1
Electricity Market Simulator to Assess High Shares
of Variable Renewable Energy
William Quiroz, Member, IEEE
Jose Koc, Member, IEEE
Faculty of Electrical and Electronic Engineering
National University of Engineering
Lima, Peru
Faculty of Electrical and Electronic Engineering
National University of Engineering
Lima, Peru
Abstract-- The high shares of renewable energy create
challenges in the operation of electrical systems and in market
designs, in this sense, the objective of this work is to develop a
model that can simulate an electricity market based on bids, which
has generation thermal, hydraulic, variable renewables and
battery energy storage systems (BESS) where energy and reserves
requirements are co-optimized. It is assumed that the firms
compete until reaching the Cournot-Nash Equilibrium in pure
strategies, which is obtained through the maximization of the
Nikaido-Isoda function and a relaxation algorithm for its solution.
In this way, with the results of generation, allocation of reserves
and prices, it will be possible to evaluate the market power in
systems that already have price offers established, as well as the
markets based on costs that wish to migrate to one of offers.
Potential mitigation of market power through the establishment of
bilateral contracts will also be evaluated. The market simulator
will be applied to a modification of the IEEE 24-bus reliability test
system and the results obtained will be analyzed.
the restrictions of the transmission system (markets in the US)
or carry out subsequent redispatches through the Transmission
System Operator (European markets). Under these schemes,
generators must internalize their direct and opportunity costs in
their offers, so that the problem of audited cost markets is
solved, if there is sufficient competition in the market.
However, in the first years in which the electricity markets
based on offers were established, in some countries the
enthusiasm led to overestimating the advantages of
competition, having little regulation in these as indicated by B.
Hobbs and S. Oren [1], which gave rise to the exercise of market
power. Among the problems identified, it is highlighted that the
exercise of market power not only depends on the market share
of the companies, but it also depends on the limits in
transmission capacity. Thus, in order to analyze the potential
exercise of power in the electricity sector, the development of
simulators is convenient, as indicated by S. Borenstein, J.
Bushnell and C. Knittel [2], since they allow predicting the
behavior of the signatures and forecast energy prices. These
electricity market simulators can also be used as a tool to
evaluate reforms in market designs, carry out market
monitoring, examine cases of mergers of business groups or
even used by generation firms for market evaluations and their
benefits.
Although the problems described in the previous paragraph
were resolved through regulatory measures, such as promoting
bilateral contracts in the case of California [3], the electricity
industry is involved in new challenges with decarbonization
policies and high penetration of variable renewable energies
(promoted through subsidies due to their high investment
costs), having an impact on spot prices and the operation of the
system, so the potential strategic behavior of these technologies
should be studied. Variable renewable plants have a very low
or even zero variable cost, thus displacing higher variable cost
plants in dispatch. However, the scarcity of solar and wind
resources throughout the day causes manageable plants such as
hydraulic and thermal generation to continuously change their
production level or be started and then stopped in order to
achieve the balance of generation demand. Thus, in periods
with an abundance of variable renewable generation and low
demand, the price of energy has low values, since the marginal
cost of production with variable renewable energies, can even
reach negative values [4] and [5], but the sudden lack of
Index Terms—Game theory, Cournot-Nash, Nikaido-Isoda
Function, variable renewable energy, market power, cooptimization, bilateral contracts.
I. INTRODUCTION
D
uring the process of deregulation and liberalization in the
electricity industry between the 80's and 90's, two ways of
organizing generation supply functions were stablished, based
on audited costs and based on bids. The first was established
mainly in Latin American markets (which have a high
percentage of hydroelectric generation), and its purpose is to
emulate a market of perfect competition, for which the
electricity system operator carries out an economic dispatch.
Despite the efforts made, this type of market can present
inefficiencies due to the asymmetry of information between the
operator of the electrical system and the generation companies,
so it is difficult for the operator to audit all the direct costs of
the generators such as the purchase of fuel and expenses
associated with maintenance, as well as the opportunity costs in
which they can incur, such as inflexibilities in fuel supply
contracts due to take or pay clauses. With regard to supply
markets, day ahead markets are usually established, so the
Power Exchange (PE) or the Independent System Operator
(ISO) is in charge of receiving the offers for each market
clearing period (generally, one hour) and close the market under
the criterion of maximum social benefit, which may consider
979-8-3503-4605-3/22/$31.00 ©2022 IEEE
2
renewable energy leads to a considerable increase in prices.
Also, due to the intermittence of the variable renewable
energies plants, in many cases there is no time to make
redispatches, then greater flexibility in the systems is required
to save operation security, which can be achieved through a
correct design of complementary services. Additionally, plants,
especially variable renewable ones, can be encouraged to invest
in BESS for better management of their available energy.
The regulatory framework must be technologically neutral
and provide adequate incentives in the short and long term, so
that variable renewable plants must also be exposed to market
signals as other generators. This is how the need arises to
produce a simulator that allows measuring the effect of
regulatory proposals regarding systems with variable renewable
energies. Among the measures recommended by IRENA [4]
and CEER [6], it is mentioned the establishment of a nodal price
system to face congestions, multiple settlement market that in
turn performs a co-optimization of energy and reserves, adapted
market settlement periods to make possible for a variable
renewable generator to adjust its offers according to changes in
weather conditions, no barriers to participation in capacity
payments and the promotion of Power Purchase Agreement
(PPA) in order to obtain certainty regarding and seeking
financing for investment.
Various types of simulators have been developed in the
literature depending on the purpose and desired scope, in which
those based on game theory stand out, however, the applications
