AC and DC aggregation effects of small-scale wind generators

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1 AC and DC aggregation effects of small-scale wind generators N. Stannard, J.R. Bumby, P. Taylor and L.M. Cipcigan Abstract: Small-scale embedded generation (SSEG) has the potential to play an important part in the future UK generation mix. If a single SSEG is considered, its environmental, commercial and network operational value is low, but if a cluster of SSEGs are considered and their outputs aggregated they have the potential to be significantly more valuable. The aggregation of small-scale wind generators is considered. Each turbine is driven by a wind that is turbulent in nature and varies from location to location. The output from an individual turbine can be very variable, but when aggregated together with the power output from a number of turbines in the same locality, can produce a power output that is much less variable. The authors examine, by simulation, how the output from a number of small turbines can aggregate together to form a more consistent power output. Aggregation of the turbine outputs both after the power inverter, at alternating current (AC), and before the inverter, at direct current (DC), are examined. In both cases power variations are shown to reduce as the number of turbines connected increases. Aggregation at DC can lead to non-optimum performance of the turbine itself, if passive rectifiers are used, but can also lead to savings in the cost of the power conversion equipment required. List of symbols A Turbine swept area, m 2 C p E phase h I c I d Coefficient of performance Induced emf per phase, V Tower height, m DC capacitor current, A DC inverter current, A J r Turbine and generator inertia, kg m 2 k Generator emf constant K F Gain of the Nichita filter L phase Generator phase inductance, H p Number of pole pairs P turbine Turbine mechanical power, W P Power transfer through the inverter, W P limit R R DC R over R ph s t T elect T F Inverter rated power, W Turbine radius, m Generator resistance referred to the DC side of the converter, V Overlap resistance, V Generator resistance per phase, V Laplace operator Time, s Turbine electrical torque, Nm Time constant of Nichita filter, s # The Institution of Engineering and Technology 2007 doi: /iet-rpg: Paper first received 17th January and in revised form 2nd April 2007 The author are with the School of Engineering, Durham University, Science Site, South Road, Durham, DH1 3LE, UK j.r.bumby@durham.ac.uk T mech Turbine mechanical torque, Nm U Average wind speed, m/s V 1 Inverter DC cut-in voltage, V V 2 DC voltage for maximum inverter power, V V d DC voltage, V V do Open circuit DC voltage, V h WB Inverter efficiency l Tip speed ratio r Density of air, kg/m 3 s u Standard deviation of wind turbulence v m Turbine rotational speed, rad/s 1 Introduction Micro-generation or small-scale embedded generation (SSEG), has a significant part to play in the future UK energy system and its importance has been recognised by the UK government in its Energy White Paper [1]. The UK government s policy on renewable energy and combined heat and power (CHP) is expected to lead to a continuous increase in distributed generation (DG). The Government have set a target that 10% of UK electricity should be sourced from renewable energy by 2010 and 20% by 2020 [2]. In order to meet the 2010 target, approximately 10 GW of additional DG will have to be connected to distribution networks. The DTI suggest that up to half of this 10% renewables target could be provided by wind energy [3]. A recent study focussing on SSEG investigated a number of potential penetration scenarios. Table 1 summarises the capacity (GW) and energy (TWh) which could be provided by micro-generation to 2020 [4]. SSEG includes small-scale wind turbines, solar photovoltaics, micro-hydro generators, micro-chp along with hydrogen fuel cells and bio-energy and is defined in IET Renew. Power Gener., 2007, 1, (2), pp

2 Table 1: Capacity and energy due to microgeneration to 2020 [4] Scenario GW TWh GW TWh GW TWh Low Mid High ER G83/1 [5] as any source of electrical energy rated up to, and including, 16 A per phase, single or multi-phase, 230/ 400 V AC. This paper is concerned with small-scale wind turbines, typically,10 kw, and although these may be less aerodynamically efficient than their large-scale counterparts, they nevertheless have an important role to play in meeting some of the country s energy needs, especially in the built environment. It is estimated that the aggregated annual energy production by 2020 from wind turbines in the built environment could be in the range of TWh (dependent on the distribution of installations with respect to optimal wind speed) resulting in