Stochastic Market Clearing With Revenue Sufficiency Constraints

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10th International Conference on Applied Energy (ICAE2018), 22-25 August 2018, Hong Kong, 10th International Conference on Applied Energy China(ICAE2018), 22-25 August 2018, Hong Kong, China

Stochastic Market Clearing With Revenue Sufficiency Constraints 15th International on DistrictSufficiency Heating and Cooling Stochastic The Market ClearingSymposium With Revenue Constraints Leena Heistreneaa, Poonam Mishraaa, Makarand Lokhandebb Leenathe Heistrene , Poonam Makarand Lokhande Assessing feasibility of Mishra using ,the heat demand-outdoor a Pandit Deendayal Petroleum University, Gandhinagar, Gujarat – 382007, India a Visvesvaraya NationalPetroleum Institute of Technology, Nagpur, Maharashtra – 440010, India Pandit Deendayal University, Gandhinagar, Gujarat – 382007, India b Visvesvaraya National Institute of Technology, Nagpur, Maharashtra – 440010, India

temperature function for a long-term district heat demand forecast b

I. Andrića,b,c*, A. Pinaa, P. Ferrãoa, J. Fournierb., B. Lacarrièrec, O. Le Correc

Abstract a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal Abstract b Veoliaproblem Recherche Innovation, 291 built Avenue Daniel, 78520 France Revenue insufficiency is a major in & market designs onDreyfous unit commitment andLimay, marginal pricing based remuneration c Département Systèmes Énergétiques et Environnement IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France Revenue insufficiency is amechanisms major problem in market designs built on have unit commitment marginal pricing based remuneration schemes. Varied financial and market clearing procedures been adoptedand by different electricity markets to address schemes. Varied mechanisms and market clearing procedures have been adopted by marketsknowledge, to address this problem. Butfinancial these schemes are based on deterministic market clearing algorithms. Todifferent the bestelectricity of the authors’ this thesedone schemes are based on deterministic market clearing algorithms. the bestinofthe themanner authors’ knowledge, thereproblem. has beenBut no work in ensuring revenue sufficiency in stochastic market clearing To algorithms adopted in the there has been work the doneincreasing in ensuring revenue sufficiency in stochastic clearing algorithms in the manner in the proposed work.noWith penetration of renewable sources market of energy, it has become imperative to adopted adopt market Abstract proposed work. Withthat the consider increasing of renewable renewable sources sourcesofofenergy energy, it has imperative to adopt market equilibrium solutions thepenetration dynamism of along withbecome a guarantee of revenue sufficiency. equilibrium solutionsa that consider the dynamism of renewable sources of energy along with aformulation guarantee of revenue includes sufficiency. This paperheating presents novel stochastic market clearing wherein the of mathematical explicitly District networks are commonly addressed inprocedure the literature as one the most effective solutions for decreasingthe the This paperpricing presents a novel market clearing procedure wherein mathematical formulation explicitly includes the marginal scheme intostochastic its modeling features. Recourse approach hasthe been adopted to include the uncertainties associated greenhouse gas emissions from the building sector. These systems require high investments which are returned through the heat marginal pricing schemeofinto its modeling features. Recoursestrategy. approachThe hasentire beenproblem adopted has to include the uncertainties associated with renewable the climate system into the market been formulated as acould mixed integer sales. Due to sources the changed conditions andclearing building renovation policies, heat demand in the future decrease, with renewable sources of theproblem system into the market pricing clearingbeing strategy. Theofentire problem has been variables. formulatedDual as a mixed integer linear programming (MILP) with marginal a part the unknown decision variables of a prolonging the investment return period. linear programming (MILP) problem marginal pricing being a part ofrevenue the unknown decision variables. Dualclearing variables of a security constrained economic dispatchwith problem is used for implementing constrained stochastic market model. The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand security constrained economic dispatch is used for implementing revenue constrained stochastic market clearing model. The proposed validated using aproblem threeinbus network with wind wind forecast presented as scenarios forecast. Themodel districtis of Alvalade, located Lisbon (Portugal), waspenetration used as athat casehas study. The district is consisted of 665 The proposed modelofisfour validated using a three bus network with wind penetration that has wind forecast presented as scenarios over a time that horizon hours. buildings vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district over a time horizon of four hours. renovation scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were Copyright ©with 2018 Elsevier All rights reserved. compared results fromLtd. a dynamic heat demand model, previously developed and validated by the authors. © 2019 The Published Elsevier Ltd. th International Conference on Applied Energy Copyright ©Authors. 2018 Elsevier Ltd. by All rights reserved. Selection and peer-review under responsibility of the scientific committee of theof 10error Theisresults showed when onlythe weather change is license considered, the margin could be acceptable for some applications This an open accessthat article under CC BY-NC-ND (http://creativecommons.org/licenses/by-nc-nd/4.0/) th International Conference on Applied Energy Selection and peer-review under responsibility of the scientific committee of the 10 (ICAE2018). Peer-review responsibility of lower the scientific committee of ICAE2018 – Theconsidered). 10th International Conference on Appliedrenovation Energy. (the error inunder annual demand was than 20% for all weather scenarios However, after introducing (ICAE2018). scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). Keywords: Revenue sufficiency constraints; recourse approach; stochastic market clearing; pool-based electricity market. The value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the Keywords: Revenue sufficiency constraints; recourse approach; stochastic market clearing; pool-based electricity market. decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the 1.coupled Introduction scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and 1.improve Introduction the accuracy of heat demand estimations.

