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Pricing Electricity in Pools with Wind Producers 1
1. Introduction To Electricity Markets
In economic terms, electricity (both power and energy) is a commodity capable
of being bought, sold and traded. An electricity market is a system for effecting
purchases, through bids to buy; sales, through offers to sell; and short-term
trades, generally in the form of financial or obligation swaps. Bids and offers
use supply and demand principles to set the price. ong-term trades are
contracts similar to power purchase agreements and generally considered
pri!ate bi-lateral transactions between counterparties. "holesale transactions
(bids and offers) in electricity are typically cleared and settled by the marketoperator or a special-purpose independent entity charged e#clusi!ely with that
function. $arket operators do not clear trades but often re%uire knowledge of
the trade in order to maintain generation and load balance. &he commodities
within an electric market generally consist of two types' power and energy.
ower is the metered net electrical transfer rate at any gi!en moment and is
measured in megawatts ($"). nergy is electricity that flows through a
metered point for a gi!en period and is measured in megawatt hours ($"h) *+.
$arkets for energy-related commodities trade net generation output for a
number of inter!als usually in increments of , + and / minutes. $arkets for
power-related commodities re%uired and managed by (and paid for by) market
operators to ensure reliability, are considered ancillary ser!ices and include such
names as spinning reser!e, non-spinning reser!e, operating reser!es, responsi!e
reser!e, regulation up, regulation down, and installed capacity. In addition, for
most ma0or operators, there are markets for transmission congestion and
electricity deri!ati!es such as electricity futures and options, which are acti!ely
traded. &hese markets de!eloped as a result of the restructuring of electric
power systems around the world. &his process has often gone on in parallel with
the restructuring of natural gas markets.
lectricity is by its nature difficult to store and has to be a!ailable on demand.
1onse%uently, unlike other products, it is not possible, under normal operating
conditions, to keep it in stock, ration it or ha!e customers %ueue for it.
2urthermore, demand and supply !ary continuously. &here is therefore a
physical re%uirement for a controlling agency, the transmission system operator ,
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to coordinate the dispatch of generating units to meet the e#pected demand of
the system across the transmission grid. If there are a mismatch between supply
and demand the generators speed up or slow down causing the system
fre%uency (either / or / hert3) to increase or decrease. If the fre%uency falls
outside a predetermined range the system operator will act to add or remo!e
either generation or load.
In addition, the laws of physics determine how electricity flows through an
electricity network . 4ence the e#tent of electricity lost in transmission and the
le!el of congestion on any particular branch of the network will influence the
economic dispatch of the generation units. &he scope of each electricity market
consists of the transmission grid or network that is a!ailable to the wholesalers,
retailers and the ultimate consumers in any geographic area. $arkets maye#tend beyond national boundaries.
1.1. Day-Ahead Electricity Market
5ay-Ahead-$arket (5A$) is a physical electricity trading market for deli!eries
for any6some6all + minute time blocks in 78 hours of ne#t day starting from
midnight. &he prices and %uantum of electricity to be traded are determined
through a double sided closed auction bidding process. &he operations are
carried out in accordance with the 9rocedure for scheduling of collecti!e
transactions: issued by the 1entral &ransmission tility (<1I), 91=1 (>pen
Access in Inter-?tate &ransmission) =egulations, 7//@, as amended from time to
time and the Bye-aws, =ules and Business =ules of the #change.*7
&rading rocess 2low
Bidding
+. articipants enter bids for sale or purchase of power for deli!ery on thefollowing day. (&+ day)
7. Bids for a total of blocks of + minute each can be entered.
C. Bidding session' +/// hrs. - +7// hrs.
8. Bids can be single and6or block including linked bids'
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a. ?ingle bids' +-$inute bids for different price and %uantity pairscan be entered through this type of order. artial e#ecution of the
bids entered is possible.
b. Block bids' =elational Block Bid for any +-min block or series of +-min blocks during the same day can be entered. Although no partial e#ecution is possible i.e. either the entire order will beselected or re0ected.
. &he bids so entered are stored in the central order book. &he bids enteredduring this phase can be re!ised or cancelled till end of bid call period(i.e.+7// hrs. of trading day)
$atching
• At the end of the bidding session, bids for each + minute time block are
matched using the price calculation algorithm. (a!ailable in ID bye-laws)
• All purchase bids and sale offers are aggregated in the unconstrained
scenario. &he aggregate supply and demand cur!es are drawn on rice-Euantity a#es. &he intersection point of the two cur!es gi!es the marketclearing price ($1) and market clearing !olume ($1F) corresponding
to price and %uantity of the intersection point.
• $1 and $1F are determined for each block of + minutes as a function
of demand and supply which is common for the selected buyers andsellers.
• ?elected members are intimated about their partially or fully e#ecuted
bids and other trade related information.
•
By +C// hrs, transmission corridor re%uired to fulfill successfultransactions are sent to G51.
&he e#ample below illustrates price calculation. Assume the price tick as below'
2or the sake of simplicity we assume only C portfolios are entered. &he %uantityentered by each portfolio A, B and 1 for the specific price tick is as shown
below'
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&he algorithm will then add the entire purchase %uantum and sell %uantum after the bidding session and look for a solution where the net transaction is 3ero i.e.the buy %uantum is e%ual to the sell %uantum.
