Multi-Time Scale Model Order Reduction and Stability Consistency Certification of Inverter-Interfaced DG System in AC Microgrid

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1 energies Article Multi-Time Scale Order Reduction and Stability Consistency Certification of Inverter-Interfaced DG System in AC Microgrid Xiaoxiao Meng, ID, Qianggang Wang, * ID, Niancheng Zhou, Shuyan Xiao and Yuan Chi 3 State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 444, China; xim@et.aau.dk X.M.; cee_nczhou@cqu.edu.cn N.Z.; 637t@cqu.edu.cn S.X. Department of Energy Technology, Aalborg University, 9 Aalborg, Denmark 3 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore , Singapore; chiy@e.ntu.edu.sg * Correspondence: yitagou@cqu.edu.cn; Tel.: Received: 3 November 7; Accepted: 8 January 8; Published: January 8 Abstract: AC microgrid mainly comprise inverter-interfaced distributed generators IIDGs, which are nonlinear complex systems with multiple time scales, including frequency control, time delay measurements, and electromagnetic transients. The droop control-based IIDG in an AC microgrid is selected as the research object in this study, which comprises power droop controller, voltage- and current-loop controllers, and filter and line. The multi-time scale characteristics of the detailed IIDG model are divided based on singular perturbation theory. In addition, the IIDG model order is reduced by neglecting the system fast dynamics. The static and transient stability consistency of the IIDG model order reduction are demonstrated by extracting features of the IIDG small signal model and using the quadratic approximation method of the stability region boundary, respectively. The dynamic response consistencies of the IIDG model order reduction are evaluated using the frequency, damping and amplitude features extracted by the Prony transformation. Results are applicable to provide a simplified model for the dynamic characteristic analysis of IIDG systems in AC microgrid. The accuracy of the proposed method is verified by using the eigenvalue comparison, the transient stability index comparison and the dynamic time-domain simulation. Keywords: inverter-interfaced distributed generator; nonlinear multi-time scale model; model order reduction; singular perturbation theory; stability consistency. Introduction Inverter-interfaced distributed generators IIDGs have been widely used in power grids as new power generation equipment 3 due to their rapid and flexible control performance. IIDGs are viewed as nonlinear systems with a long-span time constant, which contains several time scales. An effective model of IIDGs is a fundamental of dynamic and stability analysis of power systems with IIDGs. The impedance models of IIDG are usually used in the small-signal stability analysis of the system 4,5, but those models cannot explain theoretically the physical dynamic process of IIDG stability because they take the whole system as an equivalent impedance ratio. The state-space small-signal and large-signal models are general tools to describe the dynamics characteristic of IIDG in detail. However, a large-scale power system containing many IIDGs may involve several thousand state variables, and a detailed modeling of the system can lead to dimensionality curse and formidable computational burden 6, and even the extreme case of difficulty in obtaining the convergence solution. Energies 8,, 54; doi:.339/en54

2 Energies 8,, 54 of 5 Moreover, the IIDG model has a wide timescale ranging from milliseconds to seconds 7, which means that a small time step is required in simulations, so the computational effort becomes large. Due to the computing capability limit of processors, distributed computation is usually considered for large-scale power system simulations. On the other hand, some advanced control methods need dynamic model solving for real time control 8, which may not be guaranteed when a long computation time is needed. The model reduction techniques simplify complex models by eliminating the less significant states and reducing the number of equations. It is an effective approach for improving calculation efficiency and meeting the requirements of the real-time control by reducing the IIDG model to a lower-order simpler model. Therefore, research on order reduction methods for IIDG detailed models is of considerable significance. Researchers have conducted numerous comprehensive and detailed studies on the modeling methods for IIDGs, and accurate and practical mathematical models, including three control loops of power, voltage, and current, are presented 9. However, consensus on the classification standard of the multi-time scale IIDG model is not yet realized. The fast and slow dynamics are almost independent provided that their timescales are sufficiently separated, so that the state variables of a multi-time scale IIDG model can be grouped into those that participate in the fast dynamics and in the slow dynamics. Order reduction methods for the multi-time scale characteristic in traditional power grids with rotating power, such as synchronous machines, is common and mature. The multi-time scale singular perturbation model for synchronous machines and brushless doubly-fed wind turbines are presented in 3 5, and the models are simplified based on the multi-time scale theory. Different from the slow dynamics of synchronous machines due to mechanical inertia, the fast dynamics of IIDG models are bounded from the above due to the switching frequency and filters. This means that the fast and slow dynamics of IIDG models are actually quite close in timescale and the interaction between them can be significant. The fast dynamics of IIDG models can be removed when evaluating slow dynamics, whereas the slow dynamics of IIDG model can be considered stationary when evaluating fast dynamics. Reference 6 ignores the IIDG voltage and double current-loop dynamics to obtain its first-order simplified model via a small signal stability analysis. It tends to preserve the slow dynamics of the IIDG power controller and omit the fast dynamics of the PLL and current controller. The static stability is researched by comparing the system eigenvalues in 6, and there is no further verification for transient stability. References 7 9 establish an order reduction model of IIDGs adapting to different complexities and precisions, but the stability consistency is not studied for the reduction models of IIDGs. For an IIDG-dominated microgrid, investigates a six-order islanded microgrid model, and differential-algebraic order reduction models for different precision requirements are presented. The spatiotemporal model reduction of an IIDG-dominated microgrid is presented in based on singular perturbation and Kron reduction. Reference proposes an order reduction principle of neglecting fast dynamics and fixing slow dynamics for multi-time scale singular perturbation AC/DC systems by using the singular perturbation theory and matrix eigenvalue perturbation theorem. The key modes determining the stability of the system are identified in 3,4, by analyzing the parameter sensitivity of the dominant poles using small signal stability analysis. However, these reduction models of IIDGs only consider the static stability, and lack strict proof of stability consistency for the order reduction. These models cannot satisfy the static stability consistency for suffering small disturbances and the transient stability consistency for large disturbances at the same time. The static and transient stability consistencies are essential to ensure the IIDG model reduction to retain the controllability and observability of the detailed model. The paper is expected to perform an accurate IIDG model reduction to stability the static and transient stability consistencies. When the static and transient stability remain unchanged before and after the order reduction of IIDG, the dynamic response errors of the reduction models of IIDG are bounded 6. Then the reduction models can represent the performance of the detailed model for the stability analysis and the control design of the system with IIDG. Thus, the focus of this paper concentrates on the stability