Electricity markets have, for long, struggled with the problem of revenue insufficiency. US based electricity markets markets have, for long, struggled withthe theuplift problem revenue US based electricitywhile, markets have implemented financial mechanisms to settle costofoutside theinsufficiency. market equilibrium conditions on © Electricity 2017 The Authors. Published by Elsevier Ltd. have implemented financial mechanisms to settle uplift cost15th outside the market equilibrium conditions on the other hand, European electricity markets do sothe through pool based market clearing procedures [1] – [2].while, Out-ofPeer-review under responsibility of the Scientific Committee of The International Symposium on District Heating and the other hand, European do so through pool based market clearing procedures economic [1] – [2]. signals. Out-ofCooling. market financial settlementelectricity strategiesmarkets suffer from drawbacks of price discrimination and inadequate market settlement strategies suffer drawbacks price discrimination andbetter inadequate signals. One canfinancial argue that pricing signals based on from market clearing of procedures are, therefore, suited.economic But the problem Keywords: Heat demand; Forecast; Climate changeon market clearing procedures are, therefore, better suited. But the problem One can argue that pricing signals based

1876-6102 Copyright © 2018 Elsevier Ltd. All rights reserved. Selection peer-review under responsibility the scientific 1876-6102and Copyright © 2018 Elsevier Ltd. All of rights reserved. committee of the 10th International Conference on Applied Energy (ICAE2018). Selection and peer-review under responsibility of the scientific committee of the 10th International Conference on Applied Energy (ICAE2018). 1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling. 1876-6102 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of ICAE2018 – The 10th International Conference on Applied Energy. 10.1016/j.egypro.2019.01.863