to hydroelectric systems are smaller (it can be mentioned [7],
[8] and [9]) and even less for variable renewable generation,
where most of the simulators have been developed to find the
optimal offers of a firm (like in [10]). However, only in [8] and
[9] the modeling of the transmission network is considered and
in none of those mentioned are the reserves markets considered,
a simulator for purely thermal systems that includes balancing
services was proposed by [11], and uses a non-linear approach.
The main contributions of this paper are: 1) to directly apply
the Nikaido-Isoda function to find the Nash-Cournot
equilibrium in a pool-based market with co-optimization and
high shares of variable renewable energy, 2) to analyze the
bidding strategy of energy and reserves in a system with
thermal, hydro, variable renewable generators as well as BESS
and 3) to show the market power mitigating effect of bilateral
contracts.
The structure of this paper is the following. Section II
presents the Nikaido-Isoda function and the relaxation
algorithm that will be used for its solution. Then, the
mathematical formulation of the electricity market simulator is
described in detail in Section III. Section IV show the results
for two sensitivities applied to the IEEE 24-bus reliability test
system. Finally, Section V presents the conclusions obtained
from this paper.
II. DEFINITION AND CONCEPTS OF NIKAIDO-ISODA
METHODOLOGY
The developed simulator will evaluate pure strategic
behavior (as it is known in game theory), that would be adopted
by companies according to Cournot-Nash competition. In this
way, a Cournot-Nash equilibrium would be reached, when no
company has incentives to change its strategy unilaterally, since
it would be harmed. In the previous section, the importance of
the transmission network limits in the behavior of companies
has been mentioned, however, this brings a complexity to the
model, because the production of the plants jointly is involved,
this is known as coupled constraint [12]. Thus, the problem
cannot be addressed sequentially, where each company
optimizes its benefits separately. To address this problem, an
iterative solution strategy was applied in [9, 13] using the
function called Nikaido-Isoda. Another form of solution is to
treat the problem through complementary programming such as
Hobbs [14], however, due to the number of variables and
restrictions set out in this simulator, the first mentioned solution
strategy was chosen.
A. Nikaido-Isoda Function
According to game theory, a game with “n” participants can
be defined by a “triple”, in the form ⟨𝑁, (𝑋𝑖 ), (Φi ), 𝑖 𝜖 𝑁⟩,
where 𝑁 is the set of players 𝑁 = {1,2, … , 𝑁}, 𝑋𝑖 is the set of
strategies 𝑥𝑖 for player 𝑖 and Φi : 𝑋 → ℝ is the payoff function
of player 𝑖, so it assigns a real number to each element of the
cartesian product of the strategy spaces 𝑋 = 𝑋1 𝑥 𝑋2 𝑥 … 𝑥 𝑋𝑛 .
Then, let 𝑥 be the strategy profile belonging to 𝑋, containing
a list of possible individual strategies for each player 𝑖, such that
𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ), The set of elements (𝑦𝑖 |𝑥) =
(𝑥1 , … , 𝑥𝑖−1 , 𝑦𝑖 , 𝑥𝑖+1 … , 𝑥𝑛 ) is defined as the vector of strategies
where player 𝑖 decides 𝑦𝑖 , given a fixed set of strategies of the
other players.
A Nash Equilibrium in pure strategies will be 𝑥 ∗ =
∗
(𝑥1 , 𝑥2∗ , … , 𝑥𝑛∗ ) if it is true that, for each player, his payment
function is maximized, given the strategies of the rest of the
players as fixed, as represented in equation (1).
Φ𝑖 (𝑥 ∗ ) = 𝑀𝑎𝑥 Φ𝑖 (𝑥𝑖 |𝑥 ∗ )
(1)
The Nikaido-Isoda function ψ(𝑥, 𝑦) is given by equation
(3.10), which allows us to convert the equilibrium problem with
one objective function for each player into an optimization
problem with a single objective function, where the players
jointly optimize their production.
n
( x, y) =  i ( yi | x) − i ( x) 
(2)
i =1
Each summand of the Nikaido-Isoda function represents the
improvement in the profit of player 𝑖, when he changes his
strategy from 𝑥𝑖 to 𝑦𝑖 and the rest of the players keep their
strategy constant. In equilibrium, no player can increase his
benefit without harming another, therefore, it will be true that:
𝑀𝑎𝑥 ψ(𝑥 ∗ , 𝑦) = 0
(3)
Finally, the function that returns the best answers of the
players in the equilibrium is given by 𝑍(𝑥):
3
𝑍(𝑥) = 𝑎𝑟𝑔 𝑚𝑎𝑥𝑦𝜖𝑋 ψ(𝑥, 𝑦)
(4)
It is worth mentioning that in this problem the payoff function
evaluated at Φ𝑖 (𝑥) is a constant, the Nikaido-Isoda function in
(2) can be simplified to:
n
( x, y) =  i ( yi | x)
(5)
i =1
B. Relaxation Algorithm
The algorithm will converge to a Nash equilibrium if the
Nikaido-Isoda function of the problem to be solved must be
weakly convex-concave [15]. This condition is fulfilled by the
so-called smooth functions, which have continuous derivatives
in all orders. It should be noted that it is not a prerequisite for
the application of this methodology that the equilibrium be
unique, there may be various Nash equilibria, and the
convergence of the algorithm will occur at the point that is
closest to the initial condition.
To start the iterative process, an initial point 𝑥 0 must be
chosen, then it will be updated as a weighted average between
the optimal response function 𝑍(𝑥 𝑠 ) and the point 𝑥 𝑠 , according
to the following formulation:
𝑥 𝑠+1 = (1 − 𝛼𝑠 )𝑥 𝑠 + α𝑠 𝑍(𝑥 𝑠 )
𝑠 = 0,1,2,3, …
(6)
Where 𝛼𝑠 is a value between 0 ≤ 𝛼𝑠 ≤ 1, which can be a
constant or an optimized value for each iteration, however, for
the nature of the present optimization problem, a constant 𝛼𝑠
can be used.
III.
̅̅̅̅̅𝑓 ) and down (𝑅𝑈
̅̅̅̅̅𝑓 ). Prices are determined in the day ahead
(𝑅𝐷
market and a quadratic cost function is assumed for each
service. The total profit function of a firm  is given by
equation (7), where 𝐵𝐸 , 𝐵𝑅𝐷 and 𝐵𝑅𝑢 represent the net benefits
for selling energy, secondary reserve down and secondary
reserve up respectively.
̅̅̅̅̅𝑓 ) + 𝐵𝑅𝑈 (𝑅𝑈
̅̅̅̅̅𝑓 )
Φ𝑓 (𝑃̅𝑓 , ̅̅̅̅̅
𝑅𝐷𝑓 , ̅̅̅̅̅
𝑅𝑈𝑓 ) = 𝐵𝐸 (𝑃̅𝑓 ) + 𝐵𝑅𝐷 (𝑅𝐷
In addition, to represent the strategic behavior of the firms
due to the interaction with the transmission network, it is
necessary to disaggregate the power injected by a generator or
BESS into powers injected at system's purchase nodes [9,14].
̅̅̅̅̅𝑓 , ̅̅̅̅̅
In this sense, 𝑃̅𝑓 = [𝑇𝑃
𝐻𝑃𝑓 , ̅̅̅̅̅
𝑅𝑃𝑓 , ̅̅̅̅̅̅̅
𝐵𝐷𝑃𝑓 , ̅̅̅̅̅̅̅
𝐵𝐶𝑃𝑓 ] contains the
̅̅̅̅̅
̅̅̅̅̅𝑓 ), variable
power injected by thermal (𝑇𝑃𝑓 ), hydraulic (𝐻𝑃
̅̅̅̅̅
̅̅̅̅̅̅̅
renewable generators (𝑅𝑃𝑓 ) and BESS (𝐵𝐷𝑃𝑓 for discharge and
̅̅̅̅̅̅̅
𝐵𝐶𝑃𝑓 for charge) of a firm at system's purchase nodes. Then
̅̅̅̅̅
̅̅̅̅̅̅̅𝑓 , ̅̅̅̅̅̅̅
̅̅̅̅̅̅̅𝑓 , ̅̅̅̅̅̅̅
𝑅𝐷𝑓 = [𝑇𝑅𝐷
𝐻𝑅𝐷𝑓 ] and ̅̅̅̅̅
𝑅𝑈𝑓 = [𝑇𝑅𝑈
𝐻𝑅𝑈𝑓 ] contains the
̅̅̅̅̅̅̅𝑓 and
secondary reserve up and down given by thermal (𝑇𝑅𝑈
̅̅̅̅̅̅̅
̅̅̅̅̅̅̅
𝑇𝑅𝐷𝑓 ) and hydraulic generators respectively (𝑇𝑅𝑈𝑓 and ̅̅̅̅̅̅̅
𝑇𝑅𝐷𝑓 ).
Then, expressing in the most general way for firms that have
various technologies, 𝐵𝐸 is expressed as (8), while 𝐵𝑅𝐷 and 𝐵𝑅𝑈
have a similar formulation, and would be expressed as 𝐵𝑅 (9),
where ̅̅̅
𝑅𝑓 is secondary reserve up or down.
NH f
NR f
T BN 
 NT f
BE ( Pf ) =   E n ,t   TPi ,fn ,t +  HPj f, n ,t +  RPk f, n ,t
 i =1
t =1 n =1 
j =1
k =1


