Dynamic phasor based stability analysis of an inverter connected to grid

Transcription

1 Dynamic phasor based stability analysis of an inverter connected to grid A. Sawant, P. Jagtap, J. Aute, U. Mumbaikar Department of Electrical Engineering VJTI, Mumbai Abstract Increasing electricity demand has always pushed the power grid to operate at its peak. However penetration of renewable energy, though intermittent, will help in easing the stress through the development of microgrids. Since microgrids involve better and efficient control of power flow bidirectionally, intelligent devices which can provide this function are essential. The solid state transformer provide a means of serving this objective, apart from replacing the conventional low frequency transformer. This paper focuses on the analysis of the inverter stage of the SST when it is connected to the grid at the low voltage distribution side. A comparison of two modeling approaches is investigated, the averaged model and the dynamic phasor based approach. The later proves to be computationally faster and also takes into account the ripple in the states which is neglected in the former approach. By linearizing the dynamic phasor equations, a PI controller is designed which maintains the output voltage of the inverter at its desired value. Keywords-Dynamic phasors (DP), grid, inverter, PI controller, solid state transformer, stability analysis. Rfi Lfi Cfi Vdcl Rgi Lgi Vg S or D R I TABLE I NOMENCLATURE Filter Resistance Filter Inductance Filter Capacitance Input Voltage Grid Resistance Grid Inductance Grid Voltage Duty Cycle Real part Imaginary part I. INTRODUCTION Electrical Power System symbolizes a system with dynamic operating characteristics where transformer plays a major role in distribution system. To withstand different complexities in distribution system, new technology is implemented which replaces old conventional transformer with solid state transformer (SST) in microgrid[5]. SST is also known as a threeport energy router and power exchanger. The characteristics feature of a three-port solid state transformer is that it gives better performance than conventional transformer. Three stage SST is widely used at the distribution level to supply power to the consumers[5]. As shown in Figure three stages of SST are explained below: i) At rectifier stage, high voltage AC is converted to high voltage DC which is fed to the dual active bridge (DAB) converter. ii) At second stage, dual active bridge (DAB) converter, converts high voltage DC to low voltage DC. Generally this stage is called as DC/DC converter. iii) Third stage is inverter stage, which converts DC voltage to the regulated low AC voltage[5]. Fig.. Schematic diagram of three stage solid state transformer This situs also gives perfect control of voltages and currents on both primary and secondary side. Rectifier (AC to DC) stage acts as a power factor improvement device thus power factor is always at unity. Power electronic devices act as a circuit breaker thus fault detection as well as protection of a system takes place through solid state transformer[5]. Dynamic phasor (DP) representation of a system s dynamic quantity in the form of phasors which is further used in resolving system s matrix to study stability issues of the respective system. Dynamic phasor approach, in which phasor representation can be carried out from switched model system with fourier series representation. Phasor representation is a class of mathematical transformation to eliminate the fundamental frequency component from the system s equations[]. Equations are formed with reference to the state variables of that system. Performing linearization on those equations, considering small perturbations in the system s dynamical quantity, system s matrix is formed which is further used for conducting stability analysis. To maintain synchronisation in the power system, the stability of small signal model is analysed by providing small perturbations in the control signal for dynamic response. The system dynamics are controlled using a PI controller. PI controller designing includes deciding values of the gain k p and k i on a trial and error basis. Recursive loop for different values of k p and k i continues till a stable closed loop system is obtained[7].

