Mitigating Wind Power Fluctuation Using Flywheel Energy Storage Systems. Tipakorn Greigarn

Transcription

1 Mitigating Wind Power Fluctuation Using Flywheel Energy Storage Systems by Tipakorn Greigarn Submitted in partial requirements for the degree of Master of Science Thesis Adviser: Professor Mario Garcia-Sanz Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY May 2011

2 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of candidate for the degree *. (signed) (chair of the committee) (date) *We also certify that written approval has been obtained for any proprietary material contained therein. i

3 Contents Acknowledgement 1 1 Introduction 4 2 Flywheel Energy Storage Systems Introduction Model Control Feedback Linearization Robust Control MIMO QFT Controller MIMO QFT Tracking Formulation MIMO QFT Disturbance Rejection Formulation The Plant and Its Uncertainties QFT Controller Design ii

4 2.5.6 Simulation Current Coordinator Proposed Current Coordinating Scheme Simulation Conclusion FESS-Wind Farm Integration Introduction Model Control Wind Farm Configuration Disturbance Modeling Controller Design Simulation Step Disturbance Simulation Wind Fluctuation Conclusion Conclusion 71 Bibliography 73 iii

5 List of Tables 2.1 FESS Parameters iv

6 List of Figures 2.1 FESS Control Block Diagram FESS Current Controller Block Diagram Two degree-of-freedom feedback control block diagram The bounds for equivalent SISO loop Intersection of the bounds for equivalent SISO loop The bounds for equivalent SISO loop Intersection of the bounds for equivalent SISO loop Looping-shaping G Loop-shaping of controller G Pre-filter F Pre-filter F SimPowerSystems TM Simulation Torque contours and maximum torque per ampere locus on i d -i q plane Voltage ellipses at ω = 1400, 1800, 2200, 2600 rad/s v

7 2.15 (a) Gradient and tangent vectors (b) Zoomed-in gradient and tangent vectors Flow chart for MTA operation Flow chart for flux weakening operation Voltage ellipses at ω l = 1785 rad/s and ω u = 2475 rad/s Mode selector flowchart Comparison between two field weakening techniques The system responses to T = (i d, i q ) path under T = The system responses to T = (i d, i q ) path under T = Torque step response at ω = 2400 (rad/s) Normalized torque step response at ω = 2400 (rad/s) Normalized torque step response at ω = 1600, 2400, 3200 (rad/s) Normalized torque step with parameters variations Normalized torque step response with parameter variations at ω = 1600, 2400, 3200 (rad/s) Second order approximation of the torque dynamic Off-shore wind farm configuration vi

8 3.8 Wind farm power and drive train torque under a step change in wind velocity Bode plot of the transfer function from turbine torque to rotor torque QFT Bounds Loop-shaping of the controller Loop-shaping of the prefilter Wind farm behavior under step change in wind veloity Wind farm under wind fluctuation vii

9 Acknowledgement First and foremost I offer my gratitude to my adviser, Professor Mario Garcia-Sanz for his guidance, mentorship and support throughout my thesis. I have been very fortunate to have him as my adviser. Secondly, I would like to express my sincere thanks to Dr. Vira Chankong for his guidance in both academic and life. Most importantly, I owe my deepest graduate to my family. Especially my father who has been and will always be my role model, my mother for her love and understanding, and my brother who is also my best friend. I would also like to thank Ms. Sutaporn Boon-intra for her friendship and thoughtfulness. Without them I would have been lost in misery. Special thanks to all my friends here at Case Western Reserve University for making my life in the States much more pleasant. Lastly, this thesis would not have been possible without funding from The Milton and Tamar Maltz Family Foundation and Cleveland Foundation. 1

10 Mitigating Wind Power Fluctuation Using Flywheel Energy Storage Systems Abstract by TIPAKORN GREIGARN As the penetration of wind energy increases, the effect of wind fluctuations has gained considerable importance. This thesis presents wind power fluctuation mitigation using a Flywheel Energy Storage System (FESS). An Interior Permanent Magnet Synchronous Machine (IPMSM) driven by an IGBT Voltage Source Converter that was chosen for the FESS due to its attractive features such as high dynamic efficiency, high speed capability, and low electrical losses. The IPMSM was controlled by Field Oriented Control which transform the the stator equivalent circuit from abc to dq coordinate. Nonlinear coupling between direct and quadrature axis was cancelled out by feedback linearization. A robust multi-input multi-output controller designed by Quantitative Feedback Theory (QFT) was used to controlled the feedback linearized system. Due to the saliency of the IPMSM rotor, both direct and quadrature axis currents can be used to generate torque. A new current coordinating technique was developed to determined the best combination of the currents. In low speed regime, the coordinator selects the currents combination according to Maximum Torque per

11 Ampere Criterion to obtain high efficiency. High speed operation were also made possible by the coordinator s Flux Weakening Regime. A robust controller was designed using QFT to control the power of FESS injecting into the DC link of the wind farm. The FESS has proven to be effective in mitigating power fluctuation of a 10 MW wind farm in SimPowerSystems TM simulations. 3

