQ IcplD .

?

? ω0 ?Rcpl Lcpl ? ? ?, Bcplv ?

? 1 ? Lcpl =? ? 0

? ? 1 ? Lcpl

0

? (7)

Bcplw =

(8)

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B. Active Load Model A switch-mode active recti?er is used in this study as an example of an active load supplying a regulated dc bus. The recti?er is modeled using an averaging method over the switching period, as in [37]. The chosen averaging method is based on the method developed by R. D. Middlebrook and S. Cuk, which averages the circuit states [38]. Models can be readily developed by representing the switching elements as equivalent variable-ratio transformers [39]. Averaged models re?ect the key dynamics of the system that are below the switching frequency. In general, the models are nonlinear but can be linearized around an operating point [40], [41]. With linear state-space representation, frequency-domain analysis can be used to study the system stability. State-space models of recti?ers that have been previously presented have considered connection to a stiff grid [42] and have not included the dynamics arising from a supply frequency change. Not accounting for frequency change is a safe assumption when the active load is connected to a large grid or a single frequency source. In large networks, one active load will be a small percentage of the total loads and will not have enough of an effect on the network to cause a frequency disturbance. However, in a small MG with droop controllers, a load change from an active load could be a signi?cant percentage of the total load and have a signi?cant effect on the frequency. For this reason, the frequency of the network must be considered in the active load model. Switch-mode active loads require an ac-side ?lter to attenuate switching frequency harmonics. The order and design of the ?lter varies, with higher order ?lters being more common in high power equipment. In the laboratory MG, the active load used an LCL ?lter rather than an LC ?lter. This offers higher attenuation with smaller passive components. However, lightly damped resonances of LC and LCL ?lters require careful design of the power converter control loops [42], [43]. An overview of the control system, for the example active load studied here, is shown in Figs. 1, and 4. The active load must be synchronized to the network voltage. This is commonly achieved with a phase-locked loop. In modeling terms, the reference frame of the controller model is made synchronous with the local network voltage. Many different control structures are possible and the work in [44] compares different PI current controllers, reviews their advantages, and presents methods of providing active damping. The use of a linear quadratic regulator has also been reported [45]. For this study, a relatively simple structure was adopted, which is shown in Fig. 4. All controllers use a PI regulator and the outer dc-bus voltage controller forms the reference for an inner current regulator. The inner current regulator is in dq form and regulates the ac-side currents ?owing in the inductive element that is between the switching bridge and the ac-side capacitor. No active damping was incorporated but it could be with a relatively simple extension to the model. To design the current controller of the active load, the current controllers that are used within the MG are considered. For the current controllers of the MG inverters, the current is de?ned as positive when traveling from the switching bridge to the grid

Fig. 4. Active load dc voltage and ac current controller. (a). Active load dc voltage controller. (b) Active load ac current controller.

Fig. 5.

Eigenvalues of the CPL in the MG model.

connection and the following equations apply [7], [46]:
? = ?ωn Lf ilq + Kpc (i? vid ld ? ild ) + Kic ? viq = ωn Lf ild + Kpc i? lq ? ilq + Kic

(i? ld ? ild )dt (9) i? lq ? ilq dt. (10)

However, positive current in the active load is de?ned as ?owing from the grid connection to the switching bridge which is the opposite direction to an inverter. To use the current controllers of the inverter in the active load and compensate, each current measurement was negated and (11) and (12) apply. This current controller is shown in Fig. 5 and taken from [34]
? = ωn Lf ilq ? Kpc (i? vid ld ? ild ) ? Kic ? viq = ?ωn Lf ild ? Kpc (i? lq ? ilq ) ? Kic

(i? ld ? ild )dt

(11)

(i? lq ? ilq )dt. (12)