2 This work addresses application of PI controller to maintain output of grid connected inverter as constant and perform small signal stability analysis using Lyapunov s first method and control the model with PI controller. Second section introduces necessity of dynamic phasors. DP based modelling of grid connected inverter with linearization model of the inverter which is described in section three. II. NECESSITY OF DYNAMIC PHASORS To evict the fundamental frequency component from the system computation, phasor representation is carried out, which is categorize in analytic transmutation. Phasor representation of a system is classified in the analytical transmutation to evict the fundamental frequency component from the computation. For analyzing steady state operation of voltages and currents of a system, phasor representation of sinusoidal signal is realized. Phasors are widely used for the analysis of stationary case of the power system[3]. With a fourier series representation, approximate complex switched waveform x(τ)in the interval of τ (t-t, t] is given in the form of, ii) The product of two switched variables equals to the discrete time convolution of the two DP sets of variables, which is given in (4)[3]. xy k = l= (X k l Y l ) (4) III. DP BASED MODELLING OF INVERTER A. Switched modelling of inverter A full bridge single-phase inverter model, where Q, Q2, Q3 and Q4 represent IGBT switches as shown in Figure 2. The aim of developing average equivalent model is to avoid modelling of high frequency switching transients[]. With reference to Figure 2 a single phase inverter with a RLlter, V dcl with output of DAB is an input to inverter, and resistance R g, inductance L g and output voltage V g are grid parameters. x(τ) = Re{ k K X k (t)e jkωτ } () X k (t) = T t t T x(τ)e jkωτ dτ = x k (t) (2) where, ω = 2π/T X k (t) is the k th time varying fourier coefficient, which is also called dynamic phasor. K is the set of fourier coefficients. K contributes a better equivalent value of the system waveform (e.g. K =,, 2)[3]. DP approach propose a number of benefits over traditional forms. In frequency domain, selection of K gives larger bandwih. It s selection and variation also gives choice of displaying combination between various parameters and denotes specific problems at different frequencies[3]. DP makes simulation very faster than switched model because variations of DP are a lot slower than the spontaneous parameters. So they can be used to evaluate the fast electromagnetic transients with larger step size. DP approach approximates a periodically switched system with a continuous system as it permits an analytical judgment[3]. Some important characteristics of DP are: i) As given in (3), relationship between the derivatives of x(τ) and the derivatives of X k (t), where the time argument t has been neglected for accuracy. This can be easily justified by differentiating the equation given in ()[3]. dx = dx k jkωx k (3) k Fig. 2. Inverter with grid Average model of grid connected inverter is shown in Figure 3. Development of a dynamic phasor (DP) model of a singlephase inverter is achieved by average model of Figure 3. Fig. 3. Average model of inverter connected to the grid Representation of inverter in switched model by applying mesh and nodal analysis is done as, d i f L g d i g d V cf = S V dcl V cf R fi i f (5) = V cf V g R gi i g (6) = i f i g (7) Above equations are written with fundamental frequency component as,

3 d i f d V cf d i g = jωi f + S V dcl R fi i f V cf (8) = jωv cf + i f i g (9) = jωi g + V cf V g R gi i g () B. DP modelling of inverter Dynamic phasor representation of inverter obtained from switched model equations by removing fundamental frequency component is written as, d i f R = ω i f I + S R V dcl R fi i f R V cf R () d i f I = ω i f R + S I V dcl R fi i f I V cf I (2) d V cf R = ω V cf I + i f R i g R (3) d V cf I = ω V cf R + i f I i g I (4) d i g R d i g I = ω i g I + V cf R R gi i g R V g (5) = ω i g R + V cf I R gi i g I V g (6) C. Linearized model of inverter For dynamic system, linearization is the best way to achieve small signal stability. Thus linearize equations in switched and DP of the grid connected inverter with small perturbations. Linearized Equations are obtained in the form of, where, A is the state matrix of size n n ẋ = A x + B d (7) y = C x + D d (8) B is the control or input matrix of size n r C is the output matrix of size m n D is the feedforward matrix of size m r x is the state vector of dimension n y is the output vector of dimension m d is the input vector of dimension r Linearized switched model equations are written in form of (7 and 8), d i f V cf = i g + V dcl D d o R gi i f V cf i g Linearized dynamic phasor equations are written in form of (7 and 8), d i f R i f I ω V cf R V cf I = i g R i g I i f R i f I V cf R V cf I + i g R i g I ω ω ω Rgi ω Rgi V dcl cos 2πD V dcl sin(2πd) d As above switched and DP equations are written in form of eq.(7 and 8). So separate all matrices from equations and rewrite as, In switched model, A = C = [ ] T D= R gi ; B = V dcl D With dynamic phasors, ω R ω fi C A = fi ω ω Rgi ω ω Rgi ω

