## 3 Economic Dispatch Process  
The SOC controllers described in Section 2 ensure that SOC limits are always met, even in system disturbances. SOC restoration to its reference value, on the other hand, is incorporated in the real-time economic dispatch (ED) process. This is formulated as a mixed integer linear programming (MILP) optimization problem, executed almost in real time, typically every 1-15 min, resulting in dispatch orders (\(P^{*}_{i,ED}\)) and participation factors (\(pf_{i,ED}\)) for all dispatchable units. Its goal is to minimize system cost in the respective time interval, whilst respecting the units' technical constraints, commitment status and allocated reserves. The detailed description of this process for non-interconnected island systems can be found in [6,60], which formed the basis for the ED process implemented in this paper. The day-ahead or intra-day scheduling process resulting in the commitment status and reserve provision is not addressed in this paper. In terms of notation, lowercase letters are used in this section for decision variables and uppercase for parameters. Figure 5: BAU state of charge (SOC) control scheme. The objective function to be minimized in the ED process is:  
\[ \begin{array}{c}J=(\sum\limits_{g^{i}\in G}{(VCS^{gi}p_{t}^{gi})+\sum\limits_{i \in\{G,ESS\}}{(C_{up}^{sl,r}r_{up,t}^{i,sl}+C_{dn}^{sl,r}r_{dn,t}^{i,sl})}+}\\ C^{sl,eq}p_{t}^{sl,eq})\Delta t+C^{sl,soc}\sum\limits_{bi\in ESS}{((soc_{up,t} ^{bi,sl}+soc_{dn,t}^{bi,sl})E^{bi,max})}\end{array}  \tag{4}\]  
where \(G\) is the set of conventional generators, \(ESS\) is the set of storage units, \(r^{i,sl},p^{i,sl}\) and \(soc^{bi,sl}\) are the slack variables explained later in this section, \(\Delta t\) is the ED time interval (in hours) and \(E^{bi,max}\) is the ESS maximum energy capacity of each independently controlled storage unit. The first term of Equation (4) is the thermal unit variable production cost. Variable cost (\(VCS^{gi}\)) in \(\epsilon/\)MWh will depend on the production level \((p_{t}^{gi})\) of each thermal unit. Here, for simplicity, a constant \(VCS^{gi}\) is assumed for all units. The following two terms penalize deviations in available reserves and in the active power equilibrium of the system, while the last term penalizes SOC deviations from its reference value. High \(C^{sl,soc}\) will over-prioritize SOC regulation, whereas low cost values may not prevent BESS energy depletion. The following relative priority of constraints is chosen: \(C^{sl,eq}>C_{up}^{sl,r}>C_{dn}^{sl,r}>C^{sl,soc}>VCS^{gi}\) (all in \(\epsilon/\)MWh). The power equilibrium constraint is given by:  
\[\sum\limits_{gi\in G}{p_{t}^{gi}+\sum\limits_{bi\in ESS}{p_{t}^{bi,d}+p_{t}^{ res}+p_{t}^{sl,eq}}}=P_{t}^{L}+\sum\limits_{bi\in ESS}{p_{t}^{bi,c}} \tag{5}\]  
In (5), \(p_{t}^{bi,d}\) and \(p_{t}^{bi,c}\) are the ESS discharge and charge, respectively, \(p_{t}^{res}\) is the aggregate RES production and \(P_{t}^{L}\) the system load. The power equilibrium slack variable \((p_{t}^{sl,eq})\) ensures the feasibility of the solution. The technical constraints of conventional units given in Equations (6) and (7) result in retaining the reserves \((FCR_{up,dn,t}^{gi},aFRR_{up/dn,t}^{gi})\) allocated in the scheduling process, while leaving room for exceptional cases where this is not feasible, via the introduction of two slack variables \((r_{up,t}^{gi,sl},\ r_{dn,t}^{gi,sl})\). The commitment status of thermal units \((st_{t}^{gi})\) is a parameter in the ED problem, coming from the dispatch schedule. Constraint (8) ensures zero output for unsynchronized units \((st_{t}^{gi}=0\to p_{t}^{gi}=0)\), even when the slack variable \(r_{up,t}^{gi,sl}\) takes non-zero value. Ramping constraints are normally part of the ED problem, however, they are not important in small island systems, such as the study case of this paper, due to the high ramping capability of small diesel units.