Generic Wind Turbine Generator Model Comparison based on Optimal Parameter Fitting. Zhen Dai

Transcription

1 Generic Wind Turbine Generator Model Comparison based on Optimal Parameter Fitting by Zhen Dai A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2014 by Zhen Dai

2 Abstract Generic Wind Turbine Generator Model Comparison based on Optimal Parameter Fitting Zhen Dai Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2014 Parameter fitting will facilitate model validation of the generic dynamic model for type-3 WTGs. In this thesis, a test system including a single 1.5 MW DFIG has been built and tested in the PSCAD/EMTDC environment for dynamic responses. The data generated during these tests have been used as measurements for the parameter fitting which is carried out using the unscented Kalman filter. Two variations of the generic type-3 WTG model (the detailed model and the simplified model) have been compared and used for parameter estimation. The detailed model is able to capture the dynamics caused by the converter and thus has been used for parameter fitting when inputs are from a fault scenario. On the other hand, the simplified model works well for parameter fitting when a wind speed disturbance is of interest. Given measurements from PSCAD, the estimated parameters using both models are indeed improvements compared to the original belief of the parameters in terms of prediction error. ii

3 Acknowledgements First and foremost I wish to express my thanks to Professor Zeb Tate for his invaluable guidance, advice and patience. I also want to express my gratitude to my thesis committee members: Professor Peter Lehn, Professor Reza Iravani and Professor Jason Anderson, for their insights and suggestions. I would like to thank the members of Energy Systems Group at University of Toronto for their selfless help. Finally, I d like to thank my grandmother and my parents. iii

4 Contents 1 Introduction Background and Motivation Model structure Algorithm Criterion Parameter Estimation Procedure Objectives Outline Generic Models of Type-3 Wind Turbine Generators An Introduction To The Type-3 Wind Turbine Generator Generic Wind Turbine Generator (WTG) Models Type-3 Generic WTG Models Power Flow in a DFIG Aerodynamic Model Aerodynamic Model with C p Curves - Two-Dimensional Model Aerodynamic Model without C p Curves - Linear Model Comparison of the Two Models The Turbine Model iv

5 2.4 The Pitch Control Model Generator and Converter Model Generator and Converter Model Converter Controller The Scope of the Models Test System and Model Performance Test System Model Performance Type-3 WTG Generic Models for Parameter Fitting Type-3 WTG Detailed Transient Stability Model M Reduced-order Type-3 WTG Transient Stability Model M Discrete Type-3 WTG Generic Models Model Testing Equilibrium and initialization Model test results System Identifiability 72 5 UKF Parameter Estimation Parameter Estimation UKF Parameter Estimation Annealing Method Forgetting-Factor Method Robbins-Monro Method Parameter Estimation with ReBEL toolkit Parameter Estimation Results Estimation results and analysis Summary v

6 6 Conclusion Summary and Conclusions Contributions Future Work Appendices 106 A The Torque Control Model 107 B System Parameters 109 Bibliography 112 vi

7 List of Tables 2.1 Notations of the Aerodynamic Model [26] C p curve coefficients α i,j [26] Parameters of the Turbine Model [39] Parameters of the Pitch Control Model [39] Base Quantities Used in The Converter Controllers Parameters of the RSC Converter Control Model [39] Variable Notation Parameter Notation and Value Prediction Error between Models (both compared to M 0, Case 1) UKF parameter estimation Estimation Results M 1 Fault Scenario Estimation Results M 1 Fault Scenario Compared to M Estimation Results M 2 Wind Speed Jump Estimation Results M 2 Wind Speed Jump Compared to M B.1 Transmission Line Parameters B.2 Parameters of the Induction Machine B.3 Parameters of the Step-up Transformer B.4 Parameters of the Transformer (stator-converter) vii

8 List of Figures 1.1 Parameter Estimation Procedure Type-3 WTG Model Power Flow in the DFIG (Motor Convention) Power Coefficient Curves Simplified Aerodynamic Model Pitch Angle vs. Wind Speed Single-mass Turbine Model Pitch Control Model Stator-flux Model for the DFIG Grid-Side Converter Control Model in PSCAD (M 0 ) Torque Control Model Turbine Speed Setpoints (pu) vs. Real Power (pu) Rotor-side converter control in PSCAD (M 0 ) Test System with Single Type-3 WTG Real Power Under Different Wind Speeds: Results of Two Models Dynamic Response During Fault P elec ω ref curve - data fitting results System Response M 1, Wind Speed Change System Response Comparison Using M 1, M 2, noise free viii

9 3.4 Comparison of System Response using Case Comparison of System Response using Case System Response During 20% Voltage Drop, Model M 1 T g = System Response During 20% Voltage Drop, Model M I p and I pcmd during 20% voltage drop, model M I p and I pcmd during 20% voltage drop, model M 1 from 29.6 sec to 31.4 sec 71 ix

10 Abbreviations CIGRE DFIG EKF EPRI IEC L-G L-L LVRT ML MAP MMSE MSE NDE NERC ODE PCC PWM RLS RSC WECC WTG UKF Council on Large Electric Systems Doubly Fed Induction Generator Extended Kalman Filter Electrical Power Research Institute International Electrotechnical Commission Line-to-Ground Line-to-Line Low-Voltage Ride-Through Maximum Likelihood Maximum A Posterior Minimum-Mean Squared Error Mean Squared Error Nonlinear Differential Equation North American Electric Reliability Corporation Ordinary Differential Equation Point of Common Coupling Pulse-Width Modulation Recursive Least-Squares Rotor-Side Converter (of DFIG) Western Electricity Coordinating Council Wind Turbine Generator Unscented Kalman Filter x

11 Chapter 1 Introduction 1.1 Background and Motivation Dynamic models of wind turbine generators are vital since wind generation is gaining increasing importance throughout the world. Much research has been done in order to find both accurate and simple dynamic models of wind plants and their components. One important characteristic of such models should be generality. Namely the models should be general enough to represent a type of wind turbine generator regardless of its manufacturer. Undoubtedly introducing such generic models will facilitate the development of wind energy technology since exchanging research results becomes possible without disclosing specific manufacturer s data. Many organizations (e.g. WECC, IEC, CIGRE, etc.) have separate working groups and task forces for developing generic models of wind turbine generators. They are also dedicated to standardize the generic models. So far the generic models have been validated by comparing to the real-world measurements from different products. However, the model parameters may vary depending on the vendor or rating of the units. It will be very helpful if the parameters can be identified using device measurements like active and reactive power at the wind turbine unit. Based on the estimation results, different model structures can be compared and chosen. 1

