# Numerical Study on the Yaw Control for Two Wind Turbines under Different Spacings
## 4 Results
### Instantaneous Power Analysis of Two Turbines  
The power-time curves of both turbines are shown in Figure 7. It is evident from it that, shortly after the simulation begins, WT1 and WT2 operate at similar power, but when the upstream wake reaches WT2, the output power of WT2 begins to decrease. After a period of time, both turbines run steadily at the corresponding speed and maintain a relatively stable output power, and the power curves eventually present a small range of periodic fluctuations. As shown in Figure 7a, without yaw control, the output power of WT2 for 8D is not significantly higher than that for 5D, 6D and 7D, which indicates that the wake velocity still does not recover apparently even at 8D downstream. When the yaw angle of WT1 is zero, the output power of WT1 is the biggest. However, WT2 is completely located in the wake region, which makes the output power of WT2 reach its minimum number and the average power of WT2 increase with the increase of yaw angle of WT1, but the power fluctuation of WT2 becomes larger. In this study, if the inflow condition is not changed, the operating state of WT1 is only related to its own yaw angle. The yaw control makes the shaft of the turbine deviate from the direction of wind, reducing the windward area and thereby leading to the decrease in the output power of WT1. As the yaw angle of WT1 enlarges from \(0^{\circ}\) yaw to \(10^{\circ}\), \(20^{\circ}\) and \(30^{\circ}\), the output power of WT1 decreases by 3.36%, 11.83% and 25.22%, respectively, which is similar to the power loss of the first turbine in [PERSON]'s research [30], where he studied the yaw control with five turbines. However, WT1 performs similarly at various spacings with the same yaw angle. WT2 can only affect the far wake area of WT1 rather than its near wake area. The output power of turbines is mainly determined by the local wind field near the rotors, so the output power of WT1 is not affected by WT2. Figure 6: The average rotor speed of each turbine in different cases: (**a**) WT1; (**b**) WT2. The upstream wake is deflected by adopting yaw control, and the output power of WT2 is significantly improved. In addition, due to the asymmetric wake from WT1, the power fluctuation range of WT2 observably increases, and the fluctuation range positively correlates with yaw angle. This may lead to the reduction in the quality of output electricity and also increase fatigue loading of the turbine and shorten its working life. The dimensionless standard deviation of the output power of WT2 is evaluated to analyze the power fluctuation, and it is given by:  
\[SD=\frac{\sqrt{(P_{i}-P)^{2}}}{\bar{P}} \tag{8}\]  
where \(P_{i}\) is the instantaneous output power of WT2, and \(\bar{P}\) is the average value of the output power of WT2 in the stable period. The standard deviations of the power fluctuation of WT2 under various working conditions are shown in Figure 8. The power fluctuation of WT2 is related to the asymmetry of the wake caused by WT1 and the wake expansion. The wake asymmetry depends on the yaw angle and turbine spacing, but the wake expansion only depends on the turbine spacing. It can be found that when the spacing between two turbines is small, such as in the case of 5D, the output power fluctuation of WT2 increases with the yaw angle. When the yaw angle of WT1 is 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\) and 30\({}^{\circ}\), the standard deviation of the output power fluctuation of WT2 for 5D is 3.64%, 6.17%, 9.59% and 11.07%, respectively. When the spacing is 6D, 7D and 8D, the fluctuation for the yaw angle of 30\({}^{\circ}\) is smaller than that for the yaw angle of 20\({}^{\circ}\). Figure 7: Power-time curves of two turbines with different spacings and upstream yaw angles: (a) yaw = 0\({}^{\circ}\); (b) yaw = 10\({}^{\circ}\); (c) yaw = 20\({}^{\circ}\); (d) yaw = 30\({}^{\circ}\).