12 Chapter 1 Introduction As the availability of fossil fuel continues to diminish while the demand for energy and pollution levels keep rising, a shift from traditional energy source towards renewable energy is inevitable. Among many renewable energy sources, wind energy has become the fastest growing [27]. However, like most renewable energy, reliability of wind energy suffers from its variability. In order to see the effect of wind fluctuation on its power, consider the extractable power from wind given by P = 1 2 ρac pv 3, (1.1) where P is the power, ρ is the air density, A is the cross-sectional area of turbine blades, C p is the power coefficient, and v is the wind velocity. In equation (1.1) it is seen that the power is proportional to the cube of wind velocity. This means small 4

13 changes in wind will result in large variation of power. Thus, increasing wind energy penetration could cause more problems to the power systems such as flickers and faults [36]. One way to deal with the fluctuation is to integrate short-term energy storage devices to wind farms. Several storage technologies have been considered in literature including battery [2, 3, 35, 37], electrochemical capacitor [5], and flywheel [9]. In this work, the use of Flywheel Energy Storage Systems (FESS) to migitate power fluctuation of wind farm is proposed and evaluated. The main advantages of FESS include long life span (up to 100,000 cycles), high power and energy density, and depth-of-discharge-independence [18,22]. Furthermore, FESS is also very competitive economically in both price per unit power and energy [4]. Due to the its attractiveness, FESS has found applications in many areas ranging from space applications [24] to power system applications [1], [31]. FESS consist of flywheels coupled to electric machines. The energy is stored as kinetic energy in spinning masses. The rating power of FESS is primarily limited by the power electronics associated with the machine, while energy density depends upon its size, mass, and angular velocity. A combination of power electronics and electric machine has to be chosen carefully to in order obtain desired rated energy and power with high efficiency. In this work, a Permanent Magnet Synchronous Machine (PMSM) driven by an 5

14 Insulated-gate Bipolar Transistor (IGBT) was chosen for the FESS due to the following attractive features. First, PMSM is capable of supplying low to medium load with good dynamic performance [26]. Second, rotor flux of PMSM is generated by permanent magnet, thus eliminating electrical losses in the rotor circuit. High speed operation can be reached by a certain type of PMSM called an Interior Permanent Magnet Synchronous Machine (IPMSM) due to its mechanically robust rotor structure. High speed operation translates into higher energy density, a property sought after for energy storage devices. The rotor robustness is achieved by having magnets buried inside the rotor, thus the name interior magnet. The FESS was controlled using Field Oriented Control which basically transforms a three-phase sinusoidal circuit in a static frame into a DC circuit in a rotating frame. The feedback controller was designed based on the transformed model. In order to achieve robust performance over entire speed range, a method called Quantitative Feedback Theory (QFT) [21] was used to design the controller. Desired torque was achieved by coordinating the direct and quadrature currents using a current coordinator. To test the ability to mitigate power fluctuation, the FESS was integrated into a 10 MW wind farm. Simulations of the the wind farm behavior with and without the FESS are done using SimPowerSystems TM. Chapter 2 presents the modeling and control of the FESS based on an IPMSM. 6

15 The integration of the FESS to the wind farm and simulations were presented in chapter 3. 7

16 Chapter 2 Flywheel Energy Storage Systems 2.1 Introduction Among energy storage technologies, FESS has recently become one of the most competitive technology for power quality management application [22]. The main advantages of FESS include long life span, high power and energy density, and depth-ofdischarge-independent [18]. FESS has found applications in many areas ranging from space applications [24] to power system applications [1], [31]. The energy of FESS is stored as kinetic energy in spinning flywheels which are coupled to electric machines. The rating power of FESS is primarily limited by the power electronics associated with the machine, while energy density depends upon the size, mass, and angular velocity. A combination of power electronics and electric 8

17 machine has to be chosen such that desired rated energy and power are obtained with high efficiency. In this work, an IGBT driven Permanent Magnet Synchronous Machine was chosen for the FESS due to the following attractive features. First, the PMSM is capable of supplying low to medium load with good dynamic performance [26]. Second, the rotor flux of PMSM is generated by permanent magnet, thus eliminating electrical losses in the rotor circuit. A specific type of PMSM called an Interior Permanent Magnet Synchronous Machine can achieve high speed operation due to its mechanically robust rotor structure. As the name suggests, permanent magnets are buried inside the rotor of IPMSM instead of surface-mounted PMSM. High speed operation translates into higher energy density, a property sought after in all energy storage devices. Other interesting properties introduced by this rotor construction include lower effect air gap and saliency [8]. 2.2 Model The stator current was transformed to dq coordinates where the d-axis was aligned to the rotor flux direction. The system in dq coordinates can be described by (2.1). This control technique is know as Field Oriented Control [26]. The dynamics of a 9

18 permanent magnet synchronous machine in the dq coordinate is given by [30] L q di q dt = Ri q L d ωi d φω + v q, L d di d dt = Ri d + L q ωi q + v d, (2.1a) (2.1b) J dω m = Kω m + T, dt (2.1c) dε dt = ω. (2.1d) Where L d and L q are inductances in the direct and quadrature axes (H), R is stator resistance (Ω), φ is the magnetic flux (V s), K is friction coefficient, and J is moment of inertia (kg m 2 ). i d and v d are stator currents and voltage in the direct axis, while i q and v q are on the quadrature axis. ω is the electrical frequency (rad/s), ω m is the mechanical angular velocity (rad/s), and ε is the rotor angle (rad). The torque T (N m) is given by, T = 3 2 pi q[φ + (L d L q )i d ]. (2.2) Where p is the number of pole pairs in the machine. 2.3 Control The system described by (2.1) is challenging from the control perspective for two reasons. First, it is nonlinear due to the terms L q ωi q and L d ωi d in (2.1b) and (2.1a). 10