annual carbon dioxide savings in the range of Mt CO 2 [6]. The value of SSEG to the electrical supply industry depends on a number of factors including: The revenue streams that can be achieved through the sale of electricity. The degree to which SSEG can participate in future ancillary services markets [7]. The environmental impact that can be achieved and the degree to which SSEG can displace fossil fuel-based generation plant. The ability of SSEG to contribute to the deferral or avoidance of network reinforcements. (LV) network containing a mixture of SSEGs, distributed storage and load. These elements are controlled to increase the value of SSEGs [10]. This concept can be extended to consider a number of SSEZs, which can be co-ordinated through a supervisory controller. This clearly increases the potential for value improvements still further. The SSEZ concept does not have a strict definition in terms of geographical area or total generation capacity [11]. However, research being carried out to evaluate this concept at Durham University has considered a SSEZ to contain 96 customers, which means that if each customer has a 1.1 kw generator installed the total generating capacity in the zone would be approximately 100 kw. It is clear that if a significant impact in energy markets and network operation is to be achieved a number of SSEZs of this size would be required. The current SSEZ definition is derived from a UK generic LV network configuration approved by the UK distribution network operators (DNOs) [12]. This generic network would accommodate up to 192 SSEZs providing a maximum total generation capacity of approximately 20 MW. It is expected that wind will make up a significant part of the future SSEG market and therefore the aggregation of small-scale wind generators is an important part of the SSEZ concept. This paper therefore investigates, by simulation, the effect aggregating the power output of a number of small-scale wind turbines within a SSEZ has on the variability of the net generated power. Small-scale wind turbines are usually connected to the mains through a power electronic converter comprising of When considering a single SSEG it is clear that it does not have the ability to make significant impacts in the above four areas. However, if a number of SSEG outputs are aggregated, coupled with controllable energy storage devices and demand side management techniques, within a small-scale energy zone (SSEZ) their value has the potential to be significantly increased. This idea is related to the virtual power plant concept discussed in [8, 9] where aggregation and control are discussed as ways of more effective market interaction for SSEG. A SSEZ is shown in Fig. 1 and is defined as a controllable section of low voltage Fig. 1 Small-scale energy zone (SSEZ) Fig. 2 a At AC b At DC Aggregation 124 IET Renew. Power Gener., Vol. 1, No. 2, June 2007

3 a passive rectifier and mains side IGBT inverter. Aggregation of the outputs of a number of such systems can be achieved either on the AC side or DC side of the mains connect inverter as shown in Fig. 2. Aggregation at the AC side represents the normal situation where a number of customers install a single wind turbine, each with a dedicated power electronic interface. It is generally acknowledged that the aggregation of the output from several wind turbines in this way is less variable than that of a single wind turbine, and that the variability decreases as the number of wind turbines aggregated increases [13]. However, the effect of aggregating several small-scale wind turbines on the DC side of the inverter, so as to share the same inverter and hence reduce power conversion cost, is much less clear. Such an approach to aggregation could well appear attractive where, for example, a commercial building or block of flats installs a number of wind turbines which share the same inverter. The paper begins by describing the model of the wind turbine system used to investigate the AC and DC side aggregation. Aggregation case studies at AC and DC are then compared to evaluate the impact of aggregation on both the net power generated and the performance of the wind turbine itself. 