Value of Price Dependent Bidding for ThermalPower Producers

Erik B. Rudlang, Carl Fredrik Tjeransen,Stein-Erik Fleten

Department of Industrial Economics and

Technology Management

Norwegian University of Sciences and Technology

Trondheim, Norway

Email: erudlang@gmail.com, cftjeransen@hotmail.com,

stein-erik.fleten@iot.ntnu.no

Gro KlæboeDepartment of Electric Power Engineering

Norwegian University of

Sciences and Technology (NTNU)

Trondheim, Norway

Email: gro.klaeboe@powel.no

Abstract—The steady increase in the short-term trading ofelectricity through power exchanges has made the investigationof appropriate bidding strategies relevant. The goal of this paperis to quantify the difference between price-dependent and price-independent bidding strategies for thermal power producers. Thetwo bidding strategies are evaluated for trading in the German-Austrian day ahead market. Optimal bidding decisions are foundthrough two stochastic optimization models. The results indicatethat the price-dependent bids outperform the price-independentbids by utilizing the flexibility in the generators to increaseproduction in the case of price peaks and reduce productionfor price drops.

Index Terms—Bidding behavior, Day ahead markets, Electric-ity supply industry, Power markets, Stochastic programming

I. INTRODUCTION

Power producers participating in an electricity spot-market

may develop different types of bidding strategies. Two of these

are price-independent and price-dependent strategies. Price-

independent bids consist of one volume for each hour and

will be accepted regardless of the price in the market. Price-

dependent bids consist of a set of price-volume pairs which

make up a bidding curve for each hour. Price-independent bids

are promoted as a way to reduce the risk of unpredicted market

events that might leave the power producer unable to fulfill

the market obligation [1]. However, these bids are inflexible

and incapable of responding to price signals from the market.

Price-dependent bidding, on the other hand, has the potential

to better react to unforeseen market events, but might leave

the power producer unable to fulfill the market commitment

through self-scheduling. Determining optimal bidding deci-

sions is complex and depends on the success in both market

modeling, production planning and bid generation. For thermal

power producers who are restricted by temporal constraints in

the generation process, bidding is even more complex.

The Price-Based Unit Commitment Problem (PBUC) is

used to find the optimal unit commitment given a price signal,

and it is an important component in models that find bidding

strategies for power producers. In the papers [1]–[3] bids are

developed using the solution of the PBUC. In [2] bidding

curves are developed by relating the optimal quantity decisions

from the PBUC to the statistical properties of the price

estimator. In [1] the PBUC is solved a number of times for

different offsets in price to obtain bidding curves for a range of

prices. The PBUC is extended to include a multistage scenario

tree in [3] and the solution to the deterministic equivalent

problem is used as bidding curves.

The PBUC can take multiple forms depending on the market

structure that the power producer faces. In [4]–[6] the power

producer is assumed to be a price-taker and has no influence

over the market clearing price. An appropriate forecasting tool

is used to estimate the hourly electricity prices for the next

day, and the PBUC is solved based on the price expectation.

The PBUC has also been used where the power producer

holds market power. One example is [7] which presents

a formulation where the power producer can influence the

market clearing price through price quota curves.

The operation of an electricity market requires power pro-

ducers to submit bids before actual production to settle market

clearing prices and quantities. A more detailed description

of the role of power exchanges can be found in [8]. This

procedure can be modeled as a two-stage problem where

optimal bids are determined in the first stage on the basis of

unit commitment in the second stage. The PBUC can thus be

extended to a stochastic model where the first stage decisions

are made under uncertainty [6].

In [9] the bidding problem is formulated as a two-stage

problem where the spot market is explicitly modelled. The

power producer’s bids are decided in the first stage based on

the realized spot prices found by aggregating own and realized

bids of other power producers in the second stage. This

approach requires knowledge of competitors bidding functions

and can only be used where this information is available.

A two-stage stochastic formulation that assumes the power

producer to be a price-taker is found in [10], [11]. Second

stage production decisions are related to first stage bidding

decisions through a coupling constraint. By fixing a set of

price points, [11] is able to obtain a linear problem that can

be solved with standard MIP solvers.

This paper analyzes the differences between price-

dependent and price-independent bidding. Two stochastic opti-

mization models are presented in Section IV where the power

producer is assumed a price taker. These models can be used

to generate price-dependent and price-independent bids based

978-1-4799-6095-8/14/$31.00 ©2014 IEEE

on stochastic spot market prices. In Section V the numerical

results are presented. A discussion on the findings is found in

Section VI followed by a conclusion in Section VII.