12 Chapter 1. Introduction 2 There have been studies in terms of parameter estimation of DFIGs [4, 35]. However the focus of these papers is not the complete generic dynamic model. For example, they normally exclude the pitch control. So far not much research has been done in terms of parameter fitting using the complete generic dynamic model. EPRI has developed a software tool to facilitate model validation for type-3 and type-4 WTG [9]. It will optimize the user s initial guess of parameters given terminal voltage (of the WTG), active and reactive power measurements. Despite the limitation of the software, no information from the parameter estimation is given since the algorithm is not disclosed. Therefore the goal of this thesis is to compare different type-3 WTG (also known as the DFIG configuration) model structures based on parameter estimation results. Three model structures are involved in the thesis, the time-domain model M 0 and two transient stability models M 1 and M 2. M 1 is a detailed transient stability model and M 2 is a simplified version based on M 1. First the sophisticated time-domain model M 0 will be built in PSCAD to generate sample data for the purpose of identification of transient stability models M 1 and M 2. The following paragraphs will briefly introduce the source of the models, the parameter estimation procedure and how to evaluate the models, also known as model structure, algorithm and criterion of identification Model structure The development of generic WTG models is the work of many organizations and manufacturers in collaboration. WECC has been the leading force on North America [38]. So far two versions of four major types of generic WTG models have been purposed (in literature, the two versions are sometimes called two phases). The first phase of models have been implemented in several software tools including PSS/E. Recently, the second phase has been prepared by EPRI [31, 32]. Among the four types of WTGs, type-3 is

13 Chapter 1. Introduction 3 the dominant model installed in the market [3]. Type-3 WTG (the variable speed wind turbine generator with partial-scale converter) is well known as the DFIG (doubly fed induction generator) configuration. The advantages of type-3 WTGs includes separate control over active and reactive power, ability to be magnetized from the rotor circuit, capability of delivering reactive power to the stator/grid side and low cost due to the small size of the converter [3]. This thesis is based on the first phase of the type-3 WTG generic models purposed by WECC. Given some assumptions, the type-3 WTG model can be described by nonlinear differential equations which can be easily implemented for the purpose of parameter identification. The main assumption made in this thesis is that no limiters are active. The test system in the thesis consists of a single 1.5 MW DFIG unit (instead of a lumped model for wind plant) therefore the supervisory control at the plant level is not considered. It is also worth mentioning that in the generic model the DC link and related protections are not modelled. Thus they are not considered in our transient stability models either Algorithm As mentioned earlier, although the generic models of type-3 WTGs have been widely accepted, the parameters vary depending on the vendors. In search for the best parameters given certain criterion (which will be introduced in the next paragraph), various parameter estimation techniques are available. However, a crucial question before application of the identification procedure is determine if the system is identifiable. Given the particular structure of the type-3 WTG model, we are able to gain some insights about the identifiability of its subsystems. There are two commonly used estimation techniques for nonlinear estimation, the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). The EKF uses the first-order linearization of the model whereas the UKF uses the original nonlinear

14 Chapter 1. Introduction 4 functions of the model. Therefore, the EKF can only work when the Jacobian exists and it only works well when the model is not highly nonlinear. By contrast, the UKF is accurate when highly nonlinear model is involved with comparable computation cost to the EKF. In this thesis, the UKF is used for parameter estimation Criterion The criterion, model structure and the algorithm can not be separated. The criterion can be expressed as a cost function. Normally the cost function is the expected squared error between the desired measurements and the estimates (also known as prediction error). Since the WTG unit can be viewed as a controlled current source, the prediction error of the injected current will be used for evaluating the models. In this thesis, the measurements are generated using PSCAD simulation. The test system in this environment is more detailed compared the transient stability models used for parameter estimation. For example, the converter is implemented in PSCAD and the detailed aerodynamic model has been used. Given the measurements, the two transient stability models M 1 and M 2 will be tested for parameter estimation in MATLAB with the help of ReBEL toolkit [2]. The estimated parameters will be evaluated in terms of prediction error of the injected current. By comparing the measurements from the time-domain model M 0 and predictions given by M 1 and M 2 with estimated parameters, we are able to determine which candidate model structure should be chosen. 1.2 Parameter Estimation Procedure The parameter estimation procedure used in the thesis can be summarized in Fig. 1.1 and illustrated as follows:

15 Chapter 1. Introduction 5 1. Generate inputs and states. For parameter estimation, inputs and states are needed. Inputs u can be generated using M 0. However, u can also be given by M 1 and M 2. This is what is normally done when only one model structure is available or when the structure is believed to be correct and sufficient. Then the inputs are passed to numerical integration process using model M i (i can be 1 or 2 in our case). Then the states will be calculated based on the original guess of the parameter θ A1. 2. Then we pass the measurements to the UKF and get estimate θ B. 3. We evaluate the estimates by comparing M i (θ A ), M i (θ B ) to M 0 (θ A ) with respect to the prediction error. Based on the results, candidate model structures can be compared and chosen. Note that this procedure can be extended to evaluate multiple candidate model structures, i.e. i in M i can be any integral greater than Objectives The objectives of this thesis are: 1. Implement the detailed test system M 0 which consists of a type-3 WTG unit in PSCAD/EMTDC environment. Test the system under different circumstances (wind speeds, faults). And record the signals as measurements. 2. Develop models M 1,M 2 for parameter estimation. 3. Use the measurements from the PSCAD and estimate the parameters with the detailed transient stability model M 1 and the simplified model M Evaluate the estimated parameters by comparing prediction errors.