2 Small-scale wind turbine models To examine aggregation effects, detailed models of the turbine and generator have been constructed in Matlab/ Simulink. Most importantly, the model must account for the turbulence effects that exist in real winds. Such a model has been developed previously [14] and brief details about the models are given in Section 8. The impact that aggregation has on both the quality of the power supplied to the system and the operation of the turbines themselves is evaluated. Fundamental to the model is the assumption that the injection of the generated power will have negligible effect on the AC system voltage; that is a stiff AC system is assumed. Such an assumption is valid for normal operation in that the injected powers are relatively small, typically less than 20 kw, and that the turbines within the same SSEZ will be electrically close together. The wind model uses a spectral density function derived from real wind data to generate typical wind profiles around a mean wind speed as described in detail in [14] and summarised in Section 8.1. As the simulation is primarily concerned with second by second variations, the wind turbine is modelled by its C p /l performance curve such Fig. 4 Generator equivalent circuit as that shown in Fig. 3 for a Savonius wind turbine. The permanent magnet generator and associated rectifier are modelled by an equivalent circuit viewed from the DC side of the rectifier as shown in Fig. 4 [14 16]. In all the simulations, the use of a permanent magnet generator is assumed. With this model the turbine speed is computed from v m ¼ 1 J r ð (T mech T elect )dt (1) The turbine s mechanical (or aerodynamic) torque is found from the turbine performance characteristics, whilst the generator torque is determined by the power transfer characteristics of the mains-connect inverter. The output from the generator is rectified typically through a standard six-pulse rectifier before being connected to the power inverter. A widely used inverter that meets the G83 mains connect standard is the SMA Windy Boy [17]. This inverter transfers an amount of power to the mains depending on the value of the DC link voltage (Fig. 5). Fig. 5a shows the power transfer characteristic of a 500 W inverter used in the AC aggregation study and Fig. 5b the characteristic of the 2500 W inverter used in the DC aggregation study. Each of the inverter power transfer profiles is similar and, for a mains voltage of 230 V, the inverter cuts-in at a DC voltage V 1 of about 250 V. The inverter then transfers power as a linear function of the Fig. 3 Savonius turbine C p l performance characteristics Fig. 5 Windy Boy power-dc voltage characteristic a For AC aggregation of 500 W turbines b For DC aggregation of five, 500 W turbines IET Renew. Power Gener., Vol. 1, No. 2, June

4 Fig. 6 curves Windy Boy load characteristic showing Savonius power DC link voltage until the maximum power, P limit, of the inverter is reached; in this case 500 W or 2500 W, at voltage V 2. This DC voltage V 2 must be,600 V (the inverter limit) and can be programmed by the user. Values used in the study are shown in Fig. 5. For a single turbine with a PM generator, the DC voltage is approximately proportional to the turbine rotational speed, v m, so that Fig. 5 is essentially an electrical power speed characteristic. This electrical characteristic can be combined with the turbine characteristic of Fig. 3 to give the power curves of Fig. 6; in this case for a small 500 W Savonius vertical axis wind turbine (VAWT). Where the two characteristics cross determines the system operating point and power transfer to the mains for any given wind speed. Dimensions of the Savonius wind turbine are given in Table 2. Table 2: Parameters of the 500 W Savonius VAWT and generator Symbol Value Units Savonius Turbine Rotor height h m Rotor diameter d m Swept area A m 2 Rotor inertia J r Kg m 2 Cut-in wind speed V cut-in 2.5 m/s Rated wind speed V rated 12.5 m/s Cut-out wind speed V cut-out N/A Energy capture coefficient (max) C p(max) 0.2 Savonius generator Rated power P elec 500 W Rated speed N 300 rpm Rated frequency f 40 Hz Rated EMF (per coil) E coil 33.6 V Number of phases 3 Number of pole pairs p 8 Number of armature coils 12 Machine constant K 0.11 V/rpm/coil Coil resistance R coil V Coil inductance L coil 4.74 mh DC efficiency h dc 92 % Generator diameter D gen m 126 Also shown in Fig. 6 are experimental results for the permanent magnet generator described in [15] connected to the mains through a SMA inverter. This data is compared with predicted results using the wind turbine data given in Table 2. Good agreement between measured and predicted results is achieved, giving confidence in the simulation model. The Windy Boy inverter s default operating mode is at unity power factor. However, reactive power control would be possible through modulation of the amplitude and phase of the Windy Boy inverter voltage with respect to the network voltage. It is also possible that further reactive power control could be provided by the inverters associated with distributed energy storage devices within the SSEZ. Reactive power control, however, would be of limited value since the reactance/resistance, X/R, ratio of typical LV distribution networks is less than one. The key to operating a wind turbine system efficiently is in the careful matching of the different components and how the whole system is controlled. Wind turbines only work efficiently in a narrow band of speeds, which vary according to the strength of the wind. Thus, the power extracted from the generator must be carefully controlled in order to maintain the turbine s optimal rotational speed. 