TORQUE AND POWER RATING OF A WIND-POWER PM GENERATOR DRIVE FOR MAXIMUM PROFIT-TO-COST RATIO Abstract—This paper compares different solutions in micro wind-power system design. An economic analysis is carried out on the basis of the selected generator, power converter, and mechanical system. An interior permanent-magnet (PM) (IPM)generator is considered since it can be driven in the flux weakening region, exploiting higher turbine rotation speeds, without increasing proportionally the terminal voltage. The performance of the wind-power system adopting the IPM generator is compared with that adopting a surface-mounted PM generator. Several cases are considered, by imposing some constraints on both the maximum input torque of the generator and the volt-ampere rating of the power converter, as well as combining both limitations together. This paper highlights a methodology to determine the economic advantages obtained introducing such limitations, in terms of both return on investment and money profit during a given period. The study focuses on how to apply the proposed technique, such as constant torque region, flux weakening, and stalling mode, founding the optimal power and torque ratings on the basis of the maximum profit over payback time. Index Terms—Interior permanent-magnet (PM) (IPM) generator, maximum profit, wind power torque power ratio. I. INTRODUCTION WIND-POWER systems help in reducing electric energy generation by fossil fuel. Interest for low-power wind applications is growing up [1], particularly for installation in urban areas, or in isolated areas [2]. In all cases, a careful cost analysis has to be done [3], to the aim of evaluating the convenience of the system in terms of payback time. A maximum power point (MPP) tracking algorithm is necessary, to increase the energy capture, since efficiency is strongly affected by both wind speed and turbine rotation speed [4], [5]. Different speed controls can be used to manage the wind turbine [6], [7]. In large turbines, when the wind rated speed is reached, blade pitch control is used to keep the power and the rotor speed constant. To the aim of limiting the cost, in small wind turbines, this technique is not used, and blade stalling is employed [8], [9]. Furthermore, in order to limit the urban impact, the increase of the blade radius is not possible [10] to increase the power obtainable by micro generators. All these reasons imply that a proper design of the whole wind-power system has to be considered. This paper focuses on the field of low-power wind generators. Itgives a design procedure to maximize the profit-to-cost ratio in micro wind-power systems, using permanent-magnet(PM) generators [11]. Since the generator cost is generally proportional to its rated torque and the power converter cost is proportional to its volt-ampere rating, some limitations are introduced in order to reduce the initial cost of the system. Thereduction of the payback time is chosen as the optimization objective. The approach is to let the rotor speed increase when the maximum torque of the generator is reached. In this way, a slight oversizing of the turbine is needed. As far as micro windpower systems are concerned, such oversizing does practically involve only limited additional costs which are taken into accountin the overall cost estimation. It will be shown that driving the generator at constant speed (and then at constant power) yields an oversizing of the electric machine, i.e., an increase of costs and weight. Rather than only a theoretical analysis, the study refers to a practical micro wind-power generator for givenmechanical data. II. WEIBULL STATISTIC Among the various statistics used to describe the wind speed frequency curve, the Weibull statistic [12] is used hereafter as a base point for the analysis of the wind energy production. According to such statistic, the probability density