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zero is denoted by ω0 and the common system reference frame frequency is denoted by ωCOM . For simplicity, the model is split into constituent parts. The constituent parts are represented as linear state-space equations and then combined into a single state-space model. This process was presented in [34]. The model of the active load, required for the proposed system study, takes the ac network voltage as its input and returns the resulting ac network current as its output. The model has been divided into various subsystems, such as the dc bus regulator, ac current controller, and the ?lter elements. The modeling of each of these, and their assembly into a complete model of the active load, is described in [34]. The resulting state-space model consists of a state-transition matrix A, matrices for inputs from the ac network voltage Bv , the common system frequency Bω , and all other inputs Bu . The matrix Cc forms the ac current output, and the matrix C is used to observe other outputs during testing. The matrix D is the feedforward matrix that directly couples input to output. Finally, a matrix M is used to de?ne the network node, to which the active load model is connected. It also includes a high value resistive path to ground at each node which is there to de?ne the node voltages. The complete active load model is shown in (14), and has ten states, six inputs, and three outputs. All equations are shown with an i subscript to allow for Ni multiple recti?ers to be coupled to a larger circuit model
? Δvdc ˙ ] = ARECi [ ΔxRECi ] + BRECui [ ΔxRECi Δi? lq

+ BRECvi [ Δvg D Q ] + BREC ω i [ Δωcom ] + BRECdci [ Δ idc ] [ Δig D Q i ] = CRECci [ ΔxRECi ]
Fig. 6. Eigenvalues of the active load and active load in the MG. (a) Active load in isolation. (b) Active load in the MG.

(14)

[ Δvdci ] = CRECdci [ ΔxRECi ]. The states of the recti?er ΔxRECi are shown in

(15)

Transformation of the dq reference frame uses the equation, as shown in (13), at the bottom of this page. Reference inputs to the controllers are indicated with a superscript asterisk. Upper case “D” and “Q” have been used to denote variables on the common system reference frame of the MG. Lower case “d” and “q” denote variables on the individual reference frame of a particular load or inverter. The model uses various frequency variables: ω, ωn , ω0 , and ωCOM . The variable ω denotes an arbitrary time varying frequency. Nominal system frequency, with a value of 2π 50 in the example here, is denoted by ωn . The initial frequency at time ? ? ? 2? ? ? ? sin (ωt + θ) 3? ? ? 1 √ 2 cos (ωt + θ)

ΔxRECi = [ Δφdci | Δγdq i | Δildq i Δvcdq i Δig dq i | Δvdci ]T . (16)

The matrices ARECi , BRECui , BRECvi , BREC ω i , BRECdci , CRECci , CRECdci that form (14) and (15) are de?ned in (17)– (19), at the bottom of the next page. C. Active Load in the MG Model The active load is connected to node three of the three-node example MG, which is depicted in Fig. 1. By using the method 2π 3 3 2π 2π 3 3 2π ? ? ? ? ? ? ? ? ?

cos ωt + θ ?

cos ωt + θ +

[Tdq ] =

? sin ωt + θ ? 1 √ 2

? sin ωt + θ + 1 √ 2

(13)

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in [34], the matrices of the MG can be systematically built to represent any MG topology. The approach used in [34] can be used on higher dimension MG systems, and could be developed to identify the small-signal stability when integrating DG into the distribution network. The various subsystems of the example MG will be identi?ed using the subscripts listed in Table I. Each of the subsystems modeled uses the procedure presented in [7]. Bottrell and Green [34] provide further information about how the components were incorporated into the MG. One inverter was selected to provide the common system frequency and this is represented in an extra matrix “Cω .” The inverter that is the common system frequency has a nonzero ?

TABLE I SUBSCRIPTS FOR THE COMPONENTS IN THE MG

“Cω ” matrix, whereas the other inverters have a zero “Cω ” matrix. A disturbance component, not shown in Fig. 1, allows a current to be injected or drawn from any node. The state-space model of the MG that incorporates the active load is summarized in (20)–(22), at the bottom of this page. ? ? ? ? ?
10 ×10

0

0 0 BL C L u CC BSW u CC ? ,

0 BC 2 AL C L + BL C L u DC 2

BV 2 BC 1 DV 2 BL C L u DC 1 DV 2

? BC 1 CV ? ARECi = ? ? BL C L u DC 1 CV BSW u DC 1 CV ?

