Impact of Time Delays on Stability of Inertia-less Power Systems

Transcription

1 P L Power Systems Laboratory Jialun Zhan Impact of Time Delays on Stability of Inertia-less Power Systems Semester Thesis PSL 186 EEH Power Systems Laboratory Swiss Federal Institute of Technoloy (ETH) Zurich Expert: Prof. Dr. Gabriela Hu Supervisors: M.Sc Uros Markovic Zürich, Auust 15, 218

2 Abstract Time delays appear in a variety of physical systems and their effects on stability have been investiated in areas like sinal processin and circuit desin. However, there has been little study on the effects of time delays on power system stability, which result in a system of Delay Differential Alebraic Equations (DDAE). As the number of converter-interfaced (renewable) eneration increases, the impact of time delays comin from either local power electronics measurement or wide-area unit communication can play an important role in system stability. This project aims to analyze the stability properties of inertia-less power systems with inclusion of various sinal delays. The conventional mathematical representation of such systems, based on DAEs, will be extended to DDAEs, with the actual time delays adequately represented in the model. Subsequently, the stability marins will be assessed throuh Chebyshev discretization scheme and Padé approximants for both sinle and multiple (constant) time delays, as well as different system confiurations. Furthermore, the participation factor and parameters sensitivity analysis have been conducted to identify the critical states and controller parameter. Meanwhile, both two-dimensional and three-dimensional bifurcation stability mappin has been implemented to visualize the impact of controller parameter variety towards critical time delay. i

3 Contents List of Acronyms List of Symbols iv v 1 Introduction 1 2 Theoretical Analysis DDAE formulation Proposed approximation techniques Chebyshev discretization scheme Padé approximants System Modellin VSC control scheme DDAE model Electrical system modelin Phase-Locked Loop Active Power Control Other DDAEs Eienvalue Spectrum Analysis Very small time-delay: DDAE vs. DAE Same time delay for all measurement sinals Eienvalue Analysis for Sinle Time Delay Sinle time delay: e τ Sinle time delay: i τ Sinle time delay: i τ s Stability Mappin Participation factors Bifurcation analysis D bifurcation analysis D bifurcation analysis ii

4 CONTENTS iii 6 Parameter Sensitivity Analysis 37 7 Conclusion & Outlook 39 Biblioraphy 4

5 List of Acronyms DAE DDE DDAE PDE VSC DG PE SMIB SM TSO PLL APC VIE APDC LPF RPC SRF PF PS Differential Alebraic Equation Delayed Differential Equation Delay Differential Alebraic Equation Partial Differential Equations Voltae Source Converter Distributed Generators Power Electronic Sinle-Machine Infinite-Bus Synchronous Machine Transmission System Operator Phase-Locked Loop Active Power Controller Virtual Inertia Emulation Active Power Droop Control Low Pass Filter Reactive Power Controller Synchronously-rotatin Reference Frame Participation Factor Parameter Sensitivity iv

6 List of Symbols Unit Commitment Symbols Symbol f x x τ y y τ z τ λ A I n M D N e τ i τ i sτ p ki lk i rk i a n,k Description Differential equation Alebraic equation State variable Delayed state variable Alebraic variable Delayed alebraic variable Discrete event variable Constant time delay Eienvalue Characteristic matrix Identity matrix Discretization matrix Chebyshev differential matrix Delayed voltae measurement from primary side of transformer Delayed current measurement from primary side of transformer Delayed current measurement from converter Participation factor of k th state in i th mode Element of k th element in i th left eienvector matrix Element of k th element in i th riht eienvector matrix Sensitivity of n th mode to k th parameter v

7 List of Fiures 2.1 Matrix M structure for a system with a sinle time delay [1] Orinal VSC control scheme. [2] Same time on three measurement sinals Various time on three measurement sinals Eienvalues of DAE and DDAE usin Chebyshev and Padé Approximation Full eienvalue spectrum computed usin Padé approximation for time delay Full eienvalue spectrum computed usin Chebyshev discretization for time delay Critical eienvalues computed usin Padé approximation for time delay Critical eienvalues computed usin Chebyshev discretization for time delay e τ introduced in VSC control structure Sinle time delay introduced by e τ ranin from 42 to 465µs Sinle time delay introduced by e τ ranin from 42 to 465µs in a reduced rane i τ introduced in VSC control structure Sinle time delay introduced by i τ ranin from 14 to 17µs Sinle time delay introduced by i τ ranin from 14 to 17µs in a reduced rane i τ s introduced in VSC control structure Sinle time delay introduced by i τ s ranin from 2 to 215µs Sinle time delay introduced by i τ s ranin from 2 to 215µs in a reduced rane τ e, τ i affect critical time-delay ˆτ is APC in VIE mode applyin to rid-frequency followin case APC in Droop-based mode applyin to rid-frequency formin case Participation factors for ˆλ = ± j Participation factors for ˆλ = ± j Stability Map on ˆτ K v p plane Stability Map on ˆτ K v i plane vi

8 LIST OF FIGURES vii 5.5 Stability Map on ˆτ Kf i plane Stability Map on ˆτ Kp i plane Stability Map on ˆτ Ki i plane Stability Map on ˆτ Kf v plane Stability Map on ˆτ H plane Stability Surface in the ˆτ Kp v Ki v, System is stable under shaded area Stability Surface in the ˆτ Kp i Ki i, System is stable under shaded area Stability Map on ˆτ K d H plane Stability Map on ˆτ D p D q plane Stability Map on ˆτ Kp pll K pll i plane Parameter Sensitivity for critical eienvalue ˆλ = ± j Parameter Sensitivity for critical eienvalue ˆλ = ± j

