**2. Modeling**

#### *2.1. Stochastic Long-Term Power System Model*

The large-scale deployment of stochastic renewable energy sources, e.g., wind, significantly increases the random perturbations and inevitably affects the safe and stable operation of power systems [14]. As a result, the adoption of SDEs to model this randomness in power systems has gained significant interest in recent years. However, most of the literature focus on the impact of wind power variability (modeled as SDEs) on transient and small-signal stability, respectively [15]. On the other hand, stochastic long-term stability analysis of power systems has received little attention. This is of particular importance considering the expected trends regarding the integration of RES into modern power systems.

The stochastic long-term dynamic model of power systems can be represented as a set of hybrid nonlinear SDAEs [16], as follows:

$$\begin{aligned} \dot{\mathbf{x}} &= f(\mathbf{x}, y, \mathbf{u}, z, \dot{\eta}) \,, \\ \mathbf{0} &= \mathbf{g}(\mathbf{x}, y, \mathbf{u}, z, \eta) \,, \\ \dot{\eta} &= a(\mathbf{x}, y, \eta) + b(\mathbf{x}, y, \eta) \, \not\mathbf{\not\mathbf{\not\mathbf{x}}} \,, \end{aligned} \tag{1}$$

where *f* and *g* represent the differential and algebraic equations, respectively; *x* and *y* represent the state and algebraic variables, such as generator rotor speeds and bus voltage angles, respectively; *u* represents the inputs, such as the schedules of synchronous generators; *z* represents discrete variables; *η* represents the stochastic characterization of wind speed; *a* and *b* are the *drift* and *diffusion* of the SDEs, respectively; and *ζ* is the white noise. It is worth mentioning that (1) is solved using numerical integration techniques. This is possible as the algebraic equations *g*, in (1), do not explicitly depend on white noises *ζ*, or on *η*˙ [16]. In this work, an implicit trapezoidal integration method is used for functions, *f* and *a*, while the Euler–Maruyama method is used to integrate the stochastic term *b*. Further details related to the numerical integration of the SDAEs can be found in [16].

It is necessary to consider both electromechanical and long-term dynamic models, when one performs stochastic long-term dynamic simulations [17]. With this regard, (1) includes the dynamic models of conventional machines (4th order models) and their primary controllers; AGC; wind power plants (5th order Doubly-Fed Induction Generator) [18]; the model of subhourly SUC.

## *2.2. Wind Power Modeling*

Modeling wind power as a stochastic source is crucial in long-term dynamic studies [19]. With this aim, Equation (1) model wind power variations as a stochastic perturbation with respect to the wind generation forecast. More specifically, we use Ornstein–Uhlenbeck processes (modeled as SDEs) to represent the volatile nature of the wind speed variations *vs*, that feeds the wind turbines, as follows:

$$\begin{aligned} \upsilon\_s(t) &= \upsilon\_{s0} + \eta\_v(t), \\ \eta\_\upsilon(t) &= a\_\upsilon(\mu\_\upsilon - \eta\_\upsilon(t)) + b\_\upsilon \check{\varsigma}\_\upsilon, \end{aligned} \tag{2}$$

where *vs*0 represents the initial value of the wind speed; *ηv* represents the stochastic variable which depends on the drift *<sup>α</sup>v*(*μv* − *ηv*), and the diffusion term *bv* of the SDEs; *αv* represents the mean reversion speed, and shows how quickly *ηv* tends towards its mean *μv*; finally, the white noise is represented by *ζ<sup>v</sup>*.

#### *2.3. Primary and Secondary Frequency Controllers of Conventional Power Plants*

The displacement of conventional synchronous generators and their replacement with non-synchronous stochastic sources leads to a decrease in the overall inertia of the system, reduction of secondary frequency regulation and an increase of the aggregated system droop [4]. In general, this can lead to large frequency deviations and therefore it is crucial to study such an impact on the dynamic behaviour of power systems. Motivated by the above, the paper make use of a recent cosimulation platform proposed by the authors to perform a sensitivity analysis where the parameters of primary and secondary controllers are varied and their effect on the long-term dynamic behaviour of power systems is observed. In this work, the frequency of the center of inertia of the system, *ω*COI, is utilised to monitor the overall long-term dynamic behaviour of power systems. Its expression is [18]:

