## 2 Robust Co-Optimization to Energy and Reserve Joint Dispatch  
The energy and reserve co-optimization dispatch model minimizes the total costs of energy and reserve, while obtaining an economically optimal dispatch schedule within the physical constraints. However, it is difficult to up-dispatch the reserve cleared by some generators due to the uneven distribution of generation resources in the entire system. Hence, a large-scale power system is divided into several geographic areas and the minimum system spinning reserve requirements and zonal spinning reserve requirements are considered at the same time to guarantee the deliverability of reserves when the contingency occurs. The feature of the proposed model is to incorporate minimum zonal requirements to guarantee the security and reliability of the power system. The objective function is then formulated as [5]  
\[\min\ (\sum_{i=1}^{N_{g}}e_{i}P_{i}^{sch}+\sum_{a=1}^{N_{a}}b_{a}\big{(}A_{a}^{ +}+A_{a}^{-}\big{)}+\sum_{i=1}^{N_{g}}g_{i}R_{i}) \tag{1}\]  
where the three terms in the objective function represent the costs of energy, regulating reserve and spinning reserve, correspondingly. It should be noted that the coefficients (\(e_{i}\), \(b_{a}\), \(g_{i}\)) are the bidding price of energy, regulating reserve and spinning reserve of generators in the competitive deregulated power markets. The bidding prices are offered by the generation companies. The constraints of the optimization model should include the power balance, generation output limits, ramp up/down limits, transmission line limits, system and zonal reserve requirements, such that  
1. Power Balance Constraint \[\sum_{i=1}^{N_{g}}P_{i}^{sch}=\sum_{j=1}^{N_{d}}D_{j}\] (2)
2. Generation Output Limits For AGC units, the energy and reserve generation should satisfy \[P_{a}^{sch}+A_{a}^{+}+R_{a}\leq P_{\text{amax}},\ P_{a}^{sch}-A_{a}^{-}\geq P_{ \text{amin}}\] (3) \[P_{a}^{sch}+A_{a}^{+}\leq A_{\text{amax}},\ P_{a}^{sch}-A_{a}^{-}\geq A_{ \text{amin}},\ a=1,\ \cdots,N_{a}\] (4)  
For non-AGC units (dispatch-able units), the energy and spinning reserve generation should satisfy  
\[P_{i}^{sch}+R_{i}\leq P_{\text{max}},\ i=1,\cdots,N_{g} \tag{5}\]
3. The Limits of Up/Down Regulating Speed and Response Speed of Reserve  
For two adjacent dispatch intervals, the ramp up and down rates should be limited by the constraints of dispatch-able units and AGC units respectively:  
\[-T\cdot r_{i}^{-}\leq P_{i}^{sch}-P_{0i}\leq T\cdot r_{i}^{+}\ 0\leq R_{i}\leq t_{2}\cdot r_{i}^{+},\ i=1,\cdots,N_{g} \tag{6}\]  
\[0\leq A_{a}^{+}\leq t_{1}\cdot r_{a}^{+},\ 0\leq A_{a}^{-}\leq t_{1}\cdot r_{a}^{-},\ a =1,\cdots,N_{a} \tag{7}\]
4. The System/Zonal Minimum Requirements for Reserves  
Due to the uneven distribution of generation resources in actual power systems, the reserves may only locate in some specific zones if the reserve cost is very low in these regions and high in the others. This may result in difficulties in dispatching upwards the reserve cleared in some generators considering the limits of transmission line capacity and network congestion when a certain contingency really happens. In that case, there will be side effects on the security and reliability of the entire power system. Hence, it is essential to incorporate the zonal requirement constraints in the energy and reserve dispatch model. Under normal circumstances, the system minimum requirement for regulating reserve and spinning reserve is  
\[\sum_{a=1}^{N_{a}}A_{a}^{+}\geq D_{A}^{s},\sum_{i=1}^{N_{g}}R_{i}\geq D_{R}^{s} \tag{8}\]  
The zonal minimum requirement for spinning reserve can be formulated as  
\[\sum_{i=1}^{N_{g}^{z}}R_{i}\geq D_{R}^{z} \tag{9}\]  
where the given constant value \(D_{A}^{s}\), \(D_{R}^{s}\) and \(D_{R}^{z}\) can be obtained by offline analysis and calculation.