A Two-Level Optimization Problem for Analysis of Market Bidding Strategies; Weber, Overbye

8/11/2019 A Two-Level Optimization Problem for Analysis of Market Bidding Strategies; Weber, Overbye

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A Two-Level Optimization Problem for

Analysis of Market Bidding Strategies

J. D. Weber

T.

J. Overbye

overbye

@

uiuc.edu

d-weber

@

uiuc.edu

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

Abstract

Electricity markets involve suppliers gene rator s) an d

sometimes consumers loa ds) bidding fo r MWhr generation

and consumption. In this market model, a central operator

solves an Optimal Power Flow OP F) based o the

maximization of social welfare to determine the generation

and

load

dispatch and system spot prices. In this structure,

market participants will choose their bids in order to

maximize their profits. This prese nts

a

two-level

optimization problem in which participants try

to

maximize

their profit under the constraint that their dispatch and

price are determined by the OPF. This pap er presents an

efJici6nt numerical technique, using price and dispatch

sensitivity information available fr om the OP F solution, to

determine how

a

market participant should vary its bid

portfolio in order to maximize its overall profit. The pa pe r

further demonstrates the determination

of

Nash equilibrium

when all participan ts are trying to maximize their profit in

this manner.

Keywords: Electricity market, simulation, bidding

strategy, optimal power flow, Nash equilibrium

1. Introduction

Electricity markets throughout the world continue to be

opened to competitive forces. The creation of mechanisms

for suppliers, and sometimes consumers, to openly trade

electricity is at the core of these changes. Most of the

mechanisms being used and developed are based on spot

pricing ideas. Good overviews and examples of spot pricing

issues are found

in

[1]-[2]. All these spot pricing-based

markets consist of suppliers submitting bids for electricity in

the form of MW outputs and associated prices. Some

markets, such as in England and the PJM interconnection

[3], consist of only supply-side bidding. Other markets,

such as in New Zealand [4] and California [ 5 ] , also

incorporate demand-side bidding, allowing the consumers in

the market to react to pricing.

In

[6],

a market simulation was developed where

individuals submit supply andor demand bids to a central

operator. The operator then solves an optimal power flow

(OPF), with the objective of maximizing social welfare, to

determine the generation and load dispatch, as well as all the

spot prices. This paper will develop an algorithm allowing

an individual to maximize their personal welfare in such a

market. Possible equilibrium behavior can be determined

by simulating a market in which all players use this new

0-7803-5569-5/99/$10.00 1999 IEEE

algorithm. A simple example will be used to demonstrate

behavior that would be expected in the electricity market.

2. Notation

x : voltages, angles, and other variables in OPF problem

s : supply vector (generation vector)

d

:

demand vector (load vector)

p

:price vector

Ci s ) = bj+ c , sZ= supplier cost (function of supply)

B i d ) = b p + c j z= consumer benefit (function of demand)

Social Welfare =

x B i d ) -

CC, s )= B(d)-C(s)

h x,s,d)

= 0

:

equality constraints for OPF including the

power

f l ow

equations

g x,s,d) O inequality constraints for OPF including

transmission line limits and voltage limits

s,d,p) : ndividual's welfare taking into account

the price paid or received for goods

VL

he gradient of the OPF Lagrangian

H

:

he Hessian of the OPF Lagrangian

z = [ r dT

all

camurnen

all suppliers

x r

A ' :

vector of all OPF variables

3. Electricity Market Setup using OPF

Many electricity markets throughout the world include

some form of supplier bidding in which suppliers submit

MW outputs, along with associated prices. These bids

generally take one of the two forms shown in Figure

1 .

Biock Bidding Cpntinuous Bid Curve

f

Price =p

[$MWhr]

P 3 T - - - - 1 - -z- ----- q

PI-

Price =

p

[$MWhr]

v

Figure 1Supplier Block Bid and Continuous Bid Curve

Additionally, some markets also include consumer

bidding. Consumers can be modeled in a manner analogous

to the suppliers. A demand function is used, which is

mathematically similar to the supply function, except the

demand function decreases as the price increases. In this

paper, we assume a continuous bid curve model for both

suppliers and consumers such as those seen

in

Figure

2.