(17)

BRECui

BV 1 ? ? BC 1 DV 1 ? ? =? ? ? BL C L u DC 1 DV 1 ? ?

BSW + BL C L u DC 2 ASW + BSW u DC 1 DV 2 + BSW dc DDCu ? ? ? ? 0 0 ? ? 0 ? ? 0 ? ? ? ? BRECvi = ? , B = ? ? ? REC ω i ?1 ? ? BL C L v TR ? BL C L ω ? 0
10 ×2

(18)

? ? BRECdci = ? ?

BSW u DC 1 DV 1 10 ×2 ? 0 0 ? ? , CRECci = [ 0 0 ? 0 ? BSW dc
10 ×1

0

10 ×1

[ 0 0 TR ]

0 ],

CRECdci = [ 0 0

0 CSW ]

(19)

Am g = ? A INV + BINV v MINV CINVc ? + BINV ω CINV ω ? ? ? BNETv MINV CINV c ? ? + BNET ω CINV ω ? ? ? BLOADv MINV CINV c ? ? + BLOAD ω CINV ω ? ? ? BRECv MINV CINV c + BREC ω CINV ω BINV v MDIST CDISTc ? BNETv MDIST CDISTc ? =? ? BLOADv MDIST CDISTc ? ? ? Cm g = ? ? BRECv MDIST CDISTc CINV 0 0 MINV CINV c 0 0 0 ?

? BINV v MNET CNETc ANET + BNETv MNET CNETc BLOADv MNET CNETc BRECv MNET CNETc ? ? ? ? ? 0 CLOAD 0 MLOAD CLOADc 0 0 CREC MREC CRECc ? ? ? ?, ? ? ? ? Dm g = ? ? 0 0 0 MDIST CDISTc 0 ? (22) BINV v MLOAD CLOADc BNETv MLOAD CLOADc ALOAD + BLOADv MLOAD CLOADc BRECv MLOAD CLOADc BRECv MREC CRECc BNETv MREC CRECc BLOADv MREC CRECc AR E C + BRECv MREC CRECc (20) ? ? ? ? ? ? ? ? ? ? ? ? ? ?

0 0 0 BRECu

Bm g

(21)

0? ? ? 0? 0

MNET CNETc

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III. IDENTIFICATION OF MODE GROUPS Participation factors can provide a useful insight into what features of a system give rise to a given mode. The participation factor of the ith state and j th eigenvalue is de?ned as the product of the left eigenvectors (wik ) and right eigenvectors (vk i ) [47] pij = |wij | |vj i |
N k =1

(|wik | |vk i |)

.

(23)

Participation factors are a common tool for accessing the small-signal stability of networks and, as an example, have been used in [7], [48], and [49]. The participation matrix is a matrix of all the participation factors. The higher the particular participation factor, the more the state i participates in determining the mode j . Here, the highest participation factor for each mode will be used to associate that mode with a particular control subsystem within the MG. The purpose of the participation factor is to produce a state to mode mapping and to identify parameters within the active load and MG that strongly in?uence the modes of the system. A. CPL Modeled in the MG The CPL was connected to node 3 of the MG model and is a substitution for the active load in Fig. 1. Testing with the ideal power load gives an indication as to whether the MG has suf?cient damping overall given the negative damping contribution of the CPL. However, it is prudent to follow this with a study with the nonideal active load. If the eigenvalue plot demonstrates that the MG is unstable for an ideal CPL, it might be reasonable to generalize that a nonideal CPL would also be unstable. However, there is a limited extent to which this generalization is valid since the nonideal CPL might have suf?cient damping. An ideal CPL is constant power over a large bandwidth, whereas the active load (a nonideal CPL) is only constant power over a limited bandwidth and might appear as a constant current load for a proportion of its bandwidth. This is because the active load consists of two control loops: an inner current controller with a high bandwidth and an outer voltage controller with a low bandwidth. Fig. 5 shows the eigenvalue plot of the CPL connected to the MG. There are two eigenvalues which are located in on the right-hand side of the imaginary axis. It can be concluded that the MG is unstable when an ideal CPL is connected. The MG without the CPL was shown to be stable in [7]. The two eigenvalues that are unstable have a high participation factor for the coupling inductor state within inverter 3. In this MG, the CPL has caused the inverter which is connected to the node with the CPL to exhibit instabilities. B. Active Load Modeled in Isolation Fig. 6(a) shows the eigenvalue plot of the active load model in isolation, the values for which these were calculated from the A-matrix of the active load. The group of modes labeled AL.Cdc in Fig. 6(a) all have high participation factors for the states of the dc-side capacitor and the integrator of the dc-bus voltage regulator. The group labeled AL.Cf Lc are associated

