A Simplified State Model for Wind Turbines

Transcription

1 A Simplified State Model for Wind Turbines Nuno M. L. Bernardo Student nº55250 of Instituto Superior Técnico Technical University of Lisbon Lisbon, Portugal Abstract This thesis presents studies relatively for the dynamics power systems, including renewable energy sources. The control of conventional generators (steam turbine and hydro turbine) and renewable generator (wind turbines and solar panels) require control systems of terminal active and reactive power, or in other words terminal voltage and frequency control. This work analyzes a dynamic model of induction machine, as well as model of the control system of active and reactive power. Simulations are done for interconnected networks having conventional generators and renewable generators. Therefore, with these simulations will be done an evaluated the dynamic behavior of the network and the dynamic behavior of the generators in relation to disturbances. Key Words: Active Power Control, Reactive Power Control, wind Turbine, Dynamic Behavior, Doubly Fed Induction Machine. I. INTRODUCTION The operation of an Electrical Power System (EPS) has to take into account criteria of economic and security, in satisfaction the demand of the consumer market. These criteria are sometimes difficult to achieve, especially when it is integrated in the system generating units that can t be in full control. Such as wind generators. In order to a proper functioning of the EPS is need that the value of the quantities that characterize it must be near the so called nominal values, including frequency and voltage. To keep these quantities close to their nominal values, it is necessary to control the frequency and voltage. With the increasing integration of wind energy in power grid the problems in voltage a frequency control become more demanding. Due to the variation of the wind, the wind generator introduces in the EPS various disorders at both frequency and voltage. The main objective of this paper understands the behavior of this type of generators, especially the doubly fed induction generator (DFIG) during a grid fault. For study this behavior it is consider a three buses electrical grid as shown in Erro! A origem da referência não foi encontrada.. The first generator is a synchronous generator and its dynamic behavior is explained in [1]. The other generator is a DFIG wind turbine. Using a set of differentialalgebraic equations was studied the system behavior during and after a grid fault. II. STATE MODEL A wind generator has as main function transform the kinetic energy from the wind into electric power. The wind turbine as a mechanical part is designed to provide mechanical power to the electrical generator. So the dynamic model of a wind generator can be separated on two different parts, the mechanical part (turbine) and the electrical parte (electrical generator). A. Wind Turbine The wind turbine model represents the relation between the mechanical power extracted from the turbine, P [W], the wind speed V [m/s], the electrical rotor speed ω [rad/s], and the pitch angle, β[degrees]. P = 1 2 ρac V (1) Where ρ is the air density [Kg/m ], A is the wind turbine swept area [m ] and C is the power coefficient. C is dimensionless and represent the portion of power that is actually drawn by the turbine rotor blades then C is the turbine aerodynamic efficiency. The power coefficient can be calculated through the equation (2), [2]. C (λ, β) = c c c λ β c β c e The intermediate parameter λ is given by: 1 λ = 1 λ c β c β + 1 (2) (3) Fig 1: Wind system with a DFIG