&he demand-supply graph in such scenario is shown below'
&ransmission corridor and funds a!ailability
• reliminary $1 and $1F are used to calculate the pro!isional
obligation of the selected participants and the pro!isional power flow.
• 2unds a!ailable in the settlement accounts of the participants are !erified
based on the pro!isional obligation.
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• In case of insufficient funds in the account, the bids entered by such a
participant are deleted.
• =e%uired corridor capacity and pro!isional power flow is sent to G51
for scrutiny and corridor allocation is re%uisitioned based on a!ailability.
• By +8// 4rs, G51 re!erts with actual transmission corridor a!ailability
during all + minute time blocks across congestion prone bid areas.
=esults
• Based on the reser!ed transmission capacity intimated by G51, ID
recalculates $1 and $1F as well as area clearing price (A1) and area
clearing !olume (A1F).
• A1 is used for the settlement of the contracts. >n receipt of final results,
obligations are sent to the 1learing Banks for ay In from buying$embers at +8.C/ hrs and the bank is asked to confirm the same.
1onfirmation
• 2inal results for confirmation and application for scheduling of collecti!e
transactions are sent to G51.
• G51 sends the details of the schedule to respecti!e ?51s.
?cheduling
• =51s 6?51s incorporate 1ollecti!e &ransactions in the 5aily
schedule.
• A scheduled transaction is considered deemed deli!ery.
• 5e!iations from schedules are dealt under I or 5e!iation ?ettlement or
Imbalance ?ettlement regulations. &he =egional ntities (those connectedat I?&? networks) are go!erned by 1=1 =egulations and mbeddedntities (those connected to state transmission or distribution network)are go!erned by respecti!e ?tate 1ommission:s regulations.
1.2. Balancing Market !eration
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&rading on the Balancing $arket is implemented through a platform for
collecting purchase and sale bids for electricity through which the ?ystem
>perator (?) buys and sells electricity intended for the settlement of
imbalances in the electricity system. &rading on the Balancing $arket is carried
out together with Intra-day trading, that is, one hour after the closure of the
latter and until actual supply of the product. All companies included in the
Balance ?cheme of the electricity market and which acceded to trading on the
Balancing $arket and Intra-day trading can participate in trading *C.
2. "o-!ti#i$ing Energy and %eser&e "a!acity
In order to ensure that enough balancing resources are a!ailable during the real-time operation of the power system, the system operator allocates reserve
capacity in ad!ance. In practice, the procurement and scheduling of reser!e
capacity implies operating the system at less than its full capacity, while its use
or deployment usually translates into the redispatch of units pre!iously
committed in the day-ahead market, the !oluntary curtailment of loads, and6or
the %uick start-up of e#tra power plants to co!er une#pected shortages of energy
supply in real time. &here e#ist two schools of thought on how reser!e should
be traded in electricity markets. >n the one hand, reser!e capacity may be sequentially procured in a series of auctions run once the day-ahead energy
dispatch has been determined. &hese auctions are organi3ed to procure reser!es
with different acti!ation times. &he rationale behind this approach is that the
free capacity that has not been successfully placed in one market can then be
offered in the following auctions where the re%uired acti!ation time for the
traded reser!e is not as demanding. 1onse%uently, reser!e capacity offers that
are successful in one market are not considered in the subse%uent ones *8.
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>n the other hand, energy and reser!e may be simultaneously procured in the
same auction using a co-optimi3ation algorithm that captures the strong
coupling between the supply of energy and the pro!ision of reser!e capacity.
&he following illustrati!e e#ample ser!es to get a more intuiti!e understanding
of this coupling.
2.1. 'e(uential 'ettle#ent
1onsider an electricity market that solely includes two power producers, A and
B. ach of these producers runs a power plant with a capacity of +//$".
roducer A offers to sell energy at H+/ / $"h, while producer B does it at
HC/ / $"h. A demand of +C/ $"h is to be supplied. Additionally, with the aim
of dealing with unforeseen e!ents, the system operator estimates that 7/ $" of reser!e capacity are re%uired. roducer A is willing to pro!ide reser!e at no
cost, whereas producer B offers reser!e capacity at H7 / $".
&o start with, let us suppose that energy and reser!e capacity are sequentially
settled in this order. &hus, the energy-only dispatch is first determined as
follows
$in.10 P A+30 PB
s.t. P A+ PB=130,
0≤P A≤100,
0≤PB≤100,
where P A and
PB are the amounts of energy sold by producers A and B,
respecti!ely. >ptimi3ation problem is tri!ial, and its solution is gi!en by
P A
¿
=¿ +// $"h and PB
¿=¿
C/ $"h. &he clearing (marginal) price for
energy, which is defined as the dual !ariable of constraint, results in HC/ / $"h.
