#### 2.1. Modeling of Distributed Generators

The model of a wind turbine (WT) is derived from its power output curve, as illustrated in

Figure 1. Though the curve of each type of wind turbine is slightly different from others’, they can be generally represented as a piecewise function [

36].

where

P_{WT} is the output power of wind turbine (WT),

P_{r} is the rated power of WT,

v_{r} is the rated wind speed of WT,

v_{i} is the cut-in wind speed of WT,

v_{o} is the cut-out wind speed of WT,

a_{1}~

a_{4} are coefficients.

The output power of a photovoltaic (PV) panel array is determined by variables including

P_{r},

R_{c},

R_{r},

T_{c},

T_{r} and

k [

37].

where

P_{PV} is the output power of PV panel array,

P_{r} is the rated power of a single PV unit under standard conditions,

R_{c} is the current sunlight intensity,

R_{r} is the rated sunlight intensity under standard conditions,

T_{c} is the current temperature,

T_{r} is the rated temperature under standard conditions,

k is the power temperature coefficient,

n_{PV} is the number of units in the PV panel array.

#### 2.2. Modeling and Characteristic Analysis of Energy Storage Devices and Back-Up Source

ES devices and back-up source are used to compensate energy shortage caused by insufficient distributed generator (DG) generation, hence relative analysis is significant for further study on source–load coordination mechanism.

Table 1 shows the characteristics of both energy-type and power-type ES devices, including energy density, power density, investment cost, approximate cycle times and response speed.

Through observation, it is obvious that two types of ES devices are complementary to each other, and a hybrid storage system can provide enhanced power supply capability. Besides, although flywheel storage and SMES have outstanding performance, they are relatively expensive and are still in the experimental stages, which made them unsuitable for practical application. In this paper, the HESS consists of a lead-acid battery, Li-ion battery and super capacitor.

1. Energy state model of ES device

The operation status of all ES devices can be generally described via the following formula [

38,

39]:

where

E_{ES}^{(t)} is the remaining energy of the ES device at moment

t, σ is the self-discharge rate,

P_{ES}^{(t)} is the charge/discharge power during time interval

t, η is the charge/discharge efficiency, Δ

t is the duration of time interval.

The energy state of the ES device is determined by E_{ES} at the start of a time interval and charged/discharged power during that period. For different ES devices, the self-discharge rate can vary from 0.1%–0.3%/d (batteries) to 20%–40%/d (super capacitor).

2. Lead-acid battery lifetime model

The lifetime of a lead-acid battery is relatively short and is significantly influenced by operation conditions compared to super capacitors and a Li-ion battery. While the lifetime of super capacitor (SC) can extend as long as 20 years and a Li-ion battery can be charged/discharged for over 5000 cycles, substitution of a lead-acid battery may be required, which makes the lifetime calculation model of batteries essential for optimization of MG source planning.

An existing lifetime calculation model [

37] is applied to predict the lifetime of lead-acid batteries. The cycle time of lead-acid batteries is related to depth of discharge (DOD), as illustrated in

Figure 2.

The calculation formula can be expressed as:

where

D_{N} is the depth of discharge,

N is the equivalent cycle times under

D_{N},

a_{1}~

a_{5} are coefficients.

The cost of a single charge/discharge cycle is represented as:

where

C_{cap-B} is the Capital investment of batteries,

C_{cycle} is the cost of a single charge/discharge cycle.

Related studies [

40,

41] indicate that the cumulative lifetime of a battery is influenced by the state of charge (SOC) under which the battery is operating. Besides, a battery may not experience a fixed and whole charge-discharge cycle. Hence, the operation cost model for both the charging and discharging processes, considering the influence caused by SOC, is given in [

37].

• Charging process

An influence factor λ

_{c} is introduced to represent the influence on operation cost caused by SOC:

where

S_{c-start} is the SOC at the start of the charging process,

S_{c-end} is the SOC at the end of the charging process,

k_{c} is the adjustment coefficient to fix the final value of λ

_{c}.

The cost of a single charging process is then shown by the following equation:

where

C_{cl} is the cost of a single charging process, λ

_{c} is the influence factor representing SOC’s influence on

C_{cl}.

• Discharging process

An influence factor λ

_{d} is introduced to represent the influence on operation cost caused by SOC:

where

S_{d-start} is the SOC at the start of the discharging process,

S_{d-end} is the SOC at the end of the discharging process,

k_{d} is the adjustment coefficient to fix the final value of λ

_{d}.

The cost of a single discharging process is then represented as:

where

C_{dl} is the cost of a single charging process, λ

_{d} is the influence factor representing SOC’s influence on

C_{dl}.

3. Fuel consumption model of a diesel generator

The fuel consumption model of a diesel generator is obtained through fitting the fuel consumption curve [

42]:

where

C_{Diesel} is the amount of consumed fuel,

P_{G} is the generated power of diesel generator,

P_{r} is the rated power of diesel generator,

A and

B are coefficients.