19 Secondly, the nonlinear terms also act as cross couplings between the two equation which further complicated the situation. Note that, the couplings in (2.1b) and (2.1a) are proportional to angular speed, and thus are even stronger at higher speed. The saliency of IPMSM also requires a special treatment. Unlike non-salient rotor construction, IPMSM can generate torque using both field and reluctance effect [8]. The machine can be viewed as a PMSM/Reluctance Synchronous Machine hybrid. Hence, the stator current i q and i d must be coordinated controlled such that the desired power is achieved with good performance and high efficiency. To answer the above challenges, the controller of the IPMSM was divided into multiple stages. Each stage was designed to control a certain quantity of the machine such that it would be easier for the outer stages to control the machine. The structure of the controlled system is shown in figure 2.1. The current controller has two parts. First, Feedback Linearization was used to cancel out the nonlinearity of the system in dq coordinate. Then a Multi-input Multi-output (MIMO) robust linear controller was designed to ensure robust tracking of the stator current. The structure of the current controller is shown in figure 2.2. The reference stator current to be tracked was generated by the current coordinator. The coordinator took full advantage of the saliency of the rotor by scheduling d- and q-axis current such that the desired torque is achieved within the machine operating capacity in an efficient way. 11

20 T* Current Coordinator i q * v q * i d * Current Controller v d * DQ to ABC v a * v b * v c * v a v b v c IPMSM i d, i q, ω i d, i q, ω ε T, ω, ε, i d, i q Figure 2.1: FESS Control Block Diagram i q * i d * Current Pre-filter + + Current Controller u q u d v q v d Feedback Linearization i d i q ω Figure 2.2: FESS Current Controller Block Diagram 12

21 2.4 Feedback Linearization Although the system in (2.1) is in dq coordinates which is easier to work with than the original system in abc coordinates, it is still nonlinear due to the nonlinear couplings between (2.1b) and (2.1a). Feedback linearization has proven to be a successful method for eliminating nonlinearity for this type of problem [6]. The feedback linearization is given by v q = ˆL d ωi d + ˆφω + u q, v d = ˆL d ωi q + u d. (2.3a) (2.3b) The resistive terms in (2.1b) and (2.1a) were not eliminated by (2.3) because they provide stability to the system. Substituting (2.3) into (2.1b) and (2.1a) yields, L q di q dt = Ri q + (ˆL d L d )ωi d + ( ˆφ φ)ω + u q, L d di d dt = Ri d (ˆL q L q )ωi q + u d. (2.4a) (2.4b) In practice, the actual values of the system parameters such as resistance, inductance, and flux may vary with temperature and operating condition. Hence, exact cancellation of the nonlinearity by constant parameter feedback linearization cannot be obtained. It has been shown that the effect of the uncertainties can be eliminate 13

22 using adaptive backstepping technique [30]. However, this work proposed a different way to deal the uncertainties. The feedback linearization was considered as an attenuation in the nonlinear coupling between two equations where the size of the couplings in (2.4) depend only upon the the range of uncertainty in the parameters. Then, the feedback linearized system were considered as the plant for the MIMO robust controller 2.5 Robust Control The robust linear controller in this work was designed using Quantitative Feedback Theory (QFT). Originally, QFT was developed as a Single-input Single-output (SISO) controller design method for a linear plant with parametric uncertainties [19, 21]. Later on, the method was extended to Multi-input Multi-output (MIMO) case [10 14, 16, 19, 21, 32 34] MIMO QFT Controller The first step in designing a controller is to set performance specifications, which are usually given as time domain step response characteristic or frequency domain objectives. When working with QFT, the time domain specifications are translated into frequency domain specifications. The design is then carried out in the frequency domain. The benefit of working in the frequency domain is that the characteristics 14

23 of the plant become more transparent. After the set of specifications in the frequency domain is obtained. The technique in [21] was used to design the MIMO controller. The design procedure can be divided into three main steps. First, the MIMO system was decomposed into multiple equivalent SISO plants. Then the performance specifications for the original MIMO plant were translated to SISO specifications for each equivalent SISO plants. Finally, the classical SISO QFT technique are used to design controllers for the equivalent SISO plants. The main control objective for the system in (2.4) is to track reference setpoints i d and i q. u d and u q were selected to be the inputs of the plant while the back EMF ( ˆφ φ)ω was assumed to be an external disturbance at he input of the plant. d D r + F G P y Figure 2.3: Two degree-of-freedom feedback control block diagram 15

24 2.5.2 MIMO QFT Tracking Formulation The relationship between the input r and the output y in figure 2.3 is given by, y = (I + P G) 1 P GF r (2.5) Let T r be the transfer matrix from the input r to the output y, i.e., y = T r r. Simple matrix manipulation gives, (P 1 + G)T r = GF (2.6) Usually, the diagonal matrix G is considered first, since it is the most basic form of controller. If later on it is proven that diagonal controller is not enough to obtain the desired performance, then off-diagonal elements will be introduced. Hence, a diagonal G was assumed. F is generally a diagonal matrix since off-diagonal element will create extra crosstalk effect. For a 2-by-2 system such as in (2.4), the matrices are given by T r = t r11 t r21 t r12 t r22, P = p 11 p 12, G = g 1 0, F = f 1 0. p 21 p 22 0 g 2 0 f 2 16