NB f
NB f
  T  NT f
+  BDPb ,fn ,t −  BCPb ,fn ,t   −    ci (TTPi ,ft ) (8)
 t =1  i =1
k =1
k =1
 

MODELING ELECTRICITY MARKET SIMULATOR
The simulator has to be applicable in the different current
electricity markets, so it must include thermal, hydraulic and
variable renewable generation. Likewise, BESS represents an
important equipment to manage the energy of a renewable
plant, it also will be included. Finally, the importance of
increasing the flexibility of the systems was seen, and in this
sense, ancillary services have great relevance, for this reason,
the market simulator must characterize a market with cooptimization, where firms can bid energy and reserves, as it was
suggested by the international literature described in Section I.
Within the reserves, only the secondary frequency reserve will
be auctioned, since the primary frequency response is usually
established in a mandatory manner for synchronous generators
(variable renewables plants are excepted) and can be modeled
by subtracting the reserve for primary frequency response
assigned from the capacity of the generator. The tertiary
frequency response will not be considered in this simulator for
practicality; however, it could be modeled as the secondary
reserve, it is also worth mentioning that certain markets do not
have this service. Finally, the BESS system of the simulator will
only be used by the plants to arbitrate energy.
A. Pool market without financial contracts
The firms will try to maximize their profits, which will be
given by the total energy injected (𝑃̅𝑓 ) and secondary reserve up
(7)
NH f
NH f

