**5. Energy Management Strategy**

### *5.1. MPPT Control Strategy for Wind Turbine*

Perturbation and observation method with variable step is used to achieve maximum power output of the wind turbine by adjusting the duty ratio. The method needs to estimate the position of the current maximum power point by real-time monitoring of the power difference between the two times, thereby determining the size of the duty cycle. If the difference is positive, the duty cycle will be decreased. Otherwise, the duty cycle will be increased. If it is zero, it indicates that the maximum power point has been reached. Since the wind speed in nature is randomly fluctuating, it may cause the power supply system to oscillate. Two step sizes are proposed in this study. The threshold of the power difference is set. If the difference is within the threshold, a small step will be used. If the difference is outside the threshold, a large step will be used to gradually approach the maximum power point.

The mechanical energy produced by the wind turbine in this study can be expressed as follows.

$$P\_{WT} = \frac{1}{2} \mathbb{C}\_p(\lambda, \boldsymbol{\beta}) S \boldsymbol{\sigma} \boldsymbol{\nu}^3 \tag{15}$$

where *PWT* is the output of wind turbine (W); *Cp*(λ*,* β) is wind energy utilization factor; λ is tip speed ratio; β is pitch angle of blade; *S* is sweep area (m2); σ is air density (Kg/m3); *v* is wind speed (m/s).

The sweep area of wind turbine can be described as follows.

$$P\_{\rm WT} = \frac{1}{2} \mathbb{C}\_p(\lambda\_\prime \beta) S \sigma \upsilon^3 \tag{16}$$

where *R* is the impeller radius (m).

> The wind energy utilization factor can be described as follows.

$$\mathcal{C}\_{p}(\lambda,\beta) = 0.5176(116\lambda\_{c} - 0.4\beta - 5)e^{-\frac{21}{\lambda\_{c}}} + 0.0068\lambda\tag{17}$$

λ and λ*c* in Equation (16) can be described as follows.

$$\begin{cases} \lambda = \frac{\omega R}{v} = \frac{2\pi nR}{60v} \\ \lambda\_c = \frac{1}{\lambda + 0.08\beta l} - \frac{0.035}{\beta^3 + 1} \end{cases} \tag{18}$$

where ω is wind turbine angular velocity (rad/s); *n* is rotational speed of wind turbine (r/min).

In the study of the MPPT strategy of the wind turbine, β (pitch angle of blade), *v* (wind speed) and ω (wind turbine angular velocity) are set as input to the wind turbine power generation model. Air density is selected as 0.927 Kg/m3. β is 0 in this study. For wind speed *v*, we built a natural wind speed model. The wind speed model is considered as a combination of basic wind speed *Vb*, gradual wind speed *Vr*, and gus<sup>t</sup> wind speed *Vg*. The basic wind speed *Vb* is the average wind speed (7 m/s). The gradual wind speed *Vr* characterizes the slow change of the wind speed and can be expressed by follows.

$$V\_r = V\_r(\max) \frac{t\_{r1} - t}{t\_{r1} - t\_{r2}} \tag{19}$$

where *Vr*(*max*) is maximum value of gradual wind speed (10 m/s); *tr*1 is start time of gradual wind (4 s); *tr*2 is end time of gradual wind (11 s); *t* is time of gradual wind.

The gus<sup>t</sup> wind speed *Vg* can characterize the degree of abrupt change in wind speed and can be expressed by follows.

$$V\_{\mathcal{S}} = V\_{\mathcal{S}}(\max) / 2 \times \left[ 1 - \cos 2\pi \left( \frac{t - t\_{\mathcal{S}^1}}{T\_{\mathcal{S}}} \right) \right] \tag{20}$$

where *Vg*(*max*) is maximum value of gus<sup>t</sup> wind speed (6 m/s); *tr*1 is start time of gus<sup>t</sup> wind (3 s); *Tg* is period of gus<sup>t</sup> wind (6 s); *t* is time of gus<sup>t</sup> wind.

The model of natural wind speed can be described as follows.

$$v = V\_b + V\_r + V\_{\mathcal{S}} \tag{21}$$

We introduced the natural wind model into the wind power model, and the simulation results obtained by MATLAB Simulink of the wind speed, wind turbine output power and rotational speed of wind turbine are shown in Figure 8.

**Figure 8.** The simulation results of (**a**) the wind speed, (**b**) wind turbine output power and (**c**) rotational speed of wind turbine.

As can be seen in Figure 8a, at the beginning, the wind speed was low (about 2–5 m/s), thus the output power of the wind turbine was also low (Figure 8b). From 4 s to 6 s, the wind speed increased rapidly from the beginning of 2 m/s to 17 m/s, and the corresponding wind turbine output power was gradually increased, which the maximum power can reach 13 kW. After the 8 s, the wind speed began to slowly decrease and the output power decreased slowly. We also simulated the effects of abrupt wind speeds on wind turbine output power at 6 m/s, 8 m/s and 10 m/s (Figure 9). Under the influence of abrupt wind, the response time of wind turbine speed and power is less than 0.14 s.

**Figure 9.** The simulation results of (a) the wind speed, (b) wind turbine output power and (c) rotational speed of wind turbine at abrupt wind speeds.

### *5.2. MPPT Control Strategy for PV Array*

Perturbation and observation method with adaptive variable step is adopted, which achieves self-selection of the step size by adding an adaptive algorithm when setting the step size. This method not only improves the steady state performance of the power system, but also improves the dynamic performance. The step size calculation can be described as follows.

$$S(k+1) = N \frac{P(k) - P(k-1)}{S(k)} \tag{22}$$

where *S*(*k*) is step size (0 < *S*(*k*) < 1); *N* is the constant determined by the sensitivity of the adaptive variable step size adjustment; *P*(*k*) is power.

The flowchart of the perturbation and observation method with adaptive variable step is given in Figure 10.

The perturbation and observation method with adaptive variable step obtained *I*(*k* − 1), *I*(*k*), *U*(*k* − 1) and *U*(*k*) to calculate *P* (*k* − 1) and *P*(*k*). We go<sup>t</sup> the value of difference of power *dP* (Figure 10). The threshold of the power difference (*ep*) was set. The direction of perturbation can be determined by calculating *dP*-*ep*. Then the step size *S*(*k* + 1) can be adjusted by Equation (21). Until *dP* = 0, the maximum power point can be considered to be reached.

**Figure 10.** The flowchart of the perturbation and observation method with adaptive variable step.