9 List of Tables 4.1 Critical time delay for each case Critical time delay chanes with various K v p (1µs error) Critical time delay chanes with various K v i Critical time delay chanes with various K i f Critical time delay chanes with various K i p (1µs error) Critical time delay chanes with various K i i Critical time delay chanes with various K v f viii

10 Chapter 1 Introduction With the development of intermittent sources, an increasin amount of Distributed Generators has been applied to the current rid. The share of Power Electronic devices has rown dramatically in recent years. This trend has transformed the power enerators from traditional eneration sources such as synchronous machines towards Power Electronic (PE) interfaced and localized enerators. Althouh more CO 2 emissions could be avoided from the benefit of this trend as shown in [3], the power system, however, is now experiencin a low-inertia condition, resultin in detrimental consequences with respect to system stability marin [4]. As a result, a new concept called virtual inertia emulation has been evaluated in the literature [5 8], with the oal of reproducin the rotational inertia effect usin PEs. A proper Voltae Source Converter (VSC) scheme has been proposed in all four papers. Normally to analyze the stability of the power system, a widely implemented tool is smallsinal analysis, which focuses on the stability at the equilibrium operation point after a disturbance. In [9] a low-inertia system has been analyzed throuh a small-sinal method, whereas time delay sinals occurrin at power electronics are omitted. In [1] time delay sinals are considered with further analysis throuh Padé Approximants method. However, the conclusions drawn from the adopted simplified Padé method may result in defective data. Research conducted in [11] has applied the Chebyshev Discretization scheme for studyin a simplified VSC scheme without the consideration of Inner Loop Controller. To encompass time delay sinals into the existin mathematical modelin of the power system, a transition from Differential Alebraic Equation (DAE) to Delay Differential Alebraic Equations (DDAE) has to be formulated. A complete methodoloy and its derivation have been proposed in [12] which offers an insiht into the DDAE formulation from existin DAE model. Additionally, [1] further addresses the transcendental nature of the characteristic matrix with four approximation approaches includin both Chebyshev Discretization and Padé methods and compares the respective simulation results from a 1479 buses transmission system model. Due to the fact that both synchronous 1

11 CHAPTER 1. INTRODUCTION 2 machines and wind enerators are analyzed in this research, it remains unclear what impact of VSC scheme renewable sources will impose on the stability of the system independently. In this project, a complete DDAE model based on existin VSC control model in DAE form with the introduction of multiple measurement time delay sinals is derived. Meanwhile, the critical eienvalues (modes) are investiated throuh the research of eienvalue spectrum. In addition, a participation factor analysis is applied to identify the critical states that contribute to the critical eienvalues. Furthermore, a parameter sensitivity analysis is assessed for critical modes and bifurcation mappins reardin various controller parameters are displayed.

12 Chapter 2 Theoretical Analysis 2.1 DDAE formulation As mentioned in the introduction, a small sinal analysis is applied to analyze the dynamic stability of a system at a stationary workin point. Normally, DAEs are formulated as a set of equations to describe the intrinsic stability behaviour. In this section, the derivation of the DAE model is presented, followed by the next section illustratin the derived DDAE model. In [11] we see a very clear process that describes the derivation of a DDAE model from a DAE model. A concise derivation is also presented here for the sake of further clarification: ẋ = f(x, y, z) (2.1) = (x, y, z) Equation (2.1) describes the composition of the Differential Alebraic Equations. In this equation set, x is the vector of state variables, while y and z are the vector of alebraic and discrete event variables respectively. In this project discrete event variables are beyond the scope of research and thus omitted in the followin equations. f and represents differential and alebraic equations respectively. Yet the DAE model could not represent the effect brouht by the time delay variables. Hence, a set of DDAEs has to be derived from the existin DAE model for further capturin the characteristics of the system. When we consider constant time delay sinals τ bein introduced into the DAE form, we consequently obtain the followin equations as: x τ = x(t τ) (2.2) y τ = y(t τ) ẋ = f(x, y, x τ, y τ, z) = (x, y, x τ, y τ, z) (2.3) 3

13 CHAPTER 2. THEORETICAL ANALYSIS 4 From the eneral form of the DDAE model shown in (2.3), it is noticeable that both the state and alebraic variables are introduced with time delays. However, only alebraic variables introduced with time delayed sinals are considered in differential equations. A reduced DDAE model (2.4) named index-1 Hessenber form could be derived. As illustrated in [13], this form is sufficient to indicate the small-sinal stability behaviour of the system. ẋ = f(x, y, x τ, y τ, z) = (x, y, x τ, z) (2.4) Now assumin that the system is operatin at a stationary point, the above reduced DDAE model could be linearlized at this equilibrium solution which yields: ẋ = f x x + f xd x d + f y y + f yd y d = x x + xd x d + y y (2.5) If y is non-sinular, the above equations could be rewritten in the followin form: with, ν ẋ = A x(t) + A i x i (t τ i ) (2.6) i=1 A = f x f y y 1 x (2.7) A 1 = f xτ f y y 1 xτ f yτ y 1 x (2.8) This equation (2.6) is defined as linear delay differential equation (DDE). A is the state matrix of the standard DAE matrix which is not varied with time-delay variables, whereas A 1 is the matrix containin the time-delay variables. When a sample solution x = e st v proposed in [12] is substituted into the DDE, a final characteristic matrix could be obtained as: (λ) = λi n A ν A i e λτ i (2.9) Equation (2.9) is the characteristic matrix which captures the stability characteristics from its spectrum of roots [14]. When observin any positive root, the system is unstable and if all roots lie at the left-side of imainary axis, the system is stable. However, if further observation is made, equation (2.9) appears to be transcendental, meanin that there is an infinite number of roots. Nonetheless, only rihtmost eienvalues are mostly considered for identifyin the stability of the system. With three essential properties of the characteristic equation illustrated in [12], the riht-side plane roots are supposed to i=1