+  c j (THPj f,t ) +  ck (TRPk f,t ) 
j =1
k =1

NT f
NH f
T 


BR ( R f ) =   R t .   TRi f,t +  HR jf,t 


t =1 
j =1
 i =1


NH f
 NT f

−   cR i (TR jf,t ) +  cR j ( HR jf,t )  
 i =1

j =1

 
Where:
𝑖
𝑗
𝑘
b
𝑓
𝑡
n
𝑁𝑇𝑓
𝑁𝐻𝑓
:
:
:
:
:
:
:
:
:
𝑁𝑅𝑓
:
𝑁𝐵𝑓
:
𝑓
𝑇𝑃𝑖,𝑛,𝑡
:
𝑓
𝐻𝑃𝑗,𝑛,𝑡
:
(9)
Index for thermal generators of a firm
Index for hydraulic generators of a firm
Index for variable renewable generators of a firm
Index for BESS of a firm
Index for firms
Index for each period of time (one hour)
Index for nodes of the system
Number of thermal generators of firm f
Number of hydraulic generators of firm f
Number of variable renewable generators of firm
f
Number of BESS of firm f
Power injected for the thermal generator i of the
f firm, at node n, in the period t
Power injected for the hydraulic generator j of
4
𝑓
𝑅𝑃𝑘,𝑛,𝑡
the f firm, at node n, at period t
Power injected for the renewable variable
generator k of the f firm, at node n, at period t
Total power injected for the thermal generator i
of the f firm, at period t
Total power injected for the hydraulic generator
j of the f firm, at period t
Total power injected for the renewable variable
generator k of the f firm, at period t
Discharge Power for the BESS i of the f firm, at
node n, at period t
Charge Power for the BESS i of the f firm, at
node n, at period t
Secondary reserve up or down given by thermal
generator i at t period
Secondary reserve up or down given by
hydraulic generator j at t period
Price of energy at node n, in the period “t”
Price of secondary frequency response up or
down at period “t”
Cost function for energy
Cost function for providing secondary frequency
response up or down
:
f
i ,t
TTP
:
THPj f,t
:
TRPk f,t
:
𝑓
𝐵𝐷𝑃𝑖,𝑛,𝑡 :
𝑓
𝐵𝐶𝑃𝑖,𝑛,𝑡
:
𝑓
𝑇𝑅𝑖,𝑡
:
𝑓
𝐻𝑅𝑗,𝑡
:
ρ𝐸 𝑛,𝑡
:
ρ𝑅 𝑡
:
𝑐()
:
c𝑅 ()
:
Then, to find the Cournot-Nash Equilibrium, the NikaidoIsoda function must be constructed from the individual profit
functions of each firm, as it is shown in equation (10). The
strategy adopted by the firm f will be (𝑥̅𝑓 , ̅̅̅̅̅̅
𝑥𝑟𝑑𝑓 , ̅̅̅̅̅̅),
𝑥𝑟𝑢𝑓 in terms
of generation produced and secondary frequency response up
and down.
(
)
Max  x, P, RD, RU =
f
((
)(
Max  Pf , RD f , RU f | x f , xrd f , xru f