$$
\omega\_{\text{COI}} = \frac{\sum\_{\mathcal{G} \in \mathcal{G}} M\_{\mathcal{G}} \omega\_{\mathcal{G}}}{\sum\_{\mathcal{G} \in \mathcal{G}} M\_{\mathcal{G}}} \, \, \, \tag{3}
$$

where *<sup>ω</sup>g* and *Mg* are the rotor speed and the starting time (which is twice the inertia constant) of the *g*th synchronous machine, respectively. *Mg* is an intrinsic part of the machine and as such deeply impacts on the dynamic behaviour of the machine and of the system. It appears in the electromechanical swing equations of the machine, as follows [17]:

$$\begin{aligned} \frac{d\delta\_{\mathcal{S}}}{dt} &= \omega\_{\mathcal{S}} - \omega\_{\text{COI}},\\ M\_{\mathcal{S}} \frac{d\omega\_{\mathcal{S}}}{dt} &= p\_{\text{m},\emptyset} - p\_{\text{e},\emptyset}(\delta\_{\mathcal{S}}) \end{aligned} \tag{4}$$

where *<sup>δ</sup>g*, *p*m,*g* and *p*e,*g* are the rotor angle, the mechanical power and the electrical power of the *g*th synchronous machine, respectively.

A standard control scheme of a primary frequency control of a synchronous power plant is depicted in Figure 1 [18]. The model of the primary frequency control of synchronous machines includes an ensemble of the models of the turbine, the valve and the turbine governor. These regulate the mechanical power of the machine (*p*m,*g*) through the measurement of the rotor speed *<sup>ω</sup>g*. The conventional primary frequency regulator consists of a droop (*Rg*) and a lead-lag transfer function that models regulator and turbine combined dynamics. Finally *p*ord,*g* is the active power order set-point as obtained by the solution of the SUC problem (*pg*,*t*,*<sup>ξ</sup>* )—as thoroughly discussed in Section 2.4—and the signal sent by the secondary frequency control (Δ*pg*)—discussed at the end of this section. Specifically, *p*ord,*g* is given by:

$$p\_{\text{ord},\emptyset} = p\_{\mathcal{J},t,\emptyset} + \Delta p\_{\mathcal{J}} \,. \tag{5}$$

The main purpose of the primary frequency control is to restore the power system frequency at a quasi-steady state value following a disturbance into the power system. This control is locally implemented and takes place on time scales of tens of seconds.

**Figure 1.** Standard turbine governor control scheme.

A simple control scheme of a secondary frequency control or automatic generation control (AGC) is shown in Figure 2. In its simplest implementation, the AGC consists of an integrator block with gain *K*0 that coordinates the TGs of synchronous machines to nullify the frequency steady-state error originated by the primary frequency control by sending the active power corrections set-points <sup>Δ</sup>*pg*. These signals are proportional to the capacity of the machines and the TG droops *Rg*. The AGC is a centralised control and takes place on the time scales of tens of minutes.

**Figure 2.** Simple automatic generation control (AGC) control scheme.

An in-depth discussion of power system dynamics and frequency control is beyond the scope of this paper. The interested reader can find further information regarding the relevant relationship between the above variables, namely, inertia *M*, droop *R*, and gain *K*, in dedicated monographs, e.g., [17,18].

#### *2.4. Stochastic Unit Commitment*

Over the last decade, there has been a significant interest to better represent uncertainty in the UC models due to the integration of highly variable RES. Traditionally, the uncertainty in these models have been managed by scheduling some specific amount of reserves (e.g., as a percentage of the total demand). However, this may lead to suboptimal solution of the UC problem and thus impact system security. A common solution to this problem is to use stochastic optimization, in particular, two-stage stochastic programming [20]. A two-stage SUC model uses a probabilistic description of uncertain nature of parameters, e.g., wind power generation, in the form of scenarios, [21], as follows:

$$R\_{\mathcal{I}\mathcal{W}}^{\text{i\(p\)}} \sum\_{t \in \mathcal{T}} \sum\_{\mathcal{g} \in \mathcal{G}} \left( C\_{\mathcal{g}}^{F} z\_{\mathcal{g},t}^{F} + C\_{\mathcal{g}}^{SLI} z\_{\mathcal{g},t}^{SII} + C\_{\mathcal{g}}^{SD} z\_{\mathcal{g},t}^{SD} \right) \tag{6}$$

$$+ \sum\_{\tilde{\mathcal{S}} \in \Xi} \pi\_{\tilde{\mathcal{S}}} \left[ \sum\_{t \in \mathcal{T}} \sum\_{I \in \mathcal{L}} \mathbb{C}\_{\mathcal{g}}^{V} p\_{\mathcal{g},t,\tilde{\mathcal{L}}} + \sum\_{t \in \mathcal{T}} \sum\_{I \in \mathcal{L}} \mathbb{C}^{L} L\_{I,t,\tilde{\mathcal{J}}}^{SH} \right]$$

such that

$$\begin{aligned} z\_{\mathbf{g},t}^{\rm SI} - z\_{\mathbf{g},t}^{\rm SD} &= z\_{\mathbf{g},t}^{\rm F} - z\_{\mathbf{g},t-1}^{\rm F} \\ z\_{\mathbf{g},t}^{\rm SI} - z\_{\mathbf{g},t}^{\rm SD} &= z\_{\mathbf{g},t}^{\rm F} - \rm IS\_{\mathbf{g}} \end{aligned} \qquad \qquad \forall \mathbf{g} \in \mathcal{G}, \forall t \in \{2...\prime\} \tag{7}$$

$$\stackrel{\circ}{z}\_{\mathfrak{g},t}^{SL} + \stackrel{\circ}{z}\_{\mathfrak{g},t}^{SD} \le 1,\tag{9}$$

$$z\_{\mathbf{g},t}^F = IS\_{\mathbf{g}'}L\_{\mathbf{g}}^{LP} + L\_{\mathbf{g}}^{DW} > 0, \tag{10} \\ \qquad \qquad \qquad \forall \mathbf{g}, \forall t \le L\_{\mathbf{g}}^{LP} + L\_{\mathbf{g}}^{DW} \tag{10}$$

$$\sum\_{\mathbf{r}=t-\mathrm{LT}\_{\mathfrak{g}}+1}^{t} z\_{\mathbf{g},\mathbf{r}}^{SII} \le z\_{\mathbf{g},t'}^{F} \tag{11} \\ \qquad\qquad\qquad\qquad\qquad\forall \mathbf{g},\forall t > L\_{\mathfrak{g}}^{IP} + L\_{\mathfrak{g}}^{DW} \tag{11}$$

$$\sum\_{\mathbf{r}=t-DT\_{\mathcal{S}}+1}^{t} z\_{\mathbf{g},\mathbf{r}}^{SD} \le 1 - z\_{\mathbf{g},t'}^{F} \tag{12}$$

$$\sum\_{\boldsymbol{\uprho} \in \mathcal{G}\_n} p\_{\boldsymbol{\uprho}, t\_{\boldsymbol{\uprho}}^{\pi}} - \sum\_{\boldsymbol{l} \in \mathcal{L}\_n} L\_{\boldsymbol{l}, \boldsymbol{t}} + \sum\_{\boldsymbol{l} \in \mathcal{L}\_n} L\_{\boldsymbol{l}, \boldsymbol{t}\_{\boldsymbol{\uprho}}^{\pi}}^{SH} + \sum\_{\boldsymbol{f} \in \mathcal{F}\_n} W\_{f, t, \boldsymbol{t}\_{\boldsymbol{\uprho}}^{\pi}} \tag{13}$$
 
$$- \sum\_{\boldsymbol{l} \in \Xi} W\_{f, t, \boldsymbol{t}\_{\boldsymbol{\uprho}}^{\pi}}^{SD} = \sum\_{m \in M} \frac{(\theta\_{n, t, \tilde{\uprho}} - \theta\_{m, t, \tilde{\uprho}})}{X\_{n, m}}, \tag{14}$$

