# Dimensioning of Reactive Power Compensation in an Autonomous Island System
## 4 UC-ED-Oriented Cost-Benefit Analysis
### Model Fundamentals  
The mathematical model for the UC-ED used in the paper is built upon the fundamentals of the MILP generation scheduling process for island systems, as described in detail in [17; 18]. To enhance the flexibility of the system, the dual minimum loading technique proposed in [22] is adopted. This approach enables low load operation of the thermal units, below their normal technical minimum loading level, in a coordinated manner aiming to increase RES absorption during base-load hours without committing expensive peaking units [25; 26; 27].  
The problem formulation involves constraints (4) to (20). All symbols are defined in the nomenclature section in Abbreviations. The cost minimization objective function (4) includes the variable operating costs of thermal units operating within their normal operating limits (\(C_{M,I}\)), the cost attributed to the low-loading operation of thermal units (\(C_{M,II}\)) and their start-up (\(C_{su}\)) and shut-down (\(C_{sd}\)) costs. Details regarding the definition of cost terms can be found in [17; 20; 22; 28; 29]. Equation (5) defines the active power equilibrium, while constraints (6) and (7) bound the power output of the thermal units when operating in low-load mode. Equation (8) defines the final power output of thermal units, either when in normal (over) or in low-loading (under) operation. Constraints (9) to (11) impose the commitment status of thermal units, while (12) and (13) bound their operation when in normal mode (i.e., operating within their declared minimum and maximum loading levels). Constraints (14) and (15) impose the minimum up and down times for thermal units, respectively. Constraints (16) to (18) ensure the fulfillment of reserve requirements per type (primary, secondary, and tertiary) and direction (upward/downward), while (19) limits the reserves provision per type according to the respective capabilities of each unit. Equation (20) defines the primary up reserves requirements of the system, which are crucial for island system stability during severe disturbances, e.g., the sudden loss of generation. Non-linear terms in (20) are linearized according to the methodology of [30]. For the quantification of the remaining reserves requirements, the principles of [17] apply.  
\[\min\{C_{M,I}+C_{M,II}+C_{su}+C_{sd}\} \tag{4}\]  
\[\sum_{u}P_{u,t}+(P_{w,t}-x_{w,t})+P_{pv,t}=P_{L,t} \tag{5}\]  
\[P_{u,t}^{under}\geq P_{u}^{rML}\cdot s_{u,t}^{under} \tag{6}\]  
\[P_{u,t}^{under}\leq P_{u}^{TML}\cdot s_{u,t}^{under} \tag{7}\]  
\[P_{u,t}=P_{u,t}^{under}+P_{u,t}^{over} \tag{8}\]  
\[st_{u,t}=st_{u,t}^{under}+st_{u,t}^{over} \tag{9}\]  
\[su_{u,t}+sd_{u,t}\leq 1 \tag{10}\]  
\[su_{u,t}-sd_{u,t}=st_{u,t}-st_{u,t-1} \tag{11}\]  
\[P_{u,t}+\sum_{r\in\{rp,sr,tr\}}r_{u,t}^{up}\leq P_{u,t}^{\max}\cdot st_{u,t} \tag{12}\]\[P_{u,t}^{pvor}-\sum_{r\in\{pr,sr\}}r_{u,t,r}^{dn}\geq P_{u}^{TML}\cdot st_{u,t}^{ ovor} \tag{13}\]  
\[\sum_{k=t-T_{u}^{m}+1}^{t}su_{u,k}\leq st_{u,t} \tag{14}\]  
\[\sum_{k=t-T_{u}^{dvor}+1}^{t}sd_{u,k}\leq 1-st_{u,t} \tag{15}\]  
\[\sum_{u}r_{u,t,pr}^{up/dn}\geq rr_{t,pr}^{up/dn} \tag{16}\]  
\[\sum_{u}\sum_{r\in\{pr,sr\}}r_{u,t,r}^{up/dn}\geq\sum_{r\in\{pr,sr\}}rr_{t,r}^{ up/dn} \tag{17}\]  
\[\sum_{u}\sum_{r}r_{u,t,r}^{up}\geq\sum_{r}rr_{t,r}^{up} \tag{18}\]  
\[r_{u,t,r}^{up/dn}\leq R_{u,r}\cdot st_{u,t} \tag{19}\]  
\[rr_{t,pr}^{up}=\max\left(\max_{u}\left\{P_{u,t}+r_{u,t,pr}^{up}\right\},(P_{u,t}-x_{u,t})\right) \tag{20}\]