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Figure 2 Consumer and Supplier Bid Curves

The setup of the electricity market simulation is the

same as in [ 6 ] and is briefly summarized here for

convenience. For the market simulation, bids are taken as

inputs to an OPF that solves the maximization

of

social

welfare problem to determines supplies, demands and

prices. These bids are limited to linear bids' as shown in

Figure2. Incorporating these bids into the OPF formulation

involves converting the bid into a cost or benefit function,

e.g.

p s , k )= k

-s+

p,, c s , ~ )k -s + p, , , , , ,~

r:

[2:

[ 6 ] [ 8 ] .

The Lagrange multipliers of the OPF solution

determine the spot prices, p . The suppliers

in

the market are

paid the spot price at their node, and the consumers are

charged the spot price. A thorough treatment of the

mathematics involved in integrating supply and demand

bids into the OPF formulation is provided in [ 8 ] .

It should also be noted that bids are required to have a

minimum power of zero: no minimum generator outputs are

allowed. This simplifying assumption is made to more

concisely explain the algorithm. The algorithm could be

augmented in the future to account for output limits by

including a unit commitment solution prior to the OPF.

Thus far we have restricted our market participants to

either a single generator or a single load. In reality, any

combination of several generators and several loads could

constitute an economic entity; therefore, the word individual

is used to describe a set of supplies and demands whose

bidding is controlled by a single economic entity.

4. Individual Welfare Maximization

The market setup from the previous section essentially

defines the market rules for our system; however the

variation in bidding will be limited to the variation of a

single parameter, k, for each consumer

or

supplier as shown

in Figure

3.

This parameter will vary the bid around the true

marginal curve of the supplier

or

consumer. The supply

curve that reflects true marginal cost is defined as the linear

function

p s )=

-S + p , , , while for the consumer, a true

marginal curve is defined as p d )

= --d +p ,

1

m

1

m

bPrice=p .

4

rice = p

True Marginal

enefit Curve

Cost Curve

Bid

[MWI

Figure 3 Bidding Variation for Supply and Demand

While this limits market behavior, it will be shown

in

Section 7 that from an individual's viewpoint, the shape of

this curve is not important for a single market solution.

Note that modifying a bid in this manner is the same as

multiplying the cost or benefit function used in the OPF by

k.

Here we assume that each individual seeks to maximizie

its personal welfare. A single consumer's welfare is defined

as the amount of benefit received from using the power,

minus the expenses incurred in purchasing the power.

Similarly, a single supplier's welfare is defined as the

amount of revenue received from selling the power, minus

the cost of supplying the power. An individual wants to

maximize the total welfare

of

all the consumers and

suppliers that it controls.

= (-dTCdd+B:d)- )-(srC,s+B:s)+ , -1.s)

Revenuer

enefiIS Expenses CAIS;O-

where Cdand C, : are diagonal matrices of quadratic coefficients

for supply cost and demand benefit functions

BdandB,: are vectors of linear coefficients for supply

cost and demand benefit functions

Note that an individual's welfare f ls ,d,h) is not an explicit

function of its bid variable

k.

However f is an implied

function of

k

since

s ,

d,

and

h

are all determined by an OPF

solution which is a function of k. Thus

s,

d, and

h

are all

implicit functions

of

the bids k.

Assuming the individual has some estimate of what

other individuals in the market are going to bid, the

individual's goal is to maximize its welfare by choosing a

bid which is a best response to the other individuals' bids.

As a result, the maximization of an individual's welfare

forms a two-level optimization problem where the

individual maximizes its welfare subject to an OPF solution

which maximizes social welfare based on all

market. That is

Max

f s , c l k )

s.t. s,d,k)are determined by

k

bids in the

(2)

'

While only allow ing single-segmen t linear bids may

seem

imiting, the

analysis of

the

California

power

market done in

[7]

shows that over

time

supply bid curves appear to be two -segment linear curves. Two-segments

could

be

added

to

this development in

the

future, but

[7]

show s that linear

bids are reasonably accurate.