25 According to [21], the equivalent SISO plants q 11, q 12, q 21, and q 22 are defined as 1 1 q P 1 = 11 q q 21 q 22 Substituting the above expressions into (2.6) gives g 1 q 11 q g 2 q 21 q 22 t r11 t r21 t r12 t r22 = g 1 f g 2 f 2 Let L i = q ii g i. The transfer functions t rij, i = 1, 2, j = 1, 2 can be written as t r11 = L 1f L 1 q L 1 t r22 = L 2f L 2 q 22 t r12 = q L 1 t r21 = q L L 2 ( tr22 q 12 ( tr11 q 21 ( tr21 q 12 ( tr12 q 21 ), (2.7a) ), (2.7b) ), (2.7c) ). (2.7d) t r11 and t r22 are the paths from r 1 to y 1 and r 2 to y 2 respectively. Since the goal is to achieve tracking, performance specifications for t r11 and t r22 were given as two SISO tracking specifications. t r12 and t r21 were considered as crosstalk between the two channels. Hence, disturbance rejection specifications were imposed on t r12 and t r21 in 17

26 order to reduce the crosstalk effect. Define frequency domain bounds associated with the above specification as a r11 t r11 b r11, a r22 t r22 b r22, (2.8a) (2.8b) t r12 b r12, (2.8c) t r21 b r21. (2.8d) Note that, there are two terms on the right hand side of (2.7a) and (2.7b). The second term are considered as cross couplings between channels which need to be reduced. Bounds for these two quantity can be given by q L 1 q L 2 ( tr21 q 12 ( tr12 q 21 ) c r11, (2.9a) ) c r22. (2.9b) In presence of the cross coupling, the tracking bounds in (2.8a) and (2.8b) can be satisfied by assuming that the second terms are bounded by c r11 and c r22 and then bound the first term in (2.7a) and (2.7a) by smaller tracking bounds. In other words, the tracking bounds are made more restrictive by moving a ii up by τ cii and b ii down by τ cii. Thus, if the first term on the right hand sides of (2.7a) and (2.7b) can satisfy 18

27 this new smaller bounds while (2.9) holds simultaneously, then (2.8a) and (2.8b) are satisfied. The new smaller tracking bounds are given by a r11 L L 1 b r11, a r22 L L 2 b r22. (2.10a) (2.10b) Where a ii = a ii + τ cii and b ii = b ii τ cii. The equivalent bounds for the equivalent SISOs are obtained from (2.8c), (2.8d), (2.9), (2.10). The bounds on the first equivalent SISO are given by a r11 L L 1 b r11, L 1 c r11 q 12 b r21 q 11, min L 1 b r12 q 12 b r22 q 11. min (2.11a) (2.11b) (2.11c) And the bounds the second loop are a r22 L L 2 b r22, L 2 c r22 q 21 b r12 q 22, min L 2 b r21 q 21 b r11 q 22. min (2.12a) (2.12b) (2.12c) 19

28 2.5.3 MIMO QFT Disturbance Rejection Formulation The back EMF were considered as disturbance to the system. Now, let T d be the transfer matrix from the disturbance d to the output y. Solving the block diagram in figure 2.3 and performing simple matrix yield (P 1 + G)T d = P 1 D. (2.13) Where P 1, G, and F are the same as in the MIMO tracking problem. T d and D are given by T d = t d1 t d2, D = d 1 d 2. (2.14) Substituting P 1, G, F, T d, and D into (2.13) gives g 1 q 11 q g 2 q 21 q 22 t d1 t d2 d 1 + d 2 = q 11 q 12 d 1 + d 2 q 21 q 22 20

29 Let L i = q ii g i as before, the above matrix equation can be written as t d1 = q ( 11 d1 1 + L 1 t d2 = q L 2 + d 2 t d2 q 11 q 12 q 12 ( d2 + d 1 t d1 q 22 q 21 q 21 ), (2.15a) ). (2.15b) In order to reduce the effect of the disturbance to the output, the disturbance rejection bounds for the transfer functions t d1 and t d2 are defined as t d1 b d1, (2.16a) t d2 b d2. (2.16b) Notice that the third term of (2.15a) and (2.15b) are cross coupling similar to those in (2.7a) and (2.7b). The bounds for the coupling can be given by q L 1 q L 2 ( td2 q 12 ( td1 q 21 ) c d1, (2.17a) ) c d2. (2.17b) Following the same idea from tracking problem, the original disturbance bounds are made more restrictive to compensate for the cross coupling. Combining (2.15) with (2.16) and (2.17) to obtain the following equivalent SISO bounds: 21

30 Loop 1 equivalent SISO bounds, q L 1 b d1 c d1 d 1 + d 2, q 11 q 12 max q L 1 c d1 b d2, max q 12 (2.18a) (2.18b) and loop 2 equivalent SISO bounds, q L 2 b d2 c d2 d 2 + d 1, q 22 q 21 max q L 2 c d2 b d1. max q 21 (2.19a) (2.19b) The Plant and Its Uncertainties Since the dynamics of the angular velocity ω is much slower than the electric circuit s. The ω were assumed to be a constant with uncertain value. Other parameters with uncertainty include L q, L d, R. Transfer matrix from the inputs u q and u d to the output i q and i d was obtained from the feedback linearized system in (2.4) as follow, L q (L d s + R) L d Ld ω L P = d L q s 2 + R(L d + L q )s + L d Lq ω 2 L d L q s 2 + R(L d + L q )s + L d Lq ω 2 L d Ld ω L d (L q s + R). L q L q s 2 + R(L d + L q )s + L d Lq ω 2 L d L q s 2 + R(L d + L q )s + L d Lq ω 2 22 (2.20)