3 Aggregation study In this study, data for a 500 W Savonius wind turbine is used (Table 2). Such sized machines naturally lend themselves to the use of a number of turbines aggregated at DC and connected to the mains through a single inverter. With such a connection, and the use of a passive rectifier, the inverter will control the power transfer from the group of turbines rather than from each individual turbine. Larger turbines, of a few kw rating, would tend to be connected through a dedicated inverter allowing control of individual turbines. However, in order to obtain a direct comparison between the two connection systems, the same 500 W turbine system is used to study aggregation effects on both the AC and DC side of the mains connect inverter. It is recognised that AC aggregation is more suited to larger power turbines, but not exclusively so. The study is carried out assuming each wind turbine is spaced far enough apart for the turbulence at each turbine to be independent from its neighbour so that a separate turbulent wind, but with the same mean wind speed, can be used as the input for each turbine; an average wind speed of 10 m/s is used. The turbulent wind for each turbine is therefore different, but by using the same seed in the turbulent wind model [14], can be repeated to allow an effective performance comparison. The aggregation of five wind turbines is shown in Fig. 7; Fig. 7a, c and e shows the power output, speed and DC link voltage of individual turbines, whilst Fig. 7b, d and f shows the same but for DC aggregation. Fig. 7g shows the effect of aggregating the outputs of the five turbines. In both cases aggregation is seen to have a significant effect on power smoothing. At time zero all the turbines are at rest and take about 15 s to reach operating speed. Power is not produced until about 10 s after the start as the turbine speed, and hence DC link voltage, is less than the minimum required for the inverter to operate (Fig. 5). The results show that with AC aggregation the turbines operate at a lower speed, approximately 300 rpm, compared to DC aggregation, approximately 350 rpm. Speed variations are also greater with AC aggregation. Similarly, DC link voltage is higher with DC aggregation than AC IET Renew. Power Gener., Vol. 1, No. 2, June 2007

5 Fig. 7 Effect of aggregation at AC and DC a Individual speeds at AC b Individual speeds at DC c Individual power outputs at AC d Individual power outputs at DC e DC link voltage at AC f DC link voltage at DC g Total output power for turbines aggregated at AC and DC (there is only one DC link voltage with DC aggregation because of the use of a common DC bus). It is also notable from Fig. 7g that although in both cases aggregation leads to substantially smoother power delivery, the power delivered using DC aggregation is substantially less than when AC aggregation is used. Connecting multiple turbines to a single inverter creates a more complicated power transfer system than using a IET Renew. Power Gener., Vol. 1, No. 2, June

6 dedicated inverter for each individual turbine. When a single turbine is connected to a single inverter the power output follows the Windy Boy s load line according to the instantaneous turbine speed, as shown in Fig. 5, since turbine speed and the DC link voltage are tightly linked. With several turbines connected to the inverter, the turbine s power output becomes decoupled from the Windy Boy s operating line, since the individual turbine speeds and DC link voltage are set by the group. Consider Fig. 8, which shows the power transfer as a function of the DC link voltage. If all five turbines were producing the same power P then the power converter must transfer 5P to the mains. This can only be achieved if the DC link voltage rises to a higher DC voltage, V 5. This in turn corresponds to a higher turbine speed so that each turbine will operate at a higher speed than if it were transferring power individually through its own dedicated power converter. This has the effect of shifting the turbine operating curve to the right (Fig. 9a). The impact of this behaviour is well demonstrated in Fig. 9a by simulating each of the five wind turbines operating at constant wind speeds of 9, 10, 11, 12 and 14 m/s, respectively; this is shown as operating line 1 in Fig. 9a. (The operating line in Fig. 9a would be vertical if each generator was ideal; that is it had no armature resistance or inductance, and all