# A New Multi-Objective Unit Commitment Model
## 3 Methodology
### Update Tie-Line Power Flow  
The tie-line power flow is updated based on the hypothesis that for any time intervals, if a little part of load is transferred from high \(MP\) area to the other, the total MOUC cost in the whole system could be decreased, and the amount of change signifies the step size with direction defined by the difference between subsystems. The rules of updating tie-line power flow are summarized as follows.  
1. For any given time interval, the change of tie-line should decrease the total MOUC cost in the whole system. As the \(MP\) has been calculated in the previous step, the subsystem with high \(MP\) should generate less, and tie-line power flow could be updated as follows. \[\min\sum_{t}\sum_{i}\left(MP_{i,t}\left(\sum_{j}x_{i,j,t}\right)\right)\] (20) \[s.t.\] (21) \[x_{i,j,t}=0,i\in\{1,...,M\},t\in\{1,...,T\}\] (22) \[x_{i,j,t}+x_{j,t}=0,i,j\in\{1,...,M\},t\in\{1,...,T\}\] (23) \[|x_{i,j,t}|\leq P_{ij}^{max},i,j\in\{1,...,M\},t\in\{1,...,T\}\] (24) \[|\sum_{j}x_{j,i,t}|\leq Step_{i,t},i,j\in\{1,...,M\},t\in\{1,...,T\}\] (25) \[x_{i,j,t}\] is the step change which should be added to \[p_{i,j,t}^{tie}.\]\[P_{i,j,t}^{ite}=P_{i,j,t}^{tie}+x_{i,j,t},i,j\in\{1,...,M\},t\in\{1,...,T\} \tag{26}\]  
\(|\sum_{j}x_{j,j,t}|\) is the change of \(VL_{i}\) at time \(t\). We assumes that if the change of system load did not exceed \(Step_{i,t}\), the \(MP_{i,t}\) would be seen as constant. As a result, the objective of (20) represents the increase of total MOUC cost in the whole system.
2. However, \(Step\) must be chosen to update the tie-line power flow. \(Step\) could neither be too big nor too small. A big \(Step\) may cause the total MOUC cost to increase, because the direction is a decent direction only if the \(Step\) is not too big in any time interval. Furthermore, a small \(Step\) may lead to the total MOUC cost to be almost invariant, or will not encourage more units to be on. Here, \(Step\) is chosen based on the following rules. 1. The \(Step_{t}\) would be big, if the difference of margin price between the two subsystems is significant. 2. The \(Step_{t}\) shall not only meet the tie-line constraint, but also ensure that it does not exceed a certain percentage of the local load of the two subsystems to which tie-line is connected. \[0\leq Step_{t}\leq\delta\times D_{t}\] (27) \(\delta\) represents the percentage of load allowed to be changed. 3. The feasibility of UC problems related with subsystems should be guaranteed. The direct solution here is based on the observation that if system reserve constraint (5) is satisfied, then by modifying the output of units, it is easy to satisfy the demand constraint (4) [35]. The algorithm of choosing the \(Step\) could be stated as follows. Input: system index \(i\) and time \(t\) get the UC schedule during last iteration \(P_{max}=\sum_{k}P_{k,max}\), where \(k\) is the index of unit in subsystem \(i\)
3. **for each**\(j\in\{1,\ldots,M\}\)**do
4. **if**\(j=i\)**then**
5. \(steps_{j}=0\)
6. **else**
7. \(steps_{j}=\frac{|MP_{i,t}-MP_{j,t}|}{max(MP_{i,t}MP_{j,t})}\cdot\delta\cdot min (D_{i,t},D_{j,t})\)
8. **endif**
9. **endfor**
10. \(step_{i,t}=max(steps_{j})\), \(j\)\(\in\)\(\{1\ldots M\}\)
11. **if**\(P_{max}\leq D_{i,t}+R_{i,t}+step_{i,t}\)**then**
12. \(step_{i,t}=D_{i,t}+R_{i,t}-Pmax\)
13. **endif**
14. **return**\(Step_{i,t}\) ```  
**Algorithm 2** Find the \(Step\) for subsystem \(i\) at time \(t\)  
The parameter \(\delta\) varies from system to system. If the \(MP\) in subsystem \(i\) is quite low, the generation of subsystem \(i\) would increase by \(Step\). As the iteration moving on, the increased generation will encourage more units in subsystem \(i\) to be on, and \(P_{max}\) in line 10 of the Algorithm above will be bigger to encourage more generation(if the \(MP\) in the subsystem is still low).