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with such pricing mechanisms derived from market is its complexity [3]. Over and above, the presence of dynamic sources of energy such as wind, solar, etc. leads to further increase in complexity owing to the need of introducing stochasticity into the deterministic market clearing algorithm. Nomenclature Indices and sets 𝑖𝑖 Index of generators 𝑡𝑡 Index of time 𝜔𝜔 Index of scenarios ∇ Set of lines connecting to a bus 𝑛𝑛 Variables 𝑃𝑃𝑖𝑖,𝑡𝑡 Scheduled conventional generation from generator 𝑖𝑖 in day-ahead market 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 Reserve generation from generator 𝑖𝑖 under scenario 𝜔𝜔 𝑃𝑃𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 Scheduled wind energy in the day-ahead market 𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟)Power flow from bus 𝑛𝑛 to bus 𝑟𝑟 𝛿𝛿𝑛𝑛:𝑡𝑡,𝑤𝑤 Load angle at bus 𝑛𝑛 under scenario 𝜔𝜔 in time 𝑡𝑡 𝑌𝑌 Dual variables representing components of marginal pricing 𝑆𝑆𝑆𝑆𝑖𝑖,𝑡𝑡 Start-up cost of generator 𝑖𝑖 in time 𝑡𝑡 𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 Up reserve offered by generator 𝑖𝑖 in time 𝑡𝑡 𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 Down reserve offered by generator 𝑖𝑖 in time 𝑡𝑡 𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 Non-spinning reserve offered by generator 𝑖𝑖 in time 𝑡𝑡 𝑆𝑆𝑆𝑆 𝐴𝐴 𝑖𝑖,𝑡𝑡,𝑤𝑤 Start-up cost offered by generator 𝑖𝑖 in time 𝑡𝑡 under scenario 𝜔𝜔 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝑤𝑤 Additional generation to be procured from generator 𝑖𝑖 in real time 𝑡𝑡 under scenario 𝜔𝜔 𝑢𝑢𝑖𝑖,𝑡𝑡 /𝑣𝑣𝑖𝑖,𝑡𝑡,𝜔𝜔 Deterministic/stochastic binary variables representing commitment of generator 𝑖𝑖 in time 𝑡𝑡 in scenario 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑅𝑅𝑏𝑏𝑏𝑏𝑏𝑏,𝑡𝑡,𝜔𝜔 Revenue sufficiency signal Constants 𝑁𝑁𝑔𝑔 Total number of generators/players in the electricity market 𝑁𝑁𝑡𝑡 Complete time horizon under consideration 𝑁𝑁𝜔𝜔 Total number of scenarios in the forecast model 𝜆𝜆𝑖𝑖 Generation bid offered by producer/generator 𝑖𝑖 𝑃𝑃𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 Point forecast of hourly load demand 𝑃𝑃𝜔𝜔𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 Probabilistic forecast of wind power in scenario form 𝑋𝑋(𝑛𝑛, 𝑟𝑟) Reactance of line connected between bus 𝑛𝑛 and bus 𝑟𝑟 𝐿𝐿𝐿𝐿𝐿𝐿𝑏𝑏𝑏𝑏𝑏𝑏 Locational marginal price obtained from the dual formation economic dispatch problem formulation 𝜆𝜆𝑢𝑢 /𝜆𝜆𝑑𝑑 Up/down reserve bid offered by producer/generator 𝑖𝑖 𝜆𝜆𝑛𝑛𝑛𝑛 Non spinning reserve bid offered by producer/generator 𝑖𝑖 𝑃𝑃𝑃𝑃𝜔𝜔 Probability of scenario 𝜔𝜔 𝑃𝑃𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 Minimum generation limit of generator 𝑖𝑖 𝑃𝑃𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 Maximum generation limit of generator 𝑖𝑖 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤,𝑚𝑚𝑚𝑚𝑚𝑚 Minimum generation limit of wind source 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤,𝑚𝑚𝑚𝑚𝑚𝑚 Minimum generation limit of wind source 𝑚𝑚𝑚𝑚𝑚𝑚 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 Maximum up reserve generation limit of of generator 𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 Maximum down reserve generation limit of generator 𝑖𝑖 𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 Maximum non-spinning reserve generation limit of of generator 𝑖𝑖 𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚 (𝑛𝑛, 𝑟𝑟)Minimum power flow from bus 𝑛𝑛 to bus 𝑟𝑟 𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚 (𝑛𝑛, 𝑟𝑟)Maximum power flow from bus 𝑛𝑛 to bus 𝑟𝑟

