# Extracting Appropriate Nodal Marginal Prices for All Types of Committed Reserve

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## Abstract

This paper proposes a framework to extract appropriate locational marginal prices for each type of reserve (up-/down-going reserves at both generation- and demand-sides). The proposed reserve pricing scheme accounts for the lost opportunity of selling the convertible products (energy and reserve). The fair prices can be obtained for capacity reserves applying this framework, since this framework assigns the same prices to the same services provided at the same location. The proposed reserve pricing scheme provides all the market participants with the appropriate signals to modify their offers according to the system operator requirements. The pricing problem is decomposed to different hourly sub-problems considering the bounding constraints. To show the effectiveness of the proposed algorithm, it is applied to the IEEE reliability test system and the results are discussed.

## Keywords

Marginal pricing Security constraint unit commitment (SCUC) Up-/down-going demand-/generation-side reserves## List of symbols

## Functions and variables

- \(C(\ )\)
Generalized objective function

- \(com(\ )\)
Reserve commitment indicator; 1 means the regarding reserve is committed and 0 means not committed

- \(DT(\ )\)
Shutdown time counter

- \(GMP(\ )\)
Generation marginal price

*i*Index for unit

- \(ILS(\ )\)
Involuntary load shedding

*j*Index for bus

*msf*Index for segment in linearized cost function

- \(outg(\ )\)
Unit outage indicator

- \(p(\ )\)
Generation of each segment in cost function

- \(pg(\ )\)
Generation of a unit

- \(pd(\ )\)
Demand at a bus

- \(R_d^{dn} (\ )\)
Demand-side down-going reserve

- \(R_d^{up} (\ )\)
Demand-side up-going reserve

- \(R_g^{dn} (\ )\)
Generation-side down-going reserve

- \(R_g^{up} (\ )\)
Generation-side up-going reserve

*t*Index for time

- \(u(\ )\)
Unit status indicator; 1 means on and 0 means off

*x*Reactance of a line

- \(y(\ )\)
Start-up indicator

- \(z(\ )\)
Shut-down indicator

- \(\lambda _g^{up} (\ )\)
Lagrange multiplier of maximum available generation-side up-going reserve constraint

- \(\lambda _d^{up} (\ )\)
Lagrange multiplier of maximum available demand-side up-going reserve constraint

- \(\lambda _g^{dn} (\ )\)
Lagrange multiplier of maximum available generation-side down-going reserve constraint

- \(\lambda _d^{dn} (\ )\)
Lagrange multiplier of maximum available demand-side down-going reserve constraint

- \(\gamma (\ )\)
Lagrange multiplier of pre-contingency load-generation balance constraint

- \(\gamma k (\ )\)
Lagrange multiplier of post-contingency load-generation balance constraint

- \(\mu ^{\max }(\ )\)
Lagrange multiplier of maximum output limit

- \(\mu ^{\min }(\ )\)
Lagrange multiplier of minimum output limit

## Constants

- \(F_i^{min} \)
Generation cost at the minimum output of unit i

- \(IC(\ )\)
Involuntary load shedding price

- \(MSR(\ )\)
Maximum sustainable ramp rate

*Nc*Number of optimization binding constraints

*Nd*Number of buses

*Ng*Number of units

*Nl*Number of lines

*Nx*Number of optimization independent variables

*Nu*Number of optimization control variables

*NSF*Number of segments in linearized cost curves

- \(Q(\ )\)
Offered rate for reserve which takes the same subscripts and superscripts as

*R*- \(R^{of} (\ )\)
Maximum offered reserve which takes the same subscripts and superscripts as

*R*- \(RD(\ )\)
Ramping up limit of a unit

- \(RMP(\ )\)
Reserve marginal prices which takes the same subscripts and superscripts as

*R*- \(RU(\ )\)
Ramping down limit of a unit

- \(SDC(\ )\)
Shutdown cost

- \(sl(\ )\)
Slope of each segment in the linearized cost curve

- \(SUCF(\ )\)
Start-up cost function

*T*Number of hours in the time span

## Vectors and matrices

*BG*Bus-to-unit incidence matrix

*BGK*Post contingency bus-to-unit incidence matrix

- \(EMP(\ )\)
Vector of energy marginal prices

- \(f^{max}\)
Vector of upper limits of line and transformer flows

- \(fK^{max}(\ )\)
Vector of post-contingency upper limits of line and transformer flows

*GSF*Generation shift factors matrix

- \(GSFK(\ )\)
Post-contingency generation shift factors matrix

- \(Outg(\ )\)
Unit outage matrix

- \(PD(\ )\)
Demand vector

- \(PG(\ )\)
Generators real power output vector

- \(RD(\ )\)
Demand-side reserve vector

- \(RG(\ )\)
Generation-side reserve vector

*X*Inverse of DC-LF matrix

- \(\lambda \)
Vector of general Lagrange multipliers

- \(\Phi (\ )\)
Vector of Lagrange multipliers for pre-contingency line flow constraints

- \(\Phi k(\ )\)
Vector of Lagrange multipliers for post-contingency line flow constraints

- \(\delta (\ )\)
Bus voltage angle vector

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