Fig. 7.

Participation analysis of the active load in the MG.

with the voltage of the ac-side capacitor Cf and the inductor Lc . The group labeled AL.Lf are associated with the integrator of the current controller and the ac-side inductor Lf . C. Active Load Modeled in the MG Fig. 6(b) shows that when the eigenvalues of the complete system (the active load within an MG) are considered, two further mode groups are evident. Groups labeled INV are similar to those found in [7] and are associated with the LCL ?lter, voltage controller, and current controllers of the three inverters. In the group labeled AL.Cdc &DROOP , the low-frequency modes associated with the active load’s dc-bus regulator have been supplemented by the modes associated with the droop controllers of the inverters, and so, many more eigenvalues are present. A question that arises is whether the modes within the group labeled AL.Cdc &DROOP are two independent subgroups or whether the modes are jointly in?uenced by the design of the inverters and the active loads. By observing how the groups associated with the active load change from Fig. 6(a) and (b), an understanding of the effect of the MG on the recti?er can be obtained. All three groups appear to be in similar locations in Fig. 6(a) and (b), but closer inspection reveals that the groups labeled AL.Lf and AL.Cf .Lc have changed slightly. As already discussed, the groups labeled AL.Lf and AL.Cf .Lc in Fig. 6 are those associated with the LCL ?lter and the current controllers of the active load. The group labeled AL.Lf has become better damped (with the real part of the eigenvalue changing from approximately ?1750 to ?2300) and the group labeled AL.Cf .Lc has become slightly less well damped (real part moving from ?1000 to ?1200). IV. IDENTIFICATION OF MODES SHOWING LOAD-INVERTER COUPLING Participation analysis not only reveals the states that dominate in formation of the model, but can also reveal second and subsequent in?uences. Fig. 7 shows a plot of the participation factors of the eigenvalues for the active load states only and reveals the coupling between the active load and the inverters. In Fig. 7, more than

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Fig. 8.

Low-frequency MG eigenvalue trace.

Fig. 9. Participation trace of four inverter states and two active load states as a function of the active load dc voltage controller integral gain for the eigenvalue λ2 . TABLE II DESCRIPTION OF THE STATES PLOTTED IN FIG. 9