14 CHAPTER 2. THEORETICAL ANALYSIS 5 be finite. Hence, numerical approximation methods have to be implemented in order to approximate a subset of the full spectrum. In the next section, two techniques will be proposed with respect to the approximation of the results, providin a fundamental theoretical foundation for this project. 2.2 Proposed approximation techniques In this section, two approximation methods are introduced. The first one is Chebyshev discretization scheme, which can be applied to both small and lare time delays. The second one is Padé approximation method, normally used for small time delays [1]. Since the time delays related to local measurement in power electronics vary from microseconds (µs) to milliseconds (ms), both techniques could be implemented. A mathematical formulation of both techniques is conducted in the followin part Chebyshev discretization scheme To address the transcendental nature of the characteristic matrix (2.9), [1] proposes an approach to transform the oriinal computation of DDE eienvalues into calculatin roots of infinite dimension Partial Differential Equations (PDE) system. This process keeps the information needed for the stability analysis. The benefit of this transformation is that the PDE system solutions are tractable with the employment of finite element discretization approach. When only state time-delay variables are considered, an additionally simplified form of the Hessenber DDAE model could be created: ẋ = f(x, y, x τ ) = (x, y, x τ ) (2.1) Repeatin the previous procedure, the final form of the characteristic matrix with only a sinle time-delay becomes: where, (λ) = λi n A A 1 e λτ (2.11) A = f x f y y 1 x (2.12) A 1 = f xτ f y y 1 xτ (2.13)

15 CHAPTER 2. THEORETICAL ANALYSIS 6 Applyin a finite element method, discretization matrix M can be derived: M = [ Ĉ I n A 1... A ] (2.14) Here represents the Kronecker product and Ĉ indicates the N 1 rows of matrix C: C = 2D N /τ (2.15) D N is the Chebyshev differential matrix of dimensions (N + 1) (N + 1). This matrix is used to interpolate the discretization points in the rane [ 1, 1], defined throuh Chebyshev nodes as follows: x k = cos ( kπ ), k =,..., N. (2.16) N Furthermore, these components composes the elements in the differentiation matrix numerically defined as: c i ( 1) i+j i j c j (x i x j ), D (i,j) = 1 x i 2 1 x 2 i, i = j 1, N 1 2N , i = j = (2.17) 2N , i = j = N As a discretization of the PDE system, the continuum variables correlatin to the time delay are discretized alon the rid of N points. It is notable that the last n rows illustrate the boundary condition which defines the matrices bein introduced into the system due to multiple time-delay sinals. Since in this project only a sinle time delay variable is considered, only A 1 and state matrix A are placed in the boundary condition row. In addition, N stands for the size of the Chebyshev differentiation matrix and therefore controllin the precision and workload of the computation. As shown in [13] the node size is not necessarily to be hih in order to achieve an accurate result. The schematic of matrix M with the introduction of a sinle time delay case is illustrated in Fiure 2.1. The ray blocks are the Chebyshev discretization matrices which bear the property of sparsity. The dark blocks are the state and delayed matrix respectively, whereas the white blocks are null matrices.

16 CHAPTER 2. THEORETICAL ANALYSIS 7 Fiure 2.1: Matrix M structure for a system with a sinle time delay [1] Padé approximants With respect to the Padé approximation method, firstly a common time-shift property of a sinal in Laplace domain shall be presented: f(t τ)u(t τ) L e τs F (s) (2.18) where s is the Laplace domain variable, u(t τ) is the step response with time delay τ and F (s) is the Laplace transformation of f(t). One can observe that an exponential term has been introduced in Laplace domain when sinals are delayed. Moreover, if this exponential term is expanded throuh Taylor s expansion to obtain a rational polynomial fraction, an approximated time-domain function could be derived from the inverse Laplace transform of the polynomial function. As we can see from the equations shown below, where the final approximation function φ(t) is derived: e τs = 1 τs + (τs)2 (τs) ! 3! b + b 1 τs b q (τs) q (2.19) a + a 1 τs a p (τs) p e τs F (s) P (s)f (s) L 1 φ(t) (2.2) It is worthy to mention that the coefficients a 1 to a p and b 1 to b q are calculated throuh the division of the riht-side polynomials and equatin them to the first p+q counterparts

17 CHAPTER 2. THEORETICAL ANALYSIS 8 in Taylor s expansions function. Here s is the continuous Laplace domain variable, whereas in the Chebyshev discretization scheme, the variable λ is discrete. Normally p and q are chosen to be 6 due to the limitation imposed from the binary floatin point precision as explained in [1]. From this condition, the denominators and numerators coefficients could be computed throuh the followin criterion: a o = 1 a i = a i 1 p i + 1 i (2p i + 1) (2.21) b i = ( 1) i a i It is noticeable that Padé approximants do not deal with the characteristic matrix derived before, but rather chane the oriinal DDAE problem to a set of DAEs where time-delay variables are approximated throuh multiple state variables, as shown in equations (2.22) to (2.24) with the expansion of an example variable u. u d = x 1 + b 1 τ x b p 1 τ p 1 x p + b p τ p x p (2.22) x i = x i+1, i = 1, 2,..., p 1 (2.23) a p τ p x p = u (a x 1 + a 1 τ x a p 1 τ p 1 x p ) (2.24) In (2.23) we can see that each of the states bein introduced into the approximation is the differential of the previous states, and variable u is introduced with p states from x 1 to x p, with the last component x p subsequently computed throuh the equation (2.24). Apparently there are no constraints on the number of time delays that can be introduced into the system, nor a structural diverence between multiple and sinle time delay cases. Overall, both methods are approaches to approximate the full spectrum of the DDAE model. Later, two methods would be implemented into the DDAE model derived from the existin Voltae Source Converter (VSC) control scheme. Furthermore, both eienvalue spectrums would be compared with the same conditions applied.