3 Energies 8,, 54 3 of 5 Energies 8,, 54 3 of 7 consistency verification for the IIDG order reduction. Stability analysis of nonlinear system usually main includes parts, twonamely, main parts, static namely, and transient static and stability. transient The stability. former is The related former to the is related stability to of theequilibrium stability of points, equilibrium and the points, latter and is related the latter to the is related stability to region. the stability Based region. the Based mentioned thepreceding mentioned studies, preceding this paper studies, first thispresents paper first a complete presents amodel complete of a model droop ofcontrol-based a droop control-based IIDG containing IIDG containing a power a power droop controller, droop controller, voltage voltage and current and current dual dual loop loop controller, controller, and and LC LC filter filter and andoutput line impedance. Furthermore, through extracting the multi-time scale characteristics, neglecting the fast dynamics, and adopting singular perturbation theory and the quadratic approximation method of this model, the static and transient stability consistency of order reduction is proven. Finally, the effectiveness of this study is verified by presenting the eigenvalue comparisons, transient stability index comparisons and time-domain simulation results. The remainder of this paper is organized as follows. Section presents the modeling of an inverter-based DG unit in an islanded microgrid. Section 3 describes the static and transient stability consistency proof of the IIDG model order reduction, and evaluates the consistency of dynamic response by comparing the frequency, damping and amplitude between the detailed and reduction models. Section 4 provides the evaluation results to demonstrate the effectiveness of the proposed order reduction method scheme. Section 5 concludes the paper.. IIDG Multi-Time Scale Based on Droop Control IIDG in microgrid adopt the three-phase voltage source-type inverter interface that applies PQ control during the grid-connected operation and the droop control when switching to the islanding operation to regulate the the grid grid voltage and and frequency This This process process is equivalent is equivalent to a voltage to a voltage source. source. Figure Figure shows the shows control the and control interface and interface frame offrame the IIDG of the inverter. IIDG inverter. Droop Coefficient Setting Power Control U * Voltage Control i * L Current Control U * PWM Control Inverters il L R L R U C U i L U P Q Power Calculation and Fillter Figure. Control and interface block diagram of IIDG. Figure. Control and interface block diagram of IIDG. In this figure, U* and il* are the reference instruction values of the output voltage and current of the In inverter, this figure, respectively. U * and i L U* * are refers the reference to the reference instruction instruction values of value the output of the voltage output and voltage current of the of IIDG. the inverter, U and respectively. il represent U the * refers output to the voltage reference and instruction current of value the inverter, of the output respectively. voltage of the U and IIDG. il U denote and the i L represent output voltage the output and current voltage of and the current IIDG, of respectively. the inverter, U respectively. is the point U of common and i L denote coupling the PCC output voltage. voltage and current of the IIDG, respectively. U is the point of common coupling PCC voltage... IIDG Full Based on Droop Control.. IIDG Full Based on Droop Control Power Droop Controller Power Droop Controller The medium and high voltage microgrid generally has a voltage level of kv and above with The medium and high voltage microgrid generally has voltage level of kv and above with a large power supply capacity. The transmission line impedance is generally regarded as purely large power supply capacity. The transmission line impedance is generally regarded as purely inductive, that is, ωl inductive, that is, >> R ωl >> R.. In the low-voltage microgrid where the online transmission line In the low-voltage microgrid where the online transmission line impedance is resistive, realizing the conventional droop control of active power frequency and impedance is resistive, realizing the conventional droop control of active power frequency and reactive power voltage through indirect transformation is feasible by using the improved droop reactive power voltage through indirect transformation is feasible by using the improved droop control method based on virtual impedance and coordinate transformation. Thus, the output powers control method based on virtual impedance and coordinate transformation. Thus, the output powers P and Q of an IIDG can be expressed as: and of an IIDG can be expressed as:

4 Energies 8,, 54 4 of 5 { P = U U sin δ ωl Q = U U U cos δ ωl where δ is the phase angle difference between the IIDG output voltage and the PCC voltage. Generally, the value of δ is very small, and the IIDG cannot directly control the PCC voltage. Thus, P is mainly determined by the output voltage phase angle, and Q is determined by the output voltage amplitude. The phase difference between two terminal voltages can be controlled by restricting the frequency of the output voltage, that is, controlling the active power P output by IIDG. Correspondingly, the reactive power Q output by IIDG can be controlled by managing the output voltage amplitude. Thus, the droop control of IIDG can be designed as follows: { ω = ω mp U = U nq where ω and U are the reference values of the IIDG frequency and voltage, respectively; and m and n denote the active power and reactive power droop coefficients, respectively. Further, the instantaneous power of the IIDG can be calculated by its output voltage and current: { p = 3 ud i Ld u q i Lq q = 3. 3 ud i Lq u q i Ld The average power expression can be obtained through the low-pass filter: { Ps = ω c Qs = sω c p ω c sω c q 4 where ω c is the cut-off frequency of the low-pass filter. Voltage and Current Dual-Loop Controller Figure shows that the voltage outer loop of the controller is regulated by PI Proportional Integral. Meanwhile, the output current i L is introduced as the feedforward to suppress the influence of load fluctuation on the output voltage and improve the dynamic response performance of the system. The expression of the inner loop command current is as follows: { i Ld = Hi Ld K p u d u d K i u d u d dt ω n Cu q i Lq = Hi Lq K p u q u q K i u q u q dt ω n Cu d. 5 The current inner loop adopts the inductor current feedback and PI regulation mode. The reference voltage of the current inner loop is: { u d = u d K p i Ld i Ld K i i Ld i Ld dt ω n L i Lq u q = u q K p i Lq i Lq K i i Lq i Lq dt ω n L i Ld. 6 The state variables of the voltage outer and inner loops are defined to facilitate the problem analysis, as follows: { dϕd dt dϕ q dt = u d u d = u q u q 7 { dλd dt dλ q dt = i Ld i Ld = i Lq i Lq. 8

5 dλd = il d il d dt dλ = dt. q Energies 8,, 54 il q il q 5 of 5 8 u d u q - u d u q - K K s p i K K s p i i L d H L d - - ωc ωc H Ⅱ i L q Voltage Loop Controller i i Lq i L d i L q - K K p i Ⅰ K s K s p i u d - ωl ωl u q Current Loop Controller Figure. Diagram of voltage- and current loop controller. Figure. Diagram of voltage- and current loop controller. u d u q 3 Filter and Line 3 Filter and Line The dynamic influence of the switching part can be neglected at high switching frequency. The switching The dynamic tube can influence generate the of the required switching voltage part in accordance can be neglected with the at instruction, high switching that is, frequency. u* = u. The Therefore, switching the d tube and can q components generate the of the required filter inductance voltage in current accordance il are: with the instruction, that is, u * = u. Therefore, the d and q components of the filter inductance current i L are: dild R { dild = i Ld ω i Lq u d u d dt dt= R L Li Ld ωi Lq L u d u d di Lq di 9 dt Lq = ωi Ld R L i = ωild Lq L u ilq q u q. 9 u q uq dt L L. The d and q axis components of its filter capacitance voltage U are: The d and q axis components of its filter capacitance voltage U { are: dud dt = C i du Ld i Ld ωu q du q d = il d Ld ωuq dt dt =. CC i Lq i Lq d du The d and q components of the output q = current il q ii L on Lq the ωu line are shown below, where u is the d voltage of PCC: dt C. { dild The d and q components of dt = the output R L i Ld ωi current Lq il on L u the d u line are d shown below, where u is the di Lq voltage of PCC: dt = ωi Ld R L i Lq L u q u q. The full system model of the diiidg is Ld Restablished by combining Equations, as shown in Equation : = i Ld ω i Lq u d u d. dt L L x IIDG = A di IIDG x IIDG B IIDG u Fx IIDG Lq R x ωild ilq uq uq dt = L L IIDG = δ P Q ϕ dq λ dq i Ldq u dq i. Ldq T. u = u d u q T The full system model of the IIDG is established by combining Equations, as shown in The expressions and parameters of the detailed model of the preceding equation are provided in Equation : Appendix A... Multi-Time Scale Decomposition of IIDG System The linear model of the IIDG full system can be obtained after linearization at the steady-state point, as shown in Equation 3: ẋ IIDG = A S x B S u x IIDG = δ P Q ϕ dq λ dq i Ldq u dq i Ldq T u = u d u q T. 3

