Abstract
In this paper, a robust optimization approach is proposed to determine selling electricity price for a retailer who procures its obligated energy through two different resources: (1) wholesale market and (2) selfgeneration facilities. Regarding the selfgeneration facilities, two different kinds of distributed resources, including gas turbine units (GT) and rooftop photovoltaic sites (RPV) with considering energy storage systems (ESS), are addressed as the deterministic and intermittent power resources. Considering the wholesale market, the retailer can procure some parts of its obligated energy through bilateral contracts and dayahead market. To overcome the uncertainties associated with power output forecasting of solar sites, a new statistical approach is used considering the dependency of power output to the weather issues, such as irradiation, temperature and wind speed. The problem is formulated by using a robust mixedinteger quadratic program considering a confidence bound for the wholesale electricity price uncertainty. To determine the optimal selling price, a successive algorithm is developed through two iterative optimizations, including inner and outer iterative procedures. Regarding the outer optimization, the confidence bound of wholesale electricity price is portioned into subintervals to evaluate the impacts of each robust subregion of wholesale price on the offered retail selling price. Through the inner optimization, the consumers’ response to the offered price is evaluated using a complete demand function model. Finally, a case study containing the bilateral contracts, wholesale market, RPV units, GT units, ESS, flexible demands and the retailer providing demand response is considered to demonstrate the proficiency of the proposed approach.
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Abbreviations
 t :

Index of time (1, ..., \({N}_\mathrm{T})\)
 i :

Index of RPV sites (1, ..., \({N}_\mathrm{S})\)
 j :

Index of gas turbine units (1, ..., \({N}_\mathrm{GT})\)
 k :

Index of bilateral contracts (1, ..., \({N}_\mathrm{C})\)
 q :

Index of subintervals of uncertainty set (1, ..., Q)
 z :

Index of iterations for demand response (1, ..., Z)
 l :

Index of energy storage system (1, ..., \({N}_\mathrm{ES})\)
 \(\lambda _{t}^\mathrm{D}\) :

Selling electricity price ($/MWh)
 \(\lambda _{t}^\mathrm{M} \) :

Electricity price of wholesale market ($/MWh)
 \({P}_{t}^\mathrm{D} \) :

Demand level at time t (MW)
 \({P}_{t}^\mathrm{M} \) :

Traded power in wholesale market (MW)
 \({P}_{{t},{k}}^\mathrm{C} \) :

Purchased power from bilateral contract k(MW)
 \({P}_{{t},{i}}^\mathrm{S} \) :

Total forecasted power of all RPV sites (MW)
 \({P}_{{t},{j}}^{\mathrm{GT}} \) :

Generation power of gas turbine j (MW)
 \({v}_{j}({t})\) :

Binary variable of GT units, 1/0 the unit is on/off
 \({C}_{j}^\mathrm{SU}\) :

Startup cost of GT units ($)
 \({C}_{j}^\mathrm{SD}\) :

Shutdown cost of GT units ($)
 \({\lambda }_{{t,k}}^\mathrm{C}\) :

Electricity price of bilateral contract k ($/MWh)
 \({E}_{{l,t}}^{\mathrm{ESS}}\) :

Total stored energy at ESS l and hour t (MWh)
 \({P}_{\mathrm{ESS}}^{\mathrm{ch}}\) :

Charging power of energy storage system (MW)
 \({P}_{\mathrm{ESS}}^{\mathrm{dis}} \) :

Discharging power of energy storage system (MW)
 \({\xi }_{t},\,\beta \) :

Dual variables in the duality theorem
 \(\mathrm{DR}_{t}\) :

Potential of demand response at time t
 \(\gamma ^{q}\) :

Incremental rate of wholesale market price ($/MWh)
 \({C}_{i}^{\mathrm{OM}} \) :

Operation, maintenance cost of RPV sites ($/MWh)
 \({C}_{i}^{\mathrm{CI}} \) :

Capital investment cost of RPV sites ($/h)
 \(\alpha _{j},\beta _{j},\gamma _{j}\) :

Coefficients for cost function of GT units
 \(\overline{{P}_{{t,j}}^{\mathrm{GT}} } \) :

Maximum generation capacity of GT units (MW)
 \({{{P}}_{t,j}^\mathrm{GT} }\) :

Minimum generation capacity of GT units (MW)
 \(\overline{{S}_{j}^\mathrm{U}} \) :

Cost derived from startup of GT units ($)
 \({S}_{j}^\mathrm{D} \) :

Cost derived from shutdown of GT units ($)
 \({R}_{j}^\mathrm{U} \) :

Maximum rampup of GT unit j (MW/h)
 \({R}_{j}^\mathrm{D} \) :

Maximum rampdown of GT unit j (MW/h)
 \(\lambda _{{t,k}}^\mathrm{C,R} \) :

Reference price of bilateral contract k ($/MWh)
 \(\eta _\mathrm{ESS}^\mathrm{ch} \) :

Charging efficiency of energy storage system
 \(\eta _\mathrm{ESS}^\mathrm{dis} \) :

Discharging efficiency of energy storage system
 \({P}_\mathrm{ESS}^\mathrm{nom}\) :

Nominal power of energy storage system
 \({E}_\mathrm{ESS}^\mathrm{nom}\) :

Nominal capacity of energy storage system
 \({\varGamma }\) :

Robustness level
 \(\varepsilon _{X}\) :

Elasticity factor of demand function X
 \(\eta \) :

Income factor of the retailer
 \(\zeta \) :

Stopping criteria of the inner iterative procedure
 N, M :

Number of RPV sites including: N as primary and M as reduced numbers
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Golmohamadi, H., Keypour, R. Application of Robust Optimization Approach to Determine Optimal Retail Electricity Price in Presence of Intermittent and Conventional Distributed Generation Considering Demand Response. J Control Autom Electr Syst 28, 664–678 (2017). https://doi.org/10.1007/s4031301703289
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DOI: https://doi.org/10.1007/s4031301703289