$$\begin{cases} f \in \mathcal{F}\_n & m \in \mathcal{M}\_n \\ p\_{\mathbf{g}, t, \mathfrak{f}} \le P\_{\mathfrak{F}}^{\max} \mathfrak{z}\_{\mathfrak{g}, t\prime}^{\mathcal{F}} & \forall \mathfrak{g}, \forall t, \forall \mathfrak{f} \in \Xi \end{cases} \tag{14}$$

$$\begin{aligned} p\_{\mathfrak{g},t,\mathfrak{z}} & \geq P\_{\mathfrak{g}}^{\min} z\_{\mathfrak{g},t\prime}^{F} \\ p\_{\mathfrak{g},t,\mathfrak{z}} & \leq (P\_{\mathfrak{g}}^{\mathrm{IS}} + \mathrm{R}\mathrm{L}\!\!\!\_{\mathfrak{g}}) z\_{\mathfrak{g},t\prime}^{F} \end{aligned} \qquad \qquad \qquad \forall \mathfrak{g}, \forall t, \forall \mathfrak{z} \in \Xi \qquad \tag{15}$$

$$\begin{aligned} p\_{\mathfrak{g},t,\mathfrak{f}} &\geq (P\_{\mathfrak{g}}^{\mathrm{IS}} - RD\_{\mathfrak{g}}) z\_{\mathfrak{g},t'}^{\mathrm{F}} & \forall \mathfrak{g}, \forall t \in \{1\}, \forall \mathfrak{f} \in \Xi\\ p\_{\mathfrak{g},t,\mathfrak{f}} - p\_{\mathfrak{g},t-1,\mathfrak{f}} &\leq (2 - z\_{\mathfrak{g},t-1}^{\mathrm{F}} - z\_{\mathfrak{g},t}^{\mathrm{F}}) P\_{\mathfrak{g}}^{\mathrm{SI}} & \forall \end{aligned} \tag{17}$$

$$\begin{aligned} &+(1+z\_{\mathfrak{g},t-1}^{F}-z\_{\mathfrak{g},t}^{F})R I\_{\mathfrak{g}}), & \forall \mathfrak{g}, \forall t \in \{2,\ldots,T\}, \forall \mathfrak{f} \in \Xi\\ &p\_{\mathfrak{g},t-1,\mathfrak{f}}-p\_{\mathfrak{g},t,\mathfrak{f}} \leq (2-z\_{\mathfrak{g},t-1}^{F}-z\_{\mathfrak{g},t}^{F})P\_{\mathfrak{g}}^{SD} \\ &+(1-z\_{\mathfrak{g},t-1}^{F}+z\_{\mathfrak{g},t}^{F})R I\_{\mathfrak{f}}), & \forall \mathfrak{g}, \forall t \in \{2,\ldots,T\}, \forall \mathfrak{f} \in \Xi\\ &L\_{l,t,\mathfrak{f}}^{SH} \leq L\_{l,t} & \forall l, \forall t, \forall \mathfrak{f} \in \Xi \end{aligned} \tag{19}$$

$$\mathcal{W}\_{f,t,\xi}^{\text{SP}} \le \mathcal{W}\_{f,t,\xi'} \tag{21}$$

$$\begin{aligned} & \; -P\_{\mathrm{n,m}}^{\mathrm{max}} \le \frac{(\theta\_{\mathrm{n,t},\xi} - \theta\_{\mathrm{m,t},\xi})}{X\_{\mathrm{n,m}}} \le P\_{\mathrm{n,m}}^{\mathrm{max}}, \quad & \forall n, m \in M\_{\mathrm{n}}, \forall t, \forall \xi \in \Xi \qquad (22) \\ & \; p\_{\mathrm{g},t,\xi}, L\_{1,t,\xi}^{SH}, W\_{f,t,\xi}^{SD} \ge 0, \quad & \forall \mathrm{g}, \forall l, \forall f, \forall t, \forall \xi \in \Xi \qquad (23) \\ & \; z\_{\mathrm{g},t}^{\mathrm{E}}, z\_{\mathrm{g},t}^{SH}, z\_{\mathrm{g},t}^{SD} \in \{0, 1\}, \quad & \forall \mathrm{g}, \forall t \end{aligned} \tag{23}$$