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Work carried out till date is, either pertaining to deterministic market clearing procedures with revenue sufficiency feature [3]-[4], or related to stochastic market clearing with marginal pricing signals but with no revenue sufficiency [5]. This paper proposes a novel  stochastic market clearing algorithm  with locational marginal pricing signals  considering revenue sufficiency constraints  along with network constraints and other inter temporal constraints. The flow of the paper is as follows: Introduction in section 1 is followed by proposed mathematical formulation in section 2. An illustrative example is given in section 3 which includes results obtained from the proposed strategy. The paper is finally concluded with relevant discussion in section 4 followed by acknowledgement and references. 2. Mathematical Formulation of the Proposed Strategy Intensive research has been done on marginal pricing schemes based on which different market clearing strategies have been adopted by pool based electricity markets in different parts of the world. Based on these works, a locational pricing scheme has been proposed in sub-section 2.1 followed by stochastic optimization model for market clearing in sub-section 2.2. For simplicity, uncertainty pertains to wind energy alone and loads are considered to be inelastic. 2.1 Proposed marginal pricing scheme for initialization: Presence of binary variables stating the commitment status of generators make the optimization model of the mixed integer linear problem type, and hence non-convex in nature. Deriving dual variables of MILP problems is difficult and may even need special bounding strategies. In order to make it simpler, binary variables are considered only in the market clearing model whereas the pricing model is formed as linear programming problem after relaxing the binary variables. Primal problem of the pricing scheme is formed as stochastic formulation with load balance constraints, scenario based reserve constraints and network flow constraints. Mathematical formulation for the same is as shown below: 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡

𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 ∑𝑁𝑁𝑁𝑁 𝑡𝑡=1 ∑𝑖𝑖=1(𝜆𝜆𝑖𝑖 𝑃𝑃𝑖𝑖,𝑡𝑡 + ∑𝜔𝜔=1 𝜆𝜆𝑖𝑖 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝑤𝑤 )

𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ∑𝑁𝑁𝑁𝑁 = 𝑃𝑃𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 … … … … … … … … … … ∀𝑡𝑡 𝑖𝑖=1 𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑃𝑃𝑡𝑡

𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 ∑𝑁𝑁𝑁𝑁 − 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 … … … … ∀𝑡𝑡, ∀𝜔𝜔 𝑖𝑖=1(𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 ) = 𝑃𝑃𝑡𝑡

1(𝑎𝑎)

1(𝑏𝑏) 1(𝑐𝑐)

𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟) = 1⁄𝑋𝑋(𝑛𝑛, 𝑟𝑟) [𝛿𝛿𝑛𝑛:𝑡𝑡,𝑤𝑤 − 𝛿𝛿𝑟𝑟:𝑡𝑡,𝑤𝑤 ] … … … … ∀(𝑛𝑛, 𝑟𝑟) ∈ ∇, ∀𝑡𝑡, ∀𝜔𝜔

1(𝑑𝑑)

𝑃𝑃𝑖𝑖,𝑡𝑡 , 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 , 𝑃𝑃𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ≥ 0 &

1(𝑓𝑓)

∇ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑁𝑁𝑁𝑁 ∑𝑁𝑁𝑁𝑁 − 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 … … ∀(𝑛𝑛, 𝑟𝑟) ∈ ∇, ∀𝑡𝑡, ∀𝜔𝜔 𝑖𝑖=1(𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 ) − ∑𝜔𝜔=1 ∑(𝑛𝑛,𝑟𝑟) 𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟) = 𝑃𝑃𝑡𝑡

𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟), 𝛿𝛿𝑛𝑛:𝑡𝑡,𝑤𝑤 𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢

1(𝑒𝑒)

