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Table 7 Mathematical formulation of the Swissmod model

From: The future of Swiss hydropower: how to distribute the risk and the profits?

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)