Systematic Design of Virtual Component Metho for Inverter-Base Microgris Po-Hsu Huang 1, Petr Vorobev 1, Mohame Al Hosani, James L. Kirtley 1, an Konstantin Turitsyn 1 1 Massachusetts Institute of Technology Cambrige, MA 0139 Email: {pohsu, petrvoro, kirtley, turitsyn}@mit.eu Masar Institute of Science an Technology Abu Dhabi, UAE Email: mohalhosani@masar.ac.ae Abstract Control esign of inverter-base microgris plays a significant role in affecting ynamic performance of the system. Conventional roop control suffers from instability ue to higher X/R ratios an unique network characteristics as compare to large power systems. While many approaches such as virtual framework methos, virtual impeance methos, or synchronverters have been propose an proven effective, an intuitive an funamental insight into physical sources of instability has not yet completely isclose. In this paper, a systematic approach for enhancing the stability of inverter-base microgris is propose. A simplifie system is stuie to erive simple an concise stability criteria base on the propose Lyapunov function caniate. It has been iscovere that the transient susceptance B is a crucial factor in contracting the region of stable roop gains. Control schemes to minimize B are then investigate, enabling a ifferent perspective in views of the virtual component metho. Finally, simulation results are carrie out to verify the propose approach via time-omain analysis. Inex Terms Droop control, microgris, Lyapunov function, small-signal stability. I. INTRODUCTION Droop control inverter-base microgris have gaine a lot of attention recently in evelopment of future power networks ue to increasing penetration of istribute generation. In contrast to gri-connecte moes, roop control islaning operations require more sophisticate control esigns to ensure system stability [1]. Intense research works in recent years have provie comprehensive insights into the control an moeling of the inverter-base microgri systems [] [5]. Since the roop control concept originates from conventional power networks to achieve proper loa sharing, the smallsignal stability was first ientifie similar to the transmission level power gris. However, it was later realize that microgri ynamics is noticeably ifferent ue to a much lower X/R ratio [6], [7]. That is, both the phase angle an voltage couple together an contribute to active an reactive power epenently. Theore, the P ω an Q V roop controls for low-voltage microgris suffer from limite gain regions ue to eployment of highly resistive feeers to the point of coupling (PCC). Without presence of aitional coupling inuctors, system stability is significantly compromise. To aress the coupling issues, the concept of virtual inuctance has been propose an wiely appreciate ue to its effectiveness [6], [8], [9]. The emulation of inuctive ynamics helps to save bucky an costly inuctors with enhancement of both reactive power sharing an stability. These types of metho, in general, introuce aitional terms that reacts to the output currents into the erence voltages. Thus, the inverters act like an effectively controlle voltage source connecte to their terminals through the virtual inuctances. Also, the iea of synchronverters is earning a lot of attentions as they are consiere as gri frienly by mimicking the ynamics of synchronous machines [10]. Similar to the virtual inuctance methos, slower rotor an stator ynamics are emulate by using high banwith two-loop controllers that regulate the currents an voltages of output LC filters. All of these creative methos have been investigate an shown to be very effective. However, more intuitions an insights are to be carrie out from a stability perspective. This paper focuses on evelopment of virtual components for enhancing the stability margin of inverter-base microgris. Detaile explanations an clear intuitions are provie from a ifferent viewpoint than the commonly use eigenvalue analysis. Although conucting eigenvalue extraction from a etaile moel, in general, is not a computation buren for a small-scale microgri system, its numerical outcomes provie very limite information for unerstaning the system. Moreover, there exists istinct time-scale separation of system moes, which have been reporte in [11]. This leas to a more straight-forwar representation of equations using a reuceorer moel of high accuracy. Without loss of generality to a network