4 B = V dcl cos 2πD V dcl sin(2πd) C = [ ] T D= IV. SMALL SIGNAL STABILITY ANALYSIS OF GRID CONNECTED INVERTER Small signal stability analysis is to maintain synchronism of power system when system is affected by small perturbations[7]. Linearizing switched model as well as dynamic phasors (DP) equations for analysis of small signal stability of grid connected inverter. There are two methods for analysis of small signal stability. i) Lyapunov s first method ii) Lyapunov s second method or direct method In Lyapunov s first method, small signal stability is determined by the Eigen values of matrix A which is obtained by linearization of the equations. Lyapunov s second method uses state functions to obtain small signal stability. For grid connected inverter, small signal stability is achieved by using lyapunov s first method by making use of the linearized matrix A[7]. Criteria for analysis of the stability from lyapunov s first method is as follows: i) System is asymptotically stable when eigenvalues consist of negative real part. ii) If system has at least one Eigen value lying on right half of the S-plane the system will be unstable. Small signal stability can also be achieved using participation matrix. Combination of right and left eigenvectors with association between state variables and modes is called as participation matrix[7]. Φ i Ψ i P = [ P P 2... ] Φ 2i Ψ i2 P n with Pi =.. Φ ni Ψ in V. STABILITY ANALYSIS USING PI CONTROLLER PI controller is most commonly used for controlling dynamic systems as they have plain structure, easy to design and economical. Regardless of these advantages, PI controller fails when controlled element is highly nonlinear and unpredictable[6]. As grid connected inverter have semiconductor switches, PI controller eliminate forced oscillations as well as harmonics and steady state error which is resulting while on-off controller[6]. PI controller gives pessimistic effect on response speed of the system and overall stability of the system as it works with integral mode. On the other hand derivative mode has ability to predict contingencies of the errors. Thus reaction time of the controller is decreased[6]. PI controller is used when, i) Where fast response of the system is not needed. ii) If there is only one energy storage element (capacitor or inductor) in process. iii) When course of process includes large disturbances and noise. iv) There are large response delays in the system. VI. RESULTS Simulations results of all three methods are given in the following figures. A. Simulation Results Simulink simulations have been executed to demonstrate the switched model and DP based inverter connected to the grid. Figure (4), (5), (6) and (7) show output of inverter with grid, ripple is approximately 24 volts peak to peak and current 43 A peak to peak across load/grid in both switched and DP models. Fig. 4. Output voltage in switched model of a grid connected inverter where, Φ i is a right eigen vector and Ψ i is a left eigen vector Term P i = Φ i Ψ i is known as participation factor and in association with eigenvector normalization, the sum of participation factors is. Fig. 5. Load current in switched model of a grid connected inverter

