# Dimensioning of Reactive Power Compensation in an Autonomous Island System
## 4 UC-ED-Oriented Cost-Benefit Analysis
### Incorporation of Reactive Power Constraints for Thermal Units  
To include the specificities of reactive power limitations on the island of Rhodes, the UC-ED is further amended by introducing relevant constraints. Note that the mathematical formulation of the UC-ED does not account for the AC power flow analysis of the system. Rather, it focuses on the reactive power absorption capabilities of thermal units, as well as the impact that the inclusion of such constraints may have on the overall UC-ED results for power systems experiencing excess reactive power during low demand periods. As analyzed in Section 3, the reactive power compensation sizing is not an issue of voltage rise, but a question of managing the reactive power absorption capability of operating synchronous generators to reasonable and secure levels.  
Online thermal units should provide a cumulative reactive power absorption capability, as needed to meet the excess reactive power of the system. To ensure that the system will survive disturbances, such as the loss of generation, without compromising its reactive power absorption capability, VAr constraints should also be satisfied at G-1 conditions (loss of any online generator).  
Constraints (21) to (27) ensure that online thermal units will be capable of absorbing the reactive power generated by the HV network during normal and G-1 operating conditions. More specifically, (21) forces the commitment of as many units as necessary to secure reactive absorption during normal operation (\(Q_{t}^{req}\) is negative when units need to absorb reactive power). Constraint (22) aims at identifying the online thermal unit that provides maximum VAr absorption capability among the committed generators. The non-linearities of (22) are properly linearized via (23) to (25). Constraint (25) identifies the generator that contributes most by setting the corresponding binary variable \(z_{u,t}^{Q}\), equal to \"1\". Finally, (26) defines the status of all units after the loss of the unit that contributes the most in reactive power absorption, while (27) imposes the fulfillment of reactive power absorption requirements at G-1 operation. Note that constraint (21) is obsolete when (27) applies, yet it is included in the formulation for the sake of completeness.  
\[\sum_{u}Q_{u,t}^{\min}\cdot st_{u,t}\leq Q_{t}^{req} \tag{21}\]  
\[Q_{t}^{\rm gen-min}=\max_{u}\left\{-Q_{u,t}^{\min}\cdot st_{u,t}\right\} \tag{22}\]\[Q_{t}^{\text{gen-min}}\geq-Q_{u,t}^{\text{min}}\cdot st_{u,t} \tag{23}\] \[Q_{u,t}^{\text{min}}\cdot st_{u,t}\leq\left(1-z_{u,t}^{\text{Q}} \right)\cdot M-Q_{t}^{\text{gen-min}}\] (24) \[\sum_{u}z_{u,t}^{\text{Q}}=1\] (25) \[st_{u,t}^{\text{tr}z}=st_{u,t}-z_{u,t}^{\text{Q}}\] (26) \[\sum_{u}Q_{u,t}^{\text{min}}\cdot st_{u,t}^{\text{tr}z}\leq Q_{t} ^{\text{req}} \tag{27}\]  
Similarly, constraints (28) to (34) ensure the required levels of reactive power production by online thermal units to meet demand (when the system consumes reactive power).  
\[\sum_{u}Q_{u,t}^{\text{max}}\cdot st_{u,t}\geq Q_{t}^{\text{req}} \tag{28}\]  
\[Q_{t}^{\text{gen-max}}=\max_{u}\big{\{}Q_{u,t}^{\text{max}}\cdot st_{u,t}\big{\}} \tag{29}\]  
\[Q_{t}^{\text{gen-max}}\geq Q_{u,t}^{\text{max}}\cdot st_{u,t} \tag{30}\]  
\[Q_{u,t}^{\text{max}}\cdot st_{u,t}\geq Q_{t}^{\text{gen-max}}-\left(1-y_{u,t} ^{\text{Q}}\right)\cdot M \tag{31}\]  
\[\sum_{u}y_{u,t}^{\text{Q}}=1 \tag{32}\]  
\[st_{u,t}^{\text{r}y}=st_{u,t}-y_{u,t}^{\text{Q}} \tag{33}\]  
\[\sum_{u}Q_{u,t}^{\text{max}}\cdot st_{u,t}^{\text{r}y}\geq Q_{t}^{\text{req}} \tag{34}\]  
Constraints (28) to (34) have been introduced within the MILP, but they do not impact the final UC-ED solution, as the reactive power production capabilities of online units far exceed the corresponding system requirements.