31 Table 2.1: FESS Parameters Stator Resistance (R) ± 5% Ω Stator Quadrature-axis Inductance (L q ) ± 5% H Stator Direct-axis Inductance (L d ) ± 5% H Pole Pairs (p) 2 Speed (ω m ) rad/s Magnetic Flux (φ) ± 5% V s Moment of Inertia (J) 64 kg m 2 The transfer matrix from the back EMF to the outputs was also derived from (2.4). The transfer matrix is given by L q (L d s + R) L D = d L q s 2 + R(L d + L q )s + L d Lq ω 2 L d Ld ω L q L q s 2 + R(L d + L q )s + L d Lq ω 2 (2.21) Where L q = ˆL q L q and L d = ˆL d L d. The parameters were from [1, 8] where ten percent error was assumed. The parameters are given in table 2.1, 23

32 2.5.5 QFT Controller Design Tracking bounds a r11 = b r11 = a r22 = b r22 = s s s , s s , s s s , s s (2.22a) (2.22b) (2.22c) (2.22d) The crosstalk bounds are given by b r12 = b r21 = 150s (s + 100)(s + 500), (2.23a) 400s (s + 100)(s + 500). (2.23b) Cross coupling bounds for tracking problem are c r11 = c r22 = s 2 (s + 100) 3 (s ), (2.24a) s 2 (s + 100) 3 (s ). (2.24b) 24

33 The disturbance rejection bounds are b d1 = b d2 = s s + 100, s s (2.25a) (2.25b) While the disturbance rejection problem cross coupling bounds are c d1 = 0.2s s + 100, c d2 = 0.2s s (2.26a) (2.26b) Stability bound for each SISO plant are also defined as L L 1 1.1, L L (2.27a) (2.27b) QFT Computer Aided Design (CAD) tool were used to compute the bounds [7, 15, 17, 20]. From (2.11), (2.12), (2.18), (2.19), and (2.22) to (2.27). The controllers for loop 1 and 2 are (s + 5.4) g 1 =, s (2.28a) (s + 30) g 2 =. s (2.28b) 25

34 Open Loop Gain (db) e+4 5e+4 1e Open Loop Phase (deg) Figure 2.4: The bounds for equivalent SISO loop 1 Open Loop Gain (db) e+4 5e+4 1e Open Loop Phase (deg) Figure 2.5: Intersection of the bounds for equivalent SISO loop 1 26

35 Open Loop Gain (db) e+4 5e+4 1e Open Loop Phase (deg) Figure 2.6: The bounds for equivalent SISO loop 2 Open Loop Gain (db) e+4 5e+4 1e Open Loop Phase (deg) Figure 2.7: Intersection of the bounds for equivalent SISO loop 2 27

36 The pre-filter were designed after loop-shaping was done. The resulting pre-filter are Open Loop Gain (db) e+4 5e+4 1e Open Loop Phase (deg) Figure 2.8: Looping-shaping G 1 given by f 1 = 100 s + 100, f 2 = 100 s (2.29a) (2.29b) 28

37 Open Loop Gain (db) e+4 5e+4 1e Open Loop Phase (deg) Figure 2.9: Loop-shaping of controller G Magnitude (db) Frequency (rad/sec) Figure 2.10: Pre-filter F 1 29

38 Magnitude (db) Frequency (rad/sec) Simulation Figure 2.11: Pre-filter F 2 The controllers were tested in SimPowerSystems TM. Two simulations are presented. First, the simulation was carried out using average model which averages the switching cycle of the power electronics. The second simulation used detailed model of the switches. The simulations are shown in figure 2.12 where a step command was given to i q at t = 0.1 and to i d at t = 0.3. Notice how well the crosstalk were rejected at the stepping times. Moreover, the step responses of the average model was precisely those of the detailed model without switching harmonics. 30

39 i q (A) Average Model Detailed Model t (s) i d (A) 50 0 Average Model Detailed Model t (s) 150 v q (A) 100 Average Model Detailed Model t (s) 20 v d (A) Average Model Detailed Model t (s) Figure 2.12: SimPowerSystems TM Simulation 31

40 2.6 Current Coordinator In low speed operation, the main operating requirement for the machine is to generate torque equal to reference torque T, while satisfying the current constraint, i 2 q + i 2 d I, (2.30) where I is the stator current limit. On the i d i q plane, the current is represented by a circle with radius I. Figure 2.13 shows the current limit along with curves of level sets of T. Any (i d, i q ) on the curve T = T inside the circle will generate torque T. It is easy to see that there are infinitely many (i d, i q ) pairs to choose from. The question is, which one is the best? One common choice in practice is choosing the the pair that is smallest in magnitude. This criterion is know as Maximum Torque per Ampere (MTA). The MTA curve was also plotted in figure 2.13 as dash line. Every (i d, i q ) pairs on this curve achieves T with smallest current magnitude. As the angular velocity increases, the back EMF corresponded to φω in (2.1a) also increases. In order to maintain i q and hence the torque, v q has to be raised to counteract the back EMF. This approach, however, is valid only in lower speed region where the back EMF is small. When the machine operates at high speed, the back EMF can be too large such that an attempt to cancel it by raising v q may saturates 32

