3.1.2. Constraint Conditions

(1) The power balance constraint of the system

$$\sum\_{i=1}^{N} u\_{i,t} \, p\_{i,t} - P\_t^d = 0 \tag{3}$$

where *P<sup>d</sup> t*is the total system load demand at time interval *t*.

(2) The spinning reserve constraints of the system

$$\sum\_{i=1}^{N} u\_{i,t} \, \, P\_{i,\text{max}} \ge P\_t^d + S\_t^d \tag{4}$$

where *Pi*,max is the maximum output of the unit *i* at time *t* and *Sd t*is the total spare capacity at time *t*.

(3) The maximum and minimum output constrains

$$
u\_{i,t} P\_{i, \text{min}} \le p\_{i,t} \le u\_{i,t} P\_{i, \text{max}} \tag{5}$$

where *Pi*,max and *Pi*,min are the maximum and minimum output of the unit *i*, respectively.

(4) The ramp rate constraints

$$p\_{i,t+1} - p\_{i,t} \le u\_{i,t} R\_{lI,i} + p\_{i, \text{max}} (1 - u\_{i,t}) \tag{6}$$

$$p\_{i,t} - p\_{i,t+1} \le u\_{i,t+1} R\_{D,i} + p\_{i, \text{max}} (1 - u\_{i,t+1}) \tag{7}$$

where *RU*,*<sup>i</sup>* and *RD*,*<sup>i</sup>* are the maximum rate of upward ramping/downward ramping of unit *i* in each time interval, respectively.

(5) The minimum startup–shutdown time constraints

$$\begin{cases} \left(\boldsymbol{u}\_{i,t-1} - \boldsymbol{u}\_{i,t}\right) \left(\boldsymbol{T}\_{i,t-1} - \boldsymbol{T}\_{i}^{\rm con}\right) \ge 0\\ \left(\boldsymbol{u}\_{i,t} - \boldsymbol{u}\_{i,t-1}\right) \left(-\boldsymbol{T}\_{i,t-1} - \boldsymbol{T}\_{i}^{\rm off}\right) \ge 0 \end{cases} \tag{8}$$

where *<sup>T</sup>*on*i* and *<sup>T</sup>*<sup>o</sup>ff*i* are the minimum operating time and minimum offtime of the unit *i*, respectively; and *Ti*,*<sup>t</sup>* is the continuous offtime or continuous operating time of the unit *i* at time *t*.

#### (6) The startup–shutdown cost constraint

The start-up costs of cold start and hot start are different. By judging the length of continuous offtime, the specific constraint conditions for cold start and hot start can be determined as follows:

$$s\_{i,t} = \begin{cases} C\_i^{\text{hot}}, & 1 \le t\_{i,t}^{\text{off}} \le T\_i^{\text{off}} + T\_i^{\text{cold}} \\ C\_i^{\text{cold}}, & t\_{i,t}^{\text{off}} > T\_i^{\text{off}} + T\_i^{\text{cold}} \end{cases} \tag{9}$$

where *t*off*i*,*t* is the continuous offtime of unit *i* until time *t* − 1; *T*cold *i* is the cold start time of unit *i*; and *C*hot *i*and *C*cold *i*are the costs of the unit *i* in the case of cold start and hot start, respectively.

## (7) Generation fairness constraint

The Gini coefficient is introduced to model the fairness constraint of the complementation rate of electricity generation.

$$G = \frac{1}{2N(N-1)\mu} \sum\_{i=1}^{N} \sum\_{j=1}^{N} |\chi\_i - \chi\_j| \le G\_0 \tag{10}$$

where X*i* and X*j* are the scheduled power completion rates of unit *i* and unit *j*, respectively; u is the average power completion rates of all units; and *G*0 is the threshold value of the Gini coefficient, which can be set according to the actual fairness requirement, and the value range is 0~1.

$$X\_i = \frac{Q\_i}{Q\_i^0} = \sum\_{t=1}^T P\_{i,t} / \left( P\_i^{\text{max}} \times t\_{\text{sh}} \right) \tag{11}$$

where *Q*0*i* and *Qi* are the completed electric energy according to the dispatching results and the daily planned energy of unit *i*, respectively.

$$
\mu = \sum\_{i=1}^{N} \chi\_i / N \tag{12}
$$

Substituting (12) into (10), the final fairness constraint formulation (13) can be obtained after shifting and simplifying.

$$\sum\_{i=1}^{N} \sum\_{j=1}^{N} \left| \chi\_i - \chi\_j \right| \le 2G\_0(N-1) \sum\_{i=1}^{N} \chi\_i \tag{13}$$
