The optimal schedule of a VPP can be formulated for each time period as a fuzzy multiple objective optimization problem with priority on the basis of interactive satisfaction. The decision variables are the output power schedule and bus voltages of the suppliers, the storage plan and the power sent to the public grid. Other parameters include the forecasted output powers and the forecasted consumption.

#### 3.1. Multi-objective Optimization with Priority

The multi-objective optimization problem with priority based on satisfaction degrees can be defined as follows:

where

x is the decision vector;

F is the feasible solution of

x;

${f}_{i}$ is the objective function;

${\mu}_{{f}_{i}}(x)$ is the satisfaction degree of

${f}_{i}$;

${\mu}_{{f}_{i}}(x)\left\{{f}_{i}\in leve{l}^{j}\right\}>{\mu}_{{f}_{ii}^{\prime}}(x)\left\{{f}_{i}^{\prime}\in leve{l}^{j+1}\right\}$ represents that

${f}_{i}$ has precedence over

${f}_{i}^{\prime}$. To simplify the calculation, the linear method to express satisfaction degree is used (

Figure 3):

**Figure 3.**
Linear description of the satisfaction degree.

**Figure 3.**
Linear description of the satisfaction degree.

where:

${f}_{i}^{*}$
is the expected value of

${f}_{i}$
;

${f}_{i}^{\mathrm{max}}$ is the max value of

${f}_{i}$ in feasible solution. When

${\mu}_{{f}_{i}}(x)=0$,

${f}_{i}(x)$ is totally dissatisfactory; when

${\mu}_{{f}_{i}}(x)=1$,

${f}_{i}(x)$ is totally satisfactory; when

$0<{\mu}_{{f}_{i}}(x)<1$, it depends on the decision-makers (DM) to evaluate the compromised results.

#### 3.3. Two-Step Compromised Method

The objectives are grouped into different levels according to the priority order. In this paper, preemptive priorities of the multiple objective optimization problem are expressed as (for example, three levels):

where

${k}_{1}\in \left[0,1\right]$
and

${k}_{2}\in \left[0,1\right]$ present the difference of satisfying degrees between two levels. The objective in level 1 with higher satisfying degree is more important than level 2 and level 3. With the preemptive priority structure, objectives which belong to level 1 achieve their goal as much as possible.

The priority structure is established considering the special users in VPP whose power supply objectives must be satisfied foremost. Another consideration is the importance extent of different objectives, such as stability and economic objectives.

To compromise the max satisfying degree and the priority structure, a two-step interactive satisfactory method is used [

17]. This method simplifies the complicated optimization problem by dividing it into two sub-problems and solving them in sequence with high computation efficiency. The first step is to find maximum overall satisfactory degree

β^{t} treated as the given condition of the next step optimization, as shown in (7):

In the second step a decision variable

$\gamma $ is used to relax the crisp priority relationship. This method is also applied with the interaction of DM. Then, relation (6) should be represented as:

where

${\gamma}_{i}\in [\mathrm{max}(-{k}_{1},-{k}_{2}),\mathrm{min}(1-{k}_{1},1-{k}_{2})]$. When

$\gamma \to 0$, the solution satisfies the priority structure best. So the second sub-problem is modeled as:

where,

${\beta}_{r}^{t}$ is the regulated satisfactory degree and smaller than

${\beta}_{t}$. It represents the regulation of differences among the practical priorities and preemptive priorities.

#### 3.4. Objective Functions

The optimization objectives are the minimum values of the objective functions. Different objective functions are considered as following:

(1) Maximize profit. For ECCC, the economic maximization objective is considered as:

where

${f}_{1}$ is the minus profit of VPP;

ns is the number of suppliers;

nt is the number of storage devices;

nl is the number of loads;

nv is the number of sources owned and managed by VPP;

${P}_{supply}^{j}$ is the power supplied by the

jth supplier which is not totally owned by VPP;

${k}_{supply}^{j}$ is the sale price of the output power supplied by

jth supplier;

${P}_{load}^{j}$ is the

jth demanded load power;

${k}_{load}^{j}$ is the price of the power energy sold to the

jth load;

${P}_{vpp}^{j}$ is the power supplied by the

jth source which is totally owned by VPP;

${k}_{vpp}^{j}$ is the producing cost of the

jth source with consideration of the building cost;

${P}_{storage}^{j}$ is the stored power of

jth storage device;

${k}_{storage}^{j}$ is the storage cost of the

jth storage device.

