# CSO Based Solution for Load Kickback Effect in Deregulated Power Systems
## 2 Demand Response Unit Commitment Program  
The Traditional Unit Commitment (TUC) is a process of scheduling the power generation, within the system and unit operational limits. Minimizing the generation cost, along with fuel cost and startup cost, is the prime objective of the traditional unit commitment problem [19]. The primary objective of the demand response unit commitment problem is to maximize the profits of the GENCOs using time based demand response program (TBDRP).  
\[MaxP_{R}=[TR_{V}-TO_{COST}] \tag{1}\]  
where,  
\(P_{R}\)--is the total profit of the GENCOs and DRSP combined  
\(TR_{V}\)--is the total revenue calculated from GENCOs and DRSP  
\(TO_{COST}\)--is the total operating cost of GENCOs and DRSP combined  
\[TR_{v}=\left[\sum_{t=1}^{T}\sum_{i=1}^{N}P_{gen}^{i,t}\ S_{price}^{i}\ U_{ stat}^{i,t}\right] \tag{2}\]Total operating cost varies for GENCOs and consumers:  
\[TO_{COST}=TO_{\cos t}^{GENCO}+TO_{\cos t}^{DRSP} \tag{3}\]  
Total operating cost for GENCOs is represented as  
\[TO_{\cos t}^{GENCO}=\left[\sum_{t=1}^{T}\sum_{i=1}^{N}F_{cos t}^{i}\Big{(}P_{ gen}^{i,t}\Big{)}U_{stat}^{i,t}+SU_{\cos t}-I_{\cos t}+P_{\cos t}\right] \tag{4}\]  
\[F_{\cos t}=a^{i}\Big{(}P_{gen}^{i}\Big{)}^{2}+b^{i}\Big{(}P_{gen}^{i}\Big{)}+c ^{i} \tag{5}\]  
The following section illustrates the various equations associated with demand response unit commitment.  
**Equality Constraint:** The equality constraint gives the power balance between generation and load, (i.e., total generated power should be equal to the demanded power)  
\[\sum_{i=1}^{N}\Big{(}P_{gen}^{i,t}U_{stat}^{i,t}\Big{)}=P_{dem}^{t};\ (t=1\dots T) \tag{6}\]  
**Inequality Constraint:** The power generation limit should be within the specified generation limits of that unit.  
\[P_{gen}^{i,\min}\leq P_{gen}^{i,t}\leq P_{gen}^{i,\max}\ ;\ (i=1\dots N) \tag{7}\]  
**Ramp up Rate:** It is the specified limit above which the maximum power generation value cannot be increased for a unit in the next hour.  
\[P_{gen}^{i,t\ \max}=\min\Big{(}P_{gen}^{i\max},\ P_{gen}^{i(t-1)}+\psi R_{up}^{i} \Big{)} \tag{8}\]  
**Ramp down Rate:** It is the specified limit below which the minimum power generation value cannot be decreased for a unit in the next hour.  
\[P_{gen}^{i,t\ \min}=\max\Big{(}P_{gen}^{i\min}\,\ P_{gen}^{i(t-1)}-\psi R_{down}^{i} \Big{)} \tag{9}\]  
**Minimum up time:** Once a unit is committed, then for some specific number of hours, it cannot be decommitted.  
\[ON^{i}\geq M_{up}^{i} \tag{10}\]  
**Minimum down time:** Once a unit is decommitted, then for some specific number of hours it cannot be recommitted.  
\[OFF^{i}\geq M_{down}^{i} \tag{11}\]  
**Reserve Constraints:** A specified amount of power that is generated by the unit at a particular hour may be used in cases of emergencies, for which it is kept reserved.  
\[0\leq R_{gen}^{i,t}\leq\Big{(}P_{gen}^{i,\max}\ P_{gen}^{i,\min}\Big{)} \tag{12}\]  
**Spinning Reserve:** The difference between the amount of power generated from all the units synchronized to the system with its present load supplies and losses incurred in the system is called spinning reserve.  
\[S_{res}=\sum_{i=1}^{N}\left(P_{gen}^{i\;\max}-P_{gen}^{i}\right) \tag{13}\]  
**Startup Cost of Units:** Startup cost can be classified into two different types, namely, hot start cost and cold start cost. Hot start cost is the one where the generator is maintained at minimum temperature, without being shut down. Cold start cost is the one when the generator is shut down, and is started back up once again [20].  
\[SU_{\text{cost}}^{i}=\left\{\begin{array}{ll}H_{\text{cost}}^{i},&if\;M_{ down}^{i}-T^{i(t-1)}\leq CS_{\text{time}}^{i}\\ C_{\text{cost}}^{i},&if\;M_{down}^{i}-T^{i(t-1)}\geq CS_{\text{time}}^{i}\\ \end{array}\right. \tag{14}\]