f(vwind) of the wind speed vwind is f(vwind) = (1) where A, given in meters per second, and k are the scale parameter and the shape parameter. The parameter k is determined from the average data in Italy, being included in the range between k = 1.4 and 1.5. A k = 1.4 is used in the following study. The parameter A is determined from the average wind speed vavg obtained from historical measurements on the site considered. Then, vavg = AΓ(x), where x = 1+1/k and the gamma function is Γ(x) =(2) From the wind speed data of a location in South Italy, the average wind speed results to be vavg = 5 m/s. Fig1. Distribution function F(vwind) versus the wind speed vwind at 3 different values of vavg = 3, 5, &7 m/s with k = 1.4. The distribution function is the integral of the density function from zero to a given wind speed vwind, i.e., F(vwind) =wind)dv’wind =1−. (3) Fig. 1 shows three functions F(vwind) referring to vavg = 3, 5, and 7 m/s, respectively. The curves end at a maximum speed vmax, fixed to 20 m/s, which is the cutoff speed. Another key function is the differential probability function Pd (vwindi) = F (vwindi ) –F(vwindi−1 ) (4) which gives the probability of vwindi−1 < vwind < vwindi. It is used to analyze the energy productivity. III. Cp AND MPP TRACKING ALGORITHM IN WIND TURBINES As well known, the power output from practical turbines is Pm = 1/2CpρSv3wind, where Cp is the “coefficient of power,” depending on aerodynamics factors that are characteristics of any particular turbine. ρ is the air density in kilograms per cubic meter, S is the turbine cross-sectional area in square meters, and vwind is the wind speed in meters per second. Power depends strongly on the wind speed and the Cp coefficient. A horizontal axis turbine is considered. The coefficient Cp is calculated as suggested in [13]; Cp is a function of pitch angle β and the tip speed ratio λ = Rt(πnm/30)/vwind, where Rt is the blade radius and nm is the mechanical rotor speed Cp(λ, β) = c_{1}(c_{2} λ_{i}^{-1}− c_{3} β –c_{4})e^{-c}_{5} λ^{-1}_{i } + c_{6} (5) where λ_{i}^{-1}=(λ + 0.08β)^{-1}+ 0.035(β3 + 1)^{-1 }(6) Hereafter, Rt = 1 m and β = 0 are considered. The other parameters are given in Table I according to three types of turbine, i.e., three different values of Cp. They correspond to three different geometrical structures of blades, designed to optimize the turbine performance at slow, normal, and fast speeds of wind, respectively. Fig. 2 shows three Cp versus λ Characteristics. Fig. 2. Coefficient of power Cp versus the tip speed ratio λ for three different types of turbine The Cp is maximum for a given value of λ. Therefore, varying the wind speed vwind, the maximum obtainable poweris reached at different rotor speeds nm. An MPP tracking algorithm is commonly adopted in order to maximize the power at various wind speed. In this paper, the implementation proposed in [5] is used. When the wind turbine has to be operated in stalling mode, three cases can happen. 1) The control can reduce the speed by forcing the working point to move from higher speed to lower speed along the wind-power curve; an oversize of both the generator and the converter is needed. 2) The generator is slowed down by short-circuiting the phases through the converter; an oversize of both the generator and the converter could be needed depending on the generator design. 3) The generator is slowed down by a mechanical brake, and no electric oversize is needed.
TABLE I
TURBINE PARAMETERS
IV. REFERENCE CASE (SPM GENERATOR) The design of wind-power systems is based on the maximum output power of a wind turbine. It depends on the amount of wind in the site where the system is installed and on the type of wind turbine in terms of turbine radius and “coefficient of power.” Once the site and the turbine radius are fixed, different criteria can be taken for designing both the generator and power converter.