>nce the energy dispatch is determined, the reser!e capacity market is cleared
as follows
$in.0 R A+25 RB
s.t. R A+ R B=20,
0≤R A≤100− P A¿
,
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0≤RB≤100− PB,
¿
"here R A and
RB are the amounts of reser!e capacity sold by producers A
and B, respecti!ely. Gote that the reser!e scheduling takes the energy dispatch
{ P A
¿, PB
¿ } as input. &he solution to problem is also tri!ial and is gi!en by
R A
¿
/ and RB
¿
7/$". &hat is, since producer A has been dispatched at full
capacity in the energy market, reser!e needs are entirely co!ered by producer B.
&hus, the total system operation costs TC seq
, including both the procurement
costs of energy and reser!e capacity, are computed as
TC seq=10 P A
¿ +30 PB¿+0 R A
¿ +25 RB¿
H78//
&he clearing (marginal) price for reser!e capacity is H7 / $", which is the
!alue taken by the dual !ariable associated with the reser!e re%uirement
constraint. &herefore, the profits made by producers A and B, respecti!ely,
under the se%uential market organi3ation are calculated as follows
profit Aseq=(30−10 ) P A
¿ +(25−0 ) R A¿=$2000
profit Bseq=(30−30 ) PB
¿+(25−25 ) RB¿=0
2.2. 'i#ultaneous Trading
et us now consider that energy and reser!e capacity are simultaneously traded
in the same auction. &o this end, both commodities are 0ointly dispatched using
optimi3ation problem below, which minimi3es the total system operation costs.
$in.10 P A+30 PB+0 R A+25 RB
s.t. P A+ P B=130,
R A+ RB=20 ,
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must increase it by the same amount. &his action does not in!ol!e any
additional reser!e cost, but increases the cost of the energy dispatch by H7/.
2.). Pro*a*ilistic A!!roach
1onsider again the electricity market described in #ample. =ecall that this
market is a duopoly made up of producers A and B, in which reser!e
re%uirements are estimated by the system operator at 7/ $". &he reason for
this estimate is that the electricity demand may increase from +C/ $"h to +/
$"h without prior notice, and the system operator decides to protect the
electrical infrastructure against this une#pected growth of consumption by
scheduling 7/ $" of reser!e capacity in ad!ance. &he probability of this
happening is, though, relati!ely small, specifically /./. et us now rethink this
problem using a probabilistic approach. 2or this purpose, note that, in response
to a sudden increase of load, three different balancing actions may be taken,
namely
+. roducer A may increase its production from P A to
P A+r A . &he energy
increaser A is obtained from the reser!e capacity
R A scheduled beforehand
for this producer
7. ?imilarly, producer B may increase its production from PB to
PB+rB . &he
energy increaserB results from deploying the reser!e capacity
RB
dispatched beforehand for this producer.
C. A part of the load increase, Lshed
, may be simply not supplied. &his action,
howe!er, entails huge economic losses, which are estimated at H+/// / $"h.
Based on these three possible balancing measures, the energy-reser!e dispatch
problem can be reformulated as follows
M ∈.10 P A+30 PB+0 R A+25 RB+0.05 (10r A+30 rB+1000 Lshed )
s.t.
r A+rB+ Lshed=20,
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r A≤R A , rB≤RB ,
P A+ R A≤100,
PB+ RB≤100, Lshed
≤20,
P A , PB ,R A ,RB , r A , rB , Lshed
≥0,
&he solution to this problem is P A
¿
@/ $"h, R A
¿
7/ $", PB
¿
/
$"h, RB
¿
/ $",r A¿
7/ $"h, rB
¿
/, andshed∗¿
L¿ /. &herefore,
the energy and reser!e capacity dispatches, i.e., { P A
¿, PB
¿ }∧{ R A ,
¿ R B
¿ } , respecti!ely,
obtained from problem are the same as those resulting from problem in
pre!ious #ample &his is 0ust pure coincidence. Actually, these two models are
different inasmuch as the following
+. &his $arket-clearing problem takes into account e#plicitly both the
probability of occurrence of the 7/-$"h demand increase and its potential
impact on system operation costs through the utili3ation of balancing resources.
Indeed, the e#pression 0.05 (10r A+30r B+1000 Lshed
) represents the e#pected cost
incurred at the balancing stage. &his cost component is, in contrast, ignored in
pre!ious dispatch model
7. &he reser!e dispatch yielded by market-clearing model is directly determined
based on how !aluable this reser!e is to consumers by including the cost of the
e#pected load not ser!ed in ob0ecti!e function, where this cost appears as
0.05 (10 r A+30rB+1000 Lshed) . 2or the particular instance sol!ed abo!e, this cost
is e%ual to 3ero, meaning that consumers are willing to pay for 7/ $" of
reser!e capacity that can be deployed to satisfy a potential consumption
increase, if needed. In contrast, if the probability of occurrence of the 7/-$"h
demand growth is small enough, say /.//, or the !alue of lost load is
sufficiently low, e.g., H+//6$"h, no reser!e capacity is dispatched, i.e.,
{ R A ,
¿ RB
¿ }={0,0} and the whole demand increase is shed instead ( Lshed
7/
$"h), if it comes to it.
C. "hile the 7/-$" reser!e re%uirement enters pre!ious dispatch model as aninput in constraint, reser!e needs are an outcome of market-clearing in this
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model In fact, there is no reser!e re%uirement constraint in this problem.