5 Fig. 6. Output voltage in DP of a grid connected inverter Fig. 8. Switched model of a grid connected inverter without controller Fig. 7. Load current in DP of a grid connected inverter B. Eigen values and participation matrix Table II and III shows the results when stability analysis is measured through Eigen values. All eigenvalues of switched model as well as DP system are observed in left half of S-plane. Also from analysis of participation matrix, right eigenvalues and left eigenvalues have real negative part. Thus it shows that the system of an inverter connected to grid is stable. Corresponding measure of adjacency to voltage instability is provide by eigenvalues associated with a mode of voltage and reactive power variation[8]. Fig. 9. Switched model of a grid connected inverter with controller TABLE II EIGENVALUES OF SWITCHED MODEL OF GRID CONNECTED INVERTER Sr.No Eigenvalues of switched model (.e + 3) ( i) 2 (.e + 3) ( i) 3 (.e + 3) ( i) Fig.. DP model of a grid connected inverter without controller TABLE III EIGENVALUES IN DP OF GRID CONNECTED INVERTER Sr.No Eigenvalues in DP (.e + 3) ( i) 2 (.e + 3) ( i) 3 (.e + 3) ( i) 4 (.e + 3) ( i) 5 (.e + 3) ( i) 6 (.e + 3) ( i) C. Frequency analysis plot Figure (8), (9), () and () show the results of inverter without controller and inverter with controller of switched model and DP model based on gain margin and phase margin. A systematic comparision is given by simulating these differently modeled components in MATLAB based common simulation framework. Switched model and Reduced order dynamic Phasor model of component results are compared. Fig.. DP model of a grid connected inverter with controller Table IV and V shows comparison of results obtained in switched model and DP model of grid connected inverter with and without using PI controller. As the gain and phase margin increases, the stability is improved.

6 TABLE IV CONTROLLER PARAMETERS OF SWITCHED MODEL Sr.No parameter Without controller With controller Gain Margin Infinity Infinity Frequency Infinity Infinity 2 Phase Margin Frequency rad/s rad/s TABLE V CONTROLLER PARAMETERS OF DP Sr.No parameter Without controller With controller Gain Margin (db) Frequency (rad/s) Phase Margin (degree) Frequency (rad/s) [3] T. Demiray, G. Andersson and L. Busarello, Evaluation study for the simulation of power system transients using dynamic phasor models [4] Seth R. Sanders, J. Mark Noworolski, Xiaojun Z. Liu, and George C. Verghese, Generalized averaging method for power conversion circuits, IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 6. NO. 2, APRIL 99 [5] Shilpkala G. Bansode, Prasad M. Joshi, Solid state transformers: New approach and new opportunity, GCE, Karad. [6] K. Smriti Rao, Ravi. Mishra Comparative study of P, PI and PID controller for speed control of VSI-fed induction motor, IJEDR 24, Volume 2, Issue 2, ISSN: [7] P. Kundur, Textbook on Power system stability and control, McGraw- Hill,Inc. [8] Dr. Enemuoh F. O., Dr. Onuegbu J. C. and Dr. Anazia E. A. Modal based analysis and evaluation of voltage stability of bulk power system, International Journal of Engineering Research and Development, Volume 6, Issue 2 (May 23), PP VII. CONCLUSIONS A basic introduction to various methods of inverter modeling is explored in this paper. The focus of the paper is on the design of a PI controller for the linearized DP model of the grid connected inverter. A comparison between the switched model and the DP model simulations is performed which prove that the later is a better tool when ripple has to be considered. It is further proposed to use Model Predictive Control for developing the controller in the DP domain which will truly propagate the advantages of the DP approach. VIII. APPENDIX TABLE VI SYSTEM PARAMETERS OF GRID CONNECTED INVERTER f 6Hz Rfi Ω Lfi.7mH Cfi 3µF Vdcl 4V Rgi 35Ω Lgi 2mH S or D.6 V cf 24V ACKNOWLEDGMENT The authors would like to acknowledge the support of TEQIP-II through Centre of Excellence in Complex and Nonlinear Dynamic Systems(CoE-CNDS), VJTI, Matunga, Mumbai, India. We are also thankful for continuous technical inputs provided by S.R.Wagh, N.M.Singh and A.Stankovic throughout the completion of the research. REFERENCES [] Adarsh. Nagarajan and Raja. Ayyanar, Dynamic phasor model of singlephase inverters for analysis and simulation of large power distribution systems, Arizona State University Tempe, AZ [2] Tiefu. Zhao, Jie. Zeng, Subhashish. Bhattacharya, Mesut E. Baran and Alex Q. Huang, An average model of solid state transformer for dynamic system simulation, IEEE Member.