# A New Multi-Objective Unit Commitment Model
## 3 Methodology
### Stop Criterion  
The stop criterion is of great interest that we should ensure either the optimality as well as convergence. A natural idea is that the difference of \(MP\) between any pairs of subsystems attached with a tie-line is the criterion of convergence because we are taking the difference of \(MP\) as an iterative basis, but the difference of \(MP\) between subsystems may never be zero as there are other constraintslike system reserve constraint (5) and tie-line constraint (10). Therefore, the stop criterion could be formed as the following rules. 1. If \(|MP_{i,t}-MP_{j,t}|<\theta,(i\
eq j)\) is met at all time intervals and \(P_{i,j}^{max}\
eq 0\) (there exists a tie-line between subsystem \(i\) and \(j\)), or the change MOUC cost is almost invariant, the iteration process is terminated. 2. If the tie-line power flow has reached its limit or the maximum number of iterations been reached, stops updating the tie-line. 3. When the subsystem spinning reserve has been reached and the \(MP\) of that subsystem is still low, which means, if the tie-line encourages the generation of that subsystem to be decreased, the total MOUC cost would increase, but the generation should not be increased. So the attached tie-line should not be updated anymore. Theorem 1: _There exits an equilibrium so the iteration process could be terminated._  
Proof: We focus on the situation where rules 2 and 3 have not been met. The \(MP\) of subsystem is defined as the first-order derivative of one of the units in the subsystem. Assumes the \(MP\) of unit \(i\) is the lowest. So, the \(MP\) for the subsystem at time \(t\) could be written as derivative with respect to the unit output that belongs to the subsystem. \[MP=\frac{\partial F^{\prime}}{\partial p_{i,t}}=2F_{1}^{t}\frac{\partial F_{1 }^{\prime}}{\partial p_{i,t}}+2F_{2}^{t}\frac{\partial F_{2}^{\prime}}{ \partial p_{i,t}} \tag{28}\]  
Please note that the \(MP\) at time \(t\) can be influenced by not only \(p_{i,t}\), but also the output of other units at any time intervals. If the system load is increased by \(Step\), the generation of some units (especially the one with lowest \(MP\)) would be increased and vice versus. So we look at the influence on \(MP\) when the output of some units have changed. \[\frac{\partial MP}{\partial p_{i,t}}=2\left(\frac{\partial F_{1}^{\prime}}{ \partial p_{i,t}}\right)^{2}+2\left(\frac{\partial F_{2}^{\prime}}{\partial p _{i,t}}\right)^{2}+2F_{1}^{t}\frac{\partial^{2}F_{1}^{t}}{\partial p_{i,t}^{2 }}+2F_{2}^{t}\frac{\partial^{2}F_{2}^{\prime}}{\partial p_{i,t}^{2}} \tag{29}\]  
Each term in (29) is only influenced by parameters of unit \(i\). For any units, the coeffcients of fuel cost function (1) are always positive, which means the first and third term in (29) are positive. As for the first and second order derivative of (3), we refer to [36] that these derivatives are always positive when \(p_{i,t}\in\{P_{i,min},P_{i,max}\}\). Hence, for all units, (29) has always been positive when \(p_{i,t}\in\{P_{i,min},P_{i,max}\}\). If the output of any units in that subsystem is increased, the \(MP\) of the subsystem must be increased and vice versus. As the iteration process moving on, the \(MP\) between pairs of subsystem could be closing to each other. Please note that the proof above based on the unit with lowest margin price is not changed with the iteration process going on. If the margin unit has been changed, the proof could also make sense:  
1. If the margin unit in the last iteration step has been scheduled to generate at its full capacity, the margin unit will be changed, and there will be a gap of the \(MP\). The (29) could also be positive because the if the generation of new margin unit is increased, \(MP\) is increased, and if the \(VL\) of the system is decreased, margin unit would be switched to that in the last iteration step, the \(MP\) is decreased. 2. If the margin unit has been changed and the margin unit in the last iteration step has not reached its generation limit, the (29) will not be consistent, but it is still positive.