18 Chapter 3 System Modellin In this chapter a full VSC controller architecture is indicated with the inclusion of various measurement time-delay sinals. Meanwhile, a proposed DDAE model characterizin the small-sinal stability of the existin VSC control scheme at different state is displayed. 3.1 VSC control scheme The oriinal VSC control scheme is presented in Fiure 3.1. The output of the converter is connected to either a Sinle Machine Infinite Bus (SMIB) rid or simply an active power load throuh a Low-Pass Filter (LPF) and a transformer. Inside the controller there are two functionin fields which are differentiated mainly by the inner or outer control loop. Within the inner control loop there are SRF Voltae Controller and SRF R t L t Grid ω n Power Calculation Unit p q e dq i dq dq abc i abc e abc C f v R l q v Reactive Power Controller ˆv Phase Locked Loop ω = [ω n, ω pll] [ ] e dq, i dq θ apc L f R f i dq s v m p Active Power Controller Outer control loop ω apc Virtual Impedance Block v SRF ī SRF s v m m Voltae Current PWM Controller Controller v dc Inner control loop C dc v dc Fiure 3.1: Orinal VSC control scheme. [2] 9

19 CHAPTER 3. SYSTEM MODELLING 1 R t L t Grid ω n Power Calculation Unit p q e dq i dq dq abc i τ e τ C f v R l q v Reactive Power Controller ˆv Phase Locked Loop ω = [ω n, ω pll] [ ] e dq, i dq θ apc i τ s L f R f v m p Active Power Controller Outer control loop ω apc Virtual Impedance Block v SRF ī SRF s v m m Voltae Current PWM Controller Controller v dc Inner control loop C dc v dc Fiure 3.2: Same time on three measurement sinals. R t L t Grid ω n Power Calculation Unit p q e dq i dq dq abc i τ2 e τ1 C f v R l q v Reactive Power Controller ˆv Phase Locked Loop ω = [ω n, ω pll] [ ] e dq, i dq θ apc i τ3 s L f R f v m p Active Power Controller Outer control loop ω apc Virtual Impedance Block v SRF ī SRF s v m m Voltae Current PWM Controller Controller v dc Inner control loop C dc v dc Fiure 3.3: Various time on three measurement sinals. Current Controller, which are mainly used to prevent voltae and current saturation [8]. The outer control loop comprises of Active Power Controller (APC) and Reactive Power Controller (RPC) for trackin both frequency and voltae of the electrical system conditions. Within the APC there are two modes: Virtual Inertia Emulation controller and Droop-based controller. The Phase Locked Loop (PLL), which is used to synchronize a rid-followin converter to the rid. Further discussion reardin these modes will be illustrated in the followin chapter. In this project, firstly three same time-delay measurement sinals are introduced into the VSC system. As shown in Fiure 3.2, voltae and current from the primary side of the rid and the output current from the converter (e τ, i τ, i τ s) are denoted with reen colour. Subsequently a multiple time delay case is introduced with e τ, i τ and i τ s denoted in different colours, as shown in Fiure 3.3. Further analysis will be focused on the impact of critical time delay from various measurements sinal and identifyin the most

20 CHAPTER 3. SYSTEM MODELLING 11 sensitive states.

21 CHAPTER 3. SYSTEM MODELLING DDAE model The DDAE model consists of Differential equations f and Alebraic equations. In [2] a full DAE model has been established. Based on the existin DAE model, a DDAE model will be depicted briefly in the followin equations, which will also be cateorized into the specific sections dependin on their functionality. Further details reardin the DAE model can be referred in [2] Electrical system modelin As mentioned in the last chapter that only measurement sinals are introduced with time delays. Hence, the electrical states of the rid are computed without time-delay variables in three differential equations: i s = ω b l f (v m e ) i = ( rf ω b l + l t (e v ) ) ω b + jω b ω i s (3.1) l f ( ) r + r t ω b + jω b ω i (3.2) l + l t ė = ω b c f (i s i ) jω ω b e (3.3) where i s is the switchin current at the output of the converter, i and e are the current and voltae at the primary side of the transformer connectin to the rid respectively, v m indicates the modulation voltae and v represents the rid voltae. Meanwhile, the active and reactive power output of the converter are fed into the VSC controller. Consequently, time-delay variables, which are denoted in different colours, are introduced to the power calculation block as shown in the followin alebraic equations: p = e d τ i d τ + e q τ i q τ (3.4) q = e q τ i d τ e d τ i q τ (3.5) Phase-Locked Loop This synchronization unit detects and estimates the rid frequency and further maintains synchronization of VSC controller to the rid in the rid-followin mode. Mathematical expression of differential and alebraic equations in PLL are presented here: ε = ê q (3.6) ϑ pll = ν pll ω b (3.7) ν pll = ω n ω + Kp pll ê q + K pll i ε (3.8) where ϑ pll and ν pll are estimated frequency and anle, while ω n and ε are nominal frequency and interator state.