Fig. 3. Mechanical power versus rotor speed characteristics with power, speed, and torque limits (normal Cp). For the sake of comparison, the reference case is a windturbine coupled to a surface-mounted PM (SPM) machine so that no constant power operation is exhibited by the electrical drive. Operating points are located along the MPP trajectory up to a wind speed of 12 m/s. The maximum power reached at theshaft is Pn = 1600 W, according to a torque Tn = 16.4 N ・ m and a rotor speed nm = 930 rpm. For a wind speed higher than 12 m/s, the blade stalling is used up to the cutoff wind speed of 20 m/s. The reference case is illustrated in Figs. 3 and 4 by thecurve 0–T–P–A that represents the MPP trajectory. Point A indicates the operating point on such a trajectory when vwind = 12 m/s. According to the aforementioned assumptions, n this is the maximum speed at which the MPP tracking algorithms can be applied; thus, nominal power is reached in point A. The capability of such a configuration is highlighted by the SPM curves in Figs. 5–7. Fig. 4. Torque versus rotor speed characteristics with the introduced limitations (normal Cp). V. DESIGN CRITERIA OF IPM GENERATOR The potential advantage of using an interior PM (IPM) generatorinstead of a conventional SPM generator is investigated hereafter in terms of the cost of the plant, energy productivity, and payback time. The analysis is based on the assumption that the cost of the electric generator depends on its rated torque, the cost of the power converter depends on the volt-ampere rating, and the cost of the turbine depends on the product of themaximum turbine torque and the maximum speed. This costassumption is reasonably true for small variations of electricdrive torque and power rate size. The IPM machine can exhibit constant power operationsabove the base speed, at which the nominal power is reached.On the other hand, an increase of the turbine speed over thereference speed given in Section IV should be feasible. The control strategies can be described with the help ofFigs. 3 and 4. From the cut-in speed up to vwind = 12 m/s, thegenerator runs along the MPP trajectory (the 0–T–P–A curve).Then, it operates at constant power operation from A to B. Thisfeature requires no additional cost of a PM generator and of apower converter with respect to the reference case. For higher wind speeds, the control algorithm starts thestalling operations: The turbine is braked so as to reach the newequilibrium operating point characterized by lower speed. Thisis labeled as S in Figs. 3 and 4. The turbine speed has to belimited in order to limit the mechanical stress of the turbineblades. Considering micro wind systems, such a limitation isgenerally high due to the small size of the turbine. A maximumspeed nmax of 1200 rpm (130% of the reference speed) is fixed.A higher speed results in a more expensive turbine. The highercost could be balanced by the higher productivity, as the plantremains working at full power up to a higher wind speed, butWeibull statistic points out that operation at the higher speed hasa low probability. Then, an increase of the turbine cost mightbe not justified by the increase of the energy production. Thisconsideration suggests to investigate the effect of decreasingthe limit of the constant power operation with cost savings ofeach component and a resulting shorter payback time. As anexample, the operating locus 0–T–P–Q–B__ can be examined.This strategy will be referred to as “power-limited operation,”and it will be deeply described and evaluated from an economicalpoint of view. Under the “power-limited operation,” themaximum torque occurs in point P of Figs. 3 and 4. Such torqueaffects both generator and turbine costs. If such costs overcamethe power converter cost, it could be profitable to design thesystem for a “torque-limited operation” instead of a powerlimited operation, for example, along the locus 0–T–Q–B_ in Figs. 3 and 4. The use of an IPM generator allows the operating locus 0–T–Q–B__ to be also adopted where the torque limit is added to the power limit: “torque- and power-limited operation.” Such a case will be also investigated in the next section. VI. CASE STUDY The design criteria are described hereafter referring to the three different wind-power limits. The SPM generator, described in Section IV, is used as a term of comparison. For the sake of simplicity, power limit, torque limit, or both of them are applied, and the resulting performances are evaluated. • Case 1: Power limit—a limit on the volt-ampere rating of the power converter is set, i.e., a limit on the maximum output current. This yields less expensive electronic components to be used • Case 2: Torque limit—a limit on the generator torque is set so that the machine can be designed reducing materials and weight. • Case 3: Torque and power limits—both limits of torque and power are introduced. A. Case 1: Power Limit The power limit is introduced, according to a given maximum volt-ampere rating of the power converter. The power is assumed to be reduced by about 30% with respect to the reference case so that the maximum power is fixed to Pn1=1150 W. Consequently, the maximum torque is Tn1=13 N ・ m, used as the nominal torque to design the generator. Power limitation starts when the wind speed is greater than 10.7 m/s; for higher wind speeds, the operating points are along the trajectory P–B__ of Fig. 5. The maximum torque is reached when the power limit meets the MPP trajectory (point P), as shown in Fig. 5.