Instead, we enforce constrain, in which all the !ariables in!ol!ed, namely,
r A , rB
and Lshed
, represent balancing energy %uantities. But if there is no such
reser!e re%uirement constraint, how do we determine the reser!e capacity priceJ
"e will get to the answer of this %uestion in due time.
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). Introduction to 'tochastic Progra##ing
nknown data abound in decision-making problems in the real world. &his lack
of perfect information is common in problems belonging to different knowledge
areas such as engineering, economics, finances, etc. 5ecision-making problems
in electricity markets are no e#ception. In fact, uncertainty is present in most
decision-making problems faced by electricity market agents. 2or e#ample,
electricity prices are unknown when agents ha!e to submit their offers or bids to
the pool. ?imilarly, at the time of procuring the energy needed to supply client
loads, retailers do not know precisely the electricity demands of these clients.
4owe!er, decisions need to be made e!en with lack of perfect information. &his
is what moti!ates the use of stochastic programming models for decision
making under uncertainty *.
$ost decision-making problems can be ade%uately formulated as optimi3ation
problems. If the input data of an optimi3ation problem are well-defined and
deterministic, its optimal solution (decision) is achie!ed by sol!ing the problem.
&he decision is then implemented to attain the best outcome. 4owe!er, more
often than not, the input data are uncertain but describable through probability
functions. In such a situation, it is not clear how the decision-making problem
should be formulated. >ne possibility is to substitute the uncertain input data
(describable through probability functions) by their corresponding e#pected
!alues, which results in a well defined and deterministic optimi3ation problem.
4owe!er, sol!ing such a problem may lead to a solution that once implemented
does not result in the best outcome. Alternati!ely, the probability distribution of
input data can be appro#imated by a collection of plausible sets of input data
with associated probabilities of occurrence. 2or instance, three sets of input data
with three !alues of probability of occurrence adding to +.
&hen a stochastic optimi3ation problem can be formulated implicitly weighting
(with the probabilities of occurrence) the indi!idual solutions associated with
each set of input data to achie!e a single solution that is the best in some sense
for all sets of input data. &hat is, we achie!e a solution that is ade%uately pre-
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positioned with respect to all the sets of input data, but not to any one of them
particularly. As a result of the uncertain input data being described by a
collection of different sets of data, the resulting ob0ecti!e function is uncertain
and needs to be characteri3ed as a random !ariable. ?ince such ob0ecti!e
function is not a real-!alued function but a random !ariable, the problem of
establishing a specific ob0ecti!e for the decision-making problem arises. >ne
alternati!e is to ma#imi3e the e#pected !alue of the ob0ecti!e function, other
one, to ma#imi3e the e#pected !alue of such function but limiting its !ariance,
etc. Implementing the solution obtained by sol!ing the stochastic problem
abo!e pre-positions the decision-maker in the best possible manner if
considering all possible input data sets duly weighted by their respecti!e
probabilities. &his solution is not the best for each indi!idual set of input data
but it is the best if all of them, weighted with their probabilities of occurrence,
are simultaneously considered. &he price to be paid for using a stochastic
programming approach is a dramatic increase in the si3e of the problem to be
sol!ed, which if handled without care may lead to intractability.
Illustrati!e e#ample from *'
An electricity consumer is facing both uncertain electricity demand and price
for ne#t week. 2or simplicity, we consider that both price and demand are
uncertain but constant throughout the week. ?cenario data pertaining to demand
and price are pro!ided in &able. Additionally, this consumer has the possibility
of buying up to / $" at H86$"h throughout ne#t week, by signing a
bilateral contract before ne#t week, i.e., before knowing the actual demand and
pool price it has to face. &he decision-making problem of this consumer can be
formulated as a two-stage stochastic programming problem. At the first stage,
the consumer has to decide how much to buy from the contract, and the second
stage reproduces pool purchases for each of the three considered demand6price
reali3ations (scenarios).
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&he node-!ariable formulation of this two-stage stochastic programming
problem is as follows
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Fariable PC
represents the power bought through the bilateral contract, while
!ariables P
1 , P
2 , and P
3 represent the power bought in the pool for
scenarios +, 7, and C, respecti!ely. &he ob0ecti!e function is the e#pected cost
faced by the consumer to supply its uncertain demand. owers are multiplied by
the number of hours in a week (+@) to obtain the energy consumed throughout
the week. &he first three constraints enforce energy supply for the three
scenarios, while the remaining constraints are the contract bounds and non-
negati!ity declarations for all the !ariables. Gote that the !ariables of this problem are associated with the nodes of the scenario tree and so the
denomination node-!ariable formulation. &he solution to this problem isC ∗¿ P
¿
@/, P
1
¿
C/, P
2
¿
7/, P
3
¿
/, which means that, before the week, the
consumer buys @/ $" using the bilateral contract, and during the week, C/, 7/
or / $" for demands (prices) ++/ (/), +// (8) or @/ (88) $" (H6$"h),
respecti!ely.