setting, the problem is formulate using a simple two-bus system. Theore, Lyapunov function caniates can be constructe to ientify the important parameters that affect system stability. Base on the erive criteria, a new control esign using the virtual component metho is propose to further improve the stability margin. Numerical simulation for a more complicate system is then carrie out to verify the propose metho. To sum up, key contributions of this paper are as follows: 1) etaile analysis of the system ynamical behaviors that provies insights into physical sources of instability; ) erivation of simple an concise criteria to ensure system stability via the propose Lyapunov function caniates; 3) evelopment
of the virtual component metho for enhancing the stability region of the system. II. PROBLEM FORMULATION In this paper, we begin from a commonly use comprehensive moel with etaile control structures from [11]. In general, the etaile moel inclues controller states of the voltage an current loops, resulting in higher system orer. In fact, istinct time-scale separation of roop an controller moes can be easily seen; particularly, the time constant of roop moes is at aroun 100 ms, which is far greater than the controller moes. Theore, by omitting the internal controller states, a simplifie full moel that enables the terminal voltage an frequency effectively controlle can be obtaine. A. Two-Bus Moel To simplify the problem, a two-bus scenario is utilize for illustration. One can assume that the inverter is connecte to a stiff bus (PCC) regulate by a group of inverters. Then, the ODEs of the system with three inverter an two current (aggregation of coupling an line inuctances) states are given: θ = ω ω 0 (1) ω = ω ω k p ω 0 P () V = V k q Q (3) LI = V cos θ V s RI + ω 0 LI q (4) LI q = V sin θ RI q ω 0 LI (5) where k p an k q are the P ω an Q V roop gains in percentages, V an V s are the inverter an stiff bus voltages, an = wc 1 is the time constant of the low-pass filter for the power measurement. B. Reuce-orer Moel For large power systems, it is convenient to apply the quasi-steay state approximation, allowing one to treat (4) an (5) as algebraic equations by setting the erivative terms to zero. Consequentially, the system orer reuces to only three. However, for the inverter-base microgris this assumption is no longer vali ue to faster inverter states as compare to inertia an excitation time constants of the generators. Particularly, the quasi steay-state assumption fails when the X/R ratio of the microgri becomes slightly larger. Theore, a proper reuction technique has been propose to account for the electromagnetic transients, which play a critical role in the onset of instability [arvix]. To erive a reuce-orer moel, consier the ynamics of inuctive line current being expresse in the complex form: I = I + ji q = V ejθ V s R + jx + sl where X is equal to ω 0 L an ω 0 inicates the nominal frequency. Applying the Taylor series expansion on the Laplace operator s, the line current can be approximate by neglecting the high-orer terms: I I 0 (6) L R + jx si0. (7) where I 0 = (R + jx) 1 ( V e jθ V s ) an supersrcipt {0} enotes the zeroth orer term. Theore, active an reactive power can be expresse as: where P = Re[V I ] P 0 G V V B V θ (8) Q = Im[V I ] Q 0 B V V + G V θ, (9) P 0 = BV V s sin θ + G(V V V s cos θ), (10) Q 0 = B(V V s cos θ) GV V s sin θ, (11) G = R/(R + X ), B = X/(R + X ) (1) G = L(R X ) (R + X ), B = LXR (R + X ). (13) an G an B stan for the transient conuctance an susceptance. Subsituting eqs. (8) an (9) into eqs. (1)-(3), the linearize moel at the nominal operating point can be obtaine: λ p θ + (λ p B ) θ + Bθ + Gυ G υ = 0 (14a) (λ q B ) υ + (λ q + B)υ Gθ + G θ = 0 (14b) where λ p = (k p ω 0 ) 1 an λ q = kq 1. For simpler expressions, we abuse the notation of efining θ = δθ an υ = δv. Also, the assumption is mae that the operating points are close to the nominal conition as θ 0, an V s = V = V n 1 pu, which is vali as the capacity of the lines in microgris is normally greater than the rating of the inverters. III. STABILITY ACCESEMENT From eqs. (14a) an (14b), it can be observe that the quasi-steay approximation is the results of setting G an B to zeros. However, these two terms have significant impacts on preicting instability. A simple observation can be mae on the secon term (amping) of the eq. (14a) that λ p being less than B contributes