CC BY
23
UDC 519.85
Vestnik SibGAU Vol. 17, No. 1, P. 56-61
STOCHASTIC NETWORK EQUILIBRIUM SEARCH WITH APPLICATIONS IN THE GAS TRANSPORTATION MODEL