41 the stator voltage. The limit of stator is given by v 2 q + v 2 d 2 π V DC, (2.31) where V DC is the DC link voltage. An alternative method to cancel out the back EMF without saturating the voltage is to employ a technique widely known as Flux Weakening (FW) [23]. This method lets d-axis current i d to become more negative than what it is in MTA to cancel the effect of φω. To better illustrates the operation in Flux Weakening region, the voltage constraint is expressed in term of i d and i q (neglecting R for simpler presentation) as follow, i 2 q + ( L d L q ) 2 (i d + φ L d ) 2 ( 2V DC πωl q ) 2 (2.32) Equation (2.32) describe a set of ellipses centered at ( φ/l d, 0) on i d i q plane. It is easy to see that the size of the ellipses depends upon ω. At lower speed, the ellipses are larger than those of higher speed. Figures 2.14 shows these ellipses at different ω along with the torque curves and the current limit circle. Inside the ellipses are the feasible stator current at the corresponding speed. The current outside the ellipses are not feasible due to the saturation of the stator voltage. During low speed operation, the ellipse is big so that it is not necessary to take it into account, and the machine simply operate in MTA mode. When the ellipse shrink pass MTA operating point, the 33

42 current switch from MTA to Flux Weakening mode where i d becomes more negative and the operating point shifts to the left. The problem of the Flux Weakening is selecting the best (i d, i q ) pair among infinite choice such that T is achieved while the current trajectory remains inside the intersection between the voltage ellipse and the current circle. Prior non-table-lookup flux weakening technique generally fall into two categories, one relied on the solution of the optimization algorithm while the other did not. [28], [25] are the example of the first type. The approach was to partially solve the optimization problem for relationships between i d and i q in both mode, then PI controller was used to drive the current toward the solution. The other approach achieved flux weakening by using only feedback from the machine. Since the parameters of the machine are usually not involved in this technique, they tend to be more robust. An example of this method is [23]. 34

43 2.6.1 Proposed Current Coordinating Scheme The problem of determining the best (i d, i q ) in both MTA and Flux Weakening region can be formulate as a single optimization as follow, 1 min i q,i d 2 (i2 q + i 2 d), (2.33a) s.t. 3 2 pi q[φ + (L d L q )i d ] = T, (2.33b) i 2 q + i 2 d I 2, i 2 q + ( L d L q ) 2 (i d + φ L d ) 2 ( 2V DC πωl q ) 2. (2.33c) (2.33d) The machine operates in MTA mode when the constraint (2.33c) is not active at the solution, and operates in Flux Weakening mode if the constraint is active. The motivation behind the new current coordinating technique presented here comes from the fact that the constrained optimization problem in (2.33) is rather simple to solve iteratively. Since it is only a two-dimension problem with only three constraints, the problem can be solved efficiently using Active Set Algorithm. However, continuously solving the optimization problem online could be too computationally demanding. Hence, a new current coordinating technique based on iteratively solving (2.33) was proposed. The complexity of the optimization algorithm was reduced by taking advantage of the prior knowledge of the system. So, instead of solving the constrained optimization problem directly, the pair (i d, i q ) was simply driven toward 35

44 the solution whose location and property are pre-determined based on the knowledge of the system. The solution of the problem depends upon the operating mode of the system. In MTA, the goal is to find smallest (i d, i q ) that achieves T, so the voltage constraint (2.33d) is dropped. The algorithm starts by searching for the T = T curve along the gradient direction of T, denoted by T, shown in figure Once it is on the curve, then the algorithm starts moving along the curve to obtain the minimum current. The movement along the curve occurs in two steps. First, it moves along a line tangent to the T, denoted by ΨT, also shown in figure If the new point is too far away from the level set, then it moves back to the level set along the gradient direction. The algorithm continues until it reaches the optimal solution. It recognizes the solution by evaluating the dot product between the tangent of T and the gradient of the objective function. Once the dot product becomes smaller than some ɛ, the algorithm terminates. The flow chart explaining the algorithm is shown in Figure In flux weakening mode the constraint (2.33d) is active, hence the solution lies at the intersection of T curves and voltage ellipse. So, instead of minimizing the current, the algorithm searches for the intersection point. It starts the same way as in MTA by searching for T curves along T. If it ends up on the right of the voltage ellipse then it moves to the left along ΨT and vice versa. The flux weakening 36

45 algorithm has the same structure as the MTA algorithm. The flow chart explaining the algorithm is shown in Figure The switch between MTA and flux weakening was handled by mode selector. Switching from one mode to another is very important. If the switching from MTA to flux weakening happens too late, the control could saturate, while switching too soon would lower the machine efficiency. On the other hand, if the the switching from flux weakening to MTA occurs too late, the efficiency is reduced, while switching too early causes the control to saturate. The switching decision was based upon the current speed and stator voltage. Let ω l be the speed and ω u be the lowest and highest speed that the voltage ellipse intersects the MTA curve. If the speed is lower than ω l, then the machine will operate in MTA mode. When the speed is over ω u, then the system has to operate in flux weakening mode. When the speed is between ω l and ω u, the switching decision is made by comparing the stator voltage to the voltage limit in MTA mode and location of (i d, i q ) with respect to the MTA curve in flux weakening mode. The switching logic is shown in figure The voltage ellipses at ω l and ω u are shown in figure Simulation In order to demonstrate the behavior of the system over wide speed range without having to run the simulation for a very long time, the moment of inertia of the 37