6 Energies 8,, 54 6 of 5 A stable operating point x s of the IIDG is taken. By solving the state matrix A s of the linear model at the stable operating point, all characteristic roots of the IIDG model in the complex plane can be obtained, as shown in Table A and attached Figure A. Appendix B provides the detailed elements of various matrixes in Equation 3. According to Figure A and Table A, the eigenvalues of the IIDG full model are distributed in several frequency bands with evident multi-time scale characteristics. The multi-time scale model of IIDG can be obtained through model simplification and perturbation factor extraction. According to the classification criterion of the multi-time scale 6, if the original system is stable with the characteristic spectrum, then the eigenvalues of the system state matrix A s can be arranged from small to large. If the magnitude ratio of two adjacent eigenvalues, that is, the separation ratio, is less than, then this ratio can be used as the sign of time-scale division of the multi-time scale model. The time scale is inversely proportional to the natural frequency or modulus of the characteristic root. A low natural frequency leads to small characteristic root modulus and long corresponding time scale. After referring to the calculation results of the eigenvalue modulus and separation ratio of the IIDG system in Appendix B, IIDG can be considered a typical three-time scale model. The voltage and current dual loop control should be integrated into one time scale. The characteristic time scale is 3 to 4 second level. The power outer loop control and the phase angle-frequency control should be classified into different time scales, namely, and second level, respectively. The IIDG model comprises Equations to and is a 3-order differential equation set. Considering the preceding time-scale division principle, assuming: x = δ y = P Q T y = ϕ dq λ dq i Ldq u dq i Ldq T = /ω c, = /ω where x is the state variable of IIDG phase angle, y denotes the state variable of the outer loop output power of IIDG, y represents the state variable of IIDG circuit equation and dual-loop control equation, and and refer to the corresponding perturbation parameters. Thus, the three-time scale system model of IIDG can be described as: { ẋ = fx, y 4 i ẏ i = gx, y, u 5 where i =, ; f and g are the corresponding functional relations; and u represents the PCC voltage. Equation 5 is the expression form of Equation after extracting the multi-time scale features..3. Order Reduction Form of Neglecting Fast Dynamics for IIDG System According to singular perturbation theory, when the singular perturbation parameter i is sufficiently small and < <<, the following two reduction forms can be obtained: Order Reduction Form = If the electromagnetic transient characteristics of the inductor current and capacitor voltage in the LCL filter circuit are neglected, then the corresponding voltage and current values are considered capable of following the instruction values at a rapid rate. Thus, the fast dynamics of the voltage and current dual-loop control and LCL filter circuit variables can be neglected, and the original model can

7 Energies 8,, 54 7 of 5 be reduced to a third-order model with algebraic constraint equations. The constraint equations are as follows: K pr i Ld L K ph i Ld i L K pk p u d Lq i L Lq u q K p C u q L K pk i ϕ d u L K λ i d d ϕ L q K pk i u d λ L q uq = R ωl ωl ωl u d = u q ωc u d i Ld i Lq i Ld i Lq uq ω i Ld i Lq K i ω = ϕ d ϕ q ωc u d u q H ω i Lq i Ld i Ld i Lq i Ld i Lq u d u q u q u d = U = nq cos δ U nq sin δ K p u d ω C u q K p ω u d u q = u q u d. 6 The power outer loop dynamic equation is as follows: d ω c dt P Q = P Q 3 u d u q u q u d i Ld i Lq. 7 The phase angle equation is: Order Reduction Form =, = dδ dt = ω mp. 8 If the measurement delay dynamics of the low-pass filter in the power outer loop control equation is neglected, that is, the fast dynamics of the active and reactive power variables in the power outer loop control are further neglected, then the original model can be reduced to a first-order model. Therefore, only the phase angle differential equation related to the frequency control and a set of algebraic equations are preserved. The new algebraic constraint equation is as follows: P Q 3 u d u q u q u d i Ld i Lq =. 9 Thus, the two order-reduction forms of the original system model are obtained. However, the stability consistency of the system after order reduction must be further verified. 3. Stability Consistency Proof of IIDG Order Reduction The detailed mathematical model of the full IIDG system is a 3-order nonlinear model. The static and transient stability of the model after order reduction implementation are discussed. Static and transient stability are interrelated, but not equivalent; hence, providing the corresponding general proofs is necessary. The static and transient stability consistency proofs can verify the effectiveness of order reduction in theory. Besides, the stability consistency can be indirectly reflected by the dynamic response comparison between the detailed and reduction models. Equation is the general expression of a class of nonlinear multi-time scale mode corresponding to IIDG model. Considering the order reduction of neglecting fast dynamics on a rapid dynamic

8 Energies 8,, 54 8 of 5 variable ŷ, the remaining slow state variables can be combined into the normal speed variable ˆx. i can select any sufficiently small positive value : 3.. Consistency Proof of Static Stability before and after Order Reduction { ddt ˆx = fˆx, ŷ i d dt ŷ i = gˆx, ŷ, û. The general form of the small signal model of the nonlinear multi-time scale system shown in Equation at the equilibrium point can be expressed as : d dt ˆx ŷ A = A ˆx A A ŷ ˆx = A B û ŷ B B û. The preceding equation is expressed in the complex frequency-domain form, as follows: si ŷ = A ˆx A ŷ B û ŷ = si A A ˆx si A B û si ˆx = A ˆx A ŷ B û = A A si A A ˆx B A si A B û 3 where I and I are the same order unit matrix of ˆx and ŷ, respectively. Thus, the new state equation is: si ˆx = A n ˆx B n û. 4 By solving the characteristic root of Equation 4, the following equation can be obtained: si A n = si A A si A A =. 5 According to the properties of the matrix Schur complement, the characteristic root of the system after order reduction satisfies Equation 5. When si A / =, =, the corresponding characteristic roots of si A n = and si A = are the same: si A n = si A A si A A = si A A A si A si A si A si si A n = A A A si A si = A A A si A = si A. 6 Owing to the small value, the following can be set when reducing the full model of the original system: si ŷ =. 7

9 Energies 8,, 54 9 of 5 Thus, Equation is further simplified as shown below after order reduction: si ˆx = A A A ˆx B A A û si ˆx = A n ˆx B n û. 8 Subsequently, the characteristic equation of IIDG order reduction model satisfies the following equation from the properties of the matrix Schur complement: A si A n = A si A A A A si = A A A A =. 9 The following can be deduced by Equation 6: si A = si A A A si A. 3 When is close to : si A si A A A A = A si A n. 3 Therefore, the eigenvalues of the system order reduction model calculated by Equation 3 are all eigenvalues corresponding to the slow dynamic state variables in the full model of the original system, which is not related to. By combining Equations 6 and 3, when is sufficiently small, the static stability of the original system is entirely determined by the stability of the order reduction system A n and the boundary layer system A simultaneously. Hence, if and only if the order reduction and boundary layer systems are both stable, the original system can be stabilized; otherwise, the static stability may be inconsistent. 3.. Consistency Proof of Transient Stability before and after Order Reduction According to singular perturbation theory, when the singular perturbation parameter i of a state variable is sufficiently small, the system of Equation can be approximately decomposed into the following order reduction system: { ddt ˆx s = fˆx s, ŷ s 3 = gˆx s, ŷ s, û and boundary layer system: d dt ˆx s = fˆx s, ŷ s i d dt ŷ f = gˆx s t, ŷ s t ŷ f, û. 33 In Equation 3, ˆx s is a state variable with the initial value of ˆx s t = ˆx, and ŷ s denotes an algebraic variable with the initial value satisfying gˆx s t, ŷ s t =. In Equation 33, ŷ f represents a state variable. When i is sufficiently small, the action time of the boundary layer system can be approximately viewed as short, and ˆx s will remain in its original value, thereby neglecting the rapid dynamic variable of the system. Equations and 3 correspond to the system before and after order reduction, respectively. These equations have completely consistent equilibrium points. If the dominant unstable equilibrium points UEP of the system before and after order reduction are both ˆx u, ŷ u, when i selects any