16 Chapter 1. Introduction 6 Figure 1.1: Parameter Estimation Procedure

17 Chapter 1. Introduction Outline In the first part of the thesis, the type-3 WTG generic model purposed by WECC will be discussed. The whole model consists of five parts: the aerodynamic model (Section 2.2), the turbine model (Section 2.3), the pitch control model (Section 2.4), the generator/converter model and the converter controller model (Section 2.5). Three model structures will be discussed: M 0, M 1 and M 2. M 0 is the one implemented in the PSCAD/EMTDC environment and it will serve as the benchmark. This system was tested under different wind speeds and fault scenarios and the results are presented in Section 2.6. M 1 and M 2 are dynamic models represented by differential equations and were used for parameter estimation. The second part of the thesis focuses on parameter estimation. Chapter 3 gives the model descriptions of M 1 and M 2. Then both models will be discretized so that the UKF, or any other recursive methods, can be applied. The UKF algorithm for parameter estimation is covered in Chapter 5, along with a short introduction of the software toolkit ReBEL [2]. In addition, the parameter estimation results and analysis are also summarized in this chapter. Lastly, the conclusion and future work are provided in Chapter 6.

18 Chapter 2 Generic Models of Type-3 Wind Turbine Generators 8

19 Chapter 2. Generic Models of Type-3 Wind Turbine Generators An Introduction To The Type-3 Wind Turbine Generator Generic Wind Turbine Generator (WTG) Models An imminent need for dynamic models of wind power plants and their components has arisen as a result of the rapid growth of wind generation. Most dynamic models have been developed by manufacturers and they usually require the users to follow certain nondisclosure agreements. Generic WTG models, on the other hand, are non-proprietary by nature. Therefore such generic models enable exchanging data between system operators, manufacturers and researchers. Many working groups from different organizations are dedicated to develop and validate generic WTG models. The International Electrotechnical Commission (IEC) and the Western Electricity Coordinating Council (WECC) are two leading organizations on this topic. The IEC Technical Committee 88 Working Group 27 started the standardization work: the IEC since So far, the first committee draft for the dynamic wind turbine model has been presented. However, the first printed edition is still being prepared. The standard for wind plants is not expected to be available until Meanwhile, the Wind Generation Modeling Group (WGMG) of the WECC initiated the development of its generic WTG models for four major types [38]. The first generation of the generic WTG models have been developed and implemented in major commercial simulation tools, for example the GE PSLF and the Siemens PTI PSS/E. There are other detailed reports (technical brochure, special reports, subcontract reports) available prepared by the Council on Large Electric Systems (CIGRE) [6], the Electrical Power Research Institute (EPRI), the National Renewable Energy Laboratory (NREL) and the North American Electric Reliability Corporation (NERC) [28]. These groups and many industry cooperations (such as GE, ABB, Siemens, etc.) have been working in collab-

20 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 10 oration on the development, the improvement and the validation of the WECC generic WTG models. Also, in parallel with the work of WECC WGMG, the Working Group on Dynamic Performance of Wind Power Generation under the IEEE Power System Dynamic Performance Committee is also focusing on the generic dynamic modeling of the wind plants. A joint status report published in 2011 by the WECC and the IEEE [8] summarizes the first generation of the generic WTG models, their development and specifications briefly. Recently, the second generation of the type-3 and the type-4 WTG models have been proposed through the WECC Renewable Energy Modeling Task Force (REMTF) in reports prepared by the EPRI [31, 32]. Compared to the first generation, the second phase of modeling focuses on improving and completing the control strategies. However, the main structure of all parts remain the same. In order to form a relatively simpler model for parameter fitting, the first generation of the type-3 WTG model is used in this thesis Type-3 Generic WTG Models Each WTG unit in the market can be classified as one of four basic types based on their respective topologies and the grid interfaces. - Type 1: Fixed-speed induction generators - Type 2: Variable-slip induction generators with variable rotor resistance - Type 3: Doubly-fed induction generators - Type 4: Full-converter variable-speed generators Between these four types, the type-3 and the type-4 units are most commonly sold and installed. In this thesis, we focus on the type-3 WTG model which is characterized by the doubly fed induction generator.

21 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 11 A Doubly Fed Induction Generator (DFIG) or a Doubly Fed Asynchronous Generator (DFAG) system consists of two major parts: a wound rotor induction machine and a power electronic converter. The rotor is fed at slip frequency from the back-to-back inverter to the grid thereby providing a wide speed range of ±20% ±30%. In all, the application of power electronic control in the type-3 WTGs increases the generator operation speed range compared to the type-1 and the type-2 WTGs. The converter size is smaller (20% to 30% of the generator rated power) thus more budget-friendly and efficient compared to the type-4 [3]. The first generic type-3 WTG model prepared by WECC consists of four major components: the generator/converter model, the converter control model, the pitch control model and the wind turbine model. During the second phase of modeling, an updated and slightly detailed model connectivity has been presented [31]. Specifically, the converter control model can be replaced by the active (or torque) control model and the reactive control model. Meanwhile the wind turbine model can be represented by the aerodynamic model and the drive-train (or shaft-dynamic) model. The updated model is better in terms of clarity, especially when users are not interested in implementing all control blocks or intending to make changes to a certain block without altering the connectivity. Given this update, Fig. 2.1 shows the block diagrams of the type-3 WTG generic model in details. Note that in Fig. 2.1 blocks circled by dashed lines mean they are not included. For clarity, here is a list of the models which will be discussed in details in subsequent sections: 1. Aerodynamic model 2. Drivetrain model: one-mass model 3. Pitch control model 4. Electrical (converter) controls: torque control model, reactive control model

