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附錄Fixed-Speed Wind-Generator and Wind-Park Modeling for Transient Stability StudiesIncreasing levels of wind-turbine generation in modern power systems is initiating a need for accurate wind-generation transient stability models. Because many wind generators are often grouped together in wind parks， equivalence modeling of several wind generators is especially critical. In this paper， reduced-order dynamic fixed-speed wind-generator model appropriate for transient stability simulation is presented. The models derived using a model reduction technique of a high-order finite-element model. Then， an equivalency approach is presented that demonstrates how several wind generators in a wind park can be combined into a single reduced-order model. Simulation cases are presented to demonstrate several unique properties of a powersystem containing wind generators. The results in these paper focuson horizontal-axis turbines using an induction machine directly connected to the grid as the generator.Index Terms—Transient stability simulation， wind-generator modeling， wind-park modeling， wind-turbine modeling.I. INTRODUCTIONThis encompasses many modern large-scale systems. Because large wind installations consist of many wind generators， wind-park-modeling is a critical need. Consequently， the second goals to present a methodology for combining several wind generators connected to the grid through a common bus into a singleequivalent model.Wind generators are primarily classified as fixed speed or variable speed. With most fixed-speed units， the turbine drives an induction generator that is directly connected to the grid.The turbine speed varies very little due to the steep slope of the generator’s torque-speed characteristic； therefore， it is termed fixed-speed system. With a variable-speed unit， the generator is connected to the grid using power-electronic converter technology. This allows the turbine speed to be controlled to maximize performance (e.g.， power capture). Both approaches areManuscript received February 3， 2004. This work was supported in part bythe Western Area Power Administration. Paper no. TPWRS-00388-2003.The authors are with Montana Tech， University of Montana， Butte， MT59701USA (e-mail: dtrudnowski@mtech.edu).Digital Object Identifier 10.1109/TPWRS.2004.836204 common in the wind industry. In this paper， we focus on modeling the fixed-speed unit and an equivalent model of severalA wind park consists of several wind generators connected toothed transmission system through a single bus. Because modeling each individual turbine for transient stability is overly cumbersome，our goal is to lump the wind park into a minimal setoff equivalent wind-generator models. Our approach for equivalence modeling of a wind park involves combining all turbines with the same mechanical natural frequency into a single equivalent turbine. Simulation results demonstrate this approach provides accurate results.A representative example of published results for modeling wind generators for transient stability is contained in [2]–[10].Results for modeling fixed-speed wind generators have focused on two primary approaches. The first approach represents the turbine and generator rotor as a single inertia thus ignoring the system’s mechanical natural frequency [2]–[5]. The second approach represents the turbine blades and hub as one inertia connectedto the generator inertia through a spring [6]–[9]. In all of these papers， the spring stiffness is calculated from the system’s shaft.Our research indicates that representing the first-mode mechanical frequency is critical to an accurate model. Finite-element analysis has shown that the first-mode dynamics are primarily a result of the flexibility of the turbine blades not the shaft as assumed by others [11]. The modeling approach presented in this paper centers on the fact that the primary flexible mechanical component is the turbine blade. The results in [7] focus on reduced-order wind-park modeling. The authors use a standard induction generator equiva-0885-8950/04$20.00 ? 