Dual formation for the same would give the components of the marginal price for each bus in each time horizon 𝑡𝑡. 2(𝑎𝑎) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐 𝑇𝑇 𝑌𝑌 2(𝑏𝑏) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝐴𝐴𝑒𝑒𝑒𝑒 𝑌𝑌 = 𝑏𝑏𝑒𝑒𝑒𝑒 𝐴𝐴𝐴𝐴 ≤ 𝑏𝑏 2(𝑐𝑐) 𝑌𝑌 𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 2(𝑑𝑑) Objective function 𝑐𝑐 𝑇𝑇 and equality & inequality constraint matrices 𝐴𝐴𝑒𝑒𝑒𝑒 and 𝐴𝐴, seen in 2(𝑎𝑎) to 2(𝑑𝑑), are formed from the dual formulation of the primal problem mentioned in 1(𝑎𝑎) to 1(𝑓𝑓). Components of the decision variable, 𝑌𝑌, form the locational marginal price at each bus, 𝐿𝐿𝐿𝐿𝐿𝐿𝑏𝑏𝑏𝑏𝑏𝑏 .

2.2 Proposed market clearing strategy with revenue sufficiency constraint:



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Commitment variables have been relaxed in the pricing scheme mentioned in subsection 2.1, but they have been considered in the market clearing procedure, explained below, which includes both - integer and continuous variables. Secondly, in the case of a joint dispatch type market, both energy and reserve markets are cleared simultaneously in the day ahead market. Hence, simultaneous energy and reserve dispatch type of market clearing procedure has been proposed using recourse approach for stochastic formulation of the problem under consideration. Thus, the proposed stochastic model considers the scheduled generation of conventional and renewable sources of the energy market, up, down and non-spinning reserve scheduled in the reserve market and startup costs as here-and-now variables while the generation and startup costs requirements that shall be needed after the realization of wind in real time are treated as wait-and-see variables. Mathematical formulation for the same is as shown below: 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡

𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝐴𝐴 ∑𝑁𝑁𝑁𝑁 𝑡𝑡=1 ∑𝑖𝑖=1(𝑆𝑆𝑆𝑆𝑖𝑖,𝑡𝑡 + 𝜆𝜆𝑖𝑖 𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝜆𝜆𝑢𝑢 𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 + 𝜆𝜆𝑑𝑑 𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 + 𝜆𝜆𝑑𝑑 𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 + ∑𝜔𝜔=1 𝑃𝑃𝑃𝑃𝜔𝜔 (𝑆𝑆𝑆𝑆 𝑖𝑖,𝑡𝑡,𝑤𝑤 + 𝜆𝜆𝑖𝑖 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝑤𝑤 )) 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ∑𝑁𝑁𝑁𝑁 = 𝑃𝑃𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 … … … … … … … … … … ∀𝑡𝑡 𝑖𝑖=1 𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑃𝑃𝑡𝑡

𝑢𝑢𝑖𝑖,𝑡𝑡 𝑃𝑃𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑃𝑃𝑖𝑖,𝑡𝑡 ≤ 𝑢𝑢𝑖𝑖,𝑡𝑡 𝑃𝑃𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 … … … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡

3(𝑎𝑎)

3(𝑏𝑏) 3(𝑐𝑐)

𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤,𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝑃𝑃𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 ≤ 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤,𝑚𝑚𝑚𝑚𝑚𝑚 … … … … … … … ∀𝑡𝑡

3(𝑑𝑑)

𝑚𝑚𝑚𝑚𝑚𝑚 … … … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡 0 ≤ 𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 ≤ 𝑢𝑢𝑖𝑖,𝑡𝑡 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖

3(𝑓𝑓)

𝑚𝑚𝑚𝑚𝑚𝑚 … … … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡 0 ≤ 𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 ≤ 𝑢𝑢𝑖𝑖,𝑡𝑡 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖

0 ≤ 𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖,𝑡𝑡 ≤ (1 − 𝑢𝑢𝑖𝑖,𝑡𝑡 )𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 … … … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡

𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟) = 1⁄𝑋𝑋(𝑛𝑛, 𝑟𝑟) [𝛿𝛿𝑛𝑛:𝑡𝑡,𝑤𝑤 − 𝛿𝛿𝑟𝑟:𝑡𝑡,𝑤𝑤 ] … … … … ∀(𝑛𝑛, 𝑟𝑟) ∈ ∇, ∀𝑡𝑡, ∀𝜔𝜔