2 Fig 2: C curves for several values of β = 0 º, 6 º, 12 º, 18 º and 24 º The coefficient values c c are shown in Table I, these values vary from fixed speed wind turbine to variable speed wind turbine. Other parameter that is used for modeling wind turbines is the tip speed ratio, λ, which is defined as the ratio between the speed of a blade tip and the wind speed. λ = ω R V (4) Where ω is the wind turbine speed [rad/s] and R is the radius of the turbine. ω is related to ω through a proportional factor [1] as shown in (5). ω = p 2 Kω (5) but wounded so as to get access to the rotor windings and thus control the rotor currents. As shown is Fig 1Erro! A origem da referência não foi encontrada.the generator is connected to the grid in two different ways, directly through the stator and indirectly (through electronic converters) by the rotor. So the system is bidirectional because it allows power to flow either from the generator to the grid as form the grid to the generator. The electronic devices are typical two ac/dc converters using insulated-gate bipolar transistor (IGBT) linked by a dc bus. In this scheme converters decouple the rotating speed of the generator from the frequency of the grid enabling variablespeed operation of the wind turbine. In this model, the dc link is considered constant. With this the exchange of reactive power with the grid is completely performed through the stator of the generator. The rotor side converter controls rotor current in order to maintain the stator voltage and the maximum electromagnetic torque of the machine. The DFIG equations are derived from Park s equations for two axis d-q reference system, using generator convention. The dynamic behavior of the DFIG is represented by the following set of differential and algebraic equations. V = R I ω ψ + dψ V = R I + ω ψ + dψ V = R I (ω ω )ψ + dψ V = R I + (ω ω )ψ + dψ (7) (8) (9) (10) Where p is the poles of the electric generator and K is the gearbox conversion ratio. Fixed Speed Variable Speed Table I - Coefficients values c c c c c c c c c c c A torque expression represents the motion of the aerodynamic model of the rotor, the pitch angle and the shaft system. T = P = 1 ω 2 ρ C (λ, β) πr V λ Where R is the radius of the turbine and β is the pitch angle. B. Induction Generator Model The electric generator used in this work to equip the wind turbine is a doubly-fed induction machine (DFIG). The equations that describe the dynamic behavior of the DFIG have a similar structure to a squirrel-cage rotor induction machine. However in the DFIG the rotor is not short circuited (6) With V being the voltage (V), R the stator resistance, R the rotor resistance (Ω), ω and ω are the stator and rotor electrical frequency (rad/s), ψ is the flux linkage (Vs) and ds, dr, qs and qr correspond to d-axis stator and rotor and q-axis stator and rotor indices, respectively. Since the stator transients are very fast, when compared with the rotor transient, it is possible to neglect them. This correspond to consider that the stator flux is constant, so = 0 and = 0. Using this approach and substituting the linkage flux equations in order to eliminate the rotor currents, we obtain the following set of equations, which represent a voltage behind a transient reactance model. de de V = R i + X i + E (11) V = R i X i + E (12) = 1 T E + (X X )i sω E + ω ( ) V = 1 T E + (X X )i sω E + ω ( + ) V (13) (14)

3 Where X and X are the new stator reactance and the transient reactance respectively and T is the transient opencircuit time constant. These news elements are given by X = ω ( + ) = X + X (15) X = ω + + = X + X X X + X (16) T = R (17) Where = +, = + and are the stator, rotor and mutual leakage inductances, respectively. And the components of the internal voltage behind the transient reactance are defined as [1],[3]. E = ω ψ (18) E = ω ψ (19) At last the change in generator speed can be calculated using the generator equation motion. external order to reduce the output power, the normal operation is turned off and the β is defined to obtain the desired output power. When the external order is removed, the pitch angle control returns to the normal operation mode. From the analysis of the scheme of the Fig 3, we obtain the following set of equations that represent the operation of the pitch angle control. dx = K ω ω (21) dβ = K β T K β K β T T (22) dβ = β (23) Where x is and a state variable and is defined by x = β K ω ω (24) The values of the constants are K = 15, K = 5, K = 1 and T = 0.25s. To show the operation of the pitch control, β was increased its value from 0 degrees to 20 degrees at t=0 sec. 2H dω = T T (20) With H being the inertia constant and T is the electromagnetic torque in per-unit. The set of equations formed by the equations (13), (14) and (20) represent the transient behavior of the DFIG. III. PITCH ANGLE CONTROL The main propose of controlling wind turbine is limit the output power during strong winds, and thus protect the structure from damage caused by these strong winds. There are two operating regions depending on the wind speed. Below the rated power, the pitch angle is set to give the maximum power, in this work β is equal to zero in this situation because when β is zero the value of C is the maximum possible. The second region is operated when the wind is sufficiently high for the rated power, in this region the pitch angle control regulate the output power for the rated power by increasing the value of β that means decreasing the value of C. A generic control scheme for pitch angle is shown in Fig 4: Reference pitch angle Fig 3: Generic control scheme for pitch angle The angular speed of the wind generator is compared with a reference value that depends on the wind. The difference is applied in a proportional-integral (PI) controller that produce the reference value of pitch angle, β. The delay block and the integrator represent the time that takes to change the position of the blades. This is called normal operation, but there is other operation possibility. In case of receiving an Fig 5: Pitch angle As expected when we applied the step on β, β increase but not instantaneously, it takes about 4 seconds until β gets the desired value. As has mentioned before the increasing of β causes the decreasing of C, which in turn causes a decreasing the mechanical power and the electrical power.