2004 lancing method to combine several wind generator systems. But，the authors do not address the problem of combining the turbines in such a way to preserve the mechanical natural frequencies. Our research indicates this is critical to having an accurate wind park model. A thorough discussion of reduced-order modeling of variable-speed turbines is contained in [10]. The authors argue the turbine mechanics can be represented as a single inertia because the variable-speed connection decouples the mechanical dynamics from the electromechanical dynamics. Our results do not consider the variable-speed case. The work described in [2]–[10] focuses on low-order turbine models that can be easily implemented in large-scale transient stability codes. Considerable research has focused on modeling at a more detailed level. An excellent overview and literature review is contained in [17]. Detailed modeling approaches range from highly-detailed finite-element models to more simplified six-mass， five-mass， and three-mass turbine models. The majorityof these models use momentum theory [13] to calculate aerodynamic forces.III. TURBINE DYNAMICSOur approach for developing a reduced-order model consists of starting with a highly-detailed mechanical and aerodynamic turbine model and then removing all dynamic effects outside the electromechanical range. In this reduction process， all analysis is done from the perspective of the turbine shaft that drives the 325 cillation. Detailed modal analysis of the system shows that the oscillation is the result of the outer portions of the blades vibrating against both the inner portions of the blades and all other inertias on the shaft [11]，[12]. Such a result is typical， especially forlarge turbines. Modern wind-turbine blades are very large and flexible， and tend to vibrate at their first mode when excited from the hub. Pony analysis of the oscillation in Fig. 1 shows it primarily contains a 4-Hz component [12]. This is also typical of large-scale turbines， which usually have a first-mode natural mechanical frequency in the 0- to 10-Hz range. Because this range is also typical for electromechanical oscillations， it is critical to represent the mechanical oscillations of the wind-turbine as they will tend to interact with the electromechanical oscillations. The mode shape of the first-mode oscillation that dominates the response in Fig. 1 dictates that the model can be represented by a two-inertia， single spring-damper system as depicted in Fig. 2. This is the basis for the reduced-order model that follows. One inertia represents the outer portion of the blades (the blade tips in Fig. 2). The blade tips are rigidly connected as depicted in Fig. 2 with a mass less “blade ring.” The blade tips act as a single inertia because all transient disturbances equally act on all blades through the generator shaft. The other inertia represents the combined effect of the blade roots， hub， turbine shaft， gearing， generator shaft， and generator inertia. For a typical system， the inner inertia is dominated by the blade roots and generator inertia. The reduced turbine model depicted in Fig. 2 is considerably different than what other researchers have proposed [2]–[9].Many have lumped the entire turbine and generator into a single inertia and ignored the mechanical first-mode dynamics [2]–[5].Others has considered first-mode dynamics， but do not model the blade flexibility [6]–[9]. Instead， these authors have assumed the blades to be a single inertia and model the turbine shaft as a spring. But， in a typical system， the blades are much more flexible than the shaft. Our research indicates that the blades dominate the mechanical first mode and the shaft acts as a rigid body. Our research also indicates that correctly modeling the mechanics is critical to obtaining accurate transient simulation results.. SINGLE WIND-GENERATOR MODEL The single wind-generator model consists of two primary components: the reduced-order two-inertia turbine model from the previous section driven by a wind torque； and a standard TRUDNOWSKI et al.: FIXED-SPEED WIND-GENERATOR AND WIND-PARK MODELING FOR TRANSIENT STABILITY STUDIESelectric generator. For this paper， we assume the generator to be a standard induction machine directly connected to the grid as this is the most common configuration. A. Turbine ModelThe two-inertia reduced-order turbine in Fig. 2 is the basis for the turbine model. The equations of motion for the system in Fig. 2 are(1)where number of blades；effective gear ratio = /rated-turbine-speed；electrical frequency base；inertia of each blade tip；inertia of each blade root+ inertia of + inertia of turbine shaft and gearing/+ inertia of generator shaft and rotor；blade stiffness；blade damping；aerodynamic wind torque；generator electrical torque；blade tip angle reflected through the gearing；generator shaft angle. Calculating the inertias and in (1) requires knowledge of the blade break point where the spring-damper is placed (see Fig. 2). If the blade is not broken at the correct position， then the mechanical mode shape will not be correct. The break point is primarily a function of the blade mechanics and can be determined from finite-element analysis or testing of the blade and seems to occur at the second bending node of the blade. In the example cases studied in [12]， the reduced-order system’s sensitivity to improper placement of the break point is significant. This is demonstrated in the example section. Fortunately， most modern blade manufactures or blade testing facilities (such as the facility at the National RenewableEnergy Laboratory in the United States) have the required information to determine the blade break point. The power engineer simply needs to request this information. Once one has the blade break point， the inertia parameters can easily be calculated from typical manufacture’s data. The stiffness in (1) can be calculated from knowledge of the system’s first-mode mechanical natural frequency using(2)where is the first-mode mechanical lead-lag natural frequency with the system connected to infinite bus. For example，in the system in the previous section， .Typically， manufactures can provide this frequency. It can be easily calculated by applying a brake pulse on the turbine and analyzing its response (for example， Fourier analysis of the generator’s speed). In most cases the blade damping is very small and assumed to be zero. The spring stiffness is a measure of the blade’s stiffness in the rotational plane which is a combination of the blade’s edge stiffness and flat stiffness [12]. Relating to the edge and flat results in(3)where is the edge stiffness， is the flat stiffness， and is the pitch angle. Both and are constant. As can be seen in(3)， is dependent on the pitch angle . Typically， is limitedto be between zero and ten degrees. Analysis of (3) under this restriction shows that varies very little for different pitch set points. This implies， and experiments support， that the accuracy of the turbine model has very small sensitivity to variations in the system’s pitch angle [12].The wind torque is calculated assuming an ideal rotor disk from the equation [13](4)where is the velocity of the blade tip sections reflected through the gearing， is the air density， is the sweep area of the blades， is the free wind velocity， and is the turbine’s power coefficient. Unfortunately， is not a constant. However， the majority of turbine manufactures supply the owner with a curve. The curve expresses as a function caused primarily by tower shadowing and unbalanced mechanics. Typical modulation frequencies are at the 1P and 3Pmodes (note: 1P is once per revolution of a turbine blade) [6].We do not include these effects as we assume that the torque induced from the transient fault is much larger than the modulation torque. This assumption has been made by many other researchers (for example， [7]). Future research will focus on testing this assumption. In general， the two-inertia turbine model proposed here is a relatively robust model that covers many turbine operating conditions. All model parameters are relatively constant with very little sensitivity to the pitch angle. Because the main component of energy in a transient is due to turbine inertial energy，stall-controlled turbines can be accurately modeled using this approach’s. Generator Model Standard practices are well established for modeling the generator [1]. A standard detailed two-axis induction machine model is used to represent the induction generator [1]. The resulting equations are(6a) where is the transient open-circuit time constant， is the slip speed， is the synchronous reactance， is the transient reactance， and are the d-axis and q-axis stator voltages， and are the d-axis and q-axis per-unit stator currents. The torque is calculated from(6b)TRUDNOWSKI et al.: FIXED-SPEED WIND-GENERATOR AND WIND-PARK MODELING FOR TRANSIENT STABILITY STUDIES where is the sweep area， is the free wind velocity， and is the turbine’s power coefficient for turbine . B. Equivalent Generator Model The equivalence induction generator parameters are obtained using the weighted admittance averaging method in [16]. With this method， the equivalent machine parameters ，and are calculated by taking the weighted average admittances of each branch of the induction machine equivalent circuit. The weighting for the averages are calculated using the rated power of the generators. I. SIMULATION RESULTS Many example test cases have been studied to evaluate the properties of the modeling approach； these are contained in [12]，[14]， [15]. A select few are presented in this section.For this example， we compare the response of the two-inertia reduced-order turbine in (1) to the response of the finite-element model and a detailed five-inertia model. Each model is connected to an infinite bus through an induction generator. The response of the finite-element model is shown in Fig. 1.Thefive-inertia model represents each blade with edge and flap spring-dampers； the slow-speed shaft spring stiffness is also represented； and the aerodynamics are modeled using Gluer vortex momentum theory [13]. The five-inertia model also contains the centrifugal， gravity， and carioles effects. Derivation of the five-inertia model is contained in [11]， [12]. The turbine properties are described in Section III. It is directly