10 Energies 8,, 54 of 5 sufficiently small positive value, then the Jacobian matrix of the original system Equation at the UEP is as follows: fx f J = y J = J 34 where J = f x = f/ ˆx ˆxu,ŷ u ; and J = g y = g/ ŷ ˆxu,ŷ u. g x g y J J J = f y = f/ ŷ ˆxu,ŷ u ; J = g x = g/ ˆx ˆxu,ŷ u ; If the left eigenvector of the unstable eigenvalue J of µ is expressed as η = η T ηt T, in which η is n n column vector and η is n column vector, then: J T η = JT J T J T J T η η = µ η η If J is reversible, then the following can be obtained by Equation 35:. 35 lim η J T JT η = 36 J T JT J T JT lim η lim µlim η. 37 Similarly, the Jacobian matrix of the order reduction system Equation 3 at the UEP ˆx u, ŷ u can be further deduced to the following: J = J J J J. 38 Assuming µ, the unstable eigenvalues of J, which has the left eigenvector of η, then: J T η = J T JT J T JT η = µ η. 39 If J and J have the same unstable eigenvalue when is sufficiently small, that is., µ = lim µ, then η C lim η can be obtained after comparing Equations 37 and 39. C is a constant. The stability region boundary function of the original system is constructed in this study according to the quadratic approximation method of the stability region boundary of the nonlinear system 7. The function can be expressed as follows: h Q ˆx, ŷ, = η T ˆx ˆx u ŷ ŷ u ˆx ˆx u ŷ ŷ u T Ψ ˆx ˆx u ŷ ŷ u. 4 Among the preceding equations, the coefficient matrix of the quadratic term Ψ satisfies the following Lyapunov matrix equation: where Ψ = Ψ Ψ T Ψ Ψ CΨ ΨC T = H 4, in which Ψ is n n n n matrix; Ψ denotes n n n matrix; Ψ refers to n n matrix; C = µi n / J T ; I n represents n order unit matrix; and H, which is expressed below, denotes the Hessian matrix of the function at the equilibrium point ˆx u, ŷ u :

11 Energies 8,, 54 of 5 H = n n i= η i H s f i n j= η j H s g j 4 where η i is the ith element of η ; η j denotes the jth element of η ; and H s f i H s g j, which are expressed below, represent the Hessian matrixes of f i and g j at the UEP, respectively ˆx u, ŷ u ; i =,,..., n n, j =,..., n : H s f i = H s g j = H f i xx H f i xy H g j xx H g j xy H f i xy T H f i yy H g j xy T H g j yy = = f i ˆx f i ˆx ŷ g j ˆx g j ˆx ŷ f i ˆx ŷ T f i ŷ g j ˆx ŷ T g j ŷ Based on the assumption that is sufficiently small and the eigenvalue of J has strictly negative realness, Equation 4 is expanded and further expressed as follows: µin n Ψ J n n i= η i H f i n j= µin n J T Ψ J T JT η j Hg j. 45 The specific deduction proof is shown in Appendix C. In the previous equation: H f i = H f i xx H f i xyj J J T J T H f T i xy J T J T H f i xxj J H g j = H g j xx H g j xyj J J T J T H g j T xy J T J T Hg j yyj J. Similarly, the stability region boundary function of the order reduction system Equation 3 is: h Q ˆx = η T ˆx ˆxu ˆx ˆx u T Ψ ˆx ˆx u 46 where Ψ is the n n n n coefficient matrix of the quadratic coefficient. The Lyapunov matrix equation is also satisfied, which can be expressed as follows: Ψ µin n J µin n J T Ψ n n η i= i H f i n j= J T JT η j Hg j. 47 The following can be obtained after comparing the preceding equation and Equation 44: Ψ C lim Ψ. 48 Therefore, the stability region boundary functions of the original and order reduction systems have the following relations: h Q ˆx C lim h Q ˆx, ŷ,. 49 According to the definition of the transient stability index in stability region boundary theory of the nonlinear system 8, the transient stability index of the original system is defined as follows: I Q ˆx, ŷ, = h Qˆx, ŷ, h Q ˆx s, ŷ s,. 5

12 Energies 8,, 54 of 5 The transient stability index of the order reduction system is defined as: I Q ˆx = h Q ˆx h Q ˆx s. 5 ˆx s, ŷ s is the stable equilibrium point SEP of the original and order reduction systems. The transient stability indexes before and after order reduction should satisfy the following relations: I Q ˆx lim I Q ˆx, ŷ,. 5 The sufficient condition for Equation 5 is that the eigenvalue of J has the strictly negative realness. Moreover, the transient stability of the original system and the order reduction system is consistent, and the transient stability indexes are approximately equal. According to the introduction of stability region boundary theory in 8, the transient stability index reflects whether the operating point of the system is within the stability region boundary. When the system is stable, the transient stability index is positive; when the system is unstable, the transient stability index is negative; when the system is zero, the system is in the critical stable state. Table shows the transient stability law. Boundary Layer System Table. Transient state stability laws of order reduction before and after reduction. Original System Order Reduction System Transient Stability Index Transient Stability Consistency Stable Stable Stable Both positive numbers Conformity Stable Unstable Unstable Both negative numbers Conformity Unstable Unstable Stable Opposite sign Inconformity Unstable Unstable Unstable Both negative numbers Conformity 3.3. Consistency Evaluation of Dynamic Response before and after Order Reduction In order to evaluate the dynamic response consistency of the system state variables before and after the order reduction, the Prony transformation is used to extract the three crucial characteristics of dynamic response: frequency, damping and amplitude 9. The Prony transformation describes the time-series sampling data of dynamic response by a set of linear combinations of exponential function. Through the appropriate expansion 3, the frequency, damping coefficient, amplitude and initial phase angle of a given sampling data can be estimated. Three consistency indexes of frequency, damping and amplitude are applied to represent the error of dynamic response. By the Prony transformation on the dynamic response of the original detailed system, the frequency and energy can be written as: { F d = f d, f d,..., f dn 53 E d = λ d, λ d,..., λ dn where f di and λ di i =,... n are the frequency and energy of the ith element for the dynamic response of detailed system. The frequency and energy of order reduction models can also be obtained by same transformation as: { F s = f s, f s,... f sn 54 E s = λ s, λ s,..., λ sn where f si and λ si i =,... n are the frequency and energy of the ith element of the reduction model. Consistency index of frequency The frequency consistency index of the reduction model can be obtained by comparing the frequency vectors of F d and F s. The non-periodic signals are useless for analyzing frequency information, so the non-periodic signals are removed in the calculation of frequency consistency.

13 Energies 8,, 54 3 of 5 The energy signal of each element is chosen as the weight in calculating the consistency of the frequency. The frequency consistency is defined as: λ di w i = n 55 λ di i= σ i = f di f si max f di, f si 56 ϕ f = n w i σ i 57 i= where σ i and w i are the frequency consistency and weight of ith element; φ f is the frequency consistency index of dynamic response for the order reduction model. Consistency index of damping Same with the frequency consistency, the damping consistency index can be obtained as: { ξ di ξ si η i = maxξ di,ξ si, ξ di ξ si >, ξ di ξ si < 58 ϕ ξ = n w i η i, 59 i= where η i is the damping consistency of the ith element; ξ di and ξ si are the damping of the ith element for the detailed model and the reduction model calculated by the Prony transformation 3; φ ξ is the damping consistency index of the dynamic response for the order reduction model. It is noted that the η i could be zero when ξ di and ξ si having different sign, which means the dynamic response of these two models have the opposite stability trend, and the damping characteristic of these two elements are independent. 3 Consistency index of amplitude Similarly, the amplitude consistency index can be given as: i = A di A si maxa di, A si 6 ϕ A = n w i i, 6 i= where i is the amplitude consistency of the ith element; A di and A si are the amplitudes of ith element for the detailed model and the reduction model respectively; φ A is the amplitude consistency index of dynamic response for order reduction models. These three consistency indexes are between and. The more closely they approach, the more the higher similarity they have. 4. Simulation Studies The effectiveness of IIDG order reduction is verified by the static and transient stability consistency proofs in Section 3. In Section 4, the simulation results are used as well to verify the effectiveness of order reduction through a stand-alone case and a practical microgrid cases. Verifications mainly include the comparisons of eigenvalue, transient stability index and time-domain waveform between original model and different order-reduction models.