II. MEASURING THE VALUE OF PRICE-DEPENDENT

BIDDING

This paper proposes a new measure to aid the study of this

problem. The Value of Price-Dependent Bidding (VPDB) is

proposed to reflect the value of including the flexibility of

price-dependent bidding into the bidding model. The VPDB

is determined in a two-stage process. First the set of opti-

mal bidding decisions from the price-dependent model x∗PD

and the price-independent model x∗PI is found based on a

stochastic programming approach. These bidding decisions are

then fixed and the two optimization models are resolved based

on out-of-sample scenario data. The VPDB is the difference

between the optimal objective function value of the price-

dependent bidding problem and the optimal objective value

of the price-independent bidding problem (1) when fixing the

bidding decisions. To be able to evaluate the value of price-

dependent bidding between different scenarios, a relative value

of price-dependent bidding (RVPDB) is defined as the VPDB

divided by the optimal objective value of price-independent

bidding problem (2).

V PDB =V aluePD − V aluePI (1)

RV PDB =V PDB

V aluePI(2)

III. NOMENCLAUTURE

Indexess index of price scenarios

t index of time steps

i index of price points

j index of thermal units

l index of unit start up types

SetsS set of scenarios

T set of time steps for the planning period

TB set of time steps for the bidding period, TB ⊆ TT− planning horizon extended to the past

I set of price points

J set of thermal units

L set of unit temperature states L = {h,w, c}where h=hot, w=warm, c=cold

Parametersρst realized price at time step t in scenario spi bidding price at price point iπs probability of scenario scj marginal cost of unit jCl

j start up cost of start up type l in unit jCj commitment cost of unit j per hour

RLj ramping limit in unit j

UTj minimum up time in unit jDTj minimum down time in unit jPj maximum output of unit jPj minimum output of unit j

T lj number of time steps of start up

type l in unit jTmax number of time steps in the planning

horizon TR penalty for using the balance market

V aluePD value given price-dependent bidding

V aluePI value given price-independent bidding

Variablesxit volume bid at price point i in time step tyst volume committed at time step t in scenario szsjt volume produced in unit j at time step t

in scenario s+qst volume sold in the balancing market at

time step t in scenario s−qst volume bought in the balancing market at

time step t in scenario susjt binary variable that is equal to 1 if unit j

runs at time step t in scenario svsjt binary variable that is equal to 1 if unit j

shuts down at time step t in scenario sws

jt binary variable that is equal to 1 if unit jis started in time step t in scenario s

wlsjt binary variable that is equal to 1 if a type l

start up is initiated in time step t in unit jCprod aggregated variable costs

Cbal cost of using the balance market

IV. STOCHASTIC OPTIMIZATION MODEL

A. Price-dependent Bidding

The objective function (3) maximizes the power producers

profit, where profits equal the committed volume of electric-

ity at the realized electricity price minus the corresponding

operating costs and the cost for use of the balancing market.

max =

S∑s=1

πs

{T∑

t=1

ρstyst − Cprod − Cbal

}(3)

Operating Cost: The operating costs (4) consist of the

marginal production cost, the commitment cost, and the start-

up cost for the different start-up types.

Cprod =

T∑t=1

J∑j=1

cjzsjt +

T∑t=1

J∑j=1

Cjusjt +

L∑l=1

T∑t=1

J∑j=1

Cljw

lsjt

(4)

Bidding Curves: The bidding curves are piece-wise linear

convex functions where price-volume pairs determines each

line piece. The problem of finding both prices and quantities

results in a non-linear problem. To avoid non-linearities the

approach of [10] has been used where price points (pi)are fixed and the problem is solved for quantity variables

(xit). The interpolation between the price-volume points and

the realized price in each scenario (ρst ) gives the producers

committed volume (yst ) (5).

yst =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρst−p1

p2−p1x2t +

p2−ρst

p2−p1x1t if p1 ≤ ρst < p2

...ρst−pi−1

pi−pi−1xit +

pi−ρst

pi−pi−1xi−1t if pi−1 ≤ ρst < pi

...ρst−pI−1

pI−pI−1xIt +

pI−ρst

pI−pI−1xI−1t if pI−1 ≤ ρst < pI

(5)

As required by the EPEX SPOT Operational Rules, Article

1.5.1 [12] the bid has to be monotonous. This implies that

with increasing prices the volume has to equal or exceed the

previous bid volume (6).

xit ≤ xi+1,t (6)

By fixing the set of price points as in [10], bidding decisions

might have to be made for prices that are not within the set

of realized prices (ρst ) for the respective time step. To reduce

the risk of infeasible commitments quantity variables for price

points outside the range of realized prices for each hour is

set equal the quantities assigned to the last price point with

information (7) (8).

xi+1,t = xit |max(ρst ) < pi (7)

xi,t = xi+1,t |min(ρst ) > pi (8)

Balancing Market: The model includes the use of a balanc-

ing market to satisfy second stage restrictions. The balancing

market is modeled as a penalty function (11) and upper limits

on the sales and purchases in the balancing market (9), (10).

