Trade-off Between Cost and Safety To Cope with Station Blackout in A PWR in A Deregulated Electricity Market

Abstract

In this paper a close solution is presented to determine the trade-off between cost and safety to cope with station blackout (SBO) in Pressurized Water Reactors (PWRs). To compute the profit of each generation unit, a Supply Function Equilibrium (SFE) model swhich considers carbon tax is used in a uniform electricity market. A hierarchical heuristic method is applied for decision making on safety improvement of Nuclear Power Plants (NPPs). In this method the break-even point is used as a criterion to make a decision on comparing cost of safety improvement and profit of NPP in a deregulated electricity market. This method is applied in a case of adding an Emergency Water Supply (EWS) and an Emergency Diesel Generator (EDG) to a NPP where impacts of its investment cost and its profit are investigated. The achieved results show that the break-even point of investment cost and net profit of the NPP by adding the EDG is one month later than NPP with addition of EWS.

Appendix

Variables, coefficients and elements of matrices are defined as follows:

$${P}_{{S}_{i}}^{\left(t\right)}-{P}_{{S}_{i}}^{max}\le 0\perp {\mu }_{i}^{\left(t\right) max}\ge 0, \quad \forall i\in S,\forall t\in T$$

(a1)

$${P}_{{S}_{i}}^{min}-{P}_{{S}_{i}}^{\left(t\right)}\le 0\perp {\mu }_{i}^{\left(t\right) min}\ge 0, \quad \forall i\in S,\forall t\in T$$

(a2)

$${P}_{{S}_{i}}^{\left(t\right)}-{P}_{{S}_{i}}^{\left(t-1\right)}-{{U}_{rr}}_{i}.{P}_{{S}_{i}}^{max}\le 0\perp {\epsilon }_{i}^{\left(t\right) max}\ge 0, \quad \forall i\in S, http://orcid.org/0000-0002-5228-1444\forall t\in T$$

(a3)

$${P}_{{S}_{i}}^{\left(t-1\right)}-{P}_{{S}_{i}}^{\left(t\right)}-{{D}_{rr}}_{i}.{P}_{{S}_{i}}^{max}\le 0\perp {\epsilon }_{i}^{\left(t\right) min}\ge 0, \quad \forall i\in S, \forall t\in T.$$

(a4)

Computing ${P}_{{S}_{j}}^{\left(t\right)}$ and ${P}_{{D}_{l}}^{\left(t\right)}$ versus ${P}_{{S}_{i}}^{\left(t\right)}$ from (5) and (6) yields:

$$\begin{aligned} P_{{S_{j} }}^{{\left( t \right)}} & = \frac{1}{{b_{j} + \beta \rho _{j}^{2} }}\alpha _{i}^{{\left( t \right)}} + \mu _{i}^{{\left( t \right)}} + \epsilon _{i}^{{\left( t \right)}} \\ & \quad - \epsilon _{i}^{{\left( {t + 1} \right)}} ) - \alpha _{j}^{{\left( t \right)}} + \mu _{j}^{{\left( t \right)}} + \epsilon _{j}^{{\left( t \right)}} - \epsilon _{j}^{{\left( {t + 1} \right)}} \\ & \quad + \left( {b_{i} + \beta \rho _{i}^{2} } \right)P_{{S_{i} }}^{{\left( t \right)}} ,\quad \forall j \in S,j \ne i,\forall t \in T \\ \end{aligned}$$

(a5)

$${P}_{{D}_{l}}^{\left(t\right)}=\frac{1}{{d}_{l}^{\left(t\right)}}\left({c}_{l}^{(t)}-\left({\alpha }_{i}^{(t)}+{\mu }_{i}^{(t)}+{\epsilon }_{i}^{(t)}-{\epsilon }_{i}^{(t+1)}\right)-\left({b}_{i}+\beta {\rho }_{i}^{2}\right){P}_{{S}_{i}}^{\left(t\right)}\right), \quad \forall l\in D, \forall t\in T$$

(a6)

Substituting ${P}_{{S}_{j}}^{\left(t\right)}$, ${P}_{{D}_{l}}^{\left(t\right)}$, and ${\lambda }^{(t)}$ from (a5), (a6), and (5) into (2) yields

$${P}_{{S}_{i}}^{\left(t\right)}={\upsilon }_{i}^{(t)}{\mathbf{P}}_{D}^{\left(t\right)}+{\mathbf{u}}_{i}^{{(t)}^{Tr}}\left({\alpha }^{\left(t\right)}+{\mu }^{\left(t\right)}+{\epsilon }^{\left(t\right)}-{\epsilon }^{\left(t+1\right)}\right) , \quad \forall i\in S,\forall t\in T$$

(a7)

where ${\mathbf{P}}_{D}^{\left(t\right)}$, ${\upsilon }_{i}^{(t)}$ and elements of ${\mathbf{u}}_{i}^{(t)}$ are defined as follows:

$${\mathbf{P}}_{D}^{\left(t\right)}=\left\{\begin{array}{ll}total\,\, demand\, for\,\, inelastic\,\, loads\\ \sum_{i\in D}\frac{{c}_{i}^{\left(t\right)}}{{d}_{i}^{\left(t\right)}} for \,\,elastic\,\, loads\end{array}\right.$$