22 Chapter 2. Generic Models of Type-3 Wind Turbine Generators Generator/converter model Nevertheless, essentially the model components are the same despite the connectivity or name preferences in different references. Though the converter/generator model is described and included in the generic model, it will be unnecessary to implement it in the PSCAD/EMTDC environment. Since in order to implement a working test system, the converter and the generator can be modeled in details using the PSCAD/EMTDC components. Also similarly, the one-mass turbine model has been included in the generator model. Considering the amount of overlapping models, we will not list the same ones used for PSCAD simulation. From now on, if the model is only used for PSCAD simulation and not included in the generic WTG model, the section title will be followed by M 0. Snapshots from PSCAD will also be provided Power Flow in a DFIG The type-3 WTG is also known as the DFIG configuration. Because of the converter configuration, the power flow in a type-3 WTG is conceptually different compared to the other types. With the four-quadrant converter connected to the rotor, power can be extracted from or injected to the rotor. In other words, a DFIG can operate as a generator under supersynchronous conditions or subsynchronous conditions. The relationship between the rotor speed and power is showed in Fig P r is the power delivered by the rotor and P s is the power delivered to the stator. Hence the total supply power (neglecting the losses) yields P g = P s P r = P s (1 s). Here the directions are chosen to follow the motor convention (motoring positive) in accordance with [25]. Since in the PSCAD/EMTDC the motor convention is also used by default for the induction machine, the torque of the DFIG will be negative during the simulation. However, the directions of the power are not necessarily the same as defined in Fig It is more convenient with the directions in Fig thus P gen will be positive and meanwhile P m will fulfill the request of the PSCAD/EMTDC.

23 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 13 Figure 2.1: Type-3 WTG Model

24 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 14 It is possible that the rotor speed drops below the synchronous speed if the wind speed is relatively low. For the type-1 WTG, the induction machine will act as a motor as soon as the slip becomes positive. However, for a type-3 WTG, the rotor speed can vary within the range ±20% ±30%.

25 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 15 Figure 2.2: Power Flow in the DFIG (Motor Convention)

26 Chapter 2. Generic Models of Type-3 Wind Turbine Generators Aerodynamic Model The aerodynamic model purposes to calculate the turbine mechanical power based on the wind speed V wind, the blade pitch angle θ and the turbine rotor speed ω t. The relationship is defined as: P mech = 1 2 ρav wind 3 C p (λ, θ) (2.1) ω t λ = V tip = K b (2.2) V wind V wind where C p is the power coefficient and a function of the blade pitch angle θ and the tipspeed ratio λ. λ is defined as the ratio of the rotor blade tip speed to the wind speed in Eqn A is the cross-sectional area or the swept area, which is either given directly in the data sheets of the wind turbines or computed approximately using πr 2. R is the radius of the turbine i.e. the blade length. Eqn. 2.1 indicates the complexity of the aerodynamic model highly depends on the representation of the power coefficient C p. Three representations of the C p curves have been presented in [33] by Price and Sanchez-Gasca. As far as the transient stability study is concerned, the two simplified models (the two-dimensional model and the linear model) have been validated in comparison with the most detailed three-dimensional model (the three dimensions are λ, θ and C p ) [33]. As the paper mentioned, such detailed and accurate three-dimensional representation is not required for transient stability analyses. Also it s unpractical to assume this model is available to system analysts. Therefore only the two simplified models will be described in subsequent paragraphs. For clarity, Table 2.1 shows the notations for the variables and the constants used in the aerodynamic model. Some variable, for example the pitch angle θ, will also show up later in other models. The notations of these variables remain consistent throughout. Note that the unit of P mech can either be W or pu. Conventionally the per-unit system is preferred in order to facilitate analysis of the whole system. The per-unit system for the machine (in our case, induction generator) and the network are well established.

27 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 17 Table 2.1: Notations of the Aerodynamic Model [26] Notation Value Unit Description θ - degree Blade Pitch Angle λ - - Tip-speed Ratio C p - - Power Coefficient ω - pu Generator Rotor Speed ω t - pu Turbine Rotor Speed V wind - m/s Wind Speed P mech - W or pu Turbine Mechanical Power A πr 2 m 2 Swept Area K b Constant ρ kg/m 3 Air Density R 35.5 m Radius of Turbine However, it should be pointed out that the aerodynamic model (Eqn. 2.1) is described in SI forms, in which case P mech should be the actual value of the turbine mechanical power in W. Alternatively, for a given WTG unit, the mechanical power in per unit can be computed directly provided proper scaling of the constant A or ρa. For example, a typical value of 1/2ρA for the 1.5 MW GE WTG is In this case, P mech is in per unit since the rated power has been considered in the constant Aerodynamic Model with C p Curves - Two-Dimensional Model A well-known set of C p curves is presented in [26, 33] for the GE Energy s 1.5 MW and 3.6 MW wind turbines initially. The curves can be reproduced using a polynomial approx-

28 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 18 Figure 2.3: Rotor Power Coefficient C p vs. TSR λ Using Polynomial Approximation imation in Eqn The coefficients used are listed in Table. 2.2 [26]. Thus the curves were implemented in the PSCAD/EMTDC environment using a FORTRAN-based user defined component. See Fig. 2.3 for the approximation results. The accuracy of the polynomial approximation has been validated since satisfactory results in terms of the power curve have been achieved in Section In other words, this model is able to provide correct C p given the right θ and λ values. Therefore, the mechanical power can be regulated correctly using this power coefficient. Note that there is more than one way to represent the coefficient curves mathematically. Besides the polynomial approximation, there have been other attempts to represent the C p involving exponential functions [3][13]. However these functions were not developed for the type-3 WTG originally. Thus the variation can be quite large between different models. Therefore the curves should be changed accordingly if a different model of WTGs is of interest.

29 Chapter 2. Generic Models of Type-3 Wind Turbine Generators C p (λ, θ) = α i,j θ i λ j (2.3) i=0 j=0 Although the polynomial approximation is simpler compared to the original three-dimensional power coefficient curves, there are 25 parameters which need to be determined from the perspective of parameter estimation (Table. 2.2). The model will be simplified considerably if the C p representation can be replaced with a linear function Aerodynamic Model without C p Curves - Linear Model Although the two-dimensional model is much simpler compared to the actual turbine model in reality, it can be further simplified assuming constant wind speed. The simplified model is a linear approximation based on the GE s WTG parameters initially. By using this linear model, the calculation of the power coefficient in Eqn. 2.3 can be avoided altogether. Due to the substantial variations between manufacturers, the parameter values proposed in [33] can t serve for all the turbine models in the market. Nevertheless, the simplified linear aerodynamic model has been accepted as part of the generic type-3 WTG model in the WECC report and implemented in commercial softwares (for example, the PSS/E). The model is based on a linear approximation of the relation between the rate of power with respect of the pitch angle (i.e. dp /dθ) and the pitch angle θ. In other words, it s based on the linear relation below (see Table 2.3 for K aero value): which leads to Eqn dp dθ = K aeroθ P mech = P m0 K aero θ(θ θ 0 ) (2.4)