22 CHAPTER 3. SYSTEM MODELLING 13 It is noticeable that in the rid-formin mode the frequency set-point sinal is passed throuh the PLL directly to the APC block. Hence the above equations are simplified as a sinle equation: ν pll = ω n (3.9) Active Power Control For Active Power Droop Control (APDC) two differential equations and one alebraic equation are illustrated as follows: ϑ apc = ν apc ω b (3.1) p = ω c (p p) (3.11) ν apc = ν pll + D p (p p) (3.12) where ϑ apc denotes the phase anle of APC, ν apc is the output frequency of APC after SRF alinment and p stands for filtered-power-measurement sinal. A droop ain D p is adopted to adjust the output frequency dependin on the discrepancy between the measured power and the power set-point sinal p In the case of Virtual Inertia Emulation (VIE) mode, equations (3.11) and (3.12) are replaced with followin equations: ω apc = 1 2H (p p) 1 2H K d(ω apc ω ) (3.13) ν apc = ν apc + ω (3.14) This mode emulates the synchronous machine swin equation expression, where electrical power and mechanical power are substituted by set-point power p and output active power as in equation (3.13) Other DDAEs For the sake of simplicity and without loss of enerality, the remainin DDAEs are placed toether, since these equations are identical for different cases bein applied to the VSC controller. The differential equations for reactive power controller, SRF voltae and current controllers are listed in the followin equations respectively: q = ω c (q q) (3.15) ξ = v e τ (3.16) γ = īs i sτ (3.17)

23 CHAPTER 3. SYSTEM MODELLING 14 with q, ξ and γ representin LPF value of reactive power, interator state of SRF voltae and current controller, respectively. For alebraic equations derived from the other controller models: ˆv = v + D q (q q) (3.18) v d = ˆv r v i d τ + ω apc l v i q τ (3.19) v q = r v i q τ + ω apc l v i d τ (3.2) ī s = K v p ( v e τ ) + K v i ξ + jω apc c f e τ + K i f i τ (3.21) v m = K i p(īs i sτ ) + K i iγ + jω apc l f i sτ + K v f e τ (3.22) v m = v m (3.23) Now with different settins of DDAE model bein established, we can further compare the numerical results between Chebyshev discretization method and Padé Approximation technique.

24 Chapter 4 Eienvalue Spectrum Analysis A crucial criterion that the real part of rihtmost eienvalue equals to zero correspondin to the boundary condition between stable and unstable zones was established to jude the stability of the system based on the eienvalue spectrum. This offers an efficient method to distinuish between the stable and unstable system when time delays are stepped up discretely. With this conclusion, several findins of this project will be explained in the followin sections. 4.1 Very small time-delay: DDAE vs. DAE As we can see from Fiure 4.1, a very small time-delay of τ=1 µs has been applied to the DDAE system usin both Chebyshev discretization and Padé Approximants and is compared with the oriinal DAE model where time-delay sinals are not included. The three eienvalue spectrums overlap with each other and share almost the same value at each position of the roots. This result has confirmed that both approximation techniques are applicable and feasible to properly approximate the eienvalue spectrum. Moreover, the rihtmost eienvalue still lies left of the imainary axis, indicatin that the system is stable when no or low time delay is introduced. As can be seen further in the next section, when the time delay is stepped up discretely, the eienvalue spectrum experiences a shiftin trend until it passes stable boundary. 15

25 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 16 4, 2, τ = 1 µs Chebyshev Discretization τ = 1 µs Padé Approximants Oriinal DAE Model I(λ) 2, 4, 4, 3,5 3, 2,5 2, 1,5 1, 5 R(λ) Fiure 4.1: Eienvalues of DAE and DDAE usin Chebyshev and Padé Approximation 4.2 Same time delay for all measurement sinals In this section, a discrete step increase of measurement time delays are applied from 5 µs to 3 µs. Fiure 4.2 and Fiure 4.3 display full eienvalue spectrum achieved from Padé approximation and Chebyshev discretization respectively. However, we can hardly see any consistency between these two spectrums in terms of the position rane of the roots. This is due to the fact that the essence of these two methods is different. Chebyshev discretization technique transforms the oriinal characteristic matrix to a PDE system and then implemented the finite element method to discretize the matrix. Subsequently, the derived matrix is expanded and, hence, resultin in the inclusion of redundant roots alon with the oriinal solutions. While for Padé approximation only the retarded state variables have been implemented with Taylor s Expansion. Therefore, the size of the oriinal characteristic matrix is kept intact and unchaned. Consequently, the eienvalues computed throuh this method will produce the same amount of eienvalues as in the subset of the oriinal spectrum. Nevertheless, a shift of eienvalues to the riht half spectrum could be witnessed for both cases. Since the criteria for determinin the stability of the system lies at the rihtmost eienvalues. A further reduced rane of spectrum would be considered in the followin, as can been seen from Fiure 4.4 and 4.5.

26 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS τ = 5 µs τ = 1 µs τ = 15 µs τ = 2 µs τ = 25 µs τ = 3 µs I(λ) R(λ) 1 5 Fiure 4.2: delay Full eienvalue spectrum computed usin Padé approximation for time τ = 5 µs τ = 1 µs τ = 15 µs τ = 2 µs τ = 25 µs τ = 3 µs I(λ) R(λ) 1 5 Fiure 4.3: Full eienvalue spectrum computed usin Chebyshev discretization for time delay

27 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 18 5, τ = 2 µs τ = 25 µs τ = 3 µs I(λ) 5, 1,2 1, R(λ) Fiure 4.4: Critical eienvalues computed usin Padé approximation for time delay 5, τ = 2 µs τ = 25 µs τ = 3 µs I(λ) 5, 1,2 1, R(λ) Fiure 4.5: delay Critical eienvalues computed usin Chebyshev discretization for time The discrete step ranes from 2 to 3 µs for both techniques. Both methods offer