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5.

Solution of Individual Welfare

Maximization

One approach to solving

2)

s to represent the OPF

maximization sub-problem by the Kuhn-Tucker necessary

conditions as done

in

[9]. Thus the two-level optimization

problem may be written as a standard optimization problem.

This paper presents an alternative technique that makes

minimal modifications to an existing OPF to solve (2).

Consider solving the OPF problem for a given set of

bids. Then from the information available at this OPF

solution, the individuals profit sensitivity to variations in its

bid can be used to determine a Newton-step that improves

profits. This Newton-step is defined the customary way as

shown

in

(3).

af

af

Evaluation of (3) requires determination of and

ak ak

5 )

Evaluation of 4) and

5 )

requires determination of

ad

as

an

a2d a z k

,and

7

ecause s,d,and are

akakak k2ak2 ak

variables of the OPF solution, these derivatives can be

determined directly from values available from the OPF

solution. In a Newton-based OPF, an iteration of the

mismatch equation

Az

=

H

VL) s done until Az =

0,

where z =

T

dr X T r Therefore the derivatives

of

s,

d, and can be found by taking derivatives of this

mismatch equation. These can be found to be

These equations can be simplified by recalling at an OPF

solution z = 0 , and that because of the structure of our

problem, k shows up as a linear term

in

both the gradient

a2H a2vL

and the Hessian

[ IO ]

therefore?=O and

= O .

ak ak

Thus (6) and (7) may be simplified to

9)

aH az

Zz

dH -,aVL

ak ak ak ak ak

2H-I H

-2H-I

_ _

and H are extremely sparse. The partial

Both

k ak

avL

derivative- is a matrix which has exactly one entry

in

each column, while s 3-dimensional tensor that has

exactly one diagonal element in each matrix as you move in

the dimension corresponding to k. Because these are so

sparse, the amount of arithmetic is much smaller than

it

would seem.

Using 8) and

(9)

to substitute into 4) nd

5 ) ,

use

(3)

to

perform a Newton-step which improves an individuals

welfare. The following algorithm is used to determine an

individuals best bidding strategy.

ak

aH

dk

Algorithm: Individual Welfare Maximization

1.

Choose an initial guess for vector

k.

2. Solve the OPF maximization of social welfare given the

individuals assumption of other individuals bids and

the individuals guess at its own vector

k.

Use

(3)

to determine a step direction for vector k.

If Ilkncw k,II is below some tolerance stop, else go

back to step

2.

3.

4.

This algorithm will be effective as long as the binding

inequalities of the

OPF

algorithm do not change. Changes

in

binding inequalities result

in

discontinuities of

f

ak

which means that the function

f

becomes non-differentiable.

The non-differentiability can be overcome by recognizing

that a change in binding inequality can be detected from

other available information. From one side of the non-

differentiable point, the value limited by the inequality

approaches its limit. From the other side of the point, the

Lagrange multiplier associated with the inequality

approaches zero. This is shown in Figure 4.

Welfare

Limited value Lagrange multiplier

I *

kn,

Increasing

Figure

4

Binding Inequality Change

Using this information, the Individual Welfare

Maximization algorithm is modified

so

a multiplier reduces

the step direction determined by (3)

if

this step direction

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will step across a non-differentiable point. The multiplier

instead tries

to

bring the answer directly to this non-

differentiable point.

6.

Market Simulation

With the ability to determine an individuals best

response to the market, a market simulation can now be

performed to learn more about potential market behavior.

For example, the determination of economic equilibrium

points such as

Nash equilibria

[121 is of interest.

Definition:Nash Eauilibrium

An individual

looks

at

its opponents behaviors

The individual determines that its best response to its

opponents behaviors is to continue its present behavior

This

is

true FOR ALL individuals

in

the market

To determine a Nash equilibrium the individual welfare

maximization can be iteratively solved by all individuals

until a point is reach where each individuals best response

is to continue with the same vector of bids.

