Table 7 Mathematical formulation of the Swissmod model

Objective

\(\underset{{G}_{\mathrm{cp},t}}{\mathrm{min}}\sum\nolimits_{{\mathrm{cp}}.t}{{\mathrm{mc}}}_{{\mathrm{cp}}}{G}_{{\mathrm{cp}},t}\)

(1)

(Nodal) demand—supply balance

\({d}_{n,t}= \sum_{{{\rm cp}}\in n}{G}_{{{\rm cp}},t}+\sum_{{\rm hp}\in {n}^{{\rm CH}}}\left({\eta }_{{\rm hp}}^{{\rm Turb}}{{\rm Turb}}_{{\rm hp},t}^{{\rm CH}}- {{\eta }_{{\rm hp}}^{{\rm Pump}}{{\rm Pump}}}_{{\rm hp},t}^{{\rm CH}}\right)+\sum_{{\rm hp}\in {n}^{{\rm EU}}}\left({{\rm Turb}}_{{\rm hp},t}^{{\rm EU}}- {{\rm Pump}}_{{\rm hp},t}^{{\rm EU}}\right)+ {\mathrm{res}}_{n,t}- {\mathrm{Cur}}_{n,t}- \sum_{l}{\mathrm{inc}}_{n,l}{F}_{l,t}\)

\(\forall n,t\)

(2)

Capacity limit

\({\mathrm{chp}}_{{{\rm cp}},t}\le {G}_{{{\rm cp}},t}\le {g}_{{{\rm cp}}}^{{\rm max}}{{\rm avail}}_{{{\rm cp}},t}\)

\(\forall {\mathrm{cp}},t\)

(3)

Curtailment limit

\({\mathrm{Cur}}_{n,t}\le {\mathrm{res}}_{n,t}\)

\(\forall n,t\)

(4)

Hydro CH:

Turbine capacity limit

\({\eta }_{\mathrm{hp}}^{{\rm Turb}}{\mathrm{Turb}}_{{\rm hp},t}^{{\rm CH}}\le {\mathrm{turb}}_{{\rm hp}}^{\mathrm{max}}\)

\(\forall \mathrm{hp}\in \mathrm{CH},t\)

(5)

Pump capacity limit

\({{\eta }_{{\rm hp}}^{{\rm Pump}}{\rm Pump}}_{{\rm hp},t}^{{\rm CH}}\le {\mathrm{pump}}_{{\rm hp}}^{{\rm max}}\)

\(\forall \mathrm{hp}\in \mathrm{CH},t\)

(6)

Storage balance

\({S}_{{{\rm wn}},t}= {S}_{{{\rm wn}},t-1}+ {\mathrm{WI}}_{{{\rm wn}},t}- {\mathrm{WO}}_{{{\rm wn}},t}\)

\(\forall {\mathrm{wn}}\in \mathrm{CH},t\)

(7)

Water inflow

\({\mathrm{WI}}_{{{\rm wn}},t}={inj}_{{{\rm wn}},t}+\sum_{{\rm hp}\in \overline{{\mathrm{wn}}}}{\mathrm{Turb}}_{{\rm hp},t}^{\mathrm{CH}}+\sum_{{\rm hp}\in \underline{{\mathrm{wn}}}}{\mathrm{Pump}}_{{\rm hp},t}^{{\rm CH}}+ \sum_{{\rm hp}\in \overline{{\mathrm{wn}}}}{\mathrm{Spill}}_{{\rm hp},t}+\sum_{\overline{{\mathrm{wn}}}\in {\mathrm{wn}}}{\mathrm{Transfer}}_{\overline{{\mathrm{wn}}},{\mathrm{wn}},t}\)

\(\forall {\mathrm{wn}}\in \mathrm{CH},t\)

(8)

Water outflow

\({\mathrm{WO}}_{{\mathrm{wn}},t}=\sum_{\mathrm{hp}\in \underline{{\mathrm{wn}}}}{\mathrm{Turb}}_{\mathrm{hp},t}^{\mathrm{CH}} +\sum_{hp\in \overline{{\mathrm{wn}}}}{\mathrm{Pump}}_{\mathrm{hp},t}^{\mathrm{CH}}+\sum_{hp\in \underline{{\mathrm{wn}}}}{Spill}_{\mathrm{hp},t}+\sum_{\underline{{\mathrm{wn}}}\in {\mathrm{wn}}}{Transfer}_{{\mathrm{wn}},\underline{{\mathrm{wn}}},t}\)

\(\forall {\mathrm{wn}}\in \mathrm{CH},t\)

(9)

Storage limit

\({S}_{{\mathrm{wn}},t}\le {s}_{{\mathrm{wn}}}^{\mathrm{max}}\)

\(\forall {\mathrm{wn}}\in \mathrm{CH},t\)

(10)

Hydro EU (i.e., neighbors):

Turbine capacity limit

\({\mathrm{Turb}}_{\mathrm{hp},t}^{\mathrm{EU}}\le {turb}_{\mathrm{hp}}^{\mathrm{max}}\)

\(\forall \mathrm{hp}\in \mathrm{EU},t\)

(11)

Pump capacity limit

\({\mathrm{Pump}}_{\mathrm{hp},t}^{\mathrm{EU}}\le {\mathrm{pump}}_{\mathrm{hp}}^{\mathrm{max}}\)

\(\forall \mathrm{hp}\in \mathrm{EU},t\)

(12)

Storage level

\({S}_{\mathrm{co},t}= {S}_{\mathrm{co},t-1}+ \sum_{\mathrm{hp}\in \mathrm{co}}\left({{\eta }_{\mathrm{hp}}^{rt} \mathrm{Pump}}_{\mathrm{hp},t}^{\mathrm{EU}}- {\mathrm{Turb}}_{\mathrm{hp},t}^{\mathrm{EU}}\right)\)

\(\forall \mathrm{co}\in \mathrm{EU},t\)

(13)

Storage limit

\({S}_{\mathrm{co},t}\le {s}_{\mathrm{co}}^{\mathrm{max}}\)

\(\forall \mathrm{co}\in \mathrm{EU},t\)

(14)

Run-of-river generation profile

\(\sum_{\mathrm{hp}\in \mathrm{co}}{\mathrm{Turb}}_{\mathrm{hp},t}^{\mathrm{EU}} \le {\mathrm{turb}}_{\mathrm{co},t}^{\mathrm{profile}}\)

\(\forall \mathrm{co}\in \mathrm{EU},t\)

(15)

Dam’s annual generation limit

\(\sum_{t,\mathrm{hp}\in \mathrm{co}}{\mathrm{Turb}}_{\mathrm{hp},t}^{\mathrm{EU}} \le {\mathrm{turb}}_{\mathrm{co}}^{\mathrm{year}}\)

\(\forall \mathrm{co}\in \mathrm{EU}\)

(16)

Electricity grid:

Line flow

\({F}_{l,t}= {\mathrm{susceptance}}_{l} \sum_{n}{\mathrm{inc}}_{n,l}{X}_{n,t}\mathrm{voltbase}\)

\(\forall l,t\)

(17)

Slack bus

\({X}_{n=1,t}=0\)

\(\forall t\)

(18)

Line limits

\(\left|{F}_{l,t}\right|\le {f}_{l}^{\mathrm{max}}(1-\mathrm{securitymargin})\)

\(\forall l,t\)

(19)