14 of dynamic response for order reduction models. These three consistency indexes are between and. The more closely they approach, the more the higher similarity they have. 4. Simulation Studies The effectiveness of IIDG order reduction is verified by the static and transient stability Energies 8, consistency, 54 proofs in Section 3. In Section 4, the simulation results are used as well to verify the 4 of 5 effectiveness of order reduction through a stand-alone case and a practical microgrid cases. Verifications mainly include the comparisons of eigenvalue, transient stability index and timedomain System waveform System between original model and different order-reduction 4.. Stand-Alone models. The example 4.. Stand-Alone system System insystem Figure 3 is selected to verify the effectiveness of the preceding order reduction system The example presented. system Anin IIDG Figure is3 connected is selected to toverify the PCC the effectiveness via the LCL of the circuit, preceding in which order the IIDG full model reduction has 3 system orders, presented. including An IIDG three is connected orders of to outer the PCC loop via the power LCL circuit, droop in which controller, the IIDG four orders of voltagefull and model current has 3 dual-loop orders, including controller, three orders and of sixouter orders loop of power filter droop andcontroller, line. The four detailed orders of simulation voltage and current dual-loop controller, and six orders of filter and line. The detailed simulation parameters of System are consistent with the IIDG system provided in Appendix A. parameters of System are consistent with the IIDG system provided in Appendix A. IIDG U R L U R L C U i L i L R L jx L Figure 3. Structure diagram of an IIDG system. Figure 3. Structure diagram of an IIDG system. Table shows the initial values and operating equilibrium points of various state variables of IIDG. Under the operating condition of the SEP, the small signal stability of System is analyzed. Table shows the initial values and operating equilibrium points of various state variables of IIDG. Under the operating condition of the SEP, the small signal stability of System is analyzed. The eigenvalues of the system before and after order reduction can be obtained by solving the state matrix, as shown in Table 3. Table. Initial values and equilibrium points of test system. State Variables Initial Value /pu Initial Value /pu SEP/pu UEP/pu δ P Q φ d.... φ q.... λ d.... λ q.... i Ld i Lq u d u q i Ld i Lq Table 3. Eigenvalues of IIDG models at SEP. System Eigenvalues Fast subsystem σa Slow subsystem σa n Original system σa Order Reduction Form 3rd Order = st Order =, =,77 ± 33i 687± 95i 849 ± 83i 889 ± 73i ± 94i 8.4 ± 5.96i.568,77 ± 33i 689 ± 954i 85 ± 8i 888 ± 74i ± 97i 6 ± 5i 3,77 ± 33i 689 ± 954i 85 ± 8i 888 ± 74i 9 ± 98i Order Reduction Form is a third-order model that reserves the power outer loop control equation and the phase angle equation, whereas Order Reduction Form is a first-order model that reserves

15 Energies 8,, 54 5 of 5 only the phase angle equation. According to the calculation results in Table 3, the order reduction precision of the third-order model is higher than that of the first-order model. However, if the rapid dynamic reduction form is neglected, then the static stability consistency of both models before and after order reduction can be guaranteed. Table 4 shows that the transient stability index of the original system I Q x, y, is.7989 under the effect of Initial Value, which indicates the stability of the original system. At this point, the transient stability indexes of the third- and first-order systems are.84 and.64, respectively. They are of the same sign as the transient stability index of the original system, and the order reduction systems are in the stable state. In addition, J is found to guarantee the strictly negative realness of the eigenvalues under two order reduction forms through calculation, which means that the transient stability of the system before and after order reduction is consistent. Table 4. Transient stability analysis result of test system. Energies 8,, 54 6 of 7 Transient Stability Analysis Result Transient Stability Analysis Result under Initial Value 3 under Initial Value 4 Order Reduction Form Table 4. Transient stability analysis result of test system. Original 3rd Order st Order Original 3rd Order st Order Transient Stability Analysis Result Transient Stability Analysis Result under Initial Value 3 under Initial Value 4 Transient Order Stability Reduction Index Form I Q Original 3rd Order st Order Original 3rd Order st Order Stability Consistency / / Calculation Time 4 s 8 s 5 s 7 s 5 s 3 s Transient Stability Index IQ Stability Consistency / / Under the Calculation effect of Time Initial Value4, s the transient 8 s stability 5 s index7 ofs the original 5 s system3 s I Q x, y, is.33. During this time, the transient stability indexes of the third- and first-order systems are Under the effect of Initial Value, the transient stability index of the original system IQx, y,.54 and is.33..8, During respectively. this time, The the transient transient stability stability indexes indexes of the before third- and andfirst-order after order systems reduction are are of different.54 signs. and.8, The original respectively. system The transient is unstable, stability and indexes the order before reduction and after order systems reduction are in are the of stable system. different In addition, signs. JThe cannot original guarantee system is unstable, the strictly and the negative order reduction realnessystems of theare eigenvalues in the stable through calculation. system. In other In addition, words, J under cannot the guarantee precise the original strictly negative system realness instability, of the theeigenvalues transient stability through of the calculation. In other words, under the precise original system instability, the transient stability of the system before and after order reduction is inconsistent, which corresponds to the third analysis case system before and after order reduction is inconsistent, which corresponds to the third analysis case in Table. The calculation time of different models are also shown in Table 4. Due to the reduced in Table. The calculation time of different models are also shown in Table 4. Due to the reduced complexity complexity of the of model, the model, st order st order model model takes takes less less calculation time time than than 3rd 3rd order order model. model. The δ-u solution curves of the original and order reduction systems are obtained under the effect The δ-u solution curves of the original and order reduction systems are obtained under the of two initial effect of values two initial through values time-domain through time-domain simulation, simulation, as shown as shown in Figure in Figure 4. Under 4. the effect of Initial Under the effect of Value, Initial the solution Value, the curves solution ofcurves the original of the original system system model model and the order reduction reduction system system model model in in Figure 4a Figure can4a can converge to ato a SEP. However, under the the effect effect of Initial ofvalue Initial, the Value solution, the curve solution of the original curve of the originalsystem in Figure in Figure 4b cannot 4b cannot converge converge to the SEP. The to the time-domain SEP. The simulation time-domain of the curve simulation in Figure 4 indirectly of the curve in verifies the transient stability index analysis results. Figure 4 indirectly verifies the transient stability index analysis results. δ /pu.34.3 Stable Point st -order.6 3 rd -order. Orignial Initial Value U /pu a δ /pu Stable Point 3 rd -order Initial Value st -order Original U /pu b Figure 4. Simulation results of original and reduced systems under different initial values for test Figure 4. Simulation results of original and reduced systems under different initial values for test system : a at Initial Value ; b at Initial Value. system : a at Initial Value ; b at Initial Value. Figure 5a,b are the time-domain simulation results of the system under the small disturbance, where il, IL, and Δω are the instantaneous current, frequency deviation, and RMS Root-Mean- Figure 5a,b are the time-domain simulation results of the system under the small Square current of IIDG output respectively. The small disturbance form is that PCC voltage U steps disturbance, where i from. to.5. L, I The L, and ω are the instantaneous current, frequency deviation, and RMS dynamic response results in Figure 5a further verify the stability analysis results of the preceding table. At the SEP, the stability of the system remains unchanged before and after order reduction. Figure 5b corresponds to the instability of the original system. During this time, the order reduction system remains stable. Hence, the stability consistency of the system cannot be guaranteed under the condition of original system instability. Table 5 shows the calculation results of three consistency indexes of dynamic response for the IIDG output current in Figure 5a. When

16 Energies 8,, 54 6 of 5 Root-Mean-Square current of IIDG output respectively. The small disturbance form is that PCC voltage U steps from. to.5. The dynamic response results in Figure 5a further verify the stability analysis results of the preceding table. At the SEP, the stability of the system remains unchanged before and after order reduction. Figure 5b corresponds to the instability of the original system. During this time, the order reduction system remains stable. Hence, the stability consistency of the system cannot be guaranteed under the condition of original system instability. Table 5 shows the calculation results of three consistency indexes of dynamic response for the IIDG output current in Figure 5a. When operating at stable point, high values of these three consistency indexes are presented which clearly means that the difference of dynamic responses between the order reduction models and original model are relatively small. Energies 8,, 54 7 of 7 il/pu.4. Original 3 rd -order st -order Δω/pu Original 3 rd -order st -order.5.5 a IL /pu Original 3 rd -order st -order il/pu 3.5 Original 3 rd -order st -order Δω/pu.5.3. Original 3 rd -order st -order b IL /pu Original 3 rd -order st -order Figure 5. Simulation results of the test system : a at the stable point; b at the unstable point. Figure 5. Simulation results of the test system : a at the stable point; b at the unstable point. Table 5. Consistency indexes of dynamic response for reduction models in stand-alone system. IIDG Output Current at Stable Point Frequency Consistency Index 3rd Order st Order Damping Consistency Index 3rd Order st Order Amplitude Consistency Index 3rd Order st Order 98.6% 95.34% 96.4% 93.58% 95.3% 9.7%