30 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 20 Table 2.2: C p curve coefficients α i,j [26] i j α i,j E E E E E E E E E E E E E E E E E E E E E E E E E-10

31 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 21 where K aero is the aerodynamic gain; P m0 and θ 0 are the initial values of the mechanical power and the pitch angle respectively. Note that in the linear aerodynamic model, P mech is now a function of θ rather than three variables ( θ, V wind and ω t ). This is the because of the constant wind speed assumption. However, the linear aerodynamic model requires proper initialization of the power and the pitch angle. The initialization is based on the investigation in [33]. There are two parameters to be determined: P m0 is the mechanical power initial value and θ 0 the pitch angle initial value. When the pitch angle θ equals to θ 0, the mechanical power yields P mech = P m0. Normally P m0 = P elec which can be determined by the power flow condition. For example, in PSS/E after solving the steady state load flow of a system, P m0 will be initialized. If: 1. P m0 < 1.0 pu, then θ 0 = 0, V wind is initialized based on Eqn P m0 = 1.0 pu, and V wind,user > 5.75P m = (Eqn. 2.5), then θ 0 is calculated using Eqn P m0 = 1.0 pu, and V wind,user 5.75P m = (Eqn. 2.5), then θ 0 = 0. V wind = 5.75P mech (2.5) θ = 1.46V wind 11.0 (2.6) Here are some comments about the initialization steps: First, this linear aerodynamic model is developed so that it can be implemented easily as a part of the dynamic model. This initialization has to rely on the load flow solution of the system. Secondly, the linear aerodynamic model has its limitations. It is based on the linear relation dp dθ = K aeroθ which suggests it is only applicable when θ is changing. Namely, the linear relation is only true when the pitch control is active. This corresponds to the situation P m0 = 1 pu. When θ remains at its minimum, there are many possible

32 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 22 Figure 2.4: Simplified Aerodynamic Model P m0 and they are all less than 1 pu. In this case, dθ = 0 but apparently the real power may change. Therefore, θ is initialized using Eqn. 2.6 only when P m0 = 1 pu. Otherwise θ 0 = θ min = 0. Thirdly, it is not very obvious why the wind speed V wind,user is involved. As the consequence of the limitation of the linear approximation mentioned above, generally speaking when the wind speed is less than the rated speed, the pitch angle will remain at its minimum in order to get maximum power out of the wind. However the question is when the wind speed is lower than what value, θ should be set to zero. This threshold can be computed by substituting the critical P m0 (i.e. 1 pu) into Eqn Setting the threshold is crucial because in the region around the threshold which corresponding to the neighbourhood near the rated wind speed, the linear approximation becomes less and less accurate as the wind speed decreases. If the threshold is not set properly, the operating point will shift far away from what it should be provided by the 2-D aerodynamic model Comparison of the Two Models In order to compare the two models, a relationship between the wind speed and the pitch angle has been plotted in Fig The 2-D aerodynamic model was implemented in PSCAD and tested under various wind speeds. After the system reaches steady state, the pitch angle is recorded and used to generate Fig The details about the test

33 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 23 system is described in Section The discrete dots are results from the system with the polynomial representation of the aerodynamic model. The solid line is from the linear model using Eqn The figure shows that the model works as expected by comparison with results in [33]. The result indicates the pitch angle from the linear fitting is larger than its counterparts based on the coefficient curves when the wind speed exceeds 14 m/s. This is a tradeoff considering the approximation accuracy near the rated wind speed. There is an obvious outlier at 12 m/s and this is due to the limitation of the linear model. As mentioned earlier, the linear approximation does not work very well when the wind speed is around the rated value. As a result, we have the initialization steps above. For more discussion see Section In all, the simplified linear model works well regarding the overall accuracy and the simplicity.

34 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 24 Figure 2.5: Pitch Angle vs. Wind Speed Discrete Dots from 2-D Model and Solid Line is from Linear Model Eqn. 2.6

35 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 25 Table 2.3: Parameters of the Turbine Model [39] Parameter Value Unit Description K aero Aerodynamic Gain Factor H 4.94 s Equivalent Total Inertia Constant D 0 pu Shaft Damping Factor 2.3 The Turbine Model The turbine model, also called rotor mechanical model in some literature, describes the conversion from mechanical power to electrical power. It computes the generator rotor speed based on the mechanical power given by the aerodynamic/wind model and the electrical power given by the generator model. Despite a variety of existed drive train models (from the detailed six-mass model to the lumped one-mass model), the WECC presented the one-mass turbine model and the two-mass turbine model. In the twomass model, the angular velocity of the turbine differs from the angular velocity of the generator rotor since the inertia of the turbine and the generator are considered as two separate parts. Only the one-mass model is chosen in this paper. In the PSCAD/EMTDC, the multi-mass turbine model is available by adding a separate component which supports up to the six-mass model. The one-mass turbine model (see Fig. 2.6)can be implemented directly using the wound induction machine block. The parameter value H used in Eqn. 2.7 is listed in Table 2.3. dω dt = 1 2Hω (P mech P elec ) (2.7) where ω is the angular speed of the generator as well as the turbine angular speed due to the choice of the lumped model.