$$\text{with initial state conditions:}$$

 ,

$$\begin{aligned} IS\_{\mathcal{S}} &= \begin{cases} 1 & \text{if } ON\_{\mathcal{S}} > 0 \\ 0 & \text{if } ON\_{\mathcal{S}} = 0 \end{cases} \\ L\_{\mathcal{S}}^{UP} &= \min \{ T, (\mathcal{L}T\_{\mathcal{S}} - ON\_{\mathcal{S}})IS\_{\mathcal{S}} \} \\ L\_{\mathcal{S}}^{DW} &= \min \{ T, (DT\_{\mathcal{S}} - OFF\_{\mathcal{S}})(1 - IS\_{\mathcal{S}}) \} \end{aligned}$$

The SUC problem (6)–(24) is a standard model proposed in the literature. Its objective is to minimise the fixed cost (*CFg* ), start-up cost (*CSUg* ), shut-down cost (*CSDg* ), and variable cost (*CVg* ) of the synchronous generators, as well as the cost of involuntarily demand curtailment, (6). Equations (7)–(9) represent the logical expression between different binary variables (ON/OFF commitment status). Equations (10)–(12) represent the minimum and maximum up- and down-time constraints of the synchronous generators. Equation (13) model the power balance constraint. Equations (14) and (15) model the capacity limits of synchronous generators, while their respective ramping limits are modeled through (16)–(19). The limits of the demand and wind power curtailment are modeled through (20) and (21). Equation (22) represent the transmission capacity limits. Last, Equations (23) and (24) represent variable declarations.

The SUC problem (6)–(24) includes first-stage variables *<sup>z</sup>Fg*,*t*, *<sup>z</sup>SUg*,*<sup>t</sup>* , *<sup>z</sup>SDg*,*<sup>t</sup>* that model the status of the machines *g* in time period *t* (e.g., start-up/shut-down status) and second-stage variables *pg*,*t*,*ξ* , *<sup>L</sup>SHl*,*t*,*<sup>ξ</sup>* , *WSPf* ,*t*,*ξ* , *<sup>θ</sup><sup>n</sup>*,*t*,*<sup>ξ</sup>* that model the active power of the synchronous machines *g*, the power curtailment from load *l*, wind power curtailment from wind production unit *f* , and voltage angle at node *n*, in time period *t* and scenario *ξ*, respectively. The interested reader can find further details of the SUC in [22] and references therein.

The problem (6)–(24) is based on [22] and is the reference complete SUC formulation considered in the case study. Note that the cosimulation framework is general and allows to assess the impact of any other model of SUC on power system dynamics.

If one considers only one scenario, the set of Equations (6)–(24) reduces to a deterministic UC problem. In the following we use the notation "SUC" to indicate a stochastic formulation, i.e., the set Ξ consists of more than one scenario, and the notation "DUC" for the deterministic case.

## *2.5. Simplified SUC Formulation*

The level of complexity of SUC formulations proposed so far in the literature vary significantly [23]. For example, a well-assessed MILP SUC formulation is provided in [22]. Such a model takes into consideration several technical constraints, e.g., ramping limits of generators and capacity limits of transmission lines, just to mention some. These constraints improve the performance of the schedules but are not crucial in this paper, which focuses on the impact on long-term power system dynamics. Hence, a simplified model of SUC is considered, as follows:

$$\underset{\begin{subarray}{c}\mathbf{r}\_{\mathcal{S}'}^{\mathsf{F}},\mathbf{p}\_{\mathcal{S},t,\xi}\end{subarray}}{\text{minimize}}\sum\_{t\in\Xi\_{\mathbf{T}}}\sum\_{\mathcal{S}'\in\mathcal{G}}\mathsf{C}\_{\mathcal{S}}^{\mathsf{F}}\mathsf{z}\_{\mathcal{S},t}^{\mathsf{F}}+\sum\_{\xi\in\Xi}\pi\_{\xi}\left[\sum\_{t\in\Xi\_{\mathbf{T}}}\sum\_{\mathcal{S}'\in\Xi\_{\mathbf{G}}}\mathsf{C}\_{\mathcal{S}'}^{\mathsf{V}}\mathsf{p}\_{\mathcal{S},t,\xi}\right]\tag{25}$$