17 Energies 8,, 54 7 of 5 Table 5. Consistency indexes of dynamic response for reduction models in stand-alone system. IIDG Output Current at Stable Point Frequency Consistency Index 3rd Order st Order Damping Consistency Index 3rd Order st Order Amplitude Consistency Index 3rd Order st Order Energies 8,, % 95.34% 96.4% 93.58% 95.3% 9.7% 8 of 7 The 4.. distributed Three-IIDG Microgrid photovoltaic Pilot Project power System generation System and microgrid operation control pilot project at the Henan College of Finance and Taxation under practical operation is selected as a simulation example. In the pilot project, the system capacity is 38 kw PV and two kw/kwh energy storage. The network structure can be equivalent to the microgrid with three IIDGs connected in parallel two energy storage power sources and a PV power source, as shown in Figure 6. Appendix D presents the detailed parameters of System. Table 6 shows the initial values and operating equilibrium points of the state variables. The distributed photovoltaic power generation and microgrid operation control pilot project at the Henan College of Finance and Taxation under practical operation is selected as a simulation example. In the pilot project, the system capacity is 38 kw PV and two kw/kwh energy storage. The network structure can be equivalent to the microgrid with three IIDGs connected in parallel two energy storage power sources and a PV power source, as shown in Figure 6. Appendix D presents the detailed parameters of System. Table 6 shows the initial values and operating equilibrium points of the state variables. IIDG U RfjXf U RcjXc U C IIDG U RfjXf U RcjXc C IIDG3 U3 Rf3jXf3 U3 Rc3jXc3 RLjXL C3 State IV3/ IV4/ Figure Structure diagram of of the the three-iidg system. system. Table Table Initial values and and equilibrium points points of system of system.. SEP/ UEP/ State IV3/ IV4/ State IV3/ IV4/ SEP/ UEP/ State IV3/ IV4/ SEP/ UEP/ State IV3/ IV4/ SEP/ UEP/ Variable pu pu pu pu Variable pu pu pu pu Variable pu pu pu pu Variable pu pu pu pu Variable pu pu pu pu Variable pu pu pu pu δ δ δ δ.9 P δ P P 3.63 δ Q P Q Q P P φ d.... φ d.... φ d3.... Q.5 φ q Q φ q φ q3 Q λ ϕd. d.... λ.. d.... λ ϕd... d3.... ϕd3.... λ q.... λ q.... λ q3.... ϕq. i Ld ϕq i Ld i Ld3 ϕq i λd. Lq i... Lq i λd.... Lq λd3.... u d u d u d λq. u q λq u q u q3 λq i ild.554 Ld i ild Ld i Ld ild i Lq i Lq i Lq ilq ilq ilq ud ud ud The eigenvalues of the system before and after order reduction can be obtained through the small uq uq uq signal stability analysis of the microgrid system under the SEP condition. Each IIDG uses two order ild ild ild3 reduction models deducted in the paper, namely, the third- and first-order models..4 The complete ilq ilq ilq3 39-order microgrid model selects only the dominant eigenvalues to analyze the problem,.86 and all. order.78. reduction models are processed by neglecting fast dynamics to only analyze the eigenvalues of the The sloweigenvalues subsystem, asof shown the system in Table 7. before and after order reduction can be obtained through the small signal stability analysis of the microgrid system under the SEP condition. Each IIDG uses two order reduction models deducted in the paper, namely, the third- and first-order models. The complete 39-order microgrid model selects only the dominant eigenvalues to analyze the problem, and all order reduction models are processed by neglecting fast dynamics to only analyze the eigenvalues of the slow subsystem, as shown in Table 7. SEP/ UEP/ State IV3/ IV4/ SEP/ UEP/

18 Energies 8,, 54 8 of 5 Table 7. Dominant eigenvalues of test system order reduction at stable point. Order Reduction Form Original model dominant poles Adopting 3rd-order model 4.7 ± 5.i ± 5.96i.57 Eigenvalues 4.38 ± 6.9i ± 6.56i ± 6.98i ± 6.5i 9.54 Adopting st-order model From the analysis results in Table 7, the static stability consistency of the three-iidg microgrid system before and after order reduction can be guaranteed under the premise of neglecting the variable dynamics of the system. When IIDG uses the third-order model for order reduction, the eigenvalue of the slow subsystem and the static stability are relatively close to the dominant eigenvalue of the original system. The eigenvalue of the system with the first-order model is significantly different from that of the original system but can still guarantee the static stability of the system. Table 8 further analyzes the eigenvalues of System at the UEP before and after order reduction and finds that the original and order reduction systems of IIDG using the third-order model are unstable. However, the IIDG using the first-order model for order reduction remains stable. In other words, considering system instability, the order reduction of IIDG by the third-order model is suitable for the simulation results of the original system. Therefore, based on the analysis results in Table 6, the static stability of IIDG using the third-order reduction model is better than that of IIDG using the first-order model. Table 8. Dominant eigenvalues of test system order reduction at unstable point. Order Reduction Form Original model dominant poles Adopting 3rd-order model 3.3 ± 48.34i ± 53.64i.53 Eigenvalues.4 ± 39.i ± 4.98i ± 38.68i ± 4.94i.49 Adopting st-order model Table 9 further analyzes the transient stability of System at Initial Values 3 and 4 before and after the order reduction. Under the effect of Initial Value 3, the transient stability indexes of the original and order reduction systems are consistent. Under the effect of Initial Value 4, the transient stability indexes of the original and third-order systems are negative and those of the first-order system are positive. For the first-order system, under the premise that the original system is unstable, the transient stability consistency cannot be guaranteed. Thus, the transient stability of the system using the third-order reduction model is better than that of the system using the first-order reduction model. The calculation time comparison of different cases in Table 9 shows that the time saving is still impressive for the reduction models of IIDG. The finding implies that the larger the test system is, the more significant the improved computational efficiency will be. Especially for the st model, the calculation time is nearly unchanged comparing with the standalone case. However, comprehensively considering transient stability and calculation time, the 3rd order model has a better performance than adopting st order model.