∇ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑁𝑁𝑁𝑁 ∑𝑁𝑁𝑁𝑁 − 𝑃𝑃𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 … … ∀(𝑛𝑛, 𝑟𝑟) ∈ ∇, ∀𝑡𝑡, ∀𝜔𝜔 𝑖𝑖=1(𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 ) − ∑𝜔𝜔=1 ∑(𝑛𝑛,𝑟𝑟) 𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟) = 𝑃𝑃𝑡𝑡

𝑣𝑣𝑖𝑖,𝑡𝑡,𝑤𝑤 𝑃𝑃𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 ≤ (𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 ) ≤ 𝑣𝑣𝑖𝑖,𝑡𝑡,𝑤𝑤 𝑃𝑃𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 … … … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡, ∀𝜔𝜔 𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚 (𝑛𝑛, 𝑟𝑟) ≤ 𝑓𝑓𝑓𝑓𝑡𝑡,𝑤𝑤 (𝑛𝑛, 𝑟𝑟) ≤ 𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚 (𝑛𝑛, 𝑟𝑟) … … … … … ∀(𝑛𝑛, 𝑟𝑟) ∈ ∇, , ∀𝑡𝑡, ∀𝜔𝜔

𝑆𝑆𝑆𝑆𝑖𝑖,𝑡𝑡 ≥ 𝜆𝜆𝜆𝜆𝜆𝜆𝑖𝑖,𝑡𝑡 (𝑢𝑢𝑖𝑖,𝑡𝑡 − 𝑢𝑢𝑖𝑖,𝑡𝑡−1 ) … … … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡

𝑆𝑆𝑆𝑆 𝐴𝐴 𝑖𝑖,𝑡𝑡 ≥ 𝜆𝜆𝜆𝜆𝜆𝜆𝑖𝑖,𝑡𝑡 (𝑣𝑣𝑖𝑖,𝑡𝑡,𝜔𝜔 − 𝑣𝑣𝑖𝑖,𝑡𝑡−1,𝜔𝜔 ) … … … … … … … ∀𝑖𝑖, ∀𝑡𝑡, ∀𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

(𝐿𝐿𝐿𝐿𝐿𝐿 − 𝜆𝜆𝑖𝑖 )(𝑃𝑃𝑖𝑖,𝑡𝑡 + 𝑟𝑟𝑟𝑟𝑖𝑖,𝑡𝑡,𝜔𝜔 ) ≥ 𝑅𝑅𝑏𝑏𝑏𝑏𝑏𝑏,𝑡𝑡,𝜔𝜔 … … … … ∀𝑖𝑖, ∀𝑡𝑡, ∀𝜔𝜔

3(𝑒𝑒)

3(𝑔𝑔)

3(ℎ) 3(𝑖𝑖)

3(𝑗𝑗)

3(𝑘𝑘)

3(𝑙𝑙)

3(𝑚𝑚) 3(𝑛𝑛)

Here 3(𝑎𝑎) represents the expected cost function. Generation limit constraints are represented by 3(𝑐𝑐), 3(𝑑𝑑) and 3(𝑗𝑗) while load balance constraints are given by 3(𝑏𝑏). Similarly up, down and non-spinning reserve limits are represented by constraints 3(𝑒𝑒), 3(𝑓𝑓) and 3(𝑔𝑔) respectively. Power flow/network constraints are represented by 3(ℎ), 3(𝑖𝑖) and 3(𝑘𝑘) while start up costs related constraints are expressed in 3(𝑙𝑙) and 3(𝑚𝑚). Revenue sufficiency constraint function is as expressed in 3(𝑛𝑛). This constraint ensures that 𝐿𝐿𝐿𝐿𝐿𝐿 signals obtained from the decision variables of the above mathematical formulation ensure revenue sufficiency to each producer without the market operator resorting to price uplift options. 3. Illustrative example and results A three bus network has been considered whose network layout is as shown in figure 1 [6]. Network details and other details regarding the system under consideration is as given in table 1 and table 2. Flow constraints are restricted to