To promote feasible first stage decisions, both the sale and

purchase in the balancing market is related to a penalty. The

penalty is a fixed deviation (R) from the electricity price.

+qst ≤∑j∈J

Pj (9)

−qst ≤∑j∈J

Pj (10)

Cbal(+qst ,− qst ) = (ρst +R)−qst − (ρst −R)+qst (11)

Thermal Constraints: Thermal power plants have technical

constraints that couples the production in different time peri-

ods together. The formulation in [4] is used to describe these

technical constraints. The maximum and minimum output of

power for each unit is restricted in (12) and (13).

zsjt ≤ Pjusjt (12)

zsjt ≥ Pjusjt (13)

The maximum change in production in one unit from one

time step to the next is limited by the ramp-up (14) and ramp-

down (15) restrictions.

zsjt ≤ zsjt−1 +RLj (14)

zsjt ≥ zsjt−1 −RLj (15)

The limitation on minimum up and down time is enforced

by (16) and (17).

t∑τ=t−UTj+1

wsjτ ≤ us

jt (16)

t∑τ=t−DTj+1

vsjτ ≤ 1− usjt (17)

Balance Constraints: Equation (18) ensures that production,

commitment and use of the balancing market are in balance.

∑j∈J

zsjt = yts − q+st + q−s

t (18)

Logical Constraints: Equation (19) defines the start- and

stop variables. If a unit runs in one time step (usjt = 1) and

is off in the previous time step (usjt−1 = 0) the start variable

(wsjt) is set to equal 1. The stop variable (vsjt) will in the

same manner be set to 1 if the opposite is true. Equation (20)

specifies that a unit cannot start and stop in the same time

period.

wsjt − vsjt = us

jt − usjt−1 (19)

wsjt + vsjt ≤ 1 (20)

The cost of a start-up increases with the time the plant has

been shut down. This is modeled with different start-up types:

hot, warm and cold. To assign the correct start-up type to the

start-up variable, a time interval is specified that determines

each start-up type (21) [4]. The start-up variable is further

limited in (22) by the fact that there can only be one start-up

type in any time period on any generator.

wlsjt ≤

t−T lj∑

τ=t−Tl

j+1

vsjt (21)

wsjt =

∑l∈L

wlsjt (22)

Initialization Constraints: The state of the system at the

start of the planning period influences the bidding problem.

Instead of a predefined initial state the starting conditions

are model by a wrap around formulation similar to what is

done in [13]. For the production (zsjt) and commitment (usjt)

variables this is done by connecting the initial period (t = 0)

to the last period (t = Tmax) (23) ,(24). For the start (vsjt) and

stop variables (wsjt) a larger time frame is needed to determine

the start-up type and the minimum up-time and down-time

(16), (17). This is formulated in (25), (26).

zsj,0 = zsj,Tmax(23)

usj,0 = us

j,Tmax(24)

vsj,t−Tmax= vsj,t (25)

wsj,t−Tmax

= wsj,t (26)

B. Price-independent Bidding

The price-independent bidding problem is a stochastic two-

stage problem similar to the price-dependent bidding problem.

In the price-dependent model price points were used to give

volumes at different prices. In the formulation of the price-

independent model, price points are not needed since the bid

only consists of one volume per hour. The whole set I can

therefore be left out of the model. The bid variable will thus

only be dependent on time step t. The omission of price points

leaves restrictions (5), (6), (7) and (8) redundant. However, to

keep the coupling between the volume bid (xt) for each hour

and the committed volume (yst ), equation (27) is needed.

xt = yst (27)

V. NUMERICAL RESULTS

A time series model for the German-Austrian spot market

was developed and prices for Wednesday 2 October 2013

was simulated using this model (Appendix A). 5000 scenarios

were generated and then reduced through a scenario reduction

algorithm [14] to obtain a workable set of scenarios that also

ensured convergence of the model.

The portfolio of generators is based on the portfolio used in

[4] and consists of two lignite-based generators, two combined

cycle gas turbines (CCGT) and one open cycle gas turbine

(OCGT). The generator costs and thermal properties can be

found in Tables I and II.