such that

$$\mathbb{P}\_{\mathsf{g},t,\mathsf{f}} \leq P\_{\mathsf{g}}^{\max} \mathbb{Z}\_{\mathsf{g},t'}^{F} \tag{26} \\ \tag{26} \\ \mathbb{P}\_{\mathsf{t}} \forall t, \forall \mathsf{f} \in \Xi \tag{26}$$

$$p\_{\mathbf{g},t,\xi} \ge P\_{\mathbf{g}}^{\min} z\_{\mathbf{g},t,\ast}^{\mathbf{g}} \tag{27}$$

$$\sum\_{\mathbf{j}\in\Xi\_{\mathbf{G}\_{0}}} p\_{\mathbf{j},t,\mathbf{j}} - \sum\_{l\in\Xi\_{\mathbf{L}\_{0}}} L\_{l,t} + \sum\_{k\in\Xi\_{\mathbf{K}\_{0}}} W\_{k,t,\mathbf{j}} = \sum\_{m\in\Xi\_{\mathbf{M}\_{0}}} \frac{(\theta\_{n,t,\mathbf{j}} - \theta\_{m,t,\mathbf{j}})}{X\_{n,m}}, \qquad \forall n, \forall t, \forall \mathbf{j}\in\Xi \tag{28}$$

$$z\_{\mathfrak{g},t}^{\mathcal{F}} \in \{0,1\}, \tag{29}$$

where *z<sup>F</sup> g*,*t* is a first-stage decision variable that models the status (ON/OFF) of the conventional machines in time period *t*; *C<sup>F</sup> g* and *C<sup>V</sup> g* represents the fixed and variable production cost of generation unit *g*, respectively; *πξ* is the probability of wind power scenario *ξ*; *pg*,*t*,*ξ* is a variable that represents the active power of the machine *g* in scenario *ξ* and time period *t*; *P*max *g* and *P*min *g* represents the maximum and minimum active power limits of generation unit *g*, respectively; *Ll*,*<sup>t</sup>* represents the demand for load *l* at time period *t*; *Wk*,*t*,*<sup>ξ</sup>* is the wind power production of wind generation unit *k*, *<sup>θ</sup><sup>n</sup>*,*t*,*<sup>ξ</sup>* are the voltage angles at node *n* and at time period *t* in scenario *ξ*, respectively; *Xn*,*<sup>m</sup>* is the reactance of the transmission line *n* − *m*; and <sup>Ξ</sup>Kn , Ξ Mn represents the sets of wind power generation connected at node *n*, and nodes *m* ∈ *N* connected to node *n* through a line, respectively. Finally, constraints (25) to (29) model the objective function, maximum and minimum power output of generators, nodal power balance equation (DC power flow), and variable declarations, respectively.

#### *2.6. Alternative SUC Formulation*

The literature provides several different formulations of the SUC problem [24]. Therefore it is relevant to compare and study the impact that these models have on the power system dynamic behaviour. In this context, an alternative subhourly SUC model with respect to the one discussed in the previous section has been adapted based on [25], as follows:

$$\begin{aligned} &\underset{\boldsymbol{\pi}\_{\mathcal{S},t}^{\mathsf{F}},p\_{\mathcal{S},t},\boldsymbol{I}\_{\mathcal{S},t}^{\mathsf{II}},\boldsymbol{I}\_{\mathcal{S},t}^{\mathsf{D}},\boldsymbol{p}\_{\mathcal{S},t}}{\text{minimize}}\sum\_{t\in\mathsf{\Sigma}\_{\mathcal{T}}}\sum\_{\mathcal{S}\in\mathsf{\Sigma}\_{\mathcal{G}}}\mathsf{C}\_{\mathcal{S},t}^{\mathrm{SII}}+\mathsf{C}\_{\mathcal{S}}^{\mathsf{V}}p\_{\mathcal{S},t}}{\text{s.t}}\tag{30} \\ &+\sum\_{\boldsymbol{\xi}\in\mathsf{\Sigma}}\pi\_{\mathsf{\xi}}\left[\sum\_{t\in\mathsf{\Sigma}\_{\mathcal{T}}}\sum\_{\mathcal{S}\in\mathsf{\Sigma}\_{\mathcal{G}}}\mathsf{C}\_{\mathcal{S}}^{\mathrm{V}}(r\_{\mathcal{G},t,\mathsf{\tilde{\varepsilon}}}^{\mathrm{II}}-r\_{\mathcal{S},t,\mathsf{\tilde{\varepsilon}}}^{\mathrm{D}})\right] \end{aligned} \tag{30}$$

