Table 1 Constraint formula table of PB module

Non-run constraint

Logical relation constraint

\(u_{t}^{{{\text{pb}}}} - u_{t - 1}^{{{\text{pb}}}} \le r_{t}^{{{\text{pb}}}} \le \frac{{\left( {1 + u_{t}^{{{\text{pb}}}} - u_{t - 1}^{{{\text{pb}}}} } \right)}}{2}\)

Efficiency constraints of thermoelectric conversion

\(P_{t}^{{{\text{csp}}}} = \eta_{{{\text{pb}}}} P_{t}^{{{\text{ST}}}}\)

Run constraint

Maximum and minimum output constraints

\(u_{t}^{{{\text{pb}}}} P_{\min }^{{{\text{csp}}}} \le P_{t}^{{{\text{csp}}}} \le u_{t}^{{{\text{pb}}}} P_{\max }^{{{\text{csp}}}}\)

Climb up and down constraints

\(- R_{{{\text{down}}}}^{{{\text{csp}}}} \Delta t \le P_{t}^{{{\text{csp}}}} - P_{t - 1}^{{{\text{csp}}}} \le R_{{{\text{up}}}}^{{{\text{csp}}}} \Delta t\)

Minimum boot time constraint

\(\sum\limits_{{\tau = t - T_{{{\text{on}}}}^{{{\text{pb}}}} }}^{t - 1} {u_{\tau }^{{{\text{pb}}}} } \ge T_{{{\text{on}}}}^{{{\text{pb}}}} \left( {u_{t - 1}^{{{\text{pb}}}} - u_{t}^{{{\text{pb}}}} } \right)\)

Minimum downtime constraints

\(\sum\limits_{{\tau = t - T_{{{\text{off}}}}^{{{\text{pb}}}} }}^{t - 1} {\left( {1 - u_{\tau }^{{{\text{pb}}}} } \right)} \ge T_{{{\text{off}}}}^{{{\text{pb}}}} \left( {u_{t}^{{{\text{pb}}}} - u_{t - 1}^{{{\text{pb}}}} } \right)\)