4 the grid. These devices works with a frequency much higher than the grid frequency, so there have been the followings approaches, the IGBT s was considered ideal and the DC link is constant. The control of DFIG is based on the magnetic field orientation. If we chose the referential frame so that d-axis is oriented along the stator flux. Fig 6: Power coefficient Fig 9: Adapted reference frame The choice of the referential frame allows to do a simplifications that is V = V., thus obtaining a decoupling between the control of the reactive power and the control of the active power. The reactive power or voltage can be controlled by changing the d component of the rotor current, i, the active power can be controlled by changing the q component of the rotor current. Fig 7: Mechanical power A. Reactive power control Fig 10: Reactive power controller In Fig 10 show a scheme of a reactive power controller, which is a voltage source controlled by current and is composed by two proportional-integral controllers. The first PI aim is calculate the value of the reference of the d component of the rotor current, then the second PI, generate the value of the component d of the rotor voltage. Fig 8: Mechanical torque Using the pitch angle to limit the output power is an efficient method, but it is only use in case of extremely necessity because it leads to a very sharp increase in mechanical load on the structure that can reduce the useful lifetime. IV. DECUPLED CONTROL SYSTEM As shown in Fig 1, the generator is connected to the grid in the different ways. The stator is directly connected to the grid and the rotor is connected to the grid through a AC/DC/AC converter. The power electronic devices used in this converter are IGBT s (Insulated Gate Bipolar Transistor), these devices are chosen because allows bidirectional power flow and practically no lower-order harmonics are injected in The reactive power is compared with a reference Q, the result of this comparison goes to the first PI control which generates de value of I. After this the second PI will cause a change in V to compensate the change of the reactive power. The value of the Q is zero during all simulation time [3]. From the analyses of the scheme of the Fig 10, is obtained the set of equations which describes the functionality of the reactive control. V = K K (Q Q + x I ) + x (25) The d component of the rotor current is calculated as follows:

5 I = E X + X X I (26) In (25) x and x are state variables which are used to simplify the equation. dx = K Q Q (27) dx = K x + K Q Q I (28) The reference value of the d component of the rotor current is: I = x K Q Q (29) B. Active power control The q component of the rotor current is calculated as follows: I = E X + X X I (31) The state variables x and x can be calculate as in the previous case. dx = K P P dx = K x + K P P I (32) (33) The reference value of the q component of the rotor current is: I = x K P P (34) At last to complete the representation of the model is necessary to calculate the active and reactive power P = E I + E I R I + I (35) Q = E I E I X I + I (36) Fig 11: Active power controller The active power control has as main objective to extract the maximum power from the wind. As shown in Fig 12 for each wind speed there is a point at which is possible to extract the maximum power. That point can be achieved by varying the turbine speed. The turbine speeds can be controlled by changing the pitch angle or changing the q-axis rotor voltage. V. ANALYSIS MODELS The electrical grid used in this work to evaluate the dynamic behavior of the DFIG is similar to the one described in Fig 13. The grid has 3 buses and 2 generators, one synchronous generator (MS) and a DFIG generator (MIDA). Fig 12: output power vs speed This controller is quite similar to the previous one. There are two PI controllers, the first PI calculate the value of the reference of the q component of the rotor current, then the second PI, generate the value of the component q of the rotor voltage. P is the reference value of the mechanical power The dynamic equations which represent the controller are: V = K K (P P + x I ) + x (30) Fig 13: Equivalent Scheme of the Network with 3 Buses The model adopted for de synchronous generator is a sub transient model that is explained in [5], [6]. This generator is equipped with a voltage regulator and primary frequency control. The simulation was divided into two parts, in both parts is applied a short circuit on the third bus at t=0 seconds when the grid was in balance. This short circuit lasts 0,3 seconds, after