five generators would run at exactly the same speed.) As the DC voltage is set by the total power transfer, all five turbines operate at higher rotational speeds than if they were controlled separately. This new common operating line intersects the operating points of the five individual turbines, shown as circles in the figure. The power transfer mechanism is further complicated as the common operating line shifts along the x-axis depending on the average wind speed across the group of turbines. This can be seen by modifying the five wind speeds to 9.5, 10, 10.5, 11 and 12 m/s respectively, and is shown as operating line 2 on Fig. 9a. When turbulent wind inputs are used, the average wind speed fluctuates with time, and has the effect of shifting the common operating line back and forth along the x-axis. The power output of a single turbine moves up and down the common operating line as the wind speed at the individual turbine fluctuates due to turbulence, while the common operating line moves back and forth along the x-axis as the average wind speed fluctuates. This behaviour is shown in Fig. 9b where the output power and speed of one Savonius turbine from the model has been plotted during the 100 s simulation run. 4 Discussion Aggregation of a number of wind turbines rapidly smoothes the net power output compared to the power output from an individual turbine. This is true regardless of whether the Fig. 8 Windy Boy characteristic showing how DC voltage determines total exported power 128 Fig. 9 Output of Savonius VAWT a Multiple Savonius operating characteristic b Output of one Savonius turbine aggregation is performed at AC or DC. However, in DC aggregation the power transfer is determined by the behaviour of the group of turbines so that each turbine operates sub-optimally and power transfer is not as great as with AC aggregation. However, there are cost savings by using a single inverter. Current costs for mains connect inverters are tabulated in Table 3 [18]. The cost of a 2500 W inverter is about twice the cost of a 700 W inverter so that aggregating W turbines through one 2500 W inverter reduces the cost from to ; a cost saving of Further, by aggregating at DC the power variation the inverter must now cope with has been reduced; in the case of dedicated inverters each inverter may be called upon to transfer 500 W (Fig. 7c), whilst the DC aggregated maximum is under 1000 W (even the AC aggregated is less than 1500 W). This means that either more turbines could be connected to the inverter or the inverter rating could be reduced to, say 1700 W, further reducing cost (Table 3). The smoothing effect of aggregation at both AC and DC is shown in Fig. 10, where the variation in power output as a fraction of the average power output is plotted. The variation is calculated using (2) with the analysis performed for a single turbine and for the aggregated outputs of 2, 3, 4, 5, 6 and 7 turbines. As the variation in power output is dependant on the exact nature of the turbulent wind IET Renew. Power Gener., Vol. 1, No. 2, June 2007

7 Table 3: Cost of Windy Boy inverters [18] Rating (W) Cost ( ) Cost/W ( /W) profile (Fig. 7c and d), the model was run over ten different random turbulent wind profiles and the results are averaged in order to obtain a meaningful result. In all cases the same mean wind speed was used. (Maximum power Minimum power)=2 Power variation ¼ Average power (2) Fig. 10 shows that the variations in power decrease, from approximately 90% to about 20% of the average power, as the number of turbines aggregated increases to 7. It is clear that aggregation quickly smoothes the power output. Fig. 10 also suggests that better power smoothing is achieved with DC aggregation as the turbines are constrained to operate in a much narrower speed range for reasons described above. It is reasonable to assume that the reduction in variability will increase as more and more turbines are aggregated. Therefore significant smoothing can be expected within a small-scale energy zone (Fig. 1), where anything up to 50 or 100 turbines could be present and available for aggregation. As explained above, if aggregation is performed on the DC side, and a single common inverter used, then the optimum power transfer from each turbine may be compromised if the turbines are connected to a common DC link through conventional passive rectifiers as has been assumed here. This issue can be solved by modifying the operation of the power electronic interface so that the diode rectifier is replaced by a more intelligent converter system that allows each turbine to be controlled independently. Such a converter system using a standard six-pulse rectifier and DC/DC boost converter is described in [19]. This modification is worthy of further