Fig. 5. Mechanical power versus rotor speed for case 1, i.e., power is limited to Pn1 = 1150 W. B. Case 2: Torque Limit In this case, the generator is assumed to be designed with areduced torque with respect to the torque of the reference case.A torque reduction of about 30% of the reference case generator is introduced. Therefore, the rated torque is Tn2 = 11.5 N ・ m. Torque limitation starts when the wind speed becomes higher than 10.1 m/s. For higher speeds, the operating points move away from the MPP trajectory following the constant torque trajectory T–B_. The maximum output power is reached in the point B_ as shown in Fig. 6. This maximum power is Pn2 =1450Wat a rotation speed nm = 1200 rpm, and the wind speed is 12.4 m/s. Fig. 6. Mechanical power versus rotor speed for case 2, i.e., torque is limited to Tn2 = 11.5 N · m. C. Case 3: Torque and Power Limit This case is a combination of cases 1 and 2 since both torqueand power limits are introduced. The same generator of case 2 is used (with a reduction of 30% of the torque, with respect to the reference case), and the power converter of case 1 is used (with a reduction of 30% of the power, with respect to the referencecase). Therefore, the power limit is Pn3 = 1150 W, the torque limit is Tn3 = 11.5 N ・ m, and the maximum speed is nm =1200 rpm, as represented in Fig. 7. Torque limitation starts when the wind speed is higher than 10.1 m/s (point T), while power limitation starts when the wind speed is higher than 10.9 m/s (point Q). Fig. 7. Mechanical power versus rotor speed for case 3, i.e., both torque and power are limited to Tn3 = 11.5 N · m and Pn3 = 1150 W, respectively VII. ENERGY PRODUCTIVITY AND COSTS Each case of study presents an improvement in terms of the cost of the system, compared with the cost of the reference case. However, there is a loss of energy productivity, caused by the reduction of power or torque. In order to evaluate the real advantages of the limits introduced, a cost analysis is carried out. To this aim, the total energy obtained from the wind during one year is calculated. Each wind speed is considered, together with its differential probability during the year (see Fig. 1). The total energy in (Wh) obtained from the wind during one year is Etot = (7) where vcut−in is the cut-in wind speed (fixed equal to 2 m/s), vcutoff is the cutoff wind speed (fixed equal to 20 m/s), and hyear is the total number of hours in one year. Although wind data refer to a particular site in South Italy, the results can be easily applied to any other site. The design procedure to select the optimal wind energy systems with reduced initial costs and low energy productivity loss is applicable to any location. The total energy production is considered to be remunerated for a number of years yn = 20. The price due is fixed to pe = 0.3 euros/kWh. As far as the costs of the power converter, the generator, and the mechanical parts are concerned, a cost of cel = 0.65 euros/W is fixed for the power converter, and a cost of cgen = 25 euros/N ・ m is fixed for the generator, while a cost of cmec = 0.045 euros/(N ・ m ・ rpm) is fixed for the mechanical parts. A linear increase of the costs with the electromechanical rating is considered since the analysis refers to small variations with respect to the reference case. For the analysis of the return on investment, a constant wind turbine diameter is assumed. This means that the coefficients given in Table I remain constant in the comparison. The initial cost of the wind turbine, Cinitial, is determined as Cinitial = cgenTmax + celPmax + cmecTMPPnmax (8) where Tmax is the maximum torque, Pmax is the maximum power, TMPP is the higher torque value of the turbine, and nmax is the maximum rotating speed of the generator. Let us note that the cost model is intentionally simplified. A more accurate cost model implies that the generator, the converter, and the wind turbine are known in detail at this stage of the project. However, the rated quantities for their design are known only after this preliminary optimization. Since the profit is focused on the period when the wind-power generator is bought, a reduction on the profit of 5% each year (profit actualization) has been considered. The average electric prize peavg is determined as peavg =(9) where yn is the numbers of years and yindex is determined as yindex =(10) The payback time tpb results in tpb =(11) The actual profit Pactual is expressed as Pactual = peavgEtotyn − Cinitial. (12) VIII. OPTIMIZATION The optimal turbine, using an SPM generator (reference case), is first analyzed from the economical point of view according to three average wind speeds vavg = 3, 5, and 7 m/s The system cost, the profit in 20 years, and the payback time are computed by applying the cost criteria of Section VII. Two further types of turbine are studied in the same conditions: one slower and one faster than the reference case. The turbines will be identified by the corresponding Cp = slow, normal, and fast, as defined by (5) and (6) and Table I. It is assumed that theaerodynamic efficiency of the wind turbines does not change if the maximum power of 1600 W is kept constant even if the speed is changed to 70% and 130% with respect to the maximum speed of the reference case, i.e., 930 rpm. This derives from noticing that the PM machines exhibit a high efficiency in a wide operating range and it is fully controlled in the whole operating range. As far as the drive efficiency is concerned, it cannot be determined at this stage. The result of this optimization will define the optimal rated power of the converter and the rated torque of the generator. Subsequent to such analysis, it is possible to design the generator and the converter and, therefore, to determine their actual efficiency. However, neglecting the efficiency in the model is a precautionary hypothesis yielding to further reduce the payback time.