46 rotor was reduced to 0.5 kg m 2 in both simulations. Rapid changes in ω due to the reduction of the inertia could deteriorates the current regulation of the system since the controllers were designed to reject low frequency disturbance. However the effect was not apparent in the simulations and the current regulation is still achieved. First, the result from the proposed field weakening algorithm was compared to the result from active set algorithm. Figure 2.20 shows the reference currents generated by the two algorithm and the time each algorithm took to find solutions. The reference currents are almost identical in both cases. The proposed algorithm was significantly faster than active set algorithm with average of seconds as opposed to seconds. Note that the proposed algorithm take considerably longer to converge when there are jumps in the reference torque because the solutions are far from the initial points. However, even the slowest convergence time of the proposed algorithm is still approximately ten times faster than the average of the active set algorithm. The currents, torque, and speed from the same simulation are shown in Figure Approximately at ω m =1000 rad/s, the system switched from MTA to flux weakening and the reference current started to change. The effect is easier to see on the contour plot in figure 2.22 where the pair (i d, i q ), started on the intersection of the T = 40 curve and MTA curve, moves along T = 40 curve. In the second simulation, negative reference torque was applied to the system. 38

47 Figure 2.23 shows the response of the system. Note that similar to the first case, the mode changes from flux weakening to MTA when the speed dropped below 1000 rad/s. This corresponded to the pair (i d, i q ) in figure 2.24 move from left to right along T = 40 and stopped once it reached the MTA curve. 2.7 Conclusion The design of the FESS was presented. An interior permanent magnet synchronous machine was selected for the FESS due to its high speed capability and high efficiency. Multiple control stages were employ to deal with the complexity of the IPMSM. First, field oriented control was used to transform the system into dq coordinate. Then, the system was feedback linearized to reduced the effect of the nonlinear coupling between d- and q-axis. MIMO QFT controller was designed taking the remaining couplings due to inexact cancellation from the feedback linearization into account. Other performance specifications for the controller are stability, tracking, and disturbance rejection. The d- and q-axis current components were coordinated by the current coordinator. During low speed operation, the currents were coordinated such that maximum torque per ampere was achieved. Flux weakening was also achieved by the coordinator in high speed regime. 39

48 Current Limit Torque Curves MTA Curves i q (A) i (A) d Figure 2.13: Torque contours and maximum torque per ampere locus on i d -i q plane 40

49 Current Limit Torque Curves Voltage Ellipses i q (A) i (A) d Figure 2.14: Voltage ellipses at ω = 1400, 1800, 2200, 2600 rad/s 41

50 Current Limit Torque Curves MTA Curves T I Ψ T 100 i q (A) i d (A) (a) Current Limit Torque Curves MTA Curves T I Ψ T i q (A) i d (A) (b) Figure 2.15: (a) Gradient and tangent vectors 42 (b) Zoomed-in gradient and tangent vectors

51 Start x = x new Τ Τ <ϵ N x new =x (Τ Τ ) Τ Y x new =x (ΨT I )ΨT N ΨT Ι <δ Y (i d *,i q * ) = x Figure 2.16: Flow chart for MTA operation Start x = x new Τ Τ <ϵ N x new =x (Τ Τ ) Τ Y x new =x (v V+α)ΨT N v V+α <β Y (i d *,i q * ) = x Figure 2.17: Flow chart for flux weakening operation 43

52 Current Limit Torque Curves Voltage Ellipses MTA Curves i q (A) i (A) d Figure 2.18: Voltage ellipses at ω l = 1785 rad/s and ω u = 2475 rad/s 44

53 Start ω < ω l Y ΜΤΑ N ω > ω u Y FW N FW Previous Mode MTA N ψτ Ι > 0 Y v v V π d + q < DC N FW ΜΤΑ FW Figure 2.19: Mode selector flowchart 45

54 i q (A) Proposed Algorithm Active Set t (s) i d (A) Proposed Algorithm Active Set t (s) Active Set Elapsed Time (s) Proposed Algorithm Elapsed Time (s) t (s) 3 x t (s) Figure 2.20: Comparison between two field weakening techniques 46

55 200 i q (A) i q i q * t (s) 100 i d (A) i d i d * t (s) 40 T (Nm) t (s) T T * 2000 ω m (rad/s) t (s) Figure 2.21: The system responses to T = 40 47

56 t=10 Current Limit Torque Curves MTA Curves Current Path Reference Current Path 100 i q (A) 0 t= i (A) d Figure 2.22: (i d, i q ) path under T = 40 48

57 100 i q (A) i q i q * t (s) 0 i d (A) i d i d * t (s) T (Nm) T T * t (s) 2000 ω m (rad/s) t (s) Figure 2.23: The system responses to T = 40 49

58 Current Limit Torque Curves MTA Curves Current Path Reference Current Path i q (A) 0 t=0 100 t= i (A) d Figure 2.24: (i d, i q ) path under T = 40 50

59 Chapter 3 FESS-Wind Farm Integration 3.1 Introduction In the chapter, the integration of the FESS to a 10 MW wind farm to mitigate power fluctuation is presented. The FESS was assumed to consist of forty 50 kw flywheels, thus has a rated power of 2 MW. Several steps was taken in the integration process. First an approximate model of the FESS was obtained. Then, the performance specification of the FESS was determined based upon the the the effect of the wind speed variation on the wind farm. Finally, a feedback controller was designed according to the specifications using QFT to guarantee satisfactory performance. 51