investigation as the cost of G83 compliant power electronic interfaces is a significant proportion of the overall costs for small-scale wind generation systems; typically 50% of the turbine and generator cost (or 15 20% of the installed cost). This work represents the first step in aggregation in the SSEZ concept as only wind generation was considered here and with only five wind turbines. Nevertheless, the study has demonstrated the benefits of aggregation and raised some of the issues faced in the aggregation process. It must also be realised that the aggregation effect examined is due to turbulent wind and as such is limited to a relatively short period that is a few minutes. It does not examine the impact of diurnal variations nor does it account for longer term weather fluctuations. In order to realise the benefits of aggregation on these longer time scales, aggregation must be applied to zones that contain a mix of a large number of SSEG technologies, controllable loads and energy storage devices. Multiple zones must also be considered to reap the full potential of this idea. 5 Conclusions This paper has presented the aggregation of small-scale wind turbines and has shown that the aggregated output is smoothed as a result of the aggregation even for a small number of wind turbines. It is clear that aggregation at AC is straightforward but problems can be experienced with basic power electronic interfaces if aggregation takes place on the DC side. With a passive rectifier, aggregation at DC can lead to sub-optimal operation of the turbine system but can also lead to a saving in power conversion cost. The value of small-scale wind aggregation can be quantified in the following ways: 1. For the distribution network operator (DNO). If the DNO can treat a number of small-scale wind turbines as one lumped generator which has a smoother and more predictable power output profile this can assist with network operational tasks. 2. Revenue streams from the sale of electricity. If the SSEG owners can secure power purchase agreements with supply companies as an aggregated block, the increased capacity and greater predictability achieved should allow improved contractual terms to be negotiated. 3. Environmental benefits. If the aggregated SSEGs output is more predictable this will provide more confidence that fossil fuel generating plant can be displaced by the growth of SSEGs. This will therefore translate into significant environmental benefits. 6 Acknowledgments This work has been carried out as part of a project on Intelligent active energy management for small scale energy zones and the authors gratefully acknowledge the financial support from EPSRC. 7 References Fig. 10 Power output variations for different numbers of wind turbines aggregated 1 DTI Energy White Paper: Our energy future creating a low carbon economy. DTI, February Carbon Trust Report: Mott McDonald, capacity mapping and market scenarios for 2010 and 2020, November Milborrow, D.: Assimilating wind, wind energy contribution to UK energy needs, IEE Rev., 2002, 48, (1), pp DTI s Distributed Generation Programme: System integration for additional microgeneration (SIAM). DTI, Engineering Recommendation G83/1: Recommendations for the connection of small-scale embedded generators (up to 16 A per phase) in parallel with public low-voltage distribution networks (Energy Networks Association, London, September 2003), available at 6 Dutton, A.G., Halliday, J.A., and Blanch, M.J.: The feasibility of building mounted integrated wind turbines (BUWTs): achieving IET Renew. Power Gener., Vol. 1, No. 2, June

8 their potential for carbon emissions reduction. Report for the Carbon Trust, , May Ancillary service provision from distributed generation, DTI, Contract number DG/CG/00030/00/00, DTI Technology Programme: New and Renewable Energy, Gordijn, J., and Akkermans, H.: Business models for distributed energy resources in a liberalized market environment, Electr. Power Syst. Res. J., 2007 (in press) 9 Franke, M., Rolli, D., Kamper, A., Dietrich, A., Geyer-Schulz, A., Lockemann, P., Schmeck, H., and Weinhardt, C.: Impacts of distributed generation from virtual power plants. Proc. Annual Int. Sustainable Development Research Conf., June 2005, vol. 11, pp Taylor, P.: Active network management as an enabler for the proliferation of domestic combined heat and power. 