28 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 19 almost the same results for the reduced eienvalue spectrum reardin position and movement trend. The whole eienvalue spectrum is continuously shiftin to the riht until the first two eienvalues, which are conjuates of each other, first arrive at the imainary boundary, formin the critical stability condition of the whole system. In addition, the critical time delay computed from both techniques yields the same result, 233 µs. This critical time delay has also been verified throuh non-linear Electromanetic Transient simulations in Simulink on a detailed non-linear converter model. For the sake of simplicity, only Padé approximants results would be provided in the followin sections of the report. 4.3 Eienvalue Analysis for Sinle Time Delay From the last section, when same measurement time delays have been applied to the system, one can observe the critical time delay lies at 233 µs. Accordinly, a further analysis reardin each of the measurement sinal has to be conducted to identify which measurement sinal is the most critical one. Hence, three sinle time delay cases are demonstrated below. We will bein with the results obtained from e τ introduced case Sinle time delay: e τ As can be seen from Fiure 4.6, only voltae measurement has been introduced with time delays, whereas the other two measurement sinals do not involve any time delays. e τ is increased in discrete steps of 15 µs in a full eienvalue spectrum as shown in Fiure 4.7, followed by a reduced eienvalue spectrum in Fiure 4.8. R t L t Grid ω n Power Calculation Unit p q e dq i dq dq abc i abc e τ C f v R l q v Reactive Power Controller ˆv Phase Locked Loop ω = [ω n, ω pll] [ ] e dq, i dq θ apc L f R f i dq s v m p Active Power Controller Outer control loop ω apc Virtual Impedance Block v SRF ī SRF s v m m Voltae Current PWM Controller Controller v dc Inner control loop C dc v dc Fiure 4.6: e τ introduced in VSC control structure

29 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 2 6, 4, I(λ) 2, 2, τ = 4.2 ms τ = 4.35 ms τ = 4.5 ms τ = 4.65 ms 4, 6, 2,5 2, 1,5 1, 5 R(λ) Fiure 4.7: Sinle time delay introduced by e τ ranin from 42 to 465µs 6 4 I(λ) 2 2 τ = 4.2 ms τ = 4.35 ms τ = 4.5 ms τ = 4.65 ms R(λ) Fiure 4.8: Sinle time delay introduced by e τ ranin from 42 to 465µs in a reduced rane

30 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 21 As can be seen from the full eienvalue spectrum of the sinle e time delay case, the rihtmost eienvalue shifts towards riht-half plane when time delay has been increased up to 4.5 ms. The critical time delay for the VSC system is ms exactly. This time delay is an order of manitude larer than the critical time delay of 233 µs presented earlier. Hence, further research on the other measurement sinals is necessary Sinle time delay: i τ In this section only the measurement current of the rid is introduced with time delay, symbolizin with i τ in the VSC controller scheme as shown in Fiure 4.9. The current time delay i τ is increased in discrete steps of 1 µs. Same pattern could be observed for rid current measurement delay case as in the rid voltae case. The critical delay time for i τ is ms specifically, sinificantly hiher than the 233 µs achieved in the same measurement time-delay case. Hence, both e and i should not be critical states responsible for the stability marin of the system. R t L t Grid ω n Power Calculation Unit p q e dq i dq dq abc i τ e abc C f v R l q v Reactive Power Controller ˆv Phase Locked Loop ω = [ω n, ω pll] [ ] e dq, i dq θ apc L f R f i dq s v m p Active Power Controller Outer control loop ω apc Virtual Impedance Block v SRF ī SRF s v m m Voltae Current PWM Controller Controller v dc Inner control loop C dc v dc Fiure 4.9: i τ introduced in VSC control structure

31 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 22 8, 6, 4, τ = 1.4 ms τ = 1.5 ms τ = 1.6 ms τ = 1.7 ms 2, I(λ) 2, 4, 6, 8, R(λ) 1 4 Fiure 4.1: Sinle time delay introduced by i τ ranin from 14 to 17µs 2 1 τ = 1.4 ms τ = 1.5 ms τ = 1.6 ms τ = 1.7 ms I(λ) Fiure 4.11: Sinle time delay introduced by i τ ranin from 14 to 17µs in a reduced rane R(λ)

32 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 23 R t L t Grid ω n Power Calculation Unit p q e dq i dq dq abc i abc e abc C f v R l q v Reactive Power Controller ˆv Phase Locked Loop ω = [ω n, ω pll] [ ] e dq, i dq θ apc i τ s L f R f v m p Active Power Controller Outer control loop ω apc Virtual Impedance Block v SRF ī SRF s v m m Voltae Current PWM Controller Controller v dc Inner control loop C dc v dc Fiure 4.12: i τ s introduced in VSC control structure τ = 2 µs τ = 25 µs τ = 21 µs τ = 215 µs 2 I(λ) R(λ) Fiure 4.13: Sinle time delay introduced by i τ s ranin from 2 to 215µs Sinle time delay: i τ s In this case, time delay is introduced only to the converter output current measurement, as can be seen from Fiure The current time delay i τ s is increased in discrete steps of 5 µs.

33 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 24 5, τ = 2 µs τ = 25 µs τ = 21 µs τ = 215 µs I(λ) 5, 1,2 1, Fiure 4.14: Sinle time delay introduced by i τ s ranin from 2 to 215µs in a reduced rane R(λ) Table 4.1: Critical time delay for each case retarded variable e τ i τ i τ s critical delay time 4.4 ms 1.6 ms 215 µs As usual, in the reduced rane of eienvalue spectrum one can observe the shift phenomenon of the roots with the step increase of time delay. The critical time delay in this case becomes 211 µs. This is the lowest critical time delay observed so far. Three critical measurement delays have been summarized in table 4.1. It is clear that state i affects the stability marin of the system substantially comparin to the other two measurements. Besides, the sinle time delay of i τ s is unexpectedly less than the same measurement time delay case, where 233 µs is acquired. Henceforth, an analysis of the impact of critical time delay for i s with respect to the chanes of time delay sinals e τ and i τ was carried out. As can be seen from Fiure 4.15, there is a chane of critical switchin current time delay when both rid voltae and current measurement sinals are introduced with different time delays. From the perspective of τ i, the critical time delay for switchin current ˆτ is is not very sensitive towards the chane of τ i. However, the critical value of ˆτ is is affected by τ e dramatically. When all the measurement time delays are close to each other, the critical value for ˆτ is reaches a local maximum value. However, if the difference