to instability ue to negative amping. To be more legitimate, the global stability of a linear system coul always be certifie by ensuring that the real parts of all eigenvalues are strictly less than zero. While the analytical solution exists for such a simple system, their cumbersome expressions may ivert the analysis to straightforwar numerical trials without much intuition. A more qualitative way of obtaining stability certificate is to search for Lyapunov function caniates. That is, one can certify the stability of a linear system ẋ = Ax by fining a Lyapunov function, W (x) > 0 for x 0, being strictly ecaying with respect to time, W (x)/t < 0. First, we multiply eq. (14a) by θ + θ to obtain: t { λ p ( θ + θ) + (B B )θ + (λ p B ) θ + Bθ + Gθυ G θ υ + G θυ G θ υ = 0. } + λ p θ (15)
Voltage roop kq 0.00 0.015 0.005 Exact Bounary Estimate Bounary 0.005 0.015 0.00 0.05 Frequencyroopk p Fig. 1. Stability regions of the system (X = 0.008, R = 0.009) Similarly, multiplying eq. (14b) by υ + υ gives: { (3λq + B B )υ } + (G + G )θυ t + (λ q + B)υ + ( λ q + B ) υ Gθυ G θυ G θυ 4Gθ υ + G θ υ = 0. (16) Theore, the terms insie the curly brackets after the summation of eqs. (15) an (16) form a simple Lyapunov function W (x) = 1 xt Px, where x = [ θ + θ, θ, θ, υ] T an λ p 0 0 0 P = 0 λ p 0 0 0 0 B B G + G. (17) 0 0 G + G 3λ q + B B Whenever ω 0 X/R an X/R 0, it is safe to assume that G G an B B. Theore, the criteria for P being positive efinite simplify to: λ p > 0 (3λ q + B) (G)(B) 1 (G) > 0 (18a) (18b) For the ecay rate, we can then erive W (x) = y T Qy, where y = [ θ, θ, υ, υ] T an λ p B 0 0 0 Q = 0 B G G 0 0 G G λ q B 0. (19) 0 0 0 λ q + B The criteria for Q becomes: λ p > B (0a) (λ q B /) (G)B 1 (G) > 0 (0b) In fact, eqs. (18a) can be always satisfie an (0b) is stricter than (18b). Theore, the stability certificate via the propose Lyapunov function can be ensure by the following conitions: k p < 1 B (1a) B k q < G + BB (1b) Although the obtaine conitions o not lect the exact stability bounary, they illustrate the effect of B on the system stability. Particularly, the active power-frequency moe suffers significantly from the increase of B (analogy to transient susceptance). Fig. 1 shows the comparison between the exact an preicte stability regions with the preiction on k p being approximately 50% accurate (when k q is sufficiently small) while very conservative on k q. In fact, one can parametrize the multiplier terms, c θ + θ an c υ + υ, to obtain a collection of polytopes in orer to certify a wier range of roop gains. However, this paper aims to focus on the physical origin of the instability; the eication to searching of the Lyapunov caniates is beyon the scope of the work. IV. EMULATION OF VIRTUAL COMPONENTS In the previous section we have shown that the B eteriorates the amping coefficient, limiting the stability regions of the roop-controlle inverter system. This transient susceptance normally increases when the coupling between the inverter an PCC becomes stronger (ecrease of impeance). It has been reporte in [8] that aitional coupling inuctor shoul be place to enhance the satbility performance; thus the implementation of aitional coupling inuctors can be commonly seen. However, placement of bulky inuctors is not always a esirable solution, so many research works have propose the concept of virtual impeances, virtual inuctances or virtual synchronous generators that mimic the stator winings an rotor ynamics [6], [8] [10]. All these methos helps to mitigate the system instability, which will be investigate in this section. A. Virtual Inuctance The typical inverter control system utilizes the two-loop control scheme an fee-forwar terms with the inner current loop esigne to achieve higher banwith than the outer voltage loop so the controller parameters can be tune inepenently. In general, the overall time constant of the voltage regulation is far smaller than that of the roop moe. Theore, effectively we can consier a fast regulation of the inverter terminal voltage an neglect the LC filters. To mimic the virtual impeance (or equivalently generators stator winings), we can a aitional terms that react to the output currents to emulate the inuctive ynamics. That is, the moifie erence voltages are of the following forms: q = V + X m I oq sω f L m I o s + ω f = V q X m I o sω f L m I o s + ω f (a) (b) where X m = ω 0 L m enotes the virtual reactance, L m an R m are the emulate resistance an inuctance, ω f is the cut-off frequency of the high-pass filter, V,q stan for the voltage commans form the reactive power roop, an,q are the moifie erence voltage for the two-loop control scheme, an I o,oq are the output currents in q axis. We shoul highlight that the above mentione control scheme may have ifferent equivalent forms that result in the same ynamic behavior, an here we follow a similar configuration as propose in [6]. In aition, the effectiveness of the