TABLE II
REFERENCE CASE COMPARISONS (SPM GENERATORS) WITH Tn, PAND nm FOR THREE TYPES OF TURBINE (SLOW, NORMAL, AND FAST
According to the cost scheme proposed in this paper, Table II shows the results of this analysis. As expected, the higher the average wind speed, the higher the profit. However, the profit is not proportional to the wind speed, but it occurs only above a minimum speed, and it increases rapidly with the speed. Furthermore, it highlights that fast wind turbines (Cp = fast) result to be more convenient because of the lower cost, the higher profit, and the shorter payback time. The result is valid for all the wind speeds. To find out the optimal torque and power rating, a variation of both rated power and torque is applied, from 0% to 200% compared to the reference case. For each value of torque and power rating, an increase of 30% of the rotor speed is assumed, in order to exploit the constant power operations of the electric drive. This variation is considered for all the three types ofturbine and the three average wind speeds, as defined earlier. From this analysis, the map of the total system cost, profit after 20 years, and payback time is extracted. To sim plify the search of the optimum rating (Topt and Popt), the results are reported in Table II in per unit, with respect to the reference cases. From this analysis, the maps of Constant Profit and ConstantPayback time have been also computed. An example of such maps is reported in Fig. 8, considering Cp normal and wind. speed vavg = 5 m/s. Solid lines show the curves characterized by the same profit, while dash-dotted lines show the curves characterized by the same payback time.
Fig. 8. Maps of Constant Profit and Constant Fig. 9. Torque versus rotor speed characteristic Fig. 10. Power versus rotor characteristic
Payback time
The optimal turbine size is achieved when the payback time is reduced without penalizing the profit, with respect to the reference case. In other words, the optimal point for designing the wind turbine system corresponds to the maximum ratio Pactual/tpb. According to the maps shown in Fig. 8, the optimal design is found for Popt = 0.8 ・ Pn and Topt = 1.2 ・ Tn. Using this optimal design, Figs. 9 and 10 show how the working point of the IPM generator moves, for differentspeeds, highlighting the peculiar operating capabilities of the IPM generator. When the rated power is reached (at 11.1 m/s of wind speed), the operating point moves along constant power trajectory, up to the maximum turbine speed (at wind speed of 12.1 m/s). At higher wind speed, the stalling mode is started, moving the operating point to a lower speed, i.e., on the higher torque portion of the constant power trajectory. This means that the maximum power is delivered by the generator at all the speeds above 11.1 m/s up to the cutoff wind speed, 20 m/s, at which the turbine is stopped.This procedure to search the optimal turbine ratings is reported for different Cp and different vavg. The results of the analysis are reported in Table III. The Topt and Popt are indicated in percentage values. From Table III, the advantageof using fast turbines (fast Cp) is highlighted. It is worth to compare the results achieved using an SPM generator (seeTable II) and an IPM generator (see Table III). It results that the IPM machine exhibits always a better economical advantage:lower cost, higher actual profit, and lower payback time. Theseadvantages are more evident when the average wind speed is low (e.g.,3 m/s); in this case, the initial cost of the plant isalmost 2.5 times lower, and the actual profit is 2.5 times higher. In sites characterized by higher average wind speed (e.g., 7 m/s), the advantage of adopting an IPM generator remains,but the economical gap is reduced.