one state has been selected for analysis, and the participation factors shown on the z-axis are a sum of the participation factors for each state. The x-axis and the y-axis are the real and imaginary axes of the eigenvalue plot, respectively. Eigenvalues that have higher participation of the active load states have a longer stem on the xyz plot. It is observed, from Fig. 7 that there is a weak coupling between the active load and the eigenvalues shown in groups labelled inv in Fig. 6, which are principally associated with the inverters. No coupling is observed in group labelled AL.Cdc&DROOP between the eigenvalues of the active load and the eigenvalues of the inverters. The conclusion from the participation analysis is that the droop controllers of the inverters and the controllers of the active load are largely independent (at least for this example), and can be designed on that assumption. The participation analysis indicates a weak coupling between the active load and the inverter voltage and current controllers. In order to ensure that the indicated modes are stable, it would be prudent to have an element of codesign or, at the least, design rules should be agreed that allow the two to be designed by independent parties. If this is not done, the addition of an active load with a certain control characteristics could destabilize the network voltage control. V. MG EIGENVALUE TRACE It has been shown that coupling exists between the active load and some inverter states (current and voltage controller states) for the example network and with the nominal control gains. Now, it is useful to test how damping of the various modes is changed by changes in the load controller parameters. Fig. 8 shows eigenvalues for the case where the time constant K v ) was held constant at of the dc-bus voltage regulator (τ = Kp iv 1 , while the integral gain K was varied from 15 to 2700 iv 300 (the value used in the earlier plots was 150). It is seen that two eigenvalues move signi?cantly. The ?rst trace λ1 shows the low-frequency modes associated with the dc voltage controller of the active load, which one would expect to depend on Kiv . The second trace λ2 is of the modes associated with the ac voltage controller of the inverters. The second trace shows that

high gains in the load controller can lead to very low damping of the inverter voltage controller and a risk of instability. Fig. 8 also con?rms the conclusions drawn from Fig. 7 concerning coupling between the active load and inverters. The active load states do not participate in the low-frequency modes of the inverter droop controllers. The evidence for this is that eigenvalues associated with the inverter droop controllers did not move when Kiv was changed. The active load states do participate in the mid-frequency modes of the inverters as already seen in Section IV, but beyond what was expected. It was not expected that changing the controller gains of the dc voltage of the active load would have such a signi?cant effect on the mid-frequency modes associated with the voltage controller of the inverter. To investigate this feature further, the participation values were plotted against gain. Fig. 9 shows the participation of six states in eigenvalue λ2 as a function of the gain Kiv . Four states from the inverter were chosen because they have the highest participation values at low gain. Two states from the active load are chosen, one state associated with the low-frequency modes and one state associated with the mid-frequency modes. All six states are identi?ed in Table II. It is observed that the participation of the active load states grows rapidly as the gain increases. Also, high Kiv causes a further coupling between the dc voltage controller and ac current controller of the active load. This informs us that the link between eigenvalues and states may change as the design parameters change. The participation factors only provide information about the system, and its couplings, for a ?xed set of design parameters.

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TABLE III GAINS OF DC-BUS VOLTAGE REGULATOR OF THE ACTIVE LOAD

The eigenvalue trace in Fig. 8 con?rms the results shown in the participation analysis. It can be concluded that the lowfrequency modes of the active load do not couple with the lowfrequency modes of the inverters. However, there is coupling between the low-frequency modes of the active load and the mid-frequency modes of the inverters when the active load has high gains in the dc voltage controller. These couplings must be considered when designing an MG with an active load. VI. EXPERIMENTAL RESULTS The example MG of Fig. 1 was available for experimental testing. It was equipped with three 10-kVA inverters operating as droop-controlled sources with a nominal line to line voltage of 381 Vrm s (220 Vrm s phase to neutral). The active load was provided by another inverter with a further power converter for removing power from the dc bus. Tests at two different gains were conducted to illustrate the results. In each, the power consumption of the active load was stepped from 7000 to 9000 W and the transient responses of the load’s dc bus voltage, the inverters’ output powers, and the inverter’s d-axis voltage were observed. The gains used, detailed in Table III, correspond to the nominal gains used in the initial analysis and the extreme of the gain range used in Fig. 8. The purpose of the experimental work is to validate the relationship between the active load and the damping of MG modes that was found from the participation analysis. A. Transient Response With Low Gain, Nominal Gain, and High Gain In these experiments, the low-frequency modes of the MG observed in the time domain are compared to the eigenvalue plot. In Fig. 10(a), Fig. 11(a), and Fig. 12(a) two modes are identi?ed. The mode with a frequency of 10.0 Hz, 32.1 Hz, 47.6 Hz, respectively, and a damping factor of 0.208, 0.351, 0.688, respectively, are associated with the dc voltage controller and dc capacitor of the active load for the different gains used in the three experiments. The mode with a frequency of 4.13 Hz and a damping factor of 0.533 is associated with the droop controllers of the inverters. This does not change when the gains of the dc voltage controller for the active load are changed in the experiments low gain, nominal gain and high gain. Fig. 10(b), Fig. 11(b), and Fig. 12(b) show that when the load is stepped, the dc voltage oscillates with a frequency of 7 Hz, 30.3 Hz, 47.6 Hz, respectively, and have a damping factor of 0.377, 0.250, 0.224, respectively. The envelope of the damping factor is shown by the black long-dashed line. The experimental damping factor in all three experiments differs slightly than what