19 Energies 8,, 54 9 of 5 Order Reduction Form Table 9. Transient stability analysis of test system. Transient Stability Analysis Result under Initial Value 3 Original 3rd Order st Order Transient Stability Analysis Result under Initial Value 4 Original 3rd Order st Order Transient Stability Index I Q Stability Consistency / / Energies 8,, 54 of 7 Calculation Time 4 s 48 s s 8 s 5 s 5 s Figure 7 further presents the projections of the stability region boundary of System on the δ δ Figure phase 7plane further before presents and the after projections order reduction, of the stability in which region A and boundary B are the of System projections on the of Initial δ δ phase Values plane 3 and before 4, respectively. and after order The reduction, stability region in which boundary A and Bcurve are the shows projections that the ofsystem Initial Values stability 3 and region 4, respectively. is closer to that The of stability the original region system boundary compared curve with shows that of that the the first-order system stability model after region IIDG is closer adopts tothe that third-order of the original model system for order compared reduction. with As that Initial ofvalue the first-order 3 Point A model is within afterthree IIDGstability adopts the region third-order boundaries, model the for transient order reduction. stability consistency As Initial Value of this 3 Point value A before is within and after threeorder stability reduction region boundaries, must be guaranteed. the transient If Initial stability Value consistency 4 Point B of this outside value before the stability and after region order boundaries reductionof must the be original guaranteed. and third-order If Initial Value systems 4 Point and located B is outside in that the of stability the first-order region boundaries system, then ofthe thetransient original and stability third-order consistency systems of IIDG andbefore located and inafter thatorder of thereduction first-order can system, be guaranteed then theby transient taking the stability thirdorder model. ofhowever, IIDG before transient and after stability order consistency reduction cannot be be guaranteed byif taking the first-order the third-order model consistency model. is used. However, This result transient is consistent stability with consistency the transient cannot stability be guaranteed index analysis if theresults first-order in Table model 8. is used. This result is consistent with the transient stability index analysis results in Table 8. δ/pu A B δ/pu..3.6 Figure 7. Projective figures of stability region boundary of test system in δ δ Figure 7. Projective figures of stability region boundary of test system in δ phase plane. δ phase plane. Figure 8a,b are the time-domain simulation results of the microgrid system under the small Figure 8a,b are the time-domain simulation results of the microgrid system under the small disturbance happened in load, where u, U, and Δω are the instantaneous voltage, RMS voltage, and disturbance happened in load, where u, U, and ω are the instantaneous voltage, RMS voltage, and frequency deviation at PCC respectively. The small disturbance form is the load demand steps from frequency deviation at PCC respectively. The small disturbance form is the load demand steps from.. to.. When operating at SEP, the stability of the system remains unchanged with IIDG adopting to.. When operating at SEP, the stability of the system remains unchanged with IIDG adopting two two different order reduction models st and 3rd. However, at UEP in Figure 6b, the stability different order reduction models st and 3rd. However, at UEP in Figure 6b, the stability consistency consistency of the system cannot be guaranteed under the condition of original system instability. of the system cannot be guaranteed under the condition of original system instability. Table show Table show the calculation results of three consistency indexes for the microgrid PCC voltage the calculation results of three consistency indexes for the microgrid PCC voltage dynamic response in dynamic response in Figure 8a. For operating at stable point, high values demonstrate the good match Figure 8a. For operating at stable point, high values demonstrate the good match between the order between the order reduction system and original system. reduction system and original system. 3. Table. Consistency indexes of dynamic response for reduction models in microgrid system. Microgrid PCC Voltage at Stable Point u/pu Frequency.5 Consistency Index 3rd Order st Order Original 3 rd -order st -order Damping Consistency Index 3rd Order st Order Amplitude Consistency Index 3rd Order st Order 98.5% 94.87% 97.94% 96.% 96.38% 9.4%

20 Energies Energies 8, 8,,, Energies 8,, of 54 8 of 8 of 5 u/pu Original Original 3 rd -order 3 rd -order st -order st -order Δω/pu Original Original 3 rd -order 3 rd -order st -order st -order a a U/pu Original.8 Original.8 3 rd -order 3 rd -order st -order st -order u/pu Original Original 3 rd -order 3 rd -order st -order st -order Δω/pu Original Original 3 rd -order 3 rd -order st -order st -order b b U/pu Original Original.5 3 rd -order.5 3 rd -order st -order st -order Figure 8. Simulation results of test system : a at the stable point; b at the unstable point. Figure Figure Simulation results of of test system : a at the stable point; b b at atthe the unstable point. point. Table. Consistency indexes of dynamic response for reduction models in microgrid system. Table. Consistency indexes of dynamic response for reduction models in microgrid system. 5. Conclusions Frequency Consistency Damping Consistency Amplitude Consistency Frequency Consistency Damping Consistency Amplitude Consistency This study establishes the full detailed Index model for IIDG based Index on the droop control Index in microgrid, Microgrid PCC Voltage Index Index Index extracts Microgrid PCC Voltage at the Stable multi-time Point scale3rd 3rd singular Order Order perturbation st Order st Order parameters 3rd Order st Order 3rd Order 3rd Order of st Order st the Order model 3rd according Order st to Order singular Stable Point perturbation theory, implements the order reduction processing of neglecting fast dynamics, 98.5% 94.87% 97.94% 96.% 96.38% 9.4% and proposes the simplified first 98.5% and third-order 94.87% reduction 97.94% models. 96.% Then, this 96.38% study verifies 9.4% the static and transient stability consistency of the system before and after order reduction by theoretical derivation and introduction of the quadratic approximation method of the stability region boundary.

21 Energies 8,, 54 of 5 The results show that the static and transient stability consistency of the IIDG system before and after order reduction can be guaranteed when the original system operating at stable state. Through the eigenvalue comparisons, the transient stability index comparisons and time-domain waveform results of the stand-alone system and the microgrid system, the effectiveness of the time-scale division and order reduction processing of the IIDG full model are finally proved. Acknowledgments: This work was supported by the Chongqing Research Program of Basic Research and Frontier Technology cstc5jcyjbx33, National Natural Science Foundation of China and Fundamental Research Funds for the Central Universities No. 66CDJXY5. Author Contributions: Xiaoxiao Meng and Qianggang Wang contributed to paper writing and the whole revision process. Niancheng Zhou and Xiaoxiao Meng built the simulation model and analyzed the data. Shuyan Xiao and Yuan Chi helped organize the article. All the authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Appendix A.. IIDG Parameters P n = kva, U n = 8 V, f n = 5 Hz, L =.5 mh, R =.5 Ω, C = 45 µf, L =.53 mh, R =.5 Ω, m = 4 rad/s/w, n = 3 V/Var, ω c = 3 rad/s, ω n = πrad/s, K p =, K i = 5,, K p =.48, K i = 4, H =.65, P = 5 kw, Q = 5 kvar. Appendix A.. IIDG Detailed F = A CV = A IIDG = A PQ A CV A PQ = ; B = B CV m ω c ω c ; Fx = K i K p ω n C H K i ω n C K p H K p K i K L i K p K i L F F F 3 K pω nc L K p H K pk p L L K pr L ω ω n K pk p L L K i L ω n ω K pr K p ω nc L L C ω C K p H L C ω C L R L ω L ω R L... T B IIDG = L... L 3 K p K p U nq cos δ ω n 3 ω c ud i Ld u q i Lq ud i Lq u q i Ld 3 ω c ; F = U nq cos δ U nq sin δ K p U nq cos δ K p U nq sin δ ; F 3 = L K p K p L U nq sin δ A A A3 A4. A5