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55 MW while maximum wind generation capacity is 60 MW and with a load of 130 MW, 80 MW, 110 MW and 140 MW in a time horizon of 4 consecutive hours. Results were obtained as shown in table 3 and table 4. Revenue sufficiency has been achieved in each individual producer case and the LMP details are as shown in table 5. Table 1. Generator data Generator i

1

2

3

Minimum generation

10

10

10

Maximum generation

100

100

50

Offer bid by producers/generators

30

40

20

Start up cost

100

100

100

Up reserve cost

5

7

8

Down reserve cost

5

7

8

Non-spinning reserve cost

4.5

5.5

7

Period t

As point forecast

Higher bound

Lower bound

1

6

9

2

2

20

30

13

3

35

50

25

4

8

12

6

Table 2. Wind forecast details

Table 3. Expected cost obtained for different circumstances Quantity

Without wind

Perfect forecast

Scenario forecast

Expected cost

12503

10349

10389

Table 4. Wind benefits Performance metrics

in %

Average benefit

20.34

Average uncertainty cost

3.85

Net benefit

16.49

Table 5. Locational marginal prices for each bus in different time period Period t

Bus 1

Bus 2

Bus 3

1

31.10

32.61

32.64

2

22.18

22.47

23.34

3

25.10

25.22

26.58

4

31.49

33.01

33.01

4. Conclusion This paper proposes a novel stochastic algorithm model that ensures revenue sufficiency to all the players involved in the electricity market without adopting out-of-market uplift settlement schemes. The recourse method, used for



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probabilistic mathematical formulation, ensures that market is cleared in the most efficient manner possible considering dynamism of renewable sources of energy. Expected costs obtained from stochastic solution are higher than the deterministic market clearing solutions with point forecasts. But this drawback is traded off by the advantage of obtaining the feasible solutions even when actual realization of wind is different from forecasted quantity. This is not true for point forecast based deterministic solutions which may give infeasible solution for different scenarios in real time. This work can be extended to bigger systems as well. Acknowledgement The authors would like to thank Mr. Heistrene K., Mrs. Laly Hesitrene, Mr. E. K. Paulose and Mrs. Saramma Paulose for their moral support and constant guidance.

References [1] C.Wang, P. Luh, P. Gribik, L. Zhang, and T. Peng, ‘‘A study of commitment cost in approximate extended locational marginal prices,’’ in Proc. IEEE Power & Energy Soc. General Meeting, Jul. 2012, pp. 1---7. [2] V. Araoz and K. J¨ornsten, ‘‘Semi-Lagrangian approach for price discovery in markets with non-convexities,’’ Eur. J. Oper. Res., vol. 214, no. 2, pp. 411---417, 2011. [3] R. Blanco, J. Arroyo, N. Alguacil, “On the solution of revenue and network constrained day ahead market clearing under marginal pricing – part I: An exact bilevel programming approach”, IEEE transactions on transactions on power systems, vol. 32, pp.208 -219, 2017. [4] R. Blanco, J. Arroyo, N. Alguacil, “On the solution of revenue and network constrained day ahead market clearing under marginal pricing – part II: An exact bilevel programming approach”, IEEE transactions on transactions on power systems, vol. 32, pp.220 -227, 2017. [5] F. Abbaspourtorbati, A. J. Conejo, J. Wang, R. Cherkaoui, “Pricing Electricity Through a Stochastic Non-Convex Market-Clearing Model”, IEEE transactions on transactions on power systems, vol. 32, pp.208 -219, 2017 [6] A. Conejo, M. Carrion and J. Morales, “Decision making under uncertainty in electricity markets”, Springer publication, 2010.

WP Line 1 G1 G2

Bus 1 Bus 2 Line 2

Line 3

Bus 3 L3

G3

Figure 1. Three bus network layout.