36 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 26 Figure 2.6: Single-mass Turbine Model 2.4 The Pitch Control Model The pitch control model allows the blades to be pitched so that the maximum mechanical power (1.0 pu if the wind speed exceeds the rated value) can be drawn from the wind. Figure 2.7 shows the pitch control model consists of two PI controllers driven by the rotor speed error ω err and the real power error respectively. Sometimes, the former is referred to as pitch controller alone while the latter is referred to as the pitch compensator. Sometimes, the compensator is not included [3]. The blade pitch angle is both amplitude limited and rate limited. Because the pitch angle can not be changed abruptly due to the size of the rotor blades and the expensive blade drives. The rate depends on the size of the wind turbine. The maximum rate of change in the pitch angle is said to be in the order of 10 degree per second according to [36]. A time constant T pi is associated with the translation of the pitch angle to the mechanical output [8]. The parameter values are in Table 2.4. Even though the relationship between the pitch angle and the mechanical power is not linear as indicated in Eqn. 2.1 (P mech = 1ρAV 2 wind 3 C p (λ, θ)), in general as the wind speed increases the pitch angle would also increase accordingly. The upper limit of the pitch angle is 27 degrees nevertheless it might vary depending on the wind turbine model. In addition, generally speaking before the pitch angle hits its upper limit, the wind speed

37 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 27 Table 2.4: Parameters of the Pitch Control Model [39] Parameter Value Description K pp 150 Proportional gain in pitch controller K ip 25 Integrator gain in pitch controller K pc 3 Proportional gain in compensator K ic 30 Integrator gain in compensator T pi 0.3 Blade response time constant P I max 27 Pitch angle upper Limit (degree) P I min 0 Pitch angle lower limit (degree) P I rate 10 Pitch angle rate limit has past the cut-off speed. Thus the upper limit of the pitch angle is rarely a problem. In contrast, when the wind speed is lower than the rated speed (13 m/s in this case), the pitch angle decreases and remains at its minimum value in order to maximize the power that can be extracted from the wind. dx p dt = K ip(ω ω ref ) (2.8) dx c dt = K ic(p ord P cmd ) (2.9) dθ dt = 1 (θ cmd θ) T pi (2.10) θ cmd = x p + x c + K pp (ω ω ref ) + K pc (P ord P cmd ) (2.11) x p in Eqn. 2.8 represents the integral of ω err in the first integrator (the pitch control integrator, Figure 2.7). And x c represents the integration result of the pitch compensation integrator. Hence the unregulated pitch angle θ cmd is the summation of the two aforementioned integrals and the results of the proportional controllers. Note that in the generic models, the integrators are supposed to be anti-windup or non-windup integrators which means when the pitch angle is at its lower limit, if ω err is

38 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 28 negative, then the integrator should be blocked. Similarly the integrator will be blocked when ω err > 0 and θ reaches its upper limit. This mechanism aims to preventing the integrated result from going further down while the pitch angle remains zero degree. Without the anti-windup logic, the pitch angle can still remain at its limit. However, it might take much longer for x c and x p to recover from the limit. In the PSCAD simulation, all limiters can be easily applied but the anti-windup logic is not considered. To begin with, according to [15], the anti-windup logic may lead to trajectory deadlock. In short, it is possible there exist situations when x c and θ lingers at the edge of block/unblock indefinitely and infinitely. The author proposed to solve the problem by introducing hysteresis. Although this will solve the deadlock, more states as well as switching behaviours will be involved as a result. Such phenomena are much undesirable for parameter estimation since the system continuity assumption will be compromised which will lead to much complicated model description and potential numerical problems. Such systems with discrete events can be classified as hybrid dynamical systems [37]. An alternative to work around this problem without implementing the anti-wind up logic is setting up a threshold directly with respect to the wind speed. For example, despite the limiters, when the wind speed is lower than m/s (the rated speed is 13 m/s), θ is set to be zero. This is helpful when the anti-windup blocks are not implemented otherwise it may take longer for the angle to settle as the angle is approaching the limit. In addition, the threshold value m/s comes from the linear aerodynamic model, in other words, only when the wind speed exceeds the threshold, the pitch control model will take over and start to regulate the pitch angle. In summary, in this thesis, the anti-windup limited logic will not be discussed. During the PSCAD simulation, all limiters including rate of change limiters were considered. And the threshold mentioned in this section was implemented. But later as we extract the model for parameter fitting, we assume no limits are hit. Otherwise, for each limiter, there would be at least one more state. In addition, the anti-windup block in the torque

39 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 29 Figure 2.7: Pitch Controller control isn t considered. P cmd is the real power reference which can be provided by plant controller or external model. In short, through the pitch compensator, the real power will be driven to its reference P cmd by changing the pitch angle. Note that P cmd can not exceed the maximum power which can be extracted from the wind. In general, if there isn t plant controller and we want the maximum power, P cmd can be set according to the power/wind speed relation which is available in all WTG product data sheet. The power curve of our system is given later in Fig

40 Chapter 2. Generic Models of Type-3 Wind Turbine Generators Generator and Converter Model Generator and Converter Model In the PSCAD/EMTDC environment, the generator and the converter can be modelled in great details, whereas in the generic WTG model (as indicated in Fig. 2.1), the generator/converter is modelled as one block which also serves as the interface with the grid. The generator/converter model takes in two current inputs I qcmd and I pcmd and gives P elec and Q elec or injected current components I p and I q as outputs. A full generator/converter model consists of three logic/management blocks: the low voltage power logic block, the high voltage reactive current management block and the low voltage active current management block [26][31]. However, high voltage and low voltage scenarios are not considered in this thesis since our interests are mainly scenarios when the terminal voltage is near the nominal value. Moreover, adding these logic blocks will compromise the desirable continuity and differentiability of the models thus will make parameter fitting extremely difficult. Therefore, the generator/control model is: di p dt = 1 T g (I pcmd I p ) (2.12) P elec = V term I p (2.13) di q dt = 1 T g (I qcmd I q ) (2.14) Q elec = I q V term (2.15) The lags in converter action are characterized by a small time constant T g = 0.02s. In other words, the complicate electronic control is replaced by two low-pass filters approximately [7]. Thus the current commands I qcmd and I pcmd become the actual current components I q and I p after the converter. T g is also the smallest time constant in the whole model.