B' 0.18 0.17 0.16 0.15 0.14 0.13 0.1 0.6 0.8 1.0 1. 1.4 1.6 1.8.0 X/R Ratio Fig.. Variation of B with respect to X/R ratio Voltage roop kq 0.04 0.03 0.0 0.01 X m =0 X m =0.003 X m =0.006 0.00 0.005 0.015 0.00 0.05 0.030 0.035 Frequencyroopk p Voltage roop kq 0.00 0.015 0.005 XR =0.5 XR =1 XR = 0.005 0.015 0.00 0.05 Frequencyroopk p Fig. 3. Preicte stability regions for ifferent X/R ratios emulation is subject to the close-loop stability an controller banwith, proper parameter tuning of the two-loop controller is require but not isclose in this paper. With the eployment of the virtual inuctance, expansion of the stability region can be explaine by consiering B = RX ω 0Z, whose variation by 4 X/R ratio is shown in Fig., in which X = ω 0 (L m + L t ) an R = R m + R t. It can be seen that the maximum of B occurs at X/R = 1, implying that biirectional perturbation of X/R away from unity allows expansion of stability range of k p provie that k q is sufficiently small. However, ecreasing X/R may further lea to shrinkage of the k q range, which can be shown from the preicte stability regions in Fig. 3. It can also be observe that B has the same value whenever X/R is equal to 0.5 an ; that is, the corresponing preictions of k p coincie. In general, it is more beneficial to properly select the virtual inuctance to ensure X/R > 1 for further expansion of voltage roop gains. B. Virtual Reactance We have ientifie the avantages of aing the virtual inuctance into the control scheme. Another approach is to a only reactance X m without its ynamic part. In this case, we can simply set L m in eqs. (a) an (b) to zero. Let s consier now the original expression of B in (13) as XRL Z 4 with an intentional separation of L an X implying that they can be manipulate inepenently. In fact, increase of X while setting L m to zero can further reuce B. Notice that we implicitly efine X = X m + X t, R = R t an L = L t. The resulting stability regions are shown in Fig. 4, inicating that there is no clear benefits of increasing L m in stabilizing the roop moes. In fact, L m contributes to B an further limits the stability regions. This interesting fining suggests that one can tune L m an X m separately for the virtual component methos to strengthen the system stability. Voltage roop kq 0.008 0.006 0.004 0.00 Fig. 4. Preicte stability regions for ifferent X m C s =0 C s =4 C s =8 0.00 0.01 0.0 0.03 0.04 Frequencyroopk p Fig. 5. Preicte stability regions for ifferent C s (R s = 0.004) C. Virtual Capacitance It has been emonstrate that B can be manipulate by changing X an L inepenently. Although increase of X can significantly benefit the system stability, while it may reuce the voltage regulation accuracy. Inee, one may think about canceling the inuctive effect without increasing the reactance. A simple metho is propose to a instea the low pass filters, which mimic a ynamic capacitance C s in parallel with a resistance R s, into erence voltages: q 1/C s = V s + 1/(R s C s ) I o 1/C s = V q s + 1/(R s C s ) I oq (3a) (3b) By choosing 1/(R s C s ) 0, the equation correlating voltage an current can be linearize as: δv R s δi R sc s δi (4) Eq. (4) shows that the terminal voltage reacts negatively to the erivative of current, which behaves like a negative inuctance. Theore the effective B reuces, leaing to expansions of stability regions as shown in Fig. 5. V. NUMERICAL VERIFICATION In this section, simulation results are carrie out to verify the propose control metho. A system with three inverters in the cascae configuration is built base on the full moel (eqs. (1)-(5)), as shown in Fig. 6. The system parameters are given as: base peak phase voltage: 381.58 V; base inverter rating: 10 kva; nominal frequency: π 50 ra/s; coupling impeance: 0.03 + 0.11iΩ; line inuctance: 0.6 mh km 1 ; line resistance: 165 mω km 1 ; line length: [3 ] km; bus loa: [5 + 4.7i 0 + 3.77i + 3.14i] Ω.