f =1
))
(10)
Linear demand functions will be assumed, so prices will have
the form of equations (11) and (12).
E n,t = An,t − Bn,t .Qnf,t
(11)
Rt = ARt − BRt .QRtf
(12)
Where:
NH f
NR f
NB f
F  NT f
Qnf,t =    xtih, n ,t +  xh hj , n ,t +  xrkh, n ,t +  xbdbh, n ,t

h  f  i =1
j =1
k =1
b =1
NB f
NH f
NR f
 NT f
−  xbcbh, n ,t  +  TPi ,fn ,t +  HPj f, n ,t +  RPk f, n ,t
 i =1
i =1
j =1
k =1

NB f
NB f
k =1
k =1
:
𝑓
:
𝑄𝑅 𝑡
𝑓
𝑥𝑡𝑖,𝑛,𝑡
Energy demand seen by firm f at node n at
period t
Secondary reserve up or down demand seen by
firm f at node n at period t
Strategy of power injected for the thermal
generator i of the f firm, at node n, in the period
t
Strategy of power injected for the hydraulic
generator j of the f firm, at node n, at period t
Strategy of power injected for the renewable
variable generator k of the f firm, at node n, at
period t
Strategy of discharge Power for the BESS i of
the f firm, at node n, at period t
Strategy of charge Power for the BESS i of the
f firm, at node n, at period t
Strategy of secondary reserve up or down given
by thermal generator i at t period
Strategy of secondary reserve up or down given
by hydraulic generator j at t period
:
𝑓
𝑥ℎ𝑗,𝑛,𝑡
:
𝑓
𝑥𝑟𝑘,𝑛,𝑡
:
𝑓
𝑥𝑏𝑑𝑏,𝑛,𝑡
𝑓
𝑥𝑏𝑐𝑏,𝑛,𝑡
𝑓
𝑥𝑡𝑟𝑖,𝑡
:
:
:
𝑓
𝑥ℎ𝑟𝑗,𝑡
:
Finally, it is necessary to list the restrictions to the
optimization problem formulated in (10). Firstly, the balance
between the total power injected by a generator or BESS and
the power injected in each node by them is expressed in
equations (15-19). The limits on the generation capacity and
reserves for each generator are shown from (20-28), it is
highlighted that the fact of lending reserves conditions the level
of generation, so that the generator incurs opportunity costs.
Another point to consider is that hydroelectric plants have
limited energy in the optimization horizon as seen in (29). The
restrictions associated with the BESS systems are found in (3034), within these the variable called state of charge of the
battery (SOC) stands out, which would vary in each period of
time depending on the charge or discharge of the BESS,
additionally it is observed that there are energy losses in the
process of both loading and unloading. At last, the restrictions
inherent to considering the transmission system are the nodal
balance and the maximum transmission capacity, which is
observed in equations (35) and (36), respectively, according
with a DC power flow. Then, the Nikaido-Isoda maximization
problem to find a Cournot-Nash equilibrium and will be solved
by the relaxation algorithm results as shown below.
f
((
)(
Max   Pf , RD f , RU f | x f , xrd f , xru f