).1. ) + Bus 'yste# with and without %eser&e Bidding
&he proposed pricing scheme is illustrated ne#t using the three-node system
sketched in 2ig. +. ine reactances and capacities are all e%ual to /.+C p.u. and
+// $", respecti!ely. &he system includes three con!entional generators (<+,
<7, and <C) and one wind power plant ("). 5ata for the con!entional units
are pro!ided in &able I. Gote that, comparati!ely speaking, unit <+ is cheap, butinfle#ible; unit <7 is relati!ely cheap, but fle#ible; and unit <C is e#pensi!e, but
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fle#ible. &he wind plant is located at node 7. Its uncertain power output is
modeled by means of three scenarios, which are referred to as medium (C
$"), high (/ $"), and low (+/ $"), with probabilities of occurrence e%ual
to /., /.7, and /.C, in that order. &he power block offered by the wind producer
is assumed to be e%ual to its forecasted power production (i.e., C/. $"). &he
three-bus system also includes an inelastic load (C) of 7// $" located at node
C, with a !alue of lost load e%ual to H+///6$"h *.
&he market is cleared based on this information. $arket outcomes related to
dispatched %uantities and deployed reser!e are collated in &able II. &he
scheduled wind power production W qs
7/ $". &he resulting energy and
balancing prices are shown in &able III. Gote that electricity prices are the sameat all nodes in the system, because the network does not become congested in
any of the three considered wind power scenarios. <i!en the energy and
balancing prices in &able III and the dispatched %uantities in &able II, the
payments to market participants per scenario can be computed. 2or instance, the
payment to generator <C in scenario low is gi!en by 30×29+10×30=$ 1170
2urthermore, considering that the energy production cost of unit <C is e%ual to
HC/6$"h, the profit that it makes in scenario low is
1170−30×40=−30$ / Mw h
. &able IF pro!ides the benefit obtained by market participants both per scenario
and in e#pectation. >bser!e that the profit made by generator <C is indeed a
random !ariable whose e#pected !alue −30×0.5+120×0.2−30×0.3=0 .
<enerator <C can be seen then as the marginal unit in a stochastic sense. &he
randomness of its profit is inherited from the uncertain character of the reser!e
deployment ser!ice, which in turn depends on the actual wind power
reali3ation. &he proposed market settlement guarantees cost reco!ery for
generating units in e#pectation, but this does not pre!ent generator <C from
incurring economic losses in scenarios medium and low (see &able IF).
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&o reduce the risk of negati!e profits faced by market participants that are
willing to make real-time ad0ustments, reser!e capacity bids can be introduced
in the proposed market settlement as stated in ?ection II-A. 1onsider that
generator <7 offers both downward and upward reser!e capacity at a cost of
H+6$", while generator <C does it at a cost of H76$". &able F shows the day-
ahead schedule and the real-time redispatch in this case. &he wind power
production scheduled at the day-ahead stage is 7/ $" again. ikewise, &ables
FI and FII list, respecti!ely, the clearing prices and the profit made by market
participants per scenario and in e#pectation when the aforementioned reser!e
capacity bids are taken into account to clear the market. As an e#ample, obser!e
that the benefit of generator <C in scenario low is now gi!en by
40×30−40×30=$0 , where we ignore the KcostsL related to the reser!e
capacity bids inasmuch as the pro!ision of reser!e capacity does not entail
specific costs to generators. Gote, indeed, that generator <C does not incur economic losses in any of the three considered scenarios. 2urthermore, its
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Pricing Electricity in Pools with Wind Producers 20
e#pected profit is e%ual to /, i.e., greater than /. &herefore, the possibility of
bidding reser!e capacity ser!es to competiti!ely reward the capability of and
the willingness to make real-time ad0ustments, thus promoting the fle#ibility of
market participants in an efficient manner.
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,. "ase 'tudy + 1
&he pricing scheme described in ?ection II-B is further illustrated using a 78-
bus system based on the single-area !ersion of the I =eliability &est ?ystem
M+ *@. 2or simplicity, the generating units of this well-known system are
grouped by node and type. &he only purpose behind this grouping is to facilitate
the presentation and analysis of the simulation results. &hus, the simplified
system consists of C8 lines, +7 generating units, and +N loads. >n the
assumption of a perfectly competiti!e electricity market, the energy offers
submitted by generating units represent their marginal costs of energy
production, which are indicated in *@, &able FI. "e assume that nuclear and
hydro power producers offer their energy production at 3ero prices. &he amount
of reser!e capacity that each generating unit is willing to pro!ide, either
downward or upward, is listed in &able ID. "e assume that the nuclear and
hydro generators are not technically able to pro!ide reser!e. Go reser!e capacity
costs are considered.
