### Control principle

In the hybrid wind-diesel microgrid, the ACE is an important index to assure the stability of the system [29,30,31], and can be expressed as:

$$Delta ACE_{i} = Delta P_{ij} + n_{i} Delta f_{i}$$

(15)

where (n_{i}) is the frequency deviation factor.

In order to improve the LFC accuracy, the (Delta ACE) value can be divided into four control intervals according to the performance of different generators. (ACE^{N}), (ACE^{A}) and (ACE^{E}) are the lower thresholds of normal regulation zone, alert regulation zone and emergency regulation zone, respectively. Based on those thresholds, four control zones can be defined as:

(left| {Delta ACE} right| le ACE^{N}) is the dead zone.

(ACE^{N} < left| {Delta ACE} right| le ACE^{A}) is the normal regulation zone.

(ACE^{A} < left| {Delta ACE} right| le ACE^{E}) is the alert regulation zone.

(ACE^{E} < left| {Delta ACE} right|) is the emergency regulatory zone.

The system operational state is monitored by the dispatching center, which can quickly obtain the safe level of the system and the value of (Delta ACE) to keep the balance between load and source.

#### The dead zone

In order to prevent unnecessary action of the governor system, a dead zone is set near the rated frequency. The tie-line power and frequency fluctuations in the dead zone are small, so a high-power density ultra-capacitor may be the best choice to resolve the small power fluctuations in the dead zone. The required power is given as:

$$E_{ACE}^{D} = Delta ACE,{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} 0 < left| {Delta ACE} right| le ACE^{N}$$

(16)

where (E_{ACE}^{D}) is the total required power of the Dead Zone.

If (E_{ACE}^{D}) is positive, the ultra-capacitor absorbs excess power from the system, whereas the ultra-capacitor increases its output power when (E_{ACE}^{D}) is negative. Therefore, the response power of the ultra-capacitor in the dead zone is described as:

$$P_{uc}^{D} = left{ {begin{array}{*{20}c} {min left( {left| {E_{ACE}^{D} } right|,{kern 1pt} {kern 1pt} {kern 1pt} P_{uc}^{dis – max } } right),{kern 1pt} {kern 1pt} – ACE^{N} le Delta ACE < 0{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} } \ {min left( {{kern 1pt} {kern 1pt} E_{ACE}^{D} ,{kern 1pt} {kern 1pt} ;{kern 1pt} P_{uc}^{ch – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} ;;;;;0 < Delta ACE le ACE^{N} } \ end{array} } right.$$

(17)

where (P_{uc}^{dis – max }) and (P_{uc}^{dis – max }) are the ultra-capacitor’s maximum allowable discharging and charging power, respectively.

#### The normal regulation zone

When a small disturbance occurs in the power system, it is usually considered that the frequency deviation and ACE are in the normal regulation area. Here, the adjustable generator with adaptive SM LFC is used to meet the frequency modulation requirements.

In the normal state, the total demand power is:

$$E_{ACE}^{N} = Delta ACE{kern 1pt} {kern 1pt} {kern 1pt} ,{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} ACE^{N} < left| {Delta ACE} right| le ACE^{A}$$

(18)

where (E_{ACE}^{N}) is total required power of the normal regulation zone.

The response power of the adjustable generator in the normal regulation zone is:

$$P_{m}^{N} = left{ {begin{array}{*{20}c} {min left( {left| {E_{ACE}^{N} } right|,P_{m}^{in – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} – ACE^{A} {kern 1pt} le Delta ACE < – ACE^{N} } \ {min left( {E_{ACE}^{N} ,P_{m}^{de – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} ;ACE^{N} < Delta ACE le ACE^{A} } \ end{array} } right.$$

(19)

where (P_{m}^{in – max }) and (P_{m}^{de – max }) are the allowable power of the total adjustable generators, and can be increased and decreased, respectively.

#### The alert regulation zone

In the alert regulation zone, the frequency deviation has exceeded the rated frequency range, and the ACE fluctuation is also large. Thus, HESS is used to reduce the frequency deviation, where the ultra-capacitor can provide high power input or output for a short period while the batteries with higher energy density can operate for longer periods of time. In addition, the adjustable generator with adaptive SM LFC can provide stable active power. Therefore, the respective characteristics of HESS and adjustable generator can be fully utilized to complement each other for the stable operation of the system.

The total required power used for frequency control is:

$$E_{ACE}^{A} = Delta ACE,{kern 1pt} {kern 1pt} {kern 1pt} ACE^{A} < left| {Delta ACE} right| le ACE^{E}$$

(20)

where (E_{ACE}^{D}) is the total required power of the alert regulation zone.