TABLE III
OPTIMIZATION COMPARISONS (IPM GENERATORS) WITH nmax = 130% OF nm. Topt AND Popt ARE THE
OPTIMAL VALUES WITH RESPECT TO THE NOMINAL ONES
IX. CONCLUSION This paper focuses on the optimization of a direct-drive low power wind turbine. It gives a methodology for a preliminary choice of the generator and converter ratings on the basis of torque and power limitations of the wind-power system. The aim is to select the most appropriate torque and power ratings by means of economical analysis and a refinement of the wind-power system design. It is shown that the use of an IPM machine instead of an SPM machine allows a great improvement in terms of profit and payback time, particularly for sites with low average wind speed. The highest profit is achieved with the average wind speed of vavg = 5m/s. For higher wind speeds, an oversize of the torque of the electric generator and the power of the electronic drive increases the money profit, but the payback time increases. These criteria can also be used for sizing the vertical axis wind turbines. Such turbines typically have a lower rotation speed, so the generator, with the same power rating, needs to be designed with a higher torque. ACKNOWLEDGMENT The authors would like to thank Prof. G. Pavesi andProf. A. Lorenzoni, Mattia Morandin, Emanuele Fornasiero, Silverio Bolognani, and Nicola Bianchi, for their valuable suggestions on the technicaland economical aspects given during the development of this work. REFERENCES [1] H. Goto, H.-J. Guo, and O. Ichinokura, “A micro wind power generation system using permanent magnet reluctance generator,” in Proc. 13th Eur. Conf. Power Electron. Appl., 2009, pp. 1–8. [2] A. Sharaf and M. El-Sayed, “A novel hybrid integrated wind-PV micro co-generation energy scheme for village electricity,” in Proc. IEEE Int. Elect. Mach. Drives Conf., May 2009, pp. 1244–1249. [3] D. Zinger and E. Muljadi, “Annualized wind energy improvement using variable speeds,” IEEE Trans. Ind. Appl., vol. 33, no. 6, pp. 1444–1447, Nov./Dec. 1997. [4] R. Esmaili, L. Xu, and D. Nichols, “A new control method of permanent magnet generator for maximum power tracking in wind turbine application,” in Proc. IEEE Power Eng. Soc. Gen. Meeting, Jun. 2005, vol. 3, pp. 2090–2095. [5] J. Thongam, P. Bouchard, H. Ezzaidi, and M. Ouhrouche, “Wind speed sensorless maximum power point tracking control of variable speed wind energy conversion systems,” in Proc. IEEE Int. Elect. Mach. Drives Conf., May 2009, pp. 1832–1837. [6] A. Miller, E. Muljadi, and D. Zinger, “A variable speed wind turbine power control,” IEEE Trans. Energy Convers., vol. 12, no. 2, pp. 181– 186, Jun. 1997. [7] N. Horiuchi and T. Kawahito, “Torque and power limitations of variable speed wind turbines using pitch control and generator power control,” in Proc. IEEE Power Eng. Soc. Summer Meeting, 2001, vol. 1, pp. 638–643. [8] M. Khan, P. Pillay, and K. Visser, “On adapting a small PM wind generator for a multi-blade, high solidity wind turbine,” in Proc. IEEE Power Eng. Soc. Gen. Meeting, 2005, vol. 3, p. 2096. [9] D. Corbus and D. Prascher, “Analysis and comparison of test results from the small wind research turbine test project,” presented at the 43rd AIAA Aerospace Sciences Meeting Exhibit—Collection ASME Wind Energy Symp. Tech. Papers, Reno, NV, USA, 2005. [10] N. Milivojevic, I. Stamenkovic, and N. Schofield, “Power and energy analysis of commercial small wind turbine systems,” in Proc. IEEE Int. Conf. Ind. Technol., 2010, pp. 1739–1744. [11] N. Bianchi and A. Lorenzoni, “Permanent magnet generators for wind power industry: An overall comparison with traditional generators,”in Proc. Int. Conf. Opportun. Adv. Int. Elect. Power Gen., Mar. 1996
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