&wo wind farms comprising 7.-$" wind turbines Gorde# G@/67// with a
hub height of +/ m are located at nodes N and @. &he power cur!e of this
turbine model is publicly a!ailable in *. "ind speeds at both wind sites are
described by means of the same "eibull distribution with scale and shape parameters e%ual to .N and +., respecti!ely. &his probability distribution for
wind speeds, in combination with the considered wind turbine model, results in
a capacity factor for both wind farms of appro#imately 8/O. &his capacity
factor has been estimated using the Wind Turbine Power Calculator pro!ided in
*. Besides, wind speeds at both wind sites are assumed to be correlated with a
correlation coefficient of /.. 1orrelated samples are then obtained by using the
sampling procedure described in *+/. An original set of +/ /// samples is first
generated and subse%uently reduced to +// by applying the scenario reductiontechni%ue proposed in *++ and *+7. ?electing the right number of scenarios
constitutes a tradeoff between model accuracy and tractability. "e belie!e that
current computational machinery allows considering a large enough number of
scenarios. Gote that the number of scenarios should be large enough so that
adding any additional scenario does not change the market outcomes
(preferably) or minimally changes them. "e assume that wind power producers
offer their forecast production at 3ero prices. Gote that nowadays this offering
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Pricing Electricity in Pools with Wind Producers 23
strategy constitutes a common practice for wind producers participating in
electricity markets.
&he results reported below correspond to one single period characteri3ed by a
total system demand of 7@/ $". &his demand is geographically distributedamong nodes as specified in *@, &able F. oads are assumed to be inelastic
with a !alue of lost load e%ual to H7///6$"h. =esults for two different wind
penetrations le!els, +7.CO and 7.CO, are presented. &he wind penetration le!el
is gi!en as the ratio of the installed wind capacity to the total system demand.
&he number of wind turbines in the farms at nodes N and @ that are re%uired to
achie!e these penetration le!els are 8/ and +//, and +// and 7//, respecti!ely.
&he market-clearing problem has been sol!ed using 1D ./.7 under <A$?
on a "indows-based personal computer Intel(=) 1ore(&$) i with four processors clocking at 7.8 <43 and <B of =A$. &he re%uired computational
time is around fi!e seconds.
2or low wind penetration le!els, such as +7.CO, there is enough room in the
transmission network to accommodate the energy transactions settled at the
market stage plus the subse%uent energy redispatches in the form of deployed
reser!e without the occurrence of congestion e!ents. &herefore, the wind energy
in0ected at nodes N and @ is able to reach e!ery node in the system and as aresult, no differences in prices e#ist among nodes. In particular, for a wind
penetration le!el of +7.CO, the energy price ( λn) is e%ual to H+.6$"h at
e!ery node. ikewise, the probability-remo!ed balancing prices under the
highest and lowest wind production scenarios, which will be denoted,
respecti!ely, by λnw
π w and λnw /π w hereafter, are H+./6$"h and H7+.N6$"h
in that order, irrespecti!e of the node under consideration. &hese probability-
remo!ed balancing prices are obtained by di!iding each dual !ariable λnw by
its associated probability π w . &his way, the energy prices and the probability-
remo!ed balancing prices are of the same order of magnitude. 2urther, the
balancing prices so transformed are dual optimal for the real-time market model
that results from problem (+) once the wind power uncertainty is disclosed and
first-stage !ariables (scheduled %uantities) are fi#ed to their optimal !alues. >n
the contrary, for high enough wind penetration le!els, e.g., 7.CO, network
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Pricing Electricity in Pools with Wind Producers 24
bottlenecks become probable and conse%uently, the nodal prices differ. As an
e#ample, &able D shows, for this wind penetration le!el, the !alues of the
energy price and the probability-remo!ed balancing prices under the highest and
lowest wind production reali3ations at some selected nodes. In light of these
prices, the following three obser!ations are in order
+. In the scenario of highest wind power production, the balancing
prices are 3ero at the nodes where the wind farms are connected,
namely, nodes N and @. &he reason for this is that any marginal
increment of load at these nodes is satisfied, in this scenario, by the
wind energy production that would be otherwise spilled due to the
network congestion.
7. &he balancing price in the scenario of lowest wind power production is node-independent. &his is so because, under this
scenario, the network does not become congested.C. !en though the scheduled productions do not cause network
congestion at the market stage, the energy price differs among
nodes. &his highlights the coupling between energy and balancing
prices induced by the two-stage stochastic programming approach.
Intuiti!ely speaking, the energy price anticipates probable network
bottlenecks during the real-time operation of the power system.
&he payments to market participants under the proposed pricing scheme are
indicated in &able DI for the two considered wind penetration le!els. "hile the
system operator makes payments to producers, it recei!es payments from
consumers. &his is why the payments to loads in this table are e#pressed in
negati!e numbers. >bser!e that if the wind penetration le!el grows, the
payments to con!entional producers diminish, whereas the payments to wind
producers increase. ogically, there is a transfer of re!enues from con!entionalgenerators to wind producers at the same time that the payments from loads are
reduced due to the free character of the wind energy. In either case, re!enue
ade%uacy in e#pectation is guaranteed. In fact, for a wind power penetration
le!el of 7.CO (condition such that network congestion e!ents are probable) the
system operator is e#pected to incur a financial surplus of HN7@.+.
astly, &ables DII and DIII pro!ide, respecti!ely, the e#pected profits achie!ed
by con!entional and wind producers under the proposed pricing scheme.
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>bser!e that all the participants reco!er their production costs in e#pectation,
thus making an e#pected profit greater than or e%ual to 3ero. In general, the
#pected profits of con!entional producers decrease as they are displaced from
the energy supply by an increasing wind power penetration. >nly generating
units C and 8 see their e#pected profit increased due to the fact that they get
more in!ol!ed in the deployment of reser!e with the increment in wind power
penetration.