8V =F,19 =/ Freuency (ra/s) F = O 5 O /O =O =F,19 =/ =O =F,19 308.45 308.4 1 1.5.5 3 time (s) Fig. 8. Time-omain results (kp = %, kq = %) only the reactance Xm can achieve a wier stability region as compare to the conventional methos. Moreover, cancellation of inuctive ynamics is further exploite with the propose virtual capacitance metho. The effectiveness of the propose approaches is verifie by both theoretical analysis an timeomain simulation. Although the case stuy employs a simple two-bus system, the system formulation with the propose approach of constructing the Lyapunov function caniates enable possible generalization to a network setting, which is of interest for the future work. Fig. 6. Three inverter microgri configuration w/o virtual components Vitural Inuctance Virtual Reactance an Capacitance 311.8 Vitural Inuctance Virtual Reactance an Capacitance 308.5 0.5 =/ Freuency (ra/s) 311.78 311.76 311.74 311.7 R EFERENCES 0 0.5 1 1.5.5 3 time (s) Fig. 7. Time-omain results (kp = 0.9%, kq = 1%) The time-omain simulation is conucte to compare the ynamic responses between the ifferent methos, as shown in Fig. 7. For the virtual capacitance metho, the cut-off frequency of the low pass filter is set to be 1/(Rs Cs ) = 100 ra/s to capture the unstable moes, an the equivalent negative inuctance Rs Cs is selecte to cancel a fraction of the coupling inuctance. Theore, the overall virtual parameters in per-unit are chosen with Lm being 3e 4 /ω0 an Rs an Cs being 1.4e 3 an 7.07, respectively. The result clearly shows that the virtual components play a significantly role in affecting the system ynamic performance. A more aggressive case is then emonstrate in Fig. 8 with higher roop gains of kp = %, kq = % an a higher virtual inuctance of Lm = 0.006/ω0. It can be seen that the propose virtual reactance an capacitance metho can achieve better performance compare to the case when only the virtual inuctance is applie, verifying the effectiveness of the propose approach. VI. C ONCLUSION Proper control esign for roop-base inverters is crucial for enhancement of ynamic performance. Particularly, the transient suscepntacne B 0 eteriorates the amping coefficient of the embee phase angle oscillator. Base on the propose Lyapunov function caniate, the contribution of B 0 to the system instability has been clearly ientifie. A new concept of eveloping the virtual component metho is then propose to provie further enhancement of system performance against unstable roop moes. The results suggest that increase of [1] N. Hatziargyriou, H. Asano, R. Iravani, an C. Marnay, Microgris, IEEE Power Energy Mag., vol. 5, no. 4, pp. 78 94, 007. [] D. E. Olivares, A. Mehrizi-Sani, A. H. Etemai, C. A. Canizares, R. Iravani, M. Kazerani, A. H. Hajimiragha, O. Gomis-Bellmunt, M. Saeeifar, R. Palma-Behnke et al., Trens in microgri control, IEEE Trans. Smart Gri, vol. 5, no. 4, pp. 1905 1919, 014. [3] S. Parhizi, H. Lotfi, A. Khoaei, an S. Bahramira, State of the art in research on microgris: a review, Access, IEEE, vol. 3, pp. 890 95, 015. [4] Y. Han, H. Li, P. Shen, E. Coelho, an J. Guerrero, Review of active an reactive power sharing strategies in hierarchical controlle microgris, IEEE Transactions on Power Electronics, vol. PP, no. 99, pp. 1 1, 016. [5] A. Tulahar, H. Jin, T. Unger, an K. Mauch, Control of parallel inverters in istribute ac power systems with consieration of line impeance effect, IEEE Transactions on Inustry Applications, vol. 36, no. 1, pp. 131 138, Jan 000. [6] J. M. Guerrero, L. G. e Vicuna, J. Matas, M. Castilla, an J. Miret, Output impeance esign of parallel-connecte ups inverters with wireless loa-sharing control, IEEE Transactions on Inustrial Electronics, vol. 5, no. 4, pp. 116 1135, Aug 005. [7] N. Hatziargyriou, Microgris: architectures an control. John Wiley & Sons, 013. [8] J. He an Y. W. Li, Analysis, esign, an implementation of virtual impeance for power electronics interface istribute generation, IEEE Transactions on Inustry Applications, vol. 47, no. 6, pp. 55 538, Nov 011. [9] J. He, Y. W. Li, J. M. Guerrero, F. Blaabjerg, an J. C. Vasquez, An islaning microgri power sharing approach using enhance virtual impeance control scheme, IEEE Transactions on Power Electronics, vol. 8, no. 11, pp. 57 58, Nov 013. [10] Q. C. Zhong an G. Weiss, Synchronverters: Inverters that mimic synchronous generators, IEEE Transactions on Inustrial Electronics, vol. 58, no. 4, pp. 159 167, April 011. [11] N. Pogaku, M. Proanovic, an T. C. Green, Moeling, analysis an testing of autonomous operation of an inverter-base microgri, IEEE Trans. Power Electron., vol., no., pp. 613 65, 007.