34 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS ˆτ is [µs] 1 5 1, τ e [µs] 2 1,2 8 4 τ i [µs] 2 Fiure 4.15: τ e, τ i affect critical time-delay ˆτ is between τ e and τ is further increases, then ˆτ is will reduce to a lower value, resultin in a more vulnerable system. Hence, when the same measurement time delays are applied to all three sinals, the critical time delay 233 µs will be hiher than sinle time delay bein introduced to i τ only. Further observation could be made that both measurement sinals e τ, i τ are fed into all of the control units in the VSC system, while i τ s is only used as the input sinal for the inner control loop. This will lead to a hypothesis that the inner loop is the most vulnerable control unit to the measurement delays. Accordinly, a further simulation for both virtual inertia emulation and droop-based APC rid-formin case are used to verify the assumptions made for the inner control loop. Fiure 4.16 shows the eienvalue spectrum for the same time delay bein applied to three measurement sinals, while employed with VIE controlled APC in the rid-followin mode. Fiure 4.17 displays the results for the rid-formin settin of the VSC system. These two plots verify the aforementioned inner loop hypothesis, as the critical time delay of 233 µs for both confiurations remains the same. Thus, further analysis of the inner control loop would be conducted to assess the critical parameters of the controller with respect to the system stability aspect.

35 CHAPTER 4. EIGENVALUE SPECTRUM ANALYSIS 26 5, τ = 22 µs τ = 23 µs τ = 24 µs I(λ) 5, 1, R(λ) Fiure 4.16: APC in VIE mode applyin to rid-frequency followin case 5, τ = 22 µs τ = 23 µs τ = 24 µs I(λ) 5, 1, R(λ) Fiure 4.17: APC in Droop-based mode applyin to rid-frequency formin case

36 Chapter 5 Stability Mappin In the previous chapter, we observed the sinificant impact of the delayed switchin current sinal to the small-sinal stability of the VSC system. Furthermore, we noticed that most vulnerable sement of the VSC controller is the inner loop. Hence, the focus of the followin chapter is on provin that the current state variable contributes majorly to the stability marin of the system. Here several analysis tools have been proposed to testify this hypothesis, includin participation factors and bifurcation analysis. 5.1 Participation factors The contribution that each state has on a particular eienvalue is quantified throuh the participation factors. In paper [15], participation factors were deployed to identify the fastest eienvalue for Model Order Reduction process. A common formulation of the participation factors is presented in [16 18], as illustrated below: p ki = l i k ri k (5.1) where lk i denotes the kth element of the i th left eienvector, while rk i represents the k th element of the i th riht eienvector. p ki is the relative participation of the k th state variable in the i th mode (eienvalue). From the definition it is clear that the hiher value of participation of a state variable, the more contribution it has to the correspondin eienvalue. The participation factors of the most critical eienvalues are shown in Fiure 5.1 and 5.2. As can be seen from the participation factors of four critical eienvalues ˆλ = ± j and ˆλ = ± j , the delayed switchin current sinal i sτ has the hihest contribution to all of the critical modes, whereas the other two measurement states i τ and e τ contribute less. It is reasonable to conclude that the SRF current 27

37 CHAPTER 5. STABILITY MAPPING 28 i d s τ.3 e q τ i q s τ.2.1 e d τ i d τ ξ d + ξ q i q τ eps dthetapll pm dthetaapc qm xid xiq ammad ammaq ed edt1 edt2 edt3 edt4 edt5 edt6 eq eqt1 eqt2 eqt3 eqt4 eqt5 eqt6 id idt1 idt2 idt3 idt4 idt5 idt6 iq iqt1 iqt2 iqt3 iqt4 iqt5 iqt6 isd isdt1 isdt2 isdt3 isdt4 isdt5 isdt6 isq isqt1 isqt2 isqt3 isqt4 isqt5 isqt6 Fiure 5.1: Participation factors for ˆλ = ± j i q s τ.3.2 i d s τ e d τ.1 e q τ i d τ ξ d + ξ q i q τ eps dthetapll pm dthetaapc qm xid xiq ammad ammaq ed edt1 edt2 edt3 edt4 edt5 edt6 eq eqt1 eqt2 eqt3 eqt4 eqt5 eqt6 id idt1 idt2 idt3 idt4 idt5 idt6 iq iqt1 iqt2 iqt3 iqt4 iqt5 iqt6 isd isdt1 isdt2 isdt3 isdt4 isdt5 isdt6 isq isqt1 isqt2 isqt3 isqt4 isqt5 isqt6 Fiure 5.2: Participation factors for ˆλ = ± j loop is the most vulnerable controller to time delays, since it is the only controller with i sτ.

38 CHAPTER 5. STABILITY MAPPING Bifurcation analysis In this section same time delay case for three measurement sinals is used in the bifurcation analysis. First, an SRF voltae controller parameter bifurcation test is conducted. For this test an optimized sweepin method has been implemented. Initially a rouh and lare discrete steps are introduced to identify the area of a dramatic critical time chane. Subsequently, a more precise discretization is applied for further investiation and a two-dimensional bifurcation analysis example is showcased D bifurcation analysis K v p is the proportional ain of the SRF voltae controller. As can be seen from both Table 5.1 and stability map of K v p in Fiure 5.3, there is a sinificant impact on the critical time delay of the system respective of the chanes in the proportional ain parameter, especially when K v p varies from.5 to 1. In terms of the interal ains of SRF voltae controller shown in Table 5.2 and Fiure Table 5.1: Critical time delay chanes with various K v p (1µs error) Kp v ˆτ (µs) Kp v ˆτ (µs) ˆτ [µs] K v p [-] Fiure 5.3: Stability Map on ˆτ K v p plane Table 5.2: Critical time delay chanes with various K v i Ki v ˆτ (µs)