6 this time the short circuit is cleared, returning the grid to is original configuration. In the first part was not used the pitch control, i.e. the pitch angle β is kept to zero throughout the simulation. In second part was already applied the pitch angle control. In order to be able to implement the models of the two generators and their controllers is essential to know the value of voltage and current of each bus. So it is necessary to make an initial calculation of power flow by Newton-Raphson model to ascertain the initial value of voltage and current. After knowing the initial conditions, it is already possible solving the set of algebraic and differential equations referring to the generators. After each interaction is necessary recalculate the new voltage and current values. To do this it was used the method proposed in [6] which the real and the imaginary part of voltages and currents were calculated separately. It should also be noted for the synchronous generator was decided to use a high inertia constant, aiming that the synchronous generator behaves like an infinite grid. Another effect that is caused by the short circuit is the current increase, by (38) when voltage decrease the current increase inversely proportional. I = P V (38) A. Model without pitch angle control Because during the whole simulation time the wind speed remains constant and lower than the maximum speed, the pitch angle is kept to zero during the entire simulation. This is justified by the scheme of Fig 3. Where in the value of β only changed when the generator speed exceeds the reference speed. The results obtained are in the following figures. Fig 15:Voltage at bus1 Fig 16:Current at bus1 Fig 17: Voltage at bus2 Fig 18: Current at bus2 Fig 14: Generators frequency, the blue one is the frequency of the synchronous generator and the green one is the frequency of wind turbine The short circuit causes a voltage drop, as shown by the Fig 15, Fig 17 and Fig 19. The voltage drop will cause a decrease of active power as we can verify by the equation (37. P = V Z (37) Where P is the active power, V is the terminal voltage and Z is the load impedance. The drop at terminal voltage leads to a decrease of stator and rotor flux which causes a reduction of electromagnetic toque and active power as shown in Fig 21, Fig 22 and Fig 23. However the drop of mechanical power is slower than electric power. This effect will cause an imbalance which will lead to an acceleration of the generator, i.e. an increased of frequency. Fig 19: Voltage at bus3 Fig 20: Current at bus3

7 Fig 21: Mechanical power and active power in synchronous generator Fig 24: Electromotive force in synchronous generator Fig 25: Electromotive force in DFIG At the moment when short circuit occurs, the reactive power, Fig 26, increases which represent the amount that is injected into the grid in order to contribute to voltage recovery and it results in system stabilization. Fig 22: Mechanical power and active power in DFIG Fig 26: Reactive power in DFIG The following figures represent the progress of the components d and q of the rotor voltage and rotor current of DFIG. Fig 23: Mechanical and electromagnetic torque in DFIG The electromotive forces also suffer a sharp drop at the time of short circuit. After this the electromotive forces begins recovering to compensate the voltage drop. Fig 27: Components d and q of DFIG rotor voltage When the active power decreases the q component of the rotor voltage increases (30). This is because when active power decreases the first PI controller of Fig 11 will generate an increase of the reference signal I and the second PI

8 controller will cause an increase of V. On the other hand when the active power increase, the first PI controller will cause a decrease of I which will lead to decrease of V. The variation of V will allow the electromotive force recovery, which in turn will cause the recovery of the active power. On the synchronous generator the electromotive force compensation is done through the excitation control which will not be explained here because it is beyond the scope of this work. As previously stated the reactive power is controlled by the d component of the voltage and current of the rotor, V and I respectively. The control of reactive power is similar to the control of active power. Although the mechanical power being constant, the mechanical torque is not, because the mechanical torque is inversely proportional to the generator speed ω. Fig 30: Mechanical torque of DFIG B. Model with pitch angle control In this case despite the wind speed be constant throughout the simulation, it is intended that the value of the pitch angle varies in order to limit the output power. For this the pitch angle control has been slightly modified so that the angle β can vary without the generator speed exceeds its maximum value. This change is shown in Fig 31 [2]. Fig 28: Reactive power in DFIG At the moment short circuit occurs the reactive power increases, which represents the amount of reactive power that is injected into the grid to contribute for voltage recovery. Due to increased reactive power the first PI controller of Fig 10 will decreased the signal I and the second PI controller causes a decrease of V, and thus compensate the increased reactive power. Then when there is a decrease of reactive power, I increases which leads to an increase of V. This possibility of variation of V and I, DFIG does not need any extra element to compensate the reactive power. As mentioned earlier in this section, during the simulation the value of the pitch angle is constant and equals zero. It was chosen this value β = 0 because according to (2) and (3) it is the value for which the power coefficient is maximum, so the mechanical power of DFIG is always constant as shown in Fig 29. Fig 31: Generic control scheme for pitch angle The operation of this pitch angle control is equivalent to the described in section III, but the angle instead of varying with the generator speed varies with the active power. Each time the active power exceeds the reference value the pitch angle increases and consequently the mechanical power decreases as well as the active power. In this case is used the mechanical power extracted from the wind as reference value for the active power. The results obtained were as follows. Fig 32: Mechanical power and active power in DFIG Fig 29: Mechanical power of DFIG