A

similar

iterative technique for finding Nash equilibria was used in

[111, although difference individual maximization was used.

The following algorithm describes this process.

Algorithm:

Find Nash Eouilibrium

Start all individuals with a bid vector k =

1.

Run the Individual Welfare Maximization algorithm for

each individual and update all bids

Continue running this until individuals stop changing

their bids

6.1.

Market

With N o

Constraints

To demonstrate the

Find Nash Equilibrium

algorithm,

consider the two-bus example with two suppliers and one

consumer shown

in

Figure

5.

B2 d2)

k,, -0.04dz2 30dzn

g

+ jb =

-j20.6

C,(S,) = k,,(0.01s12+ O S , C,(S,) = k,, 0.01s,2 + l O S , )

Figure 5 Two-bus System: Two Suppliers and a Consumer

Only consider the supplier bidding behavior for this

example, therefore assume the consumer in this market

always bids according to it true benefit function, i.e. l ~ 2

1.00. It is important to maintain the price-dependent

demand. Otherwise, when a transmission line limit is added

to the system, which will be done shortly, supplier 2 could

have part of the constant load to serve with no competition.

The solution for supplier 2 would be to bid k . 2 equal to

infinity. This is an unreasonable result.

The Find Nash Equilibrium algorithm results in both

suppliers bidding Is =

k2

= 1.1502. This is called a pure

strategy equilibrium

[

121 because the equilibrium is at a

point where each bidder always bids that same value

of

k.

Figure 6shows the bid progression of the algorithm toward

the equilibrium point. Figure 7shows a complete solution

to the problem with the optimal response of each supplier to

any possible bid by the other supplier. The point where the

two curves in Figure

7

meet is the Nash Equilibrium point.

Iterations

Figure 6 Bid Progression with no Line Limit

z 1.4- ;Optimal Response of

.-

I

Supplier

1

to

Bids

by

2

1

o

I

1

1.0

1.1

1.2

1.3 1.4

1.5 1.6

Variation of Supplier

l

id

Figure

7

Optimal Responses with no Line Limits

6.2. Market With Constraints

To further demonstrate the algorithm, consider the

system of Figure 5 again, but now add a constraint to the

system: limit the flow on the transmission line to 80 MVA.

The bid progression that results from this system case is

shown in Figure 8.

1.6

I I I

0

5

10

15 20 25

Iterations

Figure 8 Bid Progression with 80 MVA Line Limit

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This does not result

in

a Nash equilibrium point, but

limit cycle - like behavior. To better explain why no

equilibrium point is reached, the optimal response curves

over all possible bids by each individual are determined.

These are shown in Figure 9.

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6

Variation of Supplier 1 id

Figure 9 Optimal Responses with 80 MVA Line Limit

Figure 9shows that the optimal response curves for the

two suppliers never intersect because of a discontinuity

in

the supplier 2 best response curve. In order to determine

what is causing this discontinuity, supplier 2 profit vs. bid

curves are created on either side of the discontinuity. These

are shown in Figure 10.

variat ion of Supplier

W s Bid

whan

k.,=

1.35

e

f

-

z

Supplier 2 Bid

Variat ionof Supplier

W E id when

k .40

Supplier

2

Bid

Figure 10Supplier 2 Profit vs. Bid on either side of the

Discontinuous Point

Figure 10shows that the profit function for supplier 2

is non-convex, having two local maxima. When supplier 1

bids

k1

1.3720, these local maxima have the same

magnitude. At this point, supplier 2 has no preference

between bidding either ks2 of 1.525 or 1.246. On either side

of this point the optimal response Ijumps to the other local

maximum. Ultimately, this is the reason that no pure Nash

equilibrium exists. Thus, we have shown that the

introduction of a transmission line constraint, even in a

trivial two-bus system, eliminates our pure strategy

equilibrium point.

This does not mean that no Nash equilibria exist

however. Only pure strategies have been considered.

Mixed

strategies

[121 are also possible. A brief definition of

a mixed strategy in our application follows.