12th Int. Stirling Engine Conf., Durham, September Taylor, P., and Cipcigan, L.: Small scale energy zones for the effective participation of SSEGs in energy markets. WEC Regional Energy Forum FOREN 2006, June 2006, Neptun, Romania, s Ingram, S., and Probert, S.: The impact of small scale embedded generation on the operating parameters of distribution networks. P B Power, DTI New and Renewable Energy Programme, June Manwell, J., and Hunter, R.: Wind diesel systems (Cambridge University Press, UK, 1994) 14 Bumby, J.R., and Stannard, N.: Performance aspects of mains connected small scale wind turbines, IET Gener., Transm. Distrib., 2007, 1, (2), pp Bumby, J.R., and Martin, R.: An axial flux, permanent magnet, air-cored generator for small scale wind turbines, Proc. IEE Electr. Power Appl., 2005, 152, (5), pp El Mokadem, M., Nichita, C., Dakyo, B., and Koczara, W.: Maximum wind power control using torque characteristic in a wind diesel system with battery storage. 16th Int. Conf. on Electrical Machines, Cracow, Poland, 5 8 September SMA: Windy Boy grid tied inverter operators manual addendum for the Sunny Boy 1800U/2500U. SMA America, October Wind and Sun, Inverter costs, accessed March Parker, M.A., Tavner, P.J., Ran, L., and Wilson, A.: A low cost power tracking controller for a small vertical axis wind turbine. 40th Universities Power Engineering Conf., Cork, Ireland, September Nichita, C., Luca, D., Dakyo, B., and Ceanga, E.: Large band simulation of the wind speed for real time wind turbine simulators, IEEE Trans. Energy Convers., 2002, 17, pp proposed by Nichita [20]. m W Nichita (s) ¼ K 1 T F s þ 1 F (3) T F s þ 1 m2 T F s þ 1 with m 1 ¼ 0.4 and m 2 ¼ 0.2. The gain K F and time constant T F of the filter depend on the turbulence length scale that is the typical length of eddies in the free stream wind and also on the mean wind speed and are carefully chosen so as to ensure that the standard deviation of the filtered output is equal to 1. This normalised coloured noise is then be multiplied by the respective wind-speed dependent standard deviation, s u, to provide a turbulent wind component with the correct standard deviation that is intensity. The standard deviation s u increases in proportion to the wind average speed as s u ¼ k s U (4) where k s, the proportionality constant depends on the terrain conditions. By adding the base wind speed, U, to the turbulent component, a turbulent wind profile can then be generated. This model produces a point wind speed; so before finally applying the wind speed to the turbine model it must be further filtered to take account of disc averaging. The effects of wind shear and tower shadow are not included in the model. This turbulent wind model accurately accounts for atmospheric disturbance and is valid for a time period of a few minutes. It does not, and is not intended to, account for wind variations on a longer time scale than this. 8.2 Turbine model The power produced by the turbine is related to the turbine dimensions and the wind speed by P turbine ¼ 1 2 C PrAU 3 (W) (5) with the torque produced by the turbine given by 8 Appendix 8.1 Spectral turbulent wind model A schematic representation of the turbulent wind model is shown in Fig. 11 and full details of this wind model can be found in [14]. Central to the wind model is the random number generator, which produces normally distributed white noise; this acts as the basis for the turbulent wind component. Since the wind speed cannot (for physical reasons) change instantaneously, the white noise must be smoothed using a carefully designed signal shaping filter in order to achieve the correct spectral distribution. The shaping filter must be chosen to best characterise the true spectrum of the wind and there are a number of candidates for this task. This shaping filter is well represented by a Von Karmen filter, or as used here, a second order approximation Fig. 11 Schematic representation of the turbulent wind model T mech ¼ P turbine v m (6) The wind speed is obtained at each time instant from the spectral wind model [14] whilst the air density and swept area of the turbine are constant (or at least dictated by external factors such as the weather conditions and turbine geometry), so that only the energy capture coefficient, C p, need to be determined in order to calculate the turbine power (and hence torque). C p is a function of the turbine blade pitch angle, u, and a dimensionless group, the tip-speed ratio, l. For many small turbines the turbine blades are fixed at a constant pitch angle, which constrains C p to be a function of l only. The tip-speed ratio is defined as the tip speed of the turbine blades divided by the oncoming wind speed, as l ¼ v mr (7) U The variation of C p as a function of l produces a performance curve for the turbine and is a fundamental part of the specification of any turbine, whether small or large. A typical curve for a Savonius VAWT is shown in Fig. 3. In modelling this system, l is calculated from (7), which then allows C p to be interpolated for a given value of l. The turbine torque can then be computed from (5) and (6) for use in (1). 130 IET Renew. Power Gener., Vol. 1, No. 2, June 2007