22 3 ω sin p c udilq u K U nq qi δ Ld. Energies 8,, 54 of 5 Appendix B Appendix Figure BA and Table A show that the eigenvalues of the IIDG complete model are distributed in multiple frequency bands, with significant multi-time scale characteristics. The IIDG multi-time Figure A and Table A show that the eigenvalues of the IIDG complete model are distributed scale model can be obtained by using strict model simplification and extracting perturbation factors. in multiple frequency bands, with significant multi-time scale characteristics. The IIDG multi-time According to the principle of the multi-time scale division, if the system remains stable and has a scale model can be obtained by using strict model simplification and extracting perturbation factors. characteristic spectrum, then the eigenvalues of system state matrix As can be arranged from small to According to the principle of the multi-time scale division, if the system remains stable and has a large based on their absolute values, which can be shown as follows: characteristic spectrum, then the eigenvalues of system state matrix A s can be arranged from small to large based on their < absolute 3 < 9.68 values, < 3 < which.9 4 can < be.9 shown 4 as< follows:.3 4 <.3 4 < 3 < 9.68 < 3 <.9 4 <.9 4 <.3 4 <.3 4 Table A. Eigenvalues of IIDG detailed model. Eigenvalues Values Table A. Eigenvalues of IIDG detailed model. λ,.77 4 ±.33 4i λ3,4 Eigenvalues.689 Values 4 ±.954 4i λ5,6 λ, ±.33 4 ±.8 4i 4i λ 3, ±.954 4i λ7,8 λ 5, ±.8 4 ±.7 4i 4i λ9, λ 7, ±.7 4 ±.3 4i 4i λ 9,. 4 ±.3 4i λ, λ, 6 ± 5i 6 ± 5i λ3 λ Figure A. Eigenvalue diagramof of IIDG detailed model, where the red circle indicates the dominant low-frequency eigenvalues. Among the preceding values, the separation ratios can be presented as μ = 9.68/3 =.99 << Among the preceding values, the separation ratios can be presented as µ = 9.68/3 =.99, μ = 3/9.68 =. <<, which can be viewed as the marks of the multi-time scale division. <<, µ = 3/9.68 =. <<, which can be viewed as the marks of the multi-time scale division. Therefore, the model of IIDG is standard three-time scale model. Therefore, the model of IIDG is a standard three-time scale model. State Matrix of the IIDG Complete A S = m ω c.5ω ci Ld.5ω ci Lq.5ω cu d.5ω cu q ω c.5ω ci Lq.5ω ci Ld.5ω cu q.5ω cu d U n nq sin δ n cos δ U n nq cos δ n sin δ K p U n nq sin δ nk p cos δ K i K p ω nc H K p U n nq cos δ nk p sin δ K i ω nc K p H K pkp L U n nq cos δ mi Lq n K pkp L cos δ K pk i K pkp L U n nq cos δ mi Ld n K pkp L K i K pk i L K pωnc L K ph K pkp L L K pr L ω ω n K pkp L K L sin δ i L ω n ω K pr K pωnc K ph L L L mu q C ω C mu d C ω C mi Lq L R ω L mi Ld L ω R B S = L L L L A6 T. A7

23 Energies 8,, 54 3 of 5 Appendix C Ψ µin n Ψ µin Ψ T µin n Ψ µin J µin n J J J In the preceding equations: S xx = n n η i H f i xx i= µin n µi n S yy = J T JT µi n JT n j= J T η j H g j xx; S xy = n n η i H f i yy i= Ψ Ψ J JT ΨT = S xx A8 Ψ Ψ J JT Ψ = S xy A9 Ψ T JT Ψ Ψ J Ψ Ψ T J J T Ψ = S yy. n n η i H f i i= n j= η j H g j yy. xy n j= = S T xy A η j H g j xy; A Matrix T is defined as J T I n I n J T, where can be viewed as the tensor product mapping mark of semi-tensor product algorithm. If T is invertible, then Equation A can be derived as: Ψ = V c I n µt T V c S yy Ψ T J J T Ψ A where V c is the column vector accumulation mapping in the semi-tensor algorithm. Ψ is infinitesimal of the same order with, which means lim Ψ =. Assuming that T = J T I n n, T 3 = µi n n n I n J T, and T is reversible, then Equation A3 can be obtained from Equation A9. Ψ = V c I n n n T 3 T T V c S xy Ψ J JT Ψ. A3 Plugging Equation A9 into A3 to find that Ψ is also infinitesimal of the same order with, that is, lim Ψ =. Combining Equations A8 to A3, A4 can be derived as follows: Ψ µin n J µin n J n n T Ψ = i= η i H f i Considering the relationship with Equation 35, A4 can be further represented as: n j= η j H g j. A4 µin n Ψ J µin n J T Ψ n n η i H f i n J T JT η i= j= j Hg j. A5

24 Energies 8,, 54 4 of 5 Appendix D Parameters of Three-IIDG Microgrid Pilot Project System IIDG: P n = 38 kva, U n = 8 V, f n = 5 Hz, L =.5 mh, R =.5 Ω, C = 45 µf, L =.53 mh, R =.5 Ω, m = 4 rad/s/w, n = 3 V/Var, ω c = 3 rad/s, ω n = πrad/s, K p =, K i = 5,, K p =.48, K i = 4, H =.65, P = kw, Q = kvar. IIDG: P n = kva, U n = 8 V, f n = 5 Hz, L =.5 mh, R =.5 Ω, C = 45 µf, L =.53 mh, R =.5 Ω, m =.63 5 rad/s/w, n = 5 4 V/Var, ω c = 3 rad/s, ω n = πrad/s, K p =, K i = 5,3, K p =.48, K i = 4, H =.7, P = kw, Q = 5 kvar. IIDG3: P n3 = kva, U n3 = 8 V, f n3 = 5 Hz, L 3 =.5 mh, R 3 =.5 Ω, C 3 = 45 µf, L 3 =.53 mh, R 3 =.5 Ω, m 3 =.63 5 rad/s/w, n 3 = 5 4 V/Var, ω c3 = 3 rad/s, ω n3 = πrad/s, K p3 =, K i3 = 5,3, K p3 =.48, K i3 = 4, H 3 =.75, P 3 = kw, Q 3 = 5 kvar. References. Pogaku, N.; Milan, P.; Timothy, C.G. ing, analysis and testing of autonomous operation of an inverter-based microgrid. IEEE Trans. Power Electron. 7,, CrossRef. Guerrero, J.M.; Vasquez, J.C.; Matas, J.; De Vicuña, L.G.; Castilla, M. Hierarchical control of droop-controlled AC and DC microgrids-a general approach toward standardization. IEEE Trans. Ind. Electron., 58, CrossRef 3. Hu, B.; Wang, H.; Yao, S. Optimal economic operation of isolated community micro-grid incorporating temperature controlling devices. Prot. Control Mod. Power Syst. 7,, 6. CrossRef 4. Wen, B.; Boroyevich, D.; Burgos, R.; Mattavelli, P.; Shen, Z. Analysis of D-Q small-signal impedance of grid-tied inverters. IEEE Trans. Power Electron. 6, 3, CrossRef 5. Pan, D.; Ruan, X.; Wang, X.; Yu, H.; Xing, Z. Analysis and design of current control schemes for LCL-type grid-connected inverter based on a generalmathematical model. IEEE Trans. Power Electron. 7, 3, CrossRef 6. Qi, J.; Wang, J.; Liu, H.; Dimitrovski, A.D. Nonlinear model reduction in power systems by balancing of empirical controllability and observability covariances. IEEE Trans. Power Syst. 7, 3, 4 6. CrossRef 7. Vasquez, J.C.; Guerrero, J.M.; Savaghebi, M.; Eloy-Garcia, J.; Teodorescu, R. ing, analysis, and design of stationary-reference-frame droop-controlled parallel three-phase voltage source inverters. IEEE Trans. Ind. Electron. 3, 6, 7 8. CrossRef 8. Gutierrez, B.; Kwak, S.S. Modular multilevel converters MMCs controlled by model predictive control with reduced calculation burden. IEEE Trans. Power Electron. 8, in press. CrossRef 9. Kahrobaeian, A.; Mohamed, Y.A.R.I. Analysis and mitigation of low-frequency instabilities in autonomous medium-voltage converter-based microgrids with dynamic loads. IEEE Trans. Ind. Electron. 4, 6, CrossRef. Cao, W.; Ma, Y.; Liu, Y.; Wang, F.; Tolbert, L.M. D-Q Impedance Based Stability Analysis and Parameter Design of Three-Phase Inverter-Based AC Power Systems. IEEE Trans. Ind. Electron. 7, 64, CrossRef. Arani, M.F.M.; El-Saadany, E.F. Implementing virtual inertia in DFIG-based wind power generation. IEEE Trans. Power Syst. 3, 8, CrossRef. Tang, X.; Hu, X.; Li, N.; Deng, W.; Zhang, G. A novel frequency and voltage control method for islanded microgrid based on multienergy storages. IEEE Trans. Smart Grid 6, 7, CrossRef 3. Levron, Y.; Belikov, J. Reduction of power system dynamic models using sparse representations. IEEE Trans. Power Syst. 7, 3, CrossRef 4. Li, X.Y.; Chen, X.H.; Tang, G.Q. Output feedback control of doubly-fed induction generator based on multi-time scale model. In Proceedings of the 8 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Nanjing, China, 6 9 April 8; pp

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