connected to a 60-Hz infinite bus through the 1.68-MW induction generator. Turbine and induction-generator model parameters for the reduced-order model are provided in the Appendix. The simulation is compared to the ADAMS finite-element simulation which includes highly detailed aerodynamic and mechanical modeling. The two-inertia reduced-order model is a 6th-order model [(1)， (4)， and (6)] while the finite-element model is approximately 650th-order， and the five-inertia model is 18 th order. Simulation results are shown in Fig. 4 and Fig. 5. As can be seen， the two-inertia reduced-order model closely matches the highly detailed finite-element and five-inertia models.In this example， we demonstrate the sensitivity of the two turbine model to the choice of the blade break point. The responses of three modeling cases are shown in Fig. 6. The 50% break-point places the blade spring at the center of the blade radius and is the same model used in example 1. This response is compared to a 43% break point and a 56% break point. The percentage indicates the location from the hub where the blade spring is placed along the blade radius. The differences between the responses are significant enough to merit careful selection of the blade break poiAll of the following simulations were implemented in a modified version of the Power System Toolbox (from Cherry Tree Scientific Software， Ontario，Canada). The Toolbox was modified to allow for simulation of wind generators. Fig. 8 shows the real power out of the wind generator for then two modeling cases for a disturbance consisting of a 5-cycle fault at bus 15 followed by a line opening at bus 15. Pony analysis of the two-inertia turbine response shows two modes in the oscillation: a 4.5-Hz mode and a 2.0-Hz mode. The 4.5-Hz mode is due to the mechanical mode of the turbine and the 2.0-Hz mode is the electromechanical mode. Similar analysis of the one-inertia response indicates only one mode at 2.4 Hz， which is an electromechanical mode. Because of the errors in the first-swing and oscillatory response of the single inertia system in Fig. 8， a power engineer would likely come to a different conclusion concerning the transient and small-signal stability properties of the system. The one-inertia response indicates a more stable system with a lower first-swing deviation and higher oscillatory damping. Other examples in [14] demonstrate cases where the single-inertia response is stable and the more a modeling approach. The wind park in Fig. 7 now，con-sits of 21 wind generators， each connected to bus 17 through a short transmission line. All wind generators are the same as the two-inertia system used in example 2. Two modeling cases are compared. The first case is a detailed model where each wind generator in the park is modeled individually； this effectively results in a 126th-order model for the park. The first seven wind generators are driven by a wind velocity of 14 m/s and are connected to bus 17 through a 1-km distribution line； the second seven are driven by a wind velocity of 11 m/s and are connected to bus 17 through a 2-km line； and the last seven are driven by a 8 m/s wind and connected to bus 17 through 3-km line. For the second case， the park is modeled as a single equivalent wind-generator using the method in Section V (a 6th-order model).Fig. 9 shows the wind-park real power and Fig. 10 shows the bus 17 voltage for the two cases. The disturbance is the same as in example 3. As can be seen， the equivalent model very accurately represents the detailed one. Other simulation cases also demonstrate the accuracy of the approach [15].are connected to bus 17 through a 2-km line； and the last seven are driven by a 8 m/s wind and connected to bus 17 through 3-km line. For the second case， the park is modeled as a single equivalent wind-generator using the method in Section V (a 6th-order model).Fig. 9 shows the wind-park real power and Fig. 10 shows the bus 17 voltage for the two cases. The disturbance is the same as in example 3. As can be seen， the equivalent model very accurately represents the detailed one. Other simulation cases also demonstrate the accuracy of the approach CONCLUSIONA reduced-order dynamic wind generator model appropriate for transient stability has been presented. The model represents the turbine as a 4th-order nonlinear model with wind speed as the input. The turbine equations are compatible with standard generator electrical equations used for transient stability. An equivalency approach was also presented that demonstrateshow several wind generators in a wind park can be combined into a single model. Simulation cases are presented to demonstrate the accuracy of the approaches. Future research will focus on testing the effects of modulation torques such as tower shadowing on the modeling approaches.