9 The progress of the mechanical torque is inversely proportional to the generator speed as is explicit in equation (39). T = P ω (39) Fig 33: Mechanical and electromagnetic torque in DFIG When the short circuit occurs, the active power decays faster than the mechanical power which causes an increase in generator speed and consequently a drop in torque. By observing the Fig 35 it is noted that after the torque recovery there is a new decrease, this decrease was due to the increase of the pitch angle and the decrease of the mechanical power. After the short circuit β returns to its initial value, as do the mechanical power, leading the mechanical torque to the value it had before the short circuit. Regarding frequency there is also a decrease in frequency peak of wind generator, but this decrease is very small because the difference between the electromagnetic and mechanical torque is very similar both in case where is used pitch control as in the case where is not used pitch control. Fig 34: Pitch angle of the DFIG As shown in Fig 34 when the generated active power by DFIG is higher than the mechanical power, the pitch angle control causes an increase of β in proportion to the increase of the difference between the two powers. Thereafter β returns to its initial value when the active power returns to be smaller than or equal to the mechanical power. The increase of β will cause a decrease in mechanical power and thus a reduction of the generator active power, this decrease in noticeable when comparing Fig 32 to Fig 22. With the application of the pitch angle control can be reduced the maximum value of the active power after the short circuit as well as a faster stabilization of its final value. Another parameter which suffers change with the inclusion of the pitch angle control is the mechanical torque as can be seen in Fig 35. Fig 35: Mechanical torque of DFIG Fig 36: Generators frequency, the blue one is the frequency of the synchronous generator and the green one is the frequency of wind turbine The pitch angle control method is not usually used to control the output power because a constant variation of the pitch angle causes a high wear on both the structure of the wind generator as the positioning mechanism reducing its usable lifetime. On the other hand there is always an interest to maximize the energy available on the wind and increasing β removes this possibility, as explained in section III. Whenever there is an increase of β, the power coefficient C decreases and the energy that is harnessed form the wind decreases as well. Due to it the increased of β is only used in cases of extreme necessity as the existence of strong wind capable of causing damage in the structure or if there is an order from the grid operator to stop the wind generator. VI. CONCLUSION Through this study was possible to understand the operation of a wind generator mainly how to control the active power (or generator speed) and reactive power (or terminal voltage). One of the great interests of simulates the dynamic behavior of an EPS with wind generators is the behavior of the active and reactive power controllers, and it was verified that

10 control of a wind generator can be made in a separate mode. In other words, the active power and the reactive power can be controller independently one form another if it is used the control method by stator flux orientation, which is aligning the d axis of the rotating referential dq with the stator flux reference. It is concluded then DFIG wind generators is a strong generator, able to maintain the balance between the mechanical and electromagnetic torques after the fault, which means it does not lose stability or cause instability to the system. As injects reactive current during the short circuit, it will allow a quicker voltage recovery and consequently improves the stability of the EPS. The active power compensation is achieved by varying the q component of the rotor current I and the reactive current compensation is done by varying the d component of the rotor current I. This makes unnecessary the use of auxiliary systems to compensate the reactive power. VII. DFIG generator R = pu R = pu X = pu X = 3,5859 pu X = pu H = 4.5 s s = 2% Ki s =0,001 Kp s =0,001 APPENDIX Synchronous generator X = 2,24 pu X = 1,60 pu X = pu X = pu R = 0 pu H = 100 s T = 5.8 s T = 0.62 pu R = 0.01 pu VIII. REFERENCES [1] - Héctor A. Pulgar-Painemal, Peter W. Sauer, Power System Modal Analysis Considering Doubly-Fed Induction Generators, IEEE, August, [2] - V.Akhmatov, Analysis of dynamic behavior of electric power system with large amount of wind power, PhD dissertation, Elect. Power Eng. rted-dtu, tecnical univ. Denmark, Kgs. Kyngby, Denmark, April, [3] - Marcus V. A. Nunes, J. A. Peças Lopes, Hans Helmut Zurn, Ubiratan H. Bezerra, Rogério G. Almeida, "Influence of the Variable-Speed Wind Generators in Transient Stability Margin of the Conventional Generators Integrated in Electrical Grids, IEEE, December 2004 [4] - Sucena Paiva, J. P. Rede de Energia Eléctrica Uma análise sistémica. Lisboa: IST Press, [5] - H.W. Dommel, N.Sato, Fast Transient Stability Solutions, Bonneville Power Administration Portland, Oregon.