(a8)

$${\upsilon }_{i}^{(t)}=\frac{1}{{(b}_{i}+\beta {\rho }_{i}^{2}){B}^{(t)}} \quad \forall i\in S,\forall t\in T$$

(a9)

$${u}_{{i}_{j}}^{(t)}=\frac{1}{{(b}_{i}+\beta {\rho }_{i}^{2}){(b}_{j}+\beta {\rho }_{j}^{2}){B}^{(t)}} \forall i\in S, i\ne j, \quad \forall t\in T$$

(a10)

$${B}^{(t)}=\left\{\begin{array}{c}\sum_{i\in S}\frac{1}{{b}_{i}+\beta {\rho }_{i}^{2}} for\,\, inelastic\,\, loads\\ \sum_{i\in S}\frac{1}{{b}_{i}+\beta {\rho }_{i}^{2}}+\sum_{i\in D}\frac{1}{{d}_{i}^{\left(t\right)}} \,for \,\,elastic \,\,loads\end{array}\right.$$

(a11)

Elements of ${\mathbf{P}}_{f}^{\left(t\right)}$, ${\mathbf{R}}_{f}^{\left(t\right)}$,${\mathbf{R}}_{f}^{{{\prime}}\left(t\right)}$,${\mathbf{S}}_{f}^{{{\prime}}\left(t\right)}$, and ${\mathbf{S}}_{f}^{{{\prime}}{^{\prime}}\left(t\right)}$ are defined as follows:

$${P}_{{f}_{ij}}^{\left(t\right)}=\frac{1}{2{B}^{{(t)}^{2}}}\frac{{B}_{f}}{{(b}_{i}+\beta {\rho }_{i}^{2}){(b}_{j}+\beta {\rho }_{j}^{2})} \forall i\in {S}_{f}, \quad \forall j\in {S}_{f}, i\ne j, \quad \forall t\in T$$

(a12)

$${P}_{{f}_{ii}}^{\left(t\right)}=\frac{1}{2}(\frac{{B}_{f}}{{B}^{{(t)}^{2}}{{(b}_{i}+\beta {\rho }_{i}^{2})}^{2}}-\frac{1}{{b}_{i}+\beta {\rho }_{i}^{2}}) \quad \forall i\in {S}_{f}, \forall t\in T$$

(a13)

$${P}_{{f}_{ij}}^{\left(t\right)}=\frac{1}{2{B}^{{(t)}^{2}}}\frac{{B}_{f}-{B}^{(t)}}{{(b}_{i}+\beta {\rho }_{i}^{2}){(b}_{j}+\beta {\rho }_{j}^{2})} \forall i\in {S}_{f}, \quad \forall j\in {S}_{\widehat{f}},\forall t\in T$$

(a14)

$${P}_{{f}_{ij}}^{\left(t\right)}=\frac{1}{2{B}^{{(t)}^{2}}}\frac{{B}_{f}+{B}^{(t)}}{{(b}_{i}+\beta {\rho }_{i}^{2}){(b}_{j}+\beta {\rho }_{j}^{2})} \quad \forall i\in {S}_{\widehat{f}},\forall j\in {S}_{f},\forall t\in T$$

(a15)

$${P}_{{f}_{ij}}^{\left(t\right)}=\frac{1}{2{B}^{{(t)}^{2}}}\frac{{B}_{f}}{{(b}_{i}+\beta {\rho }_{i}^{2}){(b}_{j}+\beta {\rho }_{j}^{2})} \quad \forall i\in {S}_{\widehat{f}},\forall j\in {S}_{\widehat{f}},\forall t\in T$$

(a16)

$${R}_{{f}_{i}}^{\left(t\right)}=\frac{1}{{B}^{\left(t\right)}\left({b}_{i}+\beta {\rho }_{i}^{2}\right)}\left(\left({a}_{i}+{\gamma }^{0}{\rho }_{i}\right){B}^{\left(t\right)}-{C}_{f}\right) \quad \forall i\in {S}_{f}, \forall t\in T$$

(a17)

$${R}_{{f}_{i}}^{\left(t\right)}=\frac{-{C}_{f}}{{B}^{\left(t\right)}\left({b}_{i}+\beta {\rho }_{i}^{2}\right)} \forall i\in {S}_{\widehat{f}}, {R}_{{f}_{i}}^{{{\prime}}\left(t\right)}=\frac{{B}_{f}}{{B}^{{\left(t\right)}^{2}}\left({b}_{i}+\beta {\rho }_{i}^{2}\right)} \quad \forall i\in S , \forall t\in T$$

(a18)

$${S}_{f}^{{{\prime}}\left(t\right)}=\frac{-{C}_{f}}{{B}^{\left(t\right)}}. {S}_{f}^{{{\prime}}{^{\prime}}\left(t\right)}=\frac{{B}_{f}}{2{B}^{{\left(t\right)}^{2}}} , \quad \forall t\in T$$