60 3.2 Model The torque generated by the FESS has nonlinear dynamic due to equation (2.2) which is nonlinear in i q and i d. Furthermore, the currents are also determined by the current coordinator which essentially is an optimization algorithm. Hence, determining the dynamic of the generated torque from (2.2) and (2.1) with the current controller is not an easy task. An alternate approach was to approximate the dynamics by a set of LTI systems that capture the torque dynamic over wide operating point. In order to get a better picture of the torque dynamic, several step responses of the FESS torque at ω = 2400 rad/s were plotted in figure (3.1). Notice that the step responses are very similar even though the steps are different in both magnitudes and starting points. This was confirmed when the step responses were normalized to 0 and 1 and were plotted together as shown in figure (3.2). Step responses at ω = 1600 rad/s and ω = 3200 rad/s were also plotted in figure (3.3) to see if the similarity would hold across all operating range. It turns out that the step responses are still very close to the ones at ω = 2400 rad/s. Since the current coordinator calculates the reference currents from the torque equation (2.2), any variation in the PMSM parameter would resulted in mismatches between the desired and generated torque. In order to see the effect that parameters variation have on the torque dynamic, normalized step responses of the FESS with different parameters were plotted in figure (3.4). From the plot, the predicted steady 52

61 Torque (Nm) Torque (Nm) Torque (Nm) time (s) time (s) time (s) Torque (Nm) Torque (Nm) Torque (Nm) time (s) time (s) time (s) Torque (Nm) Torque (Nm) Torque (Nm) time (s) time (s) time (s) Figure 3.1: Torque step response at ω = 2400 (rad/s) 53

62 Normalized Torque Time (s) Figure 3.2: Normalized torque step response at ω = 2400 (rad/s) Normalized Torque Time (s) Figure 3.3: Normalized torque step response at ω = 1600, 2400, 3200 (rad/s) 54

63 state errors were apparent Normalized Torque Time (s) Figure 3.4: Normalized torque step with parameters variations Finally, the variation of both parameters and speed was studied. The plot in figure 3.5 shows the step response when the parameters varied at ω = 1600, 2400, and 3200 rad/s. Notice the step responses varied only slightly as the operating point changed, while the effect of varying the parameters was more evident. The dynamic of the system is also affected by the current coordinator. The effect was modeled as pure delay since the system basically waits for the coordinator to find a solution. Test results have shown that the proposed algorithm has a worst case convergence time of approximately s seconds while the average time is around s. Worst case was assumed in this work, so the time delay was set to s. A set of linear systems was created to emulate the step responses of the original 55

64 Normalized Torque Time (s) Figure 3.5: Normalized torque step response with parameter variations at ω = 1600, 2400, 3200 (rad/s) system in figure 3.5. The linear approximations were written as a linear system with parametric uncertainties and a pure delay as follow P = kp 1 p 2 (s + p 1 )(s + p 2 ) e ds, (3.1) where k [0.88, 1.12], d = , p 1 [0.01, 0.012], and p 2 [0.001, 0.003]. Step response of the approximated system were plot together with those of the FESS in figure

65 Normalized Torque Time (s) Figure 3.6: Second order approximation of the torque dynamic 3.3 Control The power controller was designed to control the power of the FESS to mitigate the effect of wind velocity fluctuation on the output power of the wind farm. The variation of the power from the wind farm was modeled as an output disturbance of the system. Control objectives were determined from the above requirement, with the main objective was disturbance rejection in which the system has to be able to reject the disturbance from the wind. Other objectives were tracking and robust stability. The controlled system has to track the desired setpoint with zero steady states error with a good dynamic performance while maintaining robust stability. 57

66 3.3.1 Wind Farm Configuration The wind farm used in this work is a demonstration wind farm called Wind Farm - Synchronous Generator and Full Scale Converter (Type 4) from SimPowerSystems TM. The wind turbines are direct drive synchronous machines. The wind farm models five wind turbines as one aggregate wind turbine. This assumption lead to the worst case scenario for power fluctuation from the wind [29]. The generator-side and the gridside converters are connected by a 1100 V DC link. The wind farm was assumed to be an off-shore wind farm, and the grid-side converter was assumed to be an on-shore substation separated from the wind turbines. In order to reduce number of converter, the FESS was placed behind the grid-side converter, thus shared the DC link with the wind farm. The wind farm and FESS configuration is shown in figure Disturbance Modeling The power fluctuation was modeled as a disturbance at the output of the system. Behavior of the wind farm subjected to wind speed variation was studied by applying a step change in wind speed to the wind farm. The step change represents the worst case scenario of the disturbance. Figure 3.8 shows power from the wind farm under the step change. It is easy to see from the step response that mechanical torque at the turbine changed instantaneously with wind speed. However, the torque at the rotor of the generator did not. Upon further investigation, it turned out that the 58

67 Power Grid Grid-side Converter DC Link Wind Turbine Converter + v DC _ FESS Converter v a v b v c FESS Reference Power Power Power Controller T * Current Coordinator i q * Current Controller v q * dq to abc v abc * i d i q ω i d * i d i q ω v d * ε i d i q ω ε Figure 3.7: Off-shore wind farm configuration 59