The price points in the optimization model was selected

to define intervals that equally divide the range of prices in

the scenarios into 10 sorted groups. This number of price

points was found sufficient to represent the price variation

using the approach of [15]. Running the two optimization

models presented in Section V resulted in optimal quantities

for these price points for all 24 time steps. The bidding curves

for both the price-dependent and price-independent model can

be viewed in Figures 1 and 2.

TABLE IGENERATOR COSTS [4], PARAMETERS cj , Cj , Cl

j

Nr Type Marginal Cost Commitment Start Cost [EUR][EUR/MWh] Cost [EUR/h] Hot Warm Cold

1 lignite 29 1894 46600 64007 872172 lignite 31 1644 58165 79892 1088623 CCGT 55 3367 16012 24832 424724 CCGT 55 3839 19766 30476 518965 OCGT 85 965 2568 2568 2568

TABLE IIGENERATOR THERMAL PROPERTIES [4], PARAMETERS Pj , Pj , RLj , T l

j ,

UTj , DTj

Nr Power: [MW] MaxRamp Time from stop to: Min. time: [h]Max. Min. [MW/min] Warm [h] Cold [h] Start-Stop Stop-Start

1 274 160 2 5 12 8 42 342 180 2 5 12 8 43 378 200 24 5 12 4 34 476 250 24 5 12 4 35 152 63 8 5 12 1 1

Time [h]

0

5

10

15

20

Price [EUR/MW

h]

−50

0

50

100

150

200

Power [M

W]

0

500

1000

1500

Fig. 1. Bidding curves for the price-dependent model

To evaluate the quality of the bidding decisions, the bid-

ding curves were tested in an out-of-sample test. The seven

consecutive Wednesdays from EPEX SPOT starting October

2nd were used as input. The test was performed by fixing

the bidding decisions, and running the optimization models

with the new scenario set. Table III shows the result from this

analysis. The results illustrate the value of price-dependent

bidding where the price-dependent bids performs better than

the price-independent bids for all scenarios based on a penalty

of 0.5 EUR/MWh for the use of the balancing market.

VI. DISCUSSION

The optimal bids generated in this paper were based on

simulations of the behavior of the German-Austrian spot mar-

ket and on a specific portfolio of generators. In the German-

Austrian wholesale market prices have fallen as a result

Time [h]

0

5

10

15

20

Price [EUR/MW

h]

−50

0

50

100

150

200

Power [M

W]

0

500

1000

1500

Fig. 2. Bidding curves for the price-independent model

TABLE IIIPROFIT, VPDB AND RVPDB WITH SEVEN OUT-OF-SAMPLE SCENARIOS

FOR FIXED BIDDING DECISIONS FROM BOTH OPTIMIZATION MODELS,WITH PENALTY OF 0.5 EUR/MWH FOR THE USE OF THE BALANCING

MARKET

Scenario Price-dep. Price-indep. VPDB RVPDBbidding bidding

1 27233.4 26746.3 487.1 1.82 %2 250750.0 250650.0 100.0 0.04 %3 214453.0 214283.0 170.0 0.08 %4 -6627.2 -7131.0 503.8 7.06 %5 -6727.1 -7131.0 403.9 5.66 %6 -30379.0 -30871.0 492.0 1.59 %7 173380.0 173284.0 96.0 0.06 %

Mean 88869.0 88547.2 321.8 2.33 %

of the feed-in-tariff of renewable energy, this has degraded

margins for thermal power producers, and made the prospects

of turning off generators or reduce the output of generators

relevant. Price-dependent bids are able to reduce the impact of

unpredicted price drops and therefore provide superior bidding

decisions. For markets with higher prices and less volatility

improved margins might make price-independent bids suffi-

cient as generators might run continuously at maximum power.

The low prices in the market resulted in dispatch of only

the low-cost and less flexible generators. Thus, the price-

dependent and price-independent bids were based on very

similar dispatch solutions. For different market conditions the

optimal dispatch might be significantly different in the two

models. This paper shows however, that there will always exist

price-dependent bids that out-perform the price-independent

bids. These price-dependent bids are simply obtained by fixing

the unit commitment of the price-dependent model to the

unit commitment of the price-independent model and generate

price-dependent bids based on the flexibility in the committed

units.