41 Chapter 2. Generic Models of Type-3 Wind Turbine Generators Converter Controller As mentioned above, having a partial-scale converter is a key feature of the type-3 WTG. The rating of the converter is about 20% to 30% of the generator rating depending on the operating slip range. The control of the AC-DC-AC converter consists of two parts: the grid-side control and the rotor-side control. Both are outlined in this subsection. There are several different methods to implement the converter control system. Twodimensional frames, i.e. the αβ-frame and the dq-frame, are most commonly used. By controlling two subsystems, designing controllers for the converters becomes easier. Especially when the dq-frame is employed, the commands needed to be tracked will be DC signals thus simple PI controllers are sufficient to provide a satisfactory performance. Such control methods are presented frequently in literatures, and one of many resources is the book by Yazdani and Iravani [40]. In addition, there are two different choices: the reference frame aligned with the stator flux or the stator voltage. Here the stator-flux orientated frame will be used. The Stator Flux Model (M 0 ) As is known, in order to accomplish the abc-to-dq transformation or the αβ-to-dq transformation, a time-variant angle is tracked all the time. Depending on the choice of reference frame orientation, there are two options: one is the stator flux orientation, the other one is the grid voltage orientation (also known as grid flux orientation). In the stator flux orientation, the d axis is aligned with the stator flux. In contrast, in the grid voltage orientation, the q axis is aligned with the stator (grid) voltage. The flux orientated frame requires a stator flux model (flux observer) while the voltage orientated frame requires a PLL model in order to obtain the angle of the fundamental. In the type-3 WTG control, both models are implemented in the PSCAD/EMTDC. The angle θ s which is needed to accomplish the abc-to-dq transformation on the grid-side is given by computing the angle of the rotor voltage in the αβ-frame. Meanwhile, the angle

42 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 32 Figure 2.8: Stator-flux Model for the DFIG ρ for the rotor-side is given by the stator flux observer. λ s = λ s e jθ(t) θ(t) = ρ(t) + θ r (t) where λ s represents the stator flux space phasor and θ r is the rotor angle. After the transformation using ρ, the rotor side current can be presented as i rd + ji rq. In addition, i rq is proportional to the machine electrical torque in steady state (to be specific, when the amplitude of the stator flux is constant). Thus the torque as well as the real power output can be controlled through the torque control model which is discussed in the subsequent section. Grid-Side Converter Controller (M 0 ) The grid-side controller is responsible for maintaining the dc link voltage as well as the stator side reactive power. The configuration is readily available in literature. Fig. 2.9 shows the main part of the grid-side converter control, where E d and phi are the stator voltage magnitude and angle (α β frame). Thus V abcref is the reference for generating PWM gating signals.

43 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 33 Figure 2.9: Grid-Side Converter Control Model in PSCAD (M 0 )

44 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 34 Rotor-Side Converter Controller In the dq-frame, controlling over the torque and the reactive power output can be achieved by assigning the q-axis and d-axis rotor currents independently. This is also one of the advantages of type-3 WTG.The torque control model and the reactive power control model are the major two parts of the rotor-side converter controller. In Fig. 2.1, there are two blocks: Q control and P control, which follow the voltage regulation/power factor control block and the torque control block respectively. These two blocks can take in either external signals or outputs from the previous blocks as commands (Q cmd and P cmd ). Then the current commands for I p and I q will be computed and passed to generate gating signals for the converter. In other words, the torque control and the voltage regulation/power factor control block are optional. Here the torque control model has been implemented. Again, together with the P control block, they are called the real power control model. Sometimes, it is merely called the torque control model. The naming and scope of the blocks can be confusing given the various options in literature. However, essentially there are only two parts, one for the active power control corresponding to the q-axis rotor current in this case. The other one is the reactive power control corresponding to the d-axis rotor current. In the active power control part, the current reference is computed based on a relationship involving the machine torque. In contrast, the reactive power control part is a bit simpler. In this thesis, the d-axis current reference is simply computed using a PI controller driven by the reactive power error. 1. The WTG Real Power Control Model (Torque Control Model) In steady state, the electrical torque of the induction machine is proportional to the q-axis rotor current (stator flux orientation). Therefore, the torque can be controlled by assigning i rq,ref properly. Fig shows the structure of the torque control model. The PI controller acts on the rotor speed error and gives the torque reference, which is translated to the q-axis current reference in the end. Furthermore, a nonlinear relationship extracted from [39] has been used to determine the

45 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 35 turbine speed reference ω ref based on the real power P elec. A piecewise linear approximation is implemented in the PSCAD/EMTDC environment as showed in Fig In addition, the intermediate signals ω ref and P ord will be used in the pitch control model mentioned in Section 2.4. See Table 2.6 for parameter values. The torque controller presented is based on per-unit values. The base quantities are as listed in Table 2.5. More discussion about the torque control model is included in Appendix A. Note that the torque control model used in the generic WTG model is based on the model used in real world however their configurations are not the same. The most important difference is the output of the torque control I pcmd is not the same as the d-axis component of the rotor current reference used to generate gating signals for the converter. In the generic model, I pcmd will be passed to the generator/converter model. After some limiters, I pcmd will become I p. Then, through the grid interface, the injected current to the grid from the WTG can be computed (along with I q ). In other words, I p is a component of the injected current of the WTG. Meanwhile I rqcmd is the true reference used to control the real power. Fig shows the torque control used in PSCAD. In short, the torque control in PSCAD doesn t include P ord. The reason P ord is considered as a part of the torque control in the generic model is so that it can be used as a replacement of P elec. However, there is no such need to include in P ord in the torque control in PSCAD since P elec is available directly. In addition, in order to decrease the time the system takes from start-up to steady state, proper initial values should be used in PSCAD. dp ord dt dt ω dt = K itrq (ω ω ref ) (2.16) = 1 T pc (ω(t ω + K ptrq (ω ω ref )) P ord ) (2.17)

46 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 36 Table 2.5: Base Quantities Used in The Converter Controllers Expression Value Description P base 1.5 MVA The rated power of the DFIG V base kv Amplitude of the nominalline-to-neutral voltage I base ka Amplitude of the nominal line current ω base 60 Hz System frequency Table 2.6: Parameters of the RSC Converter Control Model [39] Parameter Value Unit Description K ptrq 3 - Torque control proportional gain K itrq Torque control integrator gain T pc 0.05 s Power control lag T sp 5 s Speed reference lag I pmax Current reference upper limit P max 1.12 pu Power reference upper limit P min 0.1 pu Power reference lower limit K qvp Reactive power control proportional gain K qvi 2 - Reactive power control integrator gain

47 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 37 Figure 2.10: Torque Control Model