(a19)

$${B}_{f}=\sum_{i\in {S}_{f}}\frac{1}{{b}_{i}+\beta {\rho }_{i}^{2}}, {C}_{f}=\sum_{i\in {S}_{f}}\frac{{a}_{i}+{\gamma }^{0}{\rho }_{i}}{{b}_{i}+\beta {\rho }_{i}^{2}}$$

(a20)

where subscript f indicates firm f and subscript $\widehat{f}$ indicates all firms except firm f.

s.t.: (Eq. (7))

$${{\varvec{V}}}^{(t)}{{\varvec{P}}}_{D}^{\left(t\right)}+{{\varvec{U}}}^{(t)}\left({\alpha }^{\left(t\right)}+{\mu }^{\left(t\right)}+{\epsilon }^{\left(t\right)}-{\epsilon }^{\left(t+1\right)}\right)\le {{\varvec{P}}}_{S}^{max}\perp {\mu }^{{\left(t\right)}^{max}}\ge 0, \quad \forall t\in T$$

(a21)

$${{\varvec{V}}}^{(t)}{{\varvec{P}}}_{D}^{\left(t\right)}+{{\varvec{U}}}^{(t)}\left({\alpha }^{\left(t\right)}+{\mu }^{\left(t\right)}+{\epsilon }^{\left(t\right)}-{\epsilon }^{\left(t+1\right)}\right)\ge {{\varvec{P}}}_{S}^{min}\perp {\mu }^{{\left(t\right)}^{min}}\ge 0, \quad \forall t\in T$$

(a22)

$${{\varvec{V}}}^{(t)}{{\varvec{P}}}_{D}^{\left(t\right)}-{{\varvec{V}}}^{\left(t-1\right)}{{\varvec{P}}}_{D}^{\left(t-1\right)}+{{\varvec{U}}}^{\left(t\right)}\left({\alpha }^{\left(t\right)}+{\mu }^{\left(t\right)}+{\epsilon }^{\left(t\right)}-{\epsilon }^{\left(t+1\right)}\right)-{{\varvec{U}}}^{\left(t-1\right)}\left({\alpha }^{\left(t-1\right)}+{\mu }^{\left(t-1\right)}+{\epsilon }^{\left(t-1\right)}-{\epsilon }^{\left(t\right)}\right)\le {U}_{rr}{{\varvec{P}}}_{S}^{max}\perp {\epsilon }^{{\left(t\right)}^{max}}\ge 0, \quad \forall t\in T-\{initial\,time\}$$

(a23)

$$-{{\varvec{V}}}^{\left(t\right)}{{\varvec{P}}}_{D}^{\left(t\right)}+{{\varvec{V}}}^{\left(t-1\right)}{{\varvec{P}}}_{D}^{\left(t-1\right)}-{{\varvec{U}}}^{\left(t\right)}\left({\alpha }^{\left(t\right)}+{\mu }^{\left(t\right)}+{\epsilon }^{\left(t\right)}-{\epsilon }^{\left(t+1\right)}\right)+{{\varvec{U}}}^{\left(t-1\right)}\left({\alpha }^{\left(t-1\right)}+{\mu }^{\left(t-1\right)}+{\epsilon }^{\left(t-1\right)}-{\epsilon }^{\left(t\right)}\right)\le {D}_{rr}{{\varvec{P}}}_{S}^{max}\perp {\epsilon }^{{\left(t\right)}^{min}}\ge 0, \quad \forall t\in T-\{initial\,time\}$$

(a24)

$${\varvec{V}}^{\left( t \right)} {\varvec{P}}_{D}^{\left( t \right)} + {\varvec{U}}^{\left( t \right)} \left( {\alpha^{\left( t \right)} + \mu^{\left( t \right)} +^{\left( t \right)} -^{{\left( {t + 1} \right)}} } \right) - {\varvec{P}}_{S}^{\left( 0 \right)} \le U_{rr} {\varvec{P}}_{S}^{max} \bot^{{\left( t \right)^{max} }} \ge 0, \quad \forall t \in \left\{ {initial\,time} \right\}$$

(a25)

$$- {\varvec{V}}^{\left( t \right)} {\varvec{P}}_{D}^{\left( t \right)} - {\varvec{U}}^{\left( t \right)} \left( {\alpha^{\left( t \right)} + \mu^{\left( t \right)} +^{\left( t \right)} -^{{\left( {t + 1} \right)}} } \right) + {\varvec{P}}_{S}^{\left( 0 \right)} \le D_{rr} {\varvec{P}}_{S}^{max} \bot^{{\left( t \right)^{min} }} \ge 0, \quad \forall t \in \left\{ {initial\,time} \right\}$$

(a26)

where subscript f indicates firm f, $\mu^{\left( t \right)} = \mu^{{\left( t \right)^{max} }} - \mu^{{\left( t \right)^{min} }}$, $\epsilon^{\left( t \right)} =^{{\left( t \right)^{max} }} -^{{\left( t \right)^{min} }}$.

Equations (a18) and (a19) model maximum and minimum generation constraints, respectively. Equations (a22) and (a23) model ramp-up and ramp-down limits for initial hour, and (a20) and (a21) model ramp-up and ramp-down limits for other hours, respectively.