such that

> *<sup>C</sup>SUg*,*<sup>t</sup>*

$$\mathbb{C}\_{\mathbb{g},t}^{SLI} \ge \mathbb{C}\_{\mathbb{g}}^{SLI} (z\_{\mathbb{g},t}^F - z\_{\mathbb{g},t-1}^F), \tag{31}$$

$$\geq 0,\tag{32}$$

$$\underbrace{z\_{\mathcal{S},t}^F P\_{\mathcal{S}}^{min}}\_{\mathcal{S}} \le p\_{\mathcal{S},t} \le z\_{\mathcal{S},t}^F P\_{\mathcal{S}}^{max}, \tag{33}$$

$$\sum\_{\mathbf{y}\in\Xi\_{\mathbf{G}}} p\_{\mathbf{y},t} + \sum\_{k\in\Xi\_{\mathbf{K}}} W\_{k,t} = L\_{t,\prime} \tag{34}$$

$$z\_{\mathbf{g},t}^F P\_{\mathbf{g}}^{min} \le (p\_{\mathbf{g},t} + r\_{\mathbf{g},t,\mathbf{f}}^{II} - r\_{\mathbf{g},t,\mathbf{f}}^D) \le z\_{\mathbf{g},t}^F P\_{\mathbf{g}}^{max}, \quad \forall \mathbf{g}, \forall t, \forall \mathbf{f} \tag{35}$$

$$\sum\_{\mathbf{g}\in\Xi\_{G}} (p\_{\mathbf{g},t} + r\_{\mathbf{g},t,\mathbf{f}}^{\mathrm{II}} - r\_{\mathbf{g},t,\mathbf{f}}^{\mathrm{D}}) + \sum\_{k\in\Xi\_{K}} W\_{k,t,w} = L\_{t\prime} \tag{36}$$

$$0 \le r\_{\mathbf{g},t,\mathbf{f}}^{\mathrm{LI}} \le R\_{\mathbf{g},t}^{\mathrm{LI}\max},\tag{37}$$
 
$$r\_{\mathbf{o},t,\mathbf{s}} \le D \quad \text{s.t. } \operatorname{mD}\max$$

$$\begin{aligned} 0 \le r\_{\mathbf{g},t,\mathbf{f}}^D \le R\_{\mathbf{g},t}^{D\max}, \quad & \quad \forall \mathbf{g}, \forall t, \forall \mathbf{f} \tag{38} \\ z\_{\mathbf{g},t}^F \in \{0,1\}, \quad & \quad \forall \mathbf{g}, \forall t \end{aligned} \tag{39}$$

where, apart from the variables and parameters already defined above, *<sup>C</sup>SUg*,*<sup>t</sup>* models the start-up cost of generation unit *g* incurred at the beginning of time period *t*; *<sup>r</sup>Ug*,*t*,*<sup>ξ</sup>* and *<sup>r</sup>Dg*,*t*,*<sup>ξ</sup>* are decision variables and represents the increase and decrease in the active power output of generation unit *g* at time period *t*, respectively, say, during real-time operation (e.g. to compensate wind power fluctuations); while all other parameters and variables have analogous meanings as in the previous SUC formulation.

The main difference between problem (30)–(39) and problem (25)–(29) is that in the former the active power of generation units, *pg*,*t*, is a first-stage variable and does not adapt to the uncertainty realization. Note that model (25)–(29) has a slightly different objective function, namely, it does not include the start-up cost (*CSUg*,*<sup>t</sup>*) as compared to that of model (30)–(39).