Fig. 10. Experimental results with low gain and active load operating point of 7 kW. (a) Low-frequency modes identi?ed from the model. (b) DC voltage of active load. (c) Output power of inverter sources. (d) Capacitor voltage of inverter sources.

the model predicts. The frequency observed is in reasonable agreement with the model. Fig. 10(c), Fig. 11(c), and Fig. 12(c) show that the three inverters, which have identical droop settings, share the increased power equally when the new steady state is established. The initial increase in power is all taken by inverter 3, which is

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Fig. 11. Experimental results with nominal gain and active load operating point of 7 kW. (a) Low-frequency modes identi?ed from model. (b) DC voltage of active load. (c) Output power of inverter sources. (d) Capacitor voltage of inverter sources.

Fig. 12. Experimental results with high gain and active load operating point of 7 kW. (a) Low-frequency modes identi?ed from the model. (b) DC voltage of active load. (c) Output power of inverter sources. (d) Capacitor voltage of inverter sources.

electrically closest to the load where the power step occurred. The transient of the power output from inverter 3 has a frequency of 7.14 Hz, 3.33 Hz, 3.17 Hz, respectively, and has a damping factor of 0.377, 0.500, 0.359, respectively. The damping factor envelope is shown by the black long-dashed line. The observed

frequency and damping factor are in reasonable agreement with the model. Fig. 10(d), Fig. 11(d), and Fig. 12(d) show the d-axis capacitor voltage of inverter 3. The power step of the active load change has not caused a noticeable transient in the voltage trace. This ?gure con?rms stable operation of the MG for the different active load gains when the active load is perturbed.

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B. Transient Response With Oscillatory Gain The experiment in Section VI-A was repeated for a higher gain in the dc voltage controller of the active load. The participation analysis of the MG with a high gain in the dc voltage controller of the active load showed a link between the active load dc voltage controller and the inverters’ voltage controller. The eigenvalue sweep showed that the eigenvalues, originally associated with the voltage controller of the inverter, becoming unstable and also showed the low-frequency eigenvalues remaining stable. In Fig. 13(a), two modes are identi?ed. The mode with a frequency of 55.1 Hz and a damping factor of 1 is associated with the dc voltage controller and dc capacitor of the active load. The mode with a frequency of 4.13 Hz and a damping factor of 0.533 is associated with the droop controllers of the inverters. Fig. 13(b) shows the dc voltage of the active load. The dc voltage is more oscillatory than at low, nominal or high gain. At the instant of load change, the low-frequency transient is not present. However, no obvious transient in this voltage is present when the load is stepped. This absence is explained by the change in position of eigenvalue [see Fig. 13(a)], which is seen to oscillate with a damping of 1. Fig. 13(c) shows the power output of the inverters. The droop controllers have remained stable with a damped response at a frequency of 3.1 Hz. This is a very slight change from the output of the droop controllers in the nominal gain case. The eigenvalue plot in Fig. 8 showed the low-frequency modes, associated with the droop controllers, to be unaffected by the gain change. Fig. 13(d) shows the capacitor voltage of the inverters. The gains in the inverter have not altered, yet the voltage is nonsinusoidal. Increasing the gain in the dc voltage controller of the load has caused the MG to become unstable. This experimental test con?rms the link seen in the eigenvalue trace, where the mid-frequency modes of the inverters move and become coupled with the active load when the gain of the active load dc voltage controller is increased. VII. CONCLUSION This paper has used a dynamic model of an active load as part of an overall dynamic model of a simple three-node MG. The active load was modeled as a nonlinear state-space model which was joined to an MG, with similar models of droopcontrolled inverters and a network, using a common reference frame. The model was linearized about an operating point, and participation analysis was used to identify which state variables were associated with which eigenvalues. The low-frequency eigenvalues, which are associated with either the inverter droop controllers or the dc voltage controller of the active load, had little interaction. In contrast, a medium-frequency eigenvalue, which was initially associated with the ac voltage controller of the inverters, became much less damped when the gain of the active load dc voltage controller was increased. The system model has been veri?ed against an experimental system with three 10-kVA inverters, one passive load, and one active load. Step changes of load were used to excite lowfrequency modes and allow observation of frequency and damp-