The response power of the ultra-capacitor, BES and adjustable generator in the alert regulation zone is:

$$P_{uc}^{A} = left{ {begin{array}{*{20}c} {min left( {left| {E_{ACE}^{A} } right|,{kern 1pt} {kern 1pt} {kern 1pt} P_{uc}^{dis – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} – ACE^{E} le Delta ACE < – ACE^{A} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} } \ {min left( {{kern 1pt} {kern 1pt} E_{ACE}^{A} ,{kern 1pt} {kern 1pt} {kern 1pt} P_{uc}^{ch – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} ;;;;;;ACE^{A} < Delta ACE le ACE^{E} } \ end{array} } right.$$

(21)

$$P_{m}^{A} = left{ {begin{array}{*{20}c} {min left( {left| {E_{ACE}^{A} } right| – left| {P_{uc}^{A} } right|,{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} P_{m}^{in – max } } right),{kern 1pt} {kern 1pt} – ACE^{E} le Delta ACE < – ACE^{A} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} } \ {min left( {E_{ACE}^{A} – P_{uc}^{A} ,{kern 1pt} {kern 1pt} {kern 1pt} P_{m}^{de – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} ACE^{A} < Delta ACE le ACE^{E} } \ end{array} } right.$$

(22)

$$P_{BES}^{A} = left{ {begin{array}{*{20}c} {min left( {left| {E_{ACE}^{A} } right| – left| {P_{uc}^{A} } right| – left| {P_{m}^{A} } right|,{kern 1pt} {kern 1pt} {kern 1pt} P_{BES}^{dis – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} – ACE^{E} le Delta ACE < – ACE^{A} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} } \ {min left( {E_{ACE}^{A} – P_{uc}^{A} – P_{m}^{A} ,{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} P_{BES}^{ch – max } } right),{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} ACE^{A} < Delta ACE le ACE^{E} } \ end{array} } right.$$

(23)

where (P_{BES}^{dis – max }) and (P_{BES}^{ch – max }) are the allowable maximum discharging and charging power of BES, respectively.

#### The emergency regulation zone

In the emergency regulation zone, a variety of frequency modulation devices are required to balance the active power. If the power equipment cannot significantly reduce frequency deviation, other measures need to be taken quickly, such as load shedding through power system dispatch.

From the above analysis, a diagram of the proposed control strategy is shown in Fig. 5.

### The adaptive SM LFC design

Adaptive SM LFC is used to control the adjustable generator. The design process is described in the following sub-sections.

#### The switching surface design

The vector model of the system can be obtained from (2)-(6) as:

$$dot{x}(t) = Ax(t) + Bu(t) + Fd(t)$$

(24)

where (x(t)), (A), (B), (F), (d(t)) are given in the Appendix.

In an actual power system, the changes of load and output power can change the stable operation point of the system, so the normal state model in (25) can be refined as:

$$dot{x}(t) = (A + Delta A)x(t) + (B + Delta B)u(t) + (F + Delta F)d(t)$$

(25)

Defining (wleft( t right) = Delta Ax(t) + Delta Bu(t) + (F + Delta F)d(t)) as the uncertainty, Eq. (26) can be expressed as:

$$dot{x}left( t right) = Ax(t) + Bu(t) + w(t)$$

(26)

In order to design the controller, the following hypotheses are given [32,33,34]:

### Hypothesis 1

((A,B)) is totally controllable.

### Hypothesis 2

(rank[B,w(t)] ne rank[B])

### Hypothesis 3

the disturbance (w(t)) is constrained, (left| w right| < xi), where ∥ ∥ is matrix norm and (xi) is a positive constant.

The following formula is chosen as the integral sliding mode surface:

$$eta (t) = Cx(t) – int {C(A – BH)} x(tau )dtau$$

(27)

where (C) and (H) are constant matrixes, matrix *H* has (lambda (A – BH) < 0), while matrix (C) is selected to make (CB) nonsingular. According to stability analysis, when the system falls on the sliding mode plane, it is in a steady state.

#### The adaptive SM control laws design

Reaching law theory can enhance the dynamic performance of arrival phase, and the following equation is used to meet the condition:

$$dot{eta }(t) = – hat{r}eta (t) – sigma {text{sgn}} eta (t)$$

(28)

where (sigma) is a non-negative constant, ({text{sgn}} *) is a symbolic function, and (hat{r}) is the estimation value. When multiple parameters are uncertain, for strong robustness of the controller, the parameter (hat{r}) satisfies the adaptive rule as:

$$dot{hat{r}} = aleft| eta right|$$

(29)

where (a) is the adaptive positive constant.

From (29) and (30), there is:

$$begin{gathered} dot{eta }(t) = Cdot{x}(t) – C(A – BH)x(t) \ = CAx(t) + CBu(t) + Cw(t) – C(A – BH)x(t) \ = CBHx(t) + CBu(t) + Cw(t) \ = – hat{r}eta (t) – sigma {text{sgn}} eta (t) \ end{gathered}$$

(30)

SM LFC is as follows:

$$u(t) = – Hx(t) – (CB)^{ – 1} [Cxi + hat{r}eta (t) + sigma {text{sgn}} eta (t)]$$

(31)

Since the system meets the condition of (eta_{i} dot{eta }_{i} < 0), the system is rendered stable using the designed controller. The flowchart of controller design is shown in Fig. 6.

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