VII. CONCLUSION

As prices have fallen and volatility risen in the German-

Austrian spot market thermal power producers trading on

EPEX SPOT is presented with a new reality. Prices might

fall below marginal cost of generators and even fall below

zero for a number of hours in the day-ahead market. This

might make the prospects of turning off generators for a

number of hours viable. The results from this paper show

that by submitting price-dependent bids, the power producer

is able to reduce losses in scenarios with low prices and

improve gains in scenarios with higher prices compared to

price-independent bids. Thermal power producers trading on

EPEX SPOT might incorporate some of the ideas presented in

this paper to construct price-dependent bids that will improve

their returns in the market.

APPENDIX

PRICE FORECASTING MODEL

An ARMA time series model is used to forcast the price

for 2 October 2013. This model is based on price data for

the German-Austrin spot market from 1 January 2013 to 1

October 2013.

(1− φ1B1 − φ2B

2 − φ24B24) log(yt − α)

= (1− θ24B24)εt (28)

TABLE IVESTIMATED PARAMETERS FOR THE ARMA PRICE MODEL

φ1B1 φ2B2 φ24B24 θ24B24 α

1.0454 -0.1391 0.9927 -0.9042 5.2312s.e. 0.0122 0.0122 0.0017 0.0072 0.0344

ACKNOWLEDGMENT

The authors would like to thank Christian Skar for valuable

feedback and FICO for access to their Xpress solver.

REFERENCES

[1] T. J. Larsen, “Daily scheduling of thermal power production in aderegulated electricity market,” Ph.D. dissertation, Dept. Elect. PowerEng., Faculty of Elect. Eng. and Telecommun., NTNU, Trondheim,2001.

[2] A. J. Conejo, F. J. Nogales, and J. M. Arroyo, “Price-taker biddingstrategy under price uncertainty,” IEEE Trans. Power Syst., vol. 17, no. 4,pp. 1081–1088, 2002.

[3] M. A. Plazas, A. J. Conejo, and F. J. Prieto, “Multimarket optimalbidding for a power producer,” IEEE Trans. Power Syst., vol. 20, no. 4,pp. 2041–2050, 2005.

[4] C. K. Simoglou, P. N. Biskas, and A. G. Bakirtzis, “Optimal self-scheduling of a thermal producer in short-term electricity markets byMILP,” IEEE Trans. Power Syst., vol. 25, no. 4, pp. 1965–1977, 2010.

[5] B. K. Pokharel, G. Shrestha, T. T. Lie, and S.-E. Fleten, “Price basedunit commitment for gencos in deregulated markets,” in Proc. PowerEngineering Society General Meeting. IEEE, 2005, pp. 428–433.

[6] S. W. Wallace and S.-E. Fleten, “Stochastic programming modelsin energy,” in Handbooks in Operations Research and ManagementScience, A. Ruszczynski and A. Shapiro, Eds. Elsevier, 2003, vol.Volume 10, pp. 637–677.

[7] S. de la Torre, J. M. Arroyo, A. J. Conejo, and J. Contreras, “Price makerself-scheduling in a pool-based electricity market: a mixed-integer LPapproach,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1037–1042,2002.

[8] M. M. Roggenkamp and F. Boisseleau, Regulation of Power Exchangesin Europe. Antwerpen: Intersentia, 2005, vol. 2.

[9] M. P. Nowak, R. Schultz, and M. Westphalen, “A stochastic integerprogramming model for incorporating day-ahead trading of electricityinto hydro-thermal unit commitment,” Optimization and Engineering,vol. 6, no. 2, pp. 163–176, 2005.

[10] S.-E. Fleten and E. Pettersen, “Constructing bidding curves for a price-taking retailer in the Norwegian electricity market,” IEEE Trans. PowerSyst., vol. 20, no. 2, pp. 701–708, 2005.

[11] S.-E. Fleten and T. K. Kristoffersen, “Stochastic programming for opti-mizing bidding strategies of a nordic hydropower producer,” EuropeanJournal of Operational Research, vol. 181, no. 2, pp. 916–928, 2007.

[12] EPEX SPOT SE, “EPEX Spot operational rules,” Sep 2013.[13] A. B. Philpott, M. Craddock, and H. Waterer, “Hydro-electric unit com-

mitment subject to uncertain demand,” European Journal of OperationalResearch, vol. 125, no. 2, pp. 410–424, 2000.

[14] H. Heitsch and W. Romisch, “Scenario reduction algorithms in stochasticprogramming,” Computational Optimization and Applications, vol. 24,no. 2-3, pp. 187–206, 2003.

[15] N. Lohndorf, D. Wozabal, and S. Minner, “Optimizing trading decisionsfor hydro storage systems using approximate dual dynamic program-ming,” Operations Research, vol. 61, no. 4, pp. 810–823, 2013.