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,.1. "onclusions and uture Work
+. &he proposed pricing scheme * is adapted to the specificities of wind
producers, characteri3ed by their !ariability and unpredictability. &hisconstitutes no harm to con!entional producers.
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7. &his pricing scheme is marginal and results in both cost reco!ery for
producers and re!enue reconciliation, both in e#pectation.C. &wo sets of marginal prices are deri!ed' pool prices that reflect energy
scheduling and balancing prices that reflect system operation.
8. &he proposed prices are deri!ed from the solution of an problem.&hus, they are obtained in an easy and robust manner.
. &he pricing scheme described in this paper does not embody non-
con!e#ities (e.g., start-up costs or minimum power output constraints).
2uture work is needed to incorporate such non-con!e#ities.
. "ase study + 2
=esults from a case study based on the single-area !ersion of the I
=eliability &est ?ystemM+ *@ are discussed in this section. 2or simplicity,generating units are grouped by type and node. &his way, 0ust one binary
!ariable is re%uired to determine the on6off status of each group of units.
2urther, the nuclear and hydro generators are considered must-run units. &hese
simplifications ha!e no purpose other than to alle!iate the computational burden
in!ol!ed in obtaining the results presented in this case study. By appealing to
the assumption that the electricity market is perfectly competiti!e, offers
submitted by generating units correspond to their marginal costs of energy
production, which are listed in *@, &able . &he generation mi# of the power system also includes two wind farms located at nodes N and @. &he same
"eibull distribution, with scale and shape parameters, and, e%ual to .N and +.,
respecti!ely, is used to model wind speed at both sites. &he two wind farms are
comprised of 7.-$" wind generators, model Gorde# G@/67// with a hub
height of +/ m. &he power cur!e of this turbine model can be found in *.
According to the Wind Turbine Power Calculator pro!ided in this reference, the
estimated capacity factor of both wind farms is appro#imately 8/O. "e
consider a system demand of 7@/ $", distributed among nodes as indicated in*C7, &able . oads are assumed to be inelastic. &herefore, the ma#imi3ation of
the social welfare in the market-clearing formulation *N boils down to the
minimi3ation of the operating costs.
Ge#t, we suppose a correlation coefficient between wind farms e%ual to /.@ and
we assess the impact of the wind power penetration le!el on $s in terms of
means and standard de!iations. 2or this purpose, we use +//// samples of the
wind farm power outputs in the simulation process. &his number of samples ishigh enough to pro!ide estimates for means and !ariances (the s%uare of
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standard de!iations) with a degree of precision (coefficient of !ariation of
estimates e#pressed in percentage) below +.+/ and 7.8O, respecti!ely, for all
the simulations carried out in this case study. In a!erage !alues, these numbers
drop up to /.8 and +./+O in that order. 2ig. 7(a) and (b) shows the e!olution of
the mean and standard de!iation of $s at nodes N, @, and 7/ as the
penetration le!el of wind generation in the 78-node system increases. Gote that
the wind power penetration le!el is e#pressed as a percentage of the total
system demand (7@/ $") and is augmented by increasing e%ually the number
of wind turbines installed in the two wind farms. &he choice of the nodes for
analysis is not arbitrary. Godes N and @ are those where the two considered wind
farms are placed, while node 7/ can be seen as representati!e of those buses
electrically far from the nodes where the wind generation is in0ected into the
power network. 2irst, the wind power production is used to displace part of the
energy supplied by the groups of +N-$" and +//-$" units placed at nodes
+C and N, respecti!ely. &hen, for a wind penetration le!el around +N.O, the
power produced by the wind farms in some scenarios is high enough to keep the
units at node +C shut down, pro!ided that the group of +7-$" units at bus + is
started up. &hese units are the most e#pensi!e ones in the system and as a result,
their utili3ation pushes the $s up and causes the sudden increase that can be
obser!ed in 2ig. 7(a). If the wind penetration le!el is increased a little more, up
to 7+O, the percentage of scenarios in which the +7-$" units at node + need
to be used drastically decreases and the a!erage !alues of $s start to drop
again as a conse%uence. $oreo!er, the e#clusion of these small units from the
energy dispatch also 0ustifies the sudden fall in the standard de!iations of $s
that can be appreciated in 2ig. 7(b). In addition, for low wind penetration le!els
+O , the wind energy in0ected at nodes N and @ is able to reach e!ery bus in the
system and conse%uently, the means and standard de!iations of $s are all
!ery similar. &he e#isting differences stem from the loss component of $s.
4owe!er, from a wind power penetration le!el higher than 7+O, the cur!es
represented in 2ig. 7 start to di!erge in a significant manner. &he different
beha!iors e#hibited by the means and standard de!iations of $s from this
point on are mainly due to the fact that situations in which the network becomes
congested begin to happen, or statistically speaking, begin to be probable. In
such situations, the network caps the amount of wind energy that can be
transferred from nodes N and @ to the rest of buses in the system and as a result,
the effect of wind generation on $s becomes locally accentuated. &his effect
is twofold' on the one hand, the probability of loads at nodes N and @ being
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Pricing Electricity in Pools with Wind Producers 29
supplied by free wind energy increases, with the conse%uent sharp decrease in
the means of the corresponding $s; on the other hand, the energy supply
from wind sources is inherently uncertain and such an uncertainty is passed on
to the $s in the form of a considerable increase of their standard de!iations.