Definition:

Mixed Strutenv

An individual chooses several pure strategies and assigns

a

probability to each. The individual then submits these pure

strategy bids according to their associated probabilities.

Allowing the possibility of mixed strategy equilibria,

one can easily generate a Nash equilibrium. For the

previous example, one mixed strategy equilibrium is:

Nash Equilibrium for Line Limited Case

Supplier 1: Bid k,, = 1.372 with Probability 1.00

Supplier 2: Bid ks2= 1.246 with Probability 0.56 and

ks2= 1.525 with Probability 0.44

When supplier 1 bids k,l = 1.372, supplier 2 has no

preference between either bidding ks2= 1.246or ks2= 1.525,

thus bidding these two pure strategies with arbitrary

probabilities is one optimal response.

When supplier 2

bids these two pure strategies with the probabilities shown,

the expected profit vs. bid curve for supplier 1 is as shown

in Figure 11. In Figure 11, the maximum is indeed at a bid

This shows that while pure strategy equilibria are

eliminated by the inclusion of transmission limits, mixed

strategy Nash equilibria still exist.

of = 1.372.

G m

Sup$er ;Bid 9 , I2Sup$er iBld

For IG2 E 1.246 with Prob

0.56

and ku=

1.525

with Prob 0.44

I 1 t Y I1

8,s

11 (25 ( 1

Supplier l Bid

Figure 11 Optimal Bids for Supplier 1

in

Response to a

Mixed-Strategy by Supplier 2

7. Generalizing the

Bid

Curves

To this point, bidding in our electricity market has been

limited to one dimension through changes in the multiplier,

k, for each consumer and supplier. This does not allow the

bidder to vary the slope and intercept of the bid curve

independently. It can actually be shown that varying both

slope and intercept will not result in increased personal

welfare. To demonstrate this, the example with no line

limits from Section 6.1 is considered. The optimal

intercepts for several fixed slopes are determined, and the

resulting optimal bid curves are shown in Figure 12.

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Optimal Curves for several fixed slopes

300

250

s

100

150

%

-

0

50

6 10 12 14 16

18

Price

[$/MWhr]

Figure 12 Optimal Curves for Several Fixed Slopes

These optimal bid curves all intersect at the same point

in

the supply-price space. Also, for all bids the market

solution is at this intersection point. This means that

independently varying both the slope and intercept will not

result in a higher increase in personal welfare. From this it

is also learned that when performing the individual

maximization, the individual is really trying to find a point

in the supply/demand-price space that maximizes its profit.

The individual is merely looking for a bid curve that crosses

this point. The multiplier k used throughout this paper, is a

very good variation parameter. Varying k will cover the

entire supplyldemand-price space with positive values of

slope and x-intercept, thus allowing

the

algorithm to find the

desired point.

While the bidding algorithm is only looking for a single

point in this development, a future extension of this work

could study the effect that uncertainty

in

the estimates of

other individuals bids plays in moving this optimal point

around. It is likely that the shape of the bid curve could be

chosen to maximize personal welfare for small perturbations

around the optimal point found in this paper.

8. Conclusions

The algorithm developed has been successful in

analyzing the small systems shown in the paper. It has been

shown that iteratively using the objective of maximizing

personal welfare can be an effective way of simulating

electricity markets and studying

the

equilibrium behavior of

the market. The recognition that an individual is looking for

a point in the supplyldemand-price space is also important.

A future improvement that must be achieved is allowing

for the non-convexity of the individual welfare function

as

shown in Figure 10. The calculus-based algorithm

presented here worked for small systems, but will be unable

to effectively find global maxima for individual welfare on

larger systems due to the non-convexity. Some sort of

hybrid algorithm utilizing th work done here along with a

genetic algorithm or simulated annealing technique will be

investigated later.

9.

Acknowledgements

The authors would like

to

acknowledge the support of

the Power Affiliates program of the University of Illinois at

Urbana-Champaign and the Grainger Foundation.

10.

References

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[3]

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