The following relationship is satisfied:

where

${P}_{loss}$
is the power loss of the VPP.

(2) Minimal lineloss

where

N represents the number of the power lines;

${G}_{ij}$ denotes the conductance of the line

ij connected with bus

i and bus

j;

${U}_{i}$
and

${U}_{j}$ refer to the voltages of bus

i and bus

j;

${\theta}_{ij}$ denotes the phase angle difference of the voltages of bus

i and bus

j. Therefore, less value of

${f}_{2}$ is preferred.

(3) Voltage stability index:

where:

${n}_{l}$ represents the number of the buses;

${u}_{r}^{i}$ is the voltage of the bus

i;

${u}_{rated}^{i}$
is the rated voltage of the bus

i.

(4) Ordered suppliers: Part of microgrids supply certain quantity of electric power to the VPP according to the contract. This index can be represented as the fluctuation of the power supply:

where:

and

nm represents the number of the microgrids;

${m}_{r}^{i}$
is the output power of the microgrid

i;

${m}_{set}^{i}$
is the set output power of the microgrid

i in contract.

(5) Ordered customers: Part of customers have special requirements for voltage, and the voltage supply index is

where,

${u}_{c}^{i}$
is the voltage of the bus connected to customer

i;

${u}_{cset}^{i}$
is the set voltage of the bus connected to customer

i;

${\epsilon}_{set}^{i}$
is the set voltage deviation of the bus connected to customer

i.

The criteria for the optimal problem are as follows:

- (1)
the wind turbine and solar energy are preferred for environmental protection;

- (2)
the output power to the public grid is preset one-day ahead;

- (3)
the actual lines are represented as an ideal line with impedance;

- (4)
the reliability requirements of the customers are satisfied by the suitable topology design of the VPP.

#### 3.6. Optimization Process

In this paper, the fuzzy multiple objective optimization algorithm with compromise of the satisfactory degree and the importance and priority of objectives is given based on the principle of two-step interactive satisfactory optimization. The flowchart of optimization process is shown in

Figure 6. The forecast values of each time period include the load forecast, the predicted output power of the DGs owned by the VPP, as well as the quantity and price of the electric power supplied by the ones owned independently from the VPP. The final generation schedule is established by the ECCC through the optimization process.

**Figure 6.**
Flowchart of operation.

**Figure 6.**
Flowchart of operation.

The optimization process for each time period is as follows:

Step 1: Collect the information about the original contracts and the forecasted values at a time period: the supply plan and demand plans of the DGs and the loads in VPP delivered to the ECCC.

Step 2: Determine the ranges of x. The DM gives the value c_{j} according to (5) and α-cut sets of x are obtained.

Step 3: Regulate α and calculate the initial individual objective values. DM determines objective functions and their priority structure, the desirable targets k.

Step 4: Get the power flow and calculate the individual minimum and maximum values of objective functions under the given constraints to determine the membership functions of the objectives. If needed, the DM may set the target values instead of the individual minimum and maximum values.

Step 5: Get the maximum overall degree ${\beta}^{t}=\mathrm{min}({\beta}_{1},\mathrm{...},{\beta}_{i},\mathrm{...},{\beta}_{k})$
according to (7). And ${\beta}_{i}={\mu}_{{f}_{i}}(x)$
are the satisfying degrees of the individual objectives. If it satisfies (8), the optimization can be stopped and the satisfactory solution is acquired. Otherwise, it goes to next step.

Step 6: Reduce β^{t} and solve (9) using optimal algorithm, and get γ.

Step 7: If ${\gamma}_{i}\in [\mathrm{max}(-{k}_{1},-{k}_{2}),\mathrm{min}(1-{k}_{1},1-{k}_{2})]$
, the DM decides whether the solution is satisfactory or not. If it is not satisfactory, go to step 6; otherwise, stop optimization and the optimal solution is achieved. If there is no feasible solution, go to step 6.