39 CHAPTER 5. STABILITY MAPPING ˆτ [µs] , 1,1 1,2 K v i [-] Fiure 5.4: Stability Map on ˆτ K v i plane Table 5.3: Critical time delay chanes with various K i f Kf i ˆτ (µs) ˆτ [µs] K i f [-] Fiure 5.5: Stability Map on ˆτ K i f plane 5.4, one can observe a flat line of crtical time delay as Ki v further increases. This ives an indication that the correlation between the stability of the VSC system and the interal parameter is low. A bifurcation test upon feed-forward sinal Kf i is demonstrated in Table 5.3 and Fiure 5.5. A similar pattern as in the interal ain parameter test is monitored: a flat line of critical time delay until the parameter is increased to a threshold value. Excess of this limit causes the system to collapse even without time delay bein introduced into the control scheme. Hence, the correlation between the critical time delay and the feedforward sinal is also not sinificant.

40 CHAPTER 5. STABILITY MAPPING 31 Table 5.4: Critical time delay chanes with various K i p (1µs error) Kp i ˆτ (µs) Kp i ˆτ (µs) ˆτ [µs] K i p [-] Fiure 5.6: Stability Map on ˆτ K i p plane In spite of a similar pattern bein observed for the SRF current controller, the critical time delay is more responsive to the chane in K i p than K v p. As can be seen from the results of both Table 5.4 and Fiure 5.6, the critical time delay ˆτ chanes dramatically as K i p surpasses.5. The decline of the curve is steeper than what we observe for the voltae controller as well. It is reasonable to note that the SRF current controller is more sensitive to the chane of time delay sinals. The interal ain bifurcation analysis for SRF current controller shows the pattern analoue to the interal ain test of the voltae controller. Aain, the interal ain adjustment does not affect the critical time delay sinificantly. Apparently the chanes of feed-forward sinal ain do not have impact on the critical time delay, as can be seen from Table 5.6 and Fiure 5.8. Overall, the SRF Current Controller is the most sensitive unit of the VSC controller. Meanwhile, the parametrization of interal ains and feed-forward sinals does not considerably affect the small-sinal stability of the VSC system. This result is sinificant, as most other researches resemblin [11] only consider the outer control loop while nelect- Table 5.5: Critical time delay chanes with various K i i Ki i ˆτ (µs)

41 CHAPTER 5. STABILITY MAPPING ˆτ [µs] K i i [-] Fiure 5.7: Stability Map on ˆτ K i i plane Table 5.6: Critical time delay chanes with various K v f Kf v ˆτ (µs) ˆτ [µs] K v f [-] Fiure 5.8: Stability Map on ˆτ K v f plane in the inner control loop details to investiate the small-sinal stability of the proposed VSC system. As can be seen in Fiure 5.9, the critical time delay without detailed consideration of inner loop controller is at around 25 milliseconds, drastically above what is observed in this project as microseconds. Thus, the overlook of inner control loop characteristic may cause the conclusion to be flawed. A further investiation of three-dimensional bifurcation analysis is carried out to validate our observation.

42 CHAPTER 5. STABILITY MAPPING τ [ms] H [s] Fiure 5.9: Stability Map on ˆτ H plane D bifurcation analysis Three-dimensional bifurcation analyzes the impact of two controller parameters on the critical time delay, formin an uneven plane to further indicates the effect of two variables simultaneously. The map in Fiure 5.1 illustrates the correlation of proportional and interal ains K v p, K i p to the critical delay time ˆτ. The depicted surface indicates the stability marin of the system. Similar to what is shown in Fiure 5.5, 3-D stability mappin of SRF voltae controller interal ain parameter does not influence ˆτ. On the other hand, the critical time delay is sensitive towards the chanes in proportional ain K v p. The same observation can be spotted from SRF current controller stability mappin in Fiure ˆτ [µs] , Kp v [-] Ki v [-] Fiure 5.1: Stability Surface in the ˆτ Kp v Ki v, System is stable under shaded area

43 CHAPTER 5. STABILITY MAPPING 34 4 ˆτ [µs] Ki i [-] Kp i [-] Fiure 5.11: Stability Surface in the ˆτ Kp i Ki i, System is stable under shaded area However, a steeper cliff is showcased in the SRF current controller stability mappin than in the voltae controller. When the proportional ain varies from.2 to.4 in the current controller, a sudden plummet is observed for the critical time delay compared with a more radual and smooth drop in the voltae controller. Therefore, we conclude that the proportional ain in the SRF current controller is more influential in the smallsinal stability of the system than the voltae controller, despite both havin impacts on the critical time delay.

44 CHAPTER 5. STABILITY MAPPING ˆτ [µs] K d [-] H [s] Fiure 5.12: Stability Map on ˆτ K d H plane ˆτ [µs] D p [-] D q [-] 1 2 Fiure 5.13: Stability Map on ˆτ D p D q plane With further analysis bein focused on the outer control loop, we can see a somewhat different behavior, as depicted in Fiure Here, a correlation between VIE parameters (dampin constant K d and inertia constant H) and the critical time delay ˆτ is represented with a flat plane, indicatin that the APC parameters essentially have no impact on the critical time delay of the system. Meanwhile, parameter variation for active power control in Fiure 5.13 and reactive power control in Fiure 5.14 exhibit identical characteristics. This raises a final assertion that the inner control loop is the most critical part affectin the small-sinal stability of the proposed VSC system, while the outer loop controller does not have a major impact. In the next chapter, a quantitative method to assess the impact of the parameters on the critical eienvalues is implemented to further validate the conclusion made throuh bifurcation analysis.