48 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 38 Figure 2.11: Turbine Speed Setpoints (pu) vs. Real Power (pu) 2. The WTG Reactive Power Control Model Similar to the real power control, in steady state, the reactive power of the WTG can be controlled by the d-axis rotor current. Therefore the command for the d- axis rotor current can be calculated given the desired reactive power Q cmd. As mentioned before, the voltage regulation and the power factor control block can be ignored by directly assigning the reactive power command. In our case, Q cmd is held to be constant. This is also referred as fixed reactive power control. In other cases, it can also be computed by a separate model constantly. For instance, if a certain power factor expected, the reactive power command can be computed based on the real power. Fig shows both the real and reactive control implemented in PSCAD. The parameter values are included in Table 2.6. dx q dt = K qvi (Q cmd Q elec ) (2.18) I qcmd = x q + K qvp (Q cmd Q elec ) (2.19)

49 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 39 Figure 2.12: Rotor-side converter control in PSCAD (M 0 ) The Scope of the Models So far all major components of the time-domain model and the transient stability model which will be used in the thesis have been introduced. However, it might be helpful to remind the readers of the assumptions we made on the models. 1. There are no anti-windup logic implemented and all limiters in the transient stability models are assumed to be inactive. 2. Low-voltage ride through requirement is not considered. In other words, also following the previous assumption, the low voltage and high voltage logic in the generator/converter model is not considered. 3. Historically, the DC-link and grid-side converter are not considered in the transient stability models for the current two generations of WECC models. Therefore, these two parts and related features (e.g. crow-bar model) are not considered in the transient stability models in this thesis.

50 Chapter 2. Generic Models of Type-3 Wind Turbine Generators Test System and Model Performance Test System PSCAD provides its users a very simple example of a wind farm system consisting of a single 2 MW DFIG unit. However, the system is not complete since the pitch control model and the aerodynamic model are missing. In addition, the example system uses several simplified models, for instance, the active power control model. Our test system, on the other hand, includes all major parts of generic type-3 WTG model. In the test sytsem, the generator (rated power 1.5 MVA), is connected to a 20 kv infinite bus (ideal voltage source) behind a 0.69 kv/ 20 kv transformer and the Thevenin equivalent of the grid. The back-to-back converter including its control system has been implemented in the converter module. The wind power module takes in the wind speed, the power coefficient C p and the machine rotor speed as inputs. After computation, it gives out the mechanical torque to the generator. This module won t be used if the simplified aerodynamic model is employed. The parameter values of the system are listed in Appendix B. Fault can be introduced by adding impedance at between the grid and the step-up transformer Model Performance To maintain a smooth transition between the speed control mode and the torque control mode, all initial values such as the rotor speed, real and reactive power commands should be set accordingly to match those in steady state. Furthermore, if the simplified aerodynamic model is used, the initialization of this part is also necessary. At 0.5 s, the control mode is switched.

51 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 41 Figure 2.13: Test System with Single Type-3 WTG Start-Up Settings Incorrect initialization will lead to fairly long settling time since the rate of the pitch angle is regulated. The shaft speed normally takes several seconds to settle down. The simulation duration should be set to 20 s to 30 s before any change made (for example changing the wind speed or introducing a fault). Simulation Results under Different Wind Speeds The system is tested under different wind speeds ranging from the cut-in speed (6 m/s) to the cut-off speed (20 m/s). A relationship between the wind speed and the real power output has be established based on the test results. The values are recorded after the system reaches steady state under a constant wind speed. As mentioned in Section and Section 2.2.2, there are two ways of modelling the aerodynamics. Fig shows the power curve (red/solid line) involving C p computation. By contrast the power curve of the system with the simplified linear aerodynamic model is also included (blue/dash line).

52 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 42 This well-known curve (from the 2-D aerodynamic model) shows the relation between the wind speed and active power of the WTG which is available in wind turbine product brochure. The linear relationship in Eqn. 2.5 is an approximation of the power curve when the wind speed is less than m/s. In other words, Eqn. 2.5 is a simplified version of the rising part of the curve. The result is in accordance with [33]. However, due to the sharp change of the slop when V wind is between 11 m/s and 12 m/s, the linear model is less satisfactory in this particular range given the consideration of the overall approximation accuracy. Thus if the linear aerodynamic model is used, one should make sure θ 0 is properly initialized around this region. When the wind speed is higher than the rated speed, the mechanical power is determined by the pitch angle error directly (see Eqn. 2.4). In all, the linear model works well as expected. Dynamic Response of the System To test the dynamic response of the WTG system, a voltage dip of 20% at the WTG terminal is designed to happen at 30s under the rated wind speed condition. The fault voltage is introduced by adding fault branch between the grid and the transformer. The duration of the voltage drop is 18 cycles (i.e. 0.3 s). The dynamic responses are indicated in Fig including the real and reactive power, the terminal voltage and the q-axis rotor current reference. The real power drops about 13% due to the fault and increased back to around 1.5 MW after the fault is cleared. Meanwhile, an increase of the reactive has been observed during the fault. The q-axis rotor current jumps up due to the voltage sag as an effort to increase the real power. Also during the fault, the pitch angle goes down trying to increase C p in order to increase the real power. The results agree with [33] in general. Note that since the two-mass turbine model is used in [33], the turbine shaft speed oscillates and takes longer to settle down compared to the generator shaft speed. As a result, the

53 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 43 Figure 2.14: Real Power Under Different Wind Speeds: Results of Two Models Solid/Red Line for the Two-Dimensional Model; Dash/Blue Line for the Linear Model

54 Chapter 2. Generic Models of Type-3 Wind Turbine Generators 44 (a) Real Power During Fault (b) Reactive Power During Fault (c) Voltage During Fault (d) Q-axis Current Reference During Fault Figure 2.15: Dynamic Response During Fault dynamic response when the turbine shaft speed oscillates violently (i.e. when the fault just happens) can be different from the response with the one-mass turbine model. Again, the PSCAD-type model is sophisticated time-domain model which is capable of capturing all kinds of system dynamics. The simulation results will be used to evaluate the dynamic models which will be covered in the subsequent chapters.