Fig. 13. Experimental results with oscillatory gain. (a) Low-frequency modes identi?ed from the model. (b) DC voltage of active load. (c) Output power of inverter sources. (d) Capacitor voltage of inverter sources.

ing factor for two controller gain conditions. Despite initial concerns that the negative resistance property of CPLs would destabilize the power-sharing (droop) controllers, no signi?cant reduction of damping of the low-frequency modes was observed for a range of active load voltage control parameters. However, the eigenvalue analysis showed that eigenvalues associated with the dc voltage controller of the active load moved away from the

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imaginary axis and the eigenvalues associated with the inverter ac voltage controller moved toward the imaginary axis as the gain of the recti?er dc voltage controller was varied. The degradation of damping of this mode was observed in the experiments also. Although some inverter–recti?er coupling has been identi?ed, it was not in the low-frequency power sharing eigenvalues as anticipated. In the example experimental system, unstable operation did occur for very large gains in the dc voltage loop but these were beyond the gains needed to achieve good steady-state voltage regulation. Therefore, it is possible to design the recti?er controller by using conventional frequency-domain analysis and independently of the inverter controllers, provided the design is suitably conservative. ACKNOWLEDGMENT The authors would like to thank Dr R. Silversides who provided extensive support in setting up the experimental equipment. REFERENCES
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Milan Prodanovic (M’01) received the B.Sc. degree in electrical engineering from the University of Belgrade, Belgrade, Serbia, in 1996, and the Ph.D. degree from Imperial College London, London, U.K., in 2004. From 1997 to 1999, he was with GVS engineering company, Belgrade, Serbia, developing uninterruptible power systems. From 1999 to 2010, he was a Research Associate in the Control and Power Group, Imperial College London, London. He is currently a Senior Researcher and the Head of the Electrical Processes Unit, Instituto Madrileno de Estudios Avanzados Energ? ?a Institute, Madrid, Spain. His research interests include the design and control of power electronics converters, RT control systems, microgrids, and distributed generation.

Timothy C. Green (M’89–SM’02) received the B.Sc. degree (?rst class Hons.) in electrical engineering from Imperial College London, London, U.K., in 1986, and the Ph.D. degree in electrical engineering from Heriot-Watt University, Edinburgh, U.K., in 1990. He was a Lecturer at Heriot Watt University until 1994. He is currently a Professor of Electrical Power Engineering at Imperial College London, London, U.K. where he is also the Deputy Head of the Control and Power Research Group. His research interest is in using power electronics and control to enhance power quality and power delivery. This covers interfaces and controllers for distributed generation, microgrids, active distribution networks, ?exible ac transmission system, and active power ?lters. He has an additional line of research in power microelectromechanical system and energy scavenging. Dr. Green is a Chartered Engineer in the U.K. and MIEE.

Nathaniel Bottrell (S’10) received the M.Eng. degree in electrical and electronic engineering from Imperial College London, London, UK., in 2009, where he is currently working toward the Ph.D. degree in electrical and electronic engineering. His research interests include distributed generation, microgrids, and the application of power electronics to low-voltage power systems.


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