In general terms, the impact of a growing wind generation on $s translates
into a decrease of their means, but an increase of their standard de!iations. 2or
instance, the coefficients of !ariation of $s at nodes N, @, and 7/ (defined as
the ratio of the standard de!iation to the mean) go from / for a /O wind
penetration le!el to , NN, and 7+O, respecti!ely, for a /O wind penetration
le!el.
&his subsection is intended to illustrate that correlation among wind sites can
ha!e a significant impact on $s and therefore should not be ignored when
assessing the economic repercussions of wind integration. &o this end, we
consider that the number of 7.-$" turbines installed in the wind farms at
nodes N and @ is +8/ and 7//, respecti!ely. &herefore, the total wind capacity
connected to the power grid is @/ $", which represents a wind penetration
le!el of almost C/O. 2ig. C(a) and (b) represents, respecti!ely, the means and
the standard de!iations of $s at nodes N, @, and 7/ as a function of the
correlation coefficient between wind farms. &he dashed lines ha!e been
obtained by linear regression, and their only purpose is to stress the general
trends e#hibited by the simulation outcomes. In accordance with the results
pro!ided in the pre!ious subsection, a wind penetration le!el of C/O is high
enough to produce e!entual network bottlenecks and hence the remarkable
differences e#isting among means and standard de!iations of different $s.
Gote that the correlation between wind farms has a minor impact on the means
and standard de!iations of $s at nodes N and 7/, but a considerable effect onthe mean and standard de!iation of the $ at node @. In numbers, if the
correlation coefficient is augmented from / to /., the mean of such an $
e#periences a reduction of +/.CO, whereas its standard de!iation suffers an
increase of C/.NO. ogically, power output fluctuations from wind farms fed
with uncorrelated winds cancel out and as a result, the o!erall wind generation
!ariability diminishes.
$oreo!er, due to the occurrence of network congestion, which particularlyaffects the transmission line connecting buses N and @, the impact of the
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Pricing Electricity in Pools with Wind Producers 30
correlation coefficient between wind farms on the $s should be locally
appraised. In this line, the a!erage !alue of the $ at node @ decreases as this
coefficient approaches +, because the percentage of scenarios in which the
power produced by the wind farm at this node is spilled increases considerably
from +/./NO ( ρ=0¿ to 7C.CO ( ρ=0.95¿ . &his remarkable growth of
wind spillage e!ents leads to a similar increase in the percentage of instances in
which the price at node @ is 3ero, specifically from +7.+O ( ρ=0¿ to 7N.+CO
( ρ=0.95¿ . In contrast, the mean of the $ at node N slightly increases with
the correlation coefficient between wind farms, because the a!erage power
generated by the thermal units at this bus also increases from .8 $" (
ρ=0¿ to @N.N $" (
ρ=0.95¿.
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Pricing Electricity in Pools with Wind Producers 31
.1. "onclusion
&his paper *N pro!ides a simulation methodology to assess %uantitati!ely the
impact of an increasing integration of wind power on electricity $s. &he
impact on both a!erage !alues and !olatilities is analy3ed. An increasing
amount of wind power integration results in lower $s throughout the
network until bottlenecks appear, which makes local the $ reduction
inherent to increasing wind power integration. A high correlation among wind
plants has an important impact on $ !olatilities and a reduced impact on
$ a!erage !alues. &his is a conse%uence of the fact that no-correlation
originates the statistical cancelling out of wind fluctuations and thus stable
a!erage !alues. &he methodology proposed in this paper allows !isuali3ing the
abo!e phenomena and, which is more important, calculating their actual
numerical impacts. As future works, we intend to contrast the results pro!ided
by the proposed methodology with the empirical analysis of real measurements
from di!erse power markets, and to e#tend the simulation algorithm to account
for inter-hour comple#ities such as the temporal correlations of wind speed
series and the ramping capabilities of generating units.
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* 5anish "ind Industry Association, "ind &urbine ower 1alculator.
*>nline. A!ailable' http'66guidedtour.windpower.org6en6tour6wres6pow.
*+/ 4. 4eitsch and ". =Umisch, K?cenario reduction algorithms in stochastic
programming,L Comput. !ptim. "ppl., !ol. 78, pp. +@NT7/,7//C.
*++ Q. 5upaVo!W, G. <rUwe-Suska, and ". =Umisch, K?cenario reduction in
stochastic programming' An approach using probability metrics,L #ath.
Program., !ol. , ?er. A, pp. 8CT++, 7//C.
*+7 . inson, 1. 1he!allier, and <. G. Sariniotakis, K&rading wind generation
from short-term probabilistic forecasts of wind power,L IEEE Trans. Power
yst., !ol. 77, no. C, pp. ++8@T++, Aug. 7//N.