Determination of optimal reserve contribution of thermal units to afford the wind power uncertainty

  • Mostafa EsmaeeliEmail author
  • Sajjad Golshannavaz
  • Pierluigi Siano
Original Research


Due to unforeseen variations in wind speed profiles, wind farm integrations are recognized as intermittent and uncertain energy contributors. More specifically, integration of such renewable energy resources aligned with the conventional thermal units although reduces the emissions and brings about a clean environment, it introduces serious problems in assigning optimal and reliable level of these units in load supplying and spinning reserve provision. This situation is more intensified considering the uncertainties arisen by the power system loading demand. To facilitate such operational hurdles, the ongoing study puts forward an efficient model for assigning the optimal spinning reserve which accommodates the uncertainties in both the wind speed and load profiles. Stochastic behavior of these parameters is simulated by generating a proper number of scenarios through the Monte Carlo simulation (MCS) approach. Then, each of these scenarios is evaluated based on the established linear mixed integer approach in a deterministic fashion. Accordingly, a computationally efficient approach is obtained paving the way for real-world implementations and assuring the global optimum results. The proposed approach is applied to a 12-unit test system including 10 thermal units and 2 wind farms. Results are reflected in terms of the commitment status, energy dispatches, and reserve contributions of each committed unit. A comprehensive discussion is conducted to disclose the possible improvements.


Wind farm integrations Wind speed and load uncertainties Stochastic analysis Mixed-integer linear programming Increased wind energy deployment Emission reduction. 

List of symbols

Sets and indices


Index of generating units.


Index and set of time intervals


Index of scenarios


Index of load points


Index of wind farms

Constants and parameters

\(P_{d}^{{}}\left( t \right)\)

Active power demand at load point d at time t

\({v_i}\left( t \right)\)

Wind speed at wind farm i at time t


Coefficient of the piecewise linear production cost function of unit g


Coefficients of the quadratic production cost function of unit g

\({\alpha _g},{\beta _g},{\gamma _g}\)

Coefficients of the quadratic emission function of unit g


Slope of block l of the piecewise linear production cost function of unit g


Number of segments in piecewise linearization approach

\(OFC\left( g \right)\)

Operation and maintenance fixed cost of thermal unit g

\(OFC\left( w \right)\)

Operation and maintenance fixed cost of wind farm w

\(OVC\left( g \right)\)

Operation and maintenance variable cost of thermal unit g

\(OVC\left( w \right)\)

Operation and maintenance variable cost of wind farm w

\(P_{G}^{{\hbox{max} }}\left( g \right)\)

Maximum power generating capacity of thermal unit g

\(P_{G}^{{\hbox{min} }}\left( g \right)\)

Minimum power generating limit of thermal unit g


Probability of scenario s


A fraction of total wind power considered as the reserve requirement due to wind power prediction errors


Upper limit in each segment of the linearized cost function of thermal unit g

\(n\left( t \right)\)

Number of hours at time interval t


\(P_{{GD}}^{S}\left( {g,t} \right)\)

Load contribution of thermal unit g at time t in scenario s

\(P_{{GR}}^{S}\left( {g,t} \right)\)

Reserve contribution of thermal unit g at time t in scenario s

\(P_{R}^{S}\left( t \right)\)

Fraction of total system load as the reserve requirement at time t in scenario s

\(P_{W}^{S}\left( {w,t} \right)\)

Generation of wind farm w at time t in scenario s

\({\rho ^S}\)

Probability of scenario s


Decision variables of unit i at time t in scenario s. (on = 1, off = 0)


State variables of unit i at time t in scenario s

\({\xi ^S}\)

Vector of scenario s



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Computer and Industrial EngineeringBirjand University of TechnologyBirjandIran
  2. 2.Electrical Engineering DepartmentUrmia UniversityUrmiaIran
  3. 3.Department of Management and Innovation SystemsUniversity of SalernoFiscianoItaly

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