f =1
(13)
) )
Subject to
BN
TTPi ,ft =  TPi ,fn,t
(15)
n =1
+  BDPb ,fn ,t −  BCPb ,fn ,t
NH f
NH f
 NT f
 NT f
QRf t =    xtri h,t +  xhrjh,t  +  TRi f,t +  HR jf,t
h  f  i =1
j =1
j =1
 i =1
𝑓
𝑄𝑛,𝑡
BN
THPj f,t =  HPj f, n,t
(16)
n =1
F
BN
(14)
TRPk f,t =  RPk f, n,t
(17)
n =1
BN
ℎ
:
Index for firms (same as 𝑓)
TBDPb,ft =  BDPb,fn,t
n =1
(18)
5
BN
TBCPb,ft =  BCPb,fn,t
(19)
n =1
TTP + TRU
f
j ,t
f
min j ,t
TTP
f
j ,t
 TTP
f
max j ,t
 TTP + TRD
f
j ,t
(20)
f
j ,t
THP
(22)
 THP + HRD
f
j ,t
f
i ,t
(23)
f
f
f
TRPmin
k ,t  TRPk ,t  TRPmax k ,t
f
min i ,t
 TRD  TRD
f
min i ,t
 TRU  TRU
TRD
TRU
f
i ,t
(24)
f
max i ,t
f
i ,t
(25)
f
max i ,t
(26)
f
min j ,t
 HRD  HRD
(27)
f
min i ,t
 HRU  HRU
(28)
HRD
HRU
f
j ,t
f
max j ,t
f
i ,t
T
THP
f
j ,t
t =1
f
max i ,t
t  HE jf
(29)
SOCbf,t +1 = SOCbf,t + (TBCPb ,ft cb − TBDPb ,ft db ) .t
(30)
SOCbf,1 = SOCbf,T +1 = SOC0f b
(31)
f
f
f
SOCmin
b,t  SOCb,t  SOCmax b,t
(32)
f
f
f
TBCPmin
b,t  TBCPb,t  TBCPmax b,t
(33)
f
min b,t
TBDP
NT f
 TP
f
i , n ,t
i =1
 TBDP  TBDP
f
b ,t
f
max b,t
NH f
NR f
NB f
j =1
k =1
k =1
−  BCP
k =1
f
b , n ,t
N
m ,t − n ,t
nm
zn − m
+ Qn ,t + PBase 
N
m ,t − n ,t
nm
zn − m
− PLmax m − n  PBase 
:
TBCPb,ft
:
𝑓
𝑇𝑅𝐷𝑖,𝑡
:
𝑓
:
𝑇𝑅𝑈𝑖,𝑡
𝑓
:
𝐻𝑅𝑈𝑗,𝑡
𝑓
:
SOCbf,t
:
SOC0f b
:
cb
:
d b
:
Qn , t
:
𝐻𝑅𝐷𝑗,𝑡
zn − m
:
PLmax m − n
:
Impedance of line between nodes n and m
Maximum transmission capacity of line
between nodes n and m
 PLmax m − n
B. Inclusion of financial contracts
As mentioned, one measure to mitigate market power is to
allow bilateral financial contracts to be signed between
generation companies and customers. Thus, this regulatory
measure must be considered in the simulator as it is present in
most electricity markets. To implement this in the simulator, 𝐵𝐸
must be adjusted, so that firm’s energy profits will now be the
sum of their net sales in the day ahead market and the sales by
financial contracts.
NH f
NR f
T BN 
 NT f
BE ( Pf ) =   E n ,t .   TPi ,fn ,t +  HPj f, n ,t +  RPk f, n ,t
 i =1
t =1 n =1 
j =1
k =1


NB f
NB f

+  BDPb ,fn ,t −  BCPb ,fn ,t − CQnf,t  

k =1
k =1
 
NT
NH
f
T  f
−    ci (TTPi ,ft ) +  c j (THPj f,t ) +
t =1 
j =1
 i =1
NH f
 T BN
ck (TRPk f,t )  +  CQnf,t C n ,t

k =1
 t =1 n =1
(37)
Where:
(35)
=0
(36)
Where:
TBDPb,ft
Angle at node n, at period t
(34)
+  HPj f, n ,t +  RPk f, n ,t +  BDPb ,fn ,t
NB f
:
(21)
f
THPj f,t + HRUi f,t  THPmax
j ,t
f
min j ,t
n ,t
Total discharge power for the BESS i of the f
firm, at period t
Total charge Power for the BESS i of the f firm,
at period t
Secondary Reserve Down given by thermal
generator i at t period
Secondary Reserve Down given by hydraulic
generator j at t period
Secondary Reserve Up given by thermal
generator i at t period
Secondary Reserve Up given by hydraulic
generator j at t period
State of charge of the battery b, of the firm f, at
period t
Initial state of charge of the battery b, of the
firm f
Constant of efficiency in the charge of the
BESS b
Constant of efficiency in the discharged of the
BESS b
Demand at node n, at period t. It is given by
total injected power to this node by all firms
CQnf,t
:
C n ,t
:
Amount of energy contracted by firm f, in at
the node n, at period t
Price of financial contract at node n, at period t
However, the profits from contracts turns out to be a constant
value, so these can be excluded from the objective function
since they do not influence the optimization problem.
IV. APPLICATION OF ELECTRICITY MARKET SIMULATOR
The methodology described will be applied to a modification
of the IEEE 24-bus reliability test system based on [16], the
participation of hydraulic generation, variable renewable plants
and BESS were added. At the same time, the capacity on the
transmission lines connecting the node pairs (15,21), (14,16)
and (13,23) is reduced to 400 MW, 250 MW and 200 MW,
respectively, in order to cause congestions. Within this system,
there will be four firms that will compete strategically in a pool
type market with co-optimization until the Cournot-Nash
equilibrium is reached, the detail of the generators is shown in
Table I, where it can be seen that variable renewable generation
represents 20% of the total installed capacity and its availability
in the day will depend on typical productions according to
variations in solar radiation and wind. The maximum capacity
of the thermal and hydraulic generators in Table I already
considers the margin for primary frequency response, with
respect to the variable renewable generators, they have no
obligation to provide primary frequency regulation.
Additionally, the cost function considered for energy and
reserves in the generators will be of the quadratic type. BESS
6
will be installed right next to the renewable power plants, so
that it allows them to have a better management of their
production, the characteristics of the these can be seen in Table
II.
The optimization horizon considered is one day, divided into
hourly periods. Electricity demand will be considered to be
linear in each system bus and it is built based on a typical load
diagram. Regarding the demand for reserves, operators may
require different levels of reserves throughout the day, based on
studies of typical variations in demand and variable renewable
generation, then in the present work two linear demand
functions will be considered for reserve up and down in one
day.
TABLE I
After clearing the market, it can be seen in Fig. 2 that for
each sensitivity there is a differentiation of prices in the nodes
due to the activation of congestion in the transmission system.
In the case without contracts, only line 14-16 reaches its
capacity limit, while in the case with contracts line 14-16 and
line 13-23 reach their limit. This is due to the fact that in the
case with contracts, a better allocation of resources is observed,
increasing the social benefit through a higher level of
production. From the same Fig. 2, it can be seen that in the case
with contracting, the average of the prices in each node
decreases compared to the case without contracts. In this way,
the impact of mitigating market power that the promotion of
bilateral contracts entails is verified.
Generation Plant Information by Firm
Energy Information
Plant owner Plant name
Firm 1
Firm 1
Firm 1
Firm 1
Firm 1
Firm 1
Firm 1
Firm 2
Firm 2
Firm 3
Firm 3
Firm 3
Firm 4
Firm 4
Firm 4
Firm 4
Firm 4
Firm 4
Thermal 1
Thermal 2
Thermal 3
Hydro 1
Hydro 2
Wind 1
Wind 4
Wind 2
Solar 1
Thermal 4
Thermal 5
Wind 3
Thermal 6
Thermal 7
Thermal 8
Thermal 9
Hydro 3
Solar 2
Node
1
2
7
18
21
23
16
5
21
13
15
7
15
16
18
21
22
3
TABLE III
Reserves Down/Up Information
A
RA
Maximum
B
Maximum
RB
power (MW) [$/MWh2] [$/MWh] power (MW) [$/MWh2] [$/MWh]
152
0.063
13.320
40
0.013
0.799
152
0.022
13.320
40
0.004
0.799
350
0.140
20.700
70
0.028
1.242
400
0.076
0.000
100
0.015
0.000
400
0.085
0.000
100
0.017
0.000
160
0.025
0.000
------125
0.007
0.000
------110
0.015
0.000
------180
0.035
0.000
------591
0.122
20.930
180
0.024
1.256
60
0.047
26.110
60
0.009
1.567
120
0.002
0.000
------155
0.026
10.520
30
0.005
0.631
155
0.030
10.520
30
0.006
0.631
310
0.012
10.520
60
0.002
0.631
350
0.095
10.890
40
0.019
0.653
400
0.040
0.000
80
0.008
0.000
160
0.012
0.000
-------
Bilateral Contracts Information
Firm
Peak Load
No Peak Load
Peak Load
Firm 2
No Peak Load
Peak Load
Firm 3
No Peak Load
Peak Load
Firm 4
No Peak Load
Firm 1
Firm
Firm 1
Firm 2
Firm 3
Firm 4
TABLE II
Hours
Hours
Peak Load
No Peak Load
Peak Load
No Peak Load
Peak Load
No Peak Load
Peak Load
No Peak Load
Energy selled per hour by Bilateral Contracts at Bus [MWh]
Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 7 Bus 8 Bus 9
40
80
50
80
50
100
50
100
50
30
70
50
30
50
30
50
50
50
Energy selled per hour by Bilateral Contracts at Bus [MWh]
Bus 10 Bus 13 Bus 14 Bus 15 Bus 16 Bus 18 Bus 19 Bus 20
150
20
50
200
20
50
Bus 1
50
50
50
70
70
70
60
60
100
100
150
200
40
70
BESS Information by Firm
Owner
Firm 4
Firm 2
Firm 2
Equipment
name
Node
BESS 1
BESS 2
BESS 3
5
21
3
/
,
,
40
60
50
20
35
30
0.9
0.9
0.9
0.95
0.95
0.95
Two sensitivities will be analyzed, in the first case all the
energy will be sold only in the pool market, while in the second
case bilateral contracts will be introduced in each of the firms
as shown in Table III, For the analyzed system, peak load hours
are considered to be from 4:00 p.m. to 8:00 p.m.
Fig. 2. Energy prices and power flow in relevant zones of the system
The levels of generation by technology in each sensitivity
are observed in Fig. 3 and Fig. 4. It is observed that the demand
exceeds 2400 MW in many periods in sensitivity with contracts.
The type of technology that mainly changed their production
levels is the thermal type, since they will seek to cover their
contracts through their available generation. With regard to
hydroelectric plants, a slight increase was obtained in peak hour
production.
Fig. 1. IEEE 24-bus reliability test system medicated
On the other hand, variable renewable cannot exercise
market by themselves in the proposed case of study, but they
7
can act strategically as a price maker when they have a BESS
installed. To demonstrate this, Fig. 5 shows the total production
of firm 2, which only produces from variable renewable
energies and also this firm has two BESS. This firm stores
energy in the hours of lower demand to produce it at the peak,
where the price of energy is higher and there is no solar
resource. Also, the evolution of the total SOC for the two BESS
that this firm has is shown in Fig. 6. It is observed that in the
sensitivity with contracts, the SOC tends to remain at a higher
level compared to the sensitivity without contracts.
Fig. 6. Total SOC of Firm 2’s BESS
Fig. 3. Total generation in sensibility without bilateral contracts
Regarding the up and down secondary reserve allocation, the
same results are obtained for both sensitivities. As mentioned,
with regard to the provision of reserves, decisions are given
based on the opportunity cost, in this sense, for the simulated
case it is obtained that it is mostly the hydroelectric plants that
provide service and only a thermal generator, which is one of
the most expensive in the system. This is due to the fact that, as
part of the strategy of the hydraulic power plants, it is to
properly manage their resource, and their operation is most of
the time at partial loads, thus these have the necessary margin
to provide reserves, for which the firms would prefer to operate
as much as possible its thermal units, a case that does not occur
with thermal generator 4, this also have margin to offer
reserves. It is worth mentioning that in a market of perfect
competition, the hydraulic power plants would save the energy
of their reservoirs to produce as much as possible during peak
hours of the system. In addition to this, in most of the cases,
secondary reserves of hydraulic units are slightly lower than the
thermal ones.
Fig. 4. Total generation in sensibility with bilateral contracts
Fig. 6. Total secondary reserve up assigned in both sensibilities
Fig. 5. Total generation and consumption of Firm 2
8
[9]
[10]
[11]
[12]
[13]
Fig. 7. Total secondary reserve down assigned in both sensibilities
[14]
V. CONCLUSION
The simulator developed in this paper is a useful tool for
analyzing the strategic behavior of firms in a pool-type
electricity market where there is a high amount of variable
renewable generation The simulator also considered the market
design recommendations given for this condition, which are the
nodal price system and co-optimization of energy and reserves,
so the generators can submit bids for energy and reserves. Firms
were assumed to compete with pure strategies until reaching the
Cournot-Nash Equilibrium, and the solution is obtained through
the maximization of the Nikaido-Isoda function through a
relaxation algorithm.
The effectiveness of the methodology for a modified 24 bus
IEEE system in terms of convergence was demonstrated, so it
can be applied to medium size systems despite the number of
constraints and variables involved. Through two sensitivities it
was possible to observe the influence of the transmission
system on prices, the use of water from hydroelectric plants, the
possibility of variable renewable generators to also act
strategically in the market through the use of BESS and that
reserves market clearing is given by direct and opportunity
costs. Likewise, the positive effect of the dissemination of
bilateral contracts was also corroborated, leading to a market
equilibrium where the social benefit is increased.
VI. REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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VII. BIOGRAPHIES
William Quiroz received the B.Sc. in electrical
engineering from National University of
Engineering, Lima Peru.
Currently he works at the Peruvian System
Operator and Power Exchange “COES”. His
experience and research interest include economic
operation, regulation of electricity markets and
optimization modeling.
Jose Koc received the B.Sc. in mechanics and
electrical engineering from National University of
Engineering, Lima Peru and the Master of
Engineering in Electric Power Engineering from
Rensselaer Polytechnic Institute.
Currently he is professor of electrical engineering
at National University of Engineering, Lima Peru.
His experience includes economic operation,
planning and regulation.