The μ-grid is a blend of WTG and DG, each rated at 150 kW, to realize the required power supply. DG plays a vital part in accomplishing the continuous power supply as WTG depends on the wind condition. Figure 1a show the transfer function diagram of the DG with diesel generator and turbine, and speed representative instruments. The exchange work representation is given as [11]:

$${\Delta {\text{P}}}_{{{\text{GD}}}} = \frac{{1}}{{{(1} + {\text{sT}}_{{{\text{D4}}}} {)}}}{\Delta {\text{P}}}_{{{\text{GT}}}}$$

(1)

$${\Delta {\text{P}}}_{{{\text{GT}}}} = \left( {\frac{{{\text{K}}_{{\text{D}}} {(1} + {\text{sT}}_{{{\text{D1}}}} {)}}}{{{(1} + {\text{sT}}_{{{\text{D2}}}} {)(1} + {\text{sT}}_{{{\text{D3}}}} {)}}}} \right){\Delta {\text{P}}}_{{\text{G}}}$$

(2)

$${\Delta {\text{P}}}_{{\text{G}}} = {\Delta {\text{P}}}_{{{\text{CD}}}} - \left( {\frac{{1}}{{{\text{R}}_{{\text{D}}} }}} \right){\Delta {\text{f}}}$$

(3)

where ΔP_{G} is the control flag of speed governor (p.u.), ΔP_{GT} is the incremental alteration of the speed governor (p.u.), and ΔP_{CD} is the control input to the governor of the DG unit in (p.u.). ΔF is the frequency modification, K_{D} is the gain of the governor with T_{D1}, T_{D2}, T_{D3} being the response times of the speed governor. The gain of the turbine is one and T_{D4} is the reaction time of the turbine. R_{D} is the speed control of the diesel governor. WTG converts wind energy into mechanical and electrical energy. Figure 1a demonstrates the WTG energy conversion process. The fluid coupling system regulates the speed. It compares the turbine and generator frequencies to alter the power output as [11]:

$${\Delta {\text{P}}}_{{{\text{GW}}}} = {\text{K}}_{{{\text{IG}}}} \left[ {{\Delta {\text{F}}}_{{\text{T}}} - {\Delta {\text{f}}}} \right]$$

(4)

K_{IG} is the gain of the fluid coupling with ΔF_{T} referring to the WTG speed modification. The acceptance generator speed can be written as:

$${\Delta {\text{F}}}_{{\text{T}}} = \left( {\frac{{1}}{{{1} + {\text{sTw}}}}} \right)\left[ {{\text{K}}_{{{\text{TP}}}} {\Delta {\text{F}}}_{{\text{T}}} - {\Delta {\text{P}}}_{{{\text{GW}}}} + {\text{K}}_{{{\text{PC}}}} {\Delta {\text{X}}}_{{3}} + {\Delta {\text{P}}}_{{{\text{IW}}}} } \right]$$

(5)

where K_{PC} is the gain of edge characteristics, ΔP_{IW} is the input of wind control deviation (in p.u.) and ΔX_{3} responds to the data fit pitch (DFP) framework. DFP acts as a lag compensator and its work is to match the gain value of the model. The DFP output can be written as:

$${\Delta {\text{X}}}_{{3}} = {\Delta {\text{X}}}_{{2}} \left[ {\frac{{{\text{K}}_{{{\text{P3}}}} }}{{{1} + {\text{sT}}_{{{\text{P3}}}} }}} \right]$$

(6)

In Eq. (6), ΔX_{2} is the output of the hydraulic pitch framework (HPF). T_{P3} is the reaction time of DFP with K_{P3} represent the gain of the DFP framework. The work of the hydraulic pitch framework is to control the pitch point of the edges of the wind turbine, and the output of HPF is:

$${\Delta {\text{X}}}_{{2}} = {\Delta {\text{X}}}_{{1}} \left[ {\frac{{{\text{K}}_{{{\text{P2}}}} }}{{{1} + {\text{sT}}_{{{\text{P2}}}} }}} \right]$$

(7)

In above equation, ΔX_{1} is the pitch controller output with K_{P2} as the gain of the HPF and T_{P2} represents the time constant. The pitch angle system of WTG system needs to be carefully chosen to get the maximum output from the wind turbine. The pitch angle output can be expressed as;

$${\Delta {\text{X}}}_{{1}} = {\Delta {\text{P}}}_{{{\text{CW}}}} \left[ {\frac{{{\text{K}}_{{{\text{P1}}}} \left( {{1} + {\text{sT}}_{{{\text{P1}}}} } \right)}}{{{1} + {\text{s}}}}} \right]$$

(8)

In above equation, ΔP_{CW} refers to the control flag of the pitch point framework (p.u.), K_{P1} is the gain of the pitch point component with T_{P1} being the reaction time of the pitch point instrument. The total parameters of DG and WTG are recorded in the Reference section. The frequency of μ-grid must be confined to convey quality control. Subsequently the overall control of DG and WTG must coordinate to match the load request. Any change between produced power and load may change the μ-grid frequency and consequently ought to be controlled constantly. Considering the overall power change, the frequency variation of the μ-grid can be expressed as:

$${\Delta {\text{f}}} = {\Delta {\text{P}}}_{{{\text{IHPS}}}} \left[ {\frac{{{\text{K}}_{{\text{P}}} }}{{{1} + {\text{sT}}_{{\text{P}}} }}} \right]$$

(9)

where K_{p} and T_{p} represent the gain and time constant of the μ-grid.