Traditional VPP scheduling strategy
The primary objectives of the VPP optimal scheduling problem are various, such as minimizing the cost of producing energy, or maximizing the profits of VPPs. In this paper, the priority of VPP operation is to make use of the renewable energy resources (RES), and based on this, the objective function is to minimize the costs of conventional unit generation and the interruptible load.
$$ \min \kern0.2em f={\displaystyle \sum_{t=1}^T\left\{{\displaystyle \sum_{i=1}^{N_G}\left[{J}_{Gi}(t)+{u}_i\left(t1\right){u}_i(t)S{U}_i(t)\right]}+{\displaystyle \sum_{j=1}^{N_{DR}}{J}_{DRj}(t)}\right\}} $$
(4)
$$ {J}_{Gi}(t)={u}_i(t)\left({a}_i{P_{Gi}}^2+{b}_i{P}_{Gi}+{c}_i\right) $$
(5)
$$ {J}_{DRj}(t)={d}_j{P}_{DRj}{u}_j(t) $$
(6)
where J
_{
Gi
}(t) is the operational cost of unit i at time t; SU
_{
i
}(t) is the startup cost of the unit i at time t; a
_{
i
}, b
_{
i
}, c
_{
i
} are the coefficients of the operational cost of the unit i;d
_{
j
} is the coefficient of the cost by curtailing the load j; P
_{
Gi
} is the output power of the unit i; P
_{
DRj
} is the virtual generation via curtailing the load; u
_{
i
}(t) is the binary state variable of the unit i at time t: it equals “1” if the unit i is on at time t, and equals “0” if the unit i is off at time t. Similarly, u
_{
j
}(t) is the binary state variable of the load j at time t. N
_{
G
} and N
_{
DR
} are the numbers of conventional units and loads in the demand response respectively.
The constraints considered in this model are presented as follows:

1)
Supply–demand balancing constraint
$$ {\displaystyle \sum_{i=1}^{N_G}{P}_{Gi}(t)+{\displaystyle \sum_{j=1}^{N_{DR}}{P}_{DRj}(t)+{P}_{RES}(t)={P}_L(t)}} $$
(7)

2)
System reserve constraint
$$ {\displaystyle \sum_{i=1}^{N_G}{P}_{Gi}^{\max }+{\displaystyle \sum_{j=1}^{N_{DR}}{P}_{DRj}^{\max }+{P}_{RES}(t)\ge {P}_L(t)}}+R(t) $$
(8)

3)
DG constraints
$$ {u}_i(t){P}_{Gi}^{\min}\le {P}_{Gi}(t)\le {u}_i(t){P}_{Gi}^{\max } $$
(9)
$$ R{D}_i\le {P}_{Gi}(t){P}_{Gi}\left(t1\right)\le R{U}_i $$
(10)

4)
Interruptible load constraint
$$ {u}_j(t){P}_{DRj}^{\min}\le {P}_{DRj}(t)\le {u}_j(t){P}_{DRj}^{\max } $$
(11)
where P
_{
RES
}(t) is the power generated by renewables at time t; P
_{
L
}(t) is the total load at time t; R(t) is the reserve capacity for the fluctuation of load and renewables at time t; \( {P}_{Gi}^{\min } \) and \( {P}_{Gi}^{\max } \) are the minimum and the maximum output limit of the unit i respectively; RU
_{
i
} and RD
_{
i
} are the rampup rate and the rampdown rate of the unit i; \( {P}_{DRj}^{\min } \) and \( {P}_{DRj}^{\max } \) are the minimum and maximum allowed curtailed value of the load j respectively.
Fuzzy chanceconstrained VPP scheduling
The fuzzy parameters of VPP scheduling strategy come from the unpredictable output of renewables in VPPs [24]. Accurate mathematical expressions of those uncertain characteristics should be established, which are the foundations for the uncertain scheduling strategy. The main characterizing methods include modeling output of renewables and modeling their predicted errors. In this paper, the modeling of the forecast errors is considered to describe the fuzziness of renewables instead of modeling the output. The modeling of renewables output will make three constraints containing fuzzy variables, which are the supply–demand balance constraint, the spinning reserve constraint, and the static security constraint. If the predicted value is regarded as a deterministic value and the prediction error is utilized to describe the fuzziness of renewables, only the spinning reserve constraint should be considered.
The percentage of renewables prediction error is defined as
$$ \varepsilon \%=\frac{P_{RES}{P}_{RES}^F}{P_{RES}^F}\times 100\% $$
(12)
where P
_{
RES
} and \( {P}_{RES}^F \) are the real value and the forecasted value of the output respectively.
The membership function of the prediction error is:
$$ \mu =\left\{\begin{array}{l}\frac{1}{1+\sigma {\left(\frac{\varepsilon }{E_{+}}\right)}^2},\kern0.5em \varepsilon >0\\ {}\frac{1}{1+\sigma {\left(\frac{\varepsilon }{E_{\hbox{} }}\right)}^2},\kern0.5em \varepsilon \le 0\end{array}\right. $$
(13)
where E
_{+} and E
_{−} are the average values of positive and negative prediction errors obtained from statistics, respectively; σ is the weighting factor.
According to credibility theory, the credibility measure of the fuzzy parameter ξ can be expressed as:
$$ \mathrm{C}\mathrm{r}\left(\xi \le \varepsilon \right)=\left\{\begin{array}{l}1\frac{1}{2\left[1+\sigma {\left(\frac{\varepsilon }{E_{+}}\right)}^2\right]},\kern0.5em \varepsilon >0\\ {}\frac{1}{2\left[1+\sigma \left(\frac{\varepsilon }{E_{}}\right)\right]},\kern2em \varepsilon \le 0\end{array}\right. $$
(14)
Thus, according to credibility theory, the spinning reserve constraint of the traditional scheduling model should be changed into chance constraints containing fuzzy variables.
$$ \mathrm{C}\mathrm{r}\left\{{\displaystyle \sum_{i=1}^{N_G}{u}_i(t){P}_{Gi}^{\max }+{\displaystyle \sum_{j=1}^{N_{DR}}{u}_j(t){P}_{DRj}^{\max }+{P}_{RES}^F(t)\left(1+\xi \right)\ge }}\;{P}_L(t)+R(t)\right\}\ge \alpha $$
(15)
where α is the credibility level, which represents the reliability level of the reserve constraint. In the realistic problem, α should be more than 0.5.
According to the equivalent theorem of credibility theory, (15) can be transformed into an equivalent form as indicated in (16):
$$ \frac{P_L(t)+R(t){P}_{RES}^F(t){\displaystyle \sum_{i=1}^{N_G}{u}_i(t){P}_{Gi}^{\max }{\displaystyle \sum_{j=1}^{N_{DR}}{u}_j(t){P}_{DRj}^{\max }}}}{P_{RES}^F(t)}\le {K}_{\alpha } $$
(16)
$$ {K}_{\alpha }= \inf \left\{K\BigK={\mu}^{1}\left(2\left(1\alpha \right)\right)\right\},\forall \alpha \ge 0.5 $$
(17)
The equivalent form of the fuzzy chance constraint can be further deduced to the following forms:
$$ {\displaystyle \sum_{i=1}^{N_G}{u}_i(t){P}_{Gi}^{\max }+{\displaystyle \sum_{j=1}^{N_{DR}}{u}_j(t){P}_{DRj}^{\max }+\left(1+{K}_{\alpha}\right){P}_{RES}^F(t)\ge }}{P}_L(t)+R(t) $$
(18)
$$ {K}_{\alpha }=\left{E}_{}\%\right\sqrt{\frac{2\alpha 1}{2\sigma \left(1\alpha \right)}} $$
(19)
It can be proved that K
_{
α
} is a monotonously increasing function. Equation (18) illustrates the links between the confidence level of the fuzzy constraint and spinning reserve allocation when taking the fuzziness of RES predicted error into account. It shows that to improve the confidence level of the spinning reserve constraint, a greater spinning reserve should be allocated. The form of the formula is similar to the traditional spinning reserve constraint. Thus, the model can be easily solved by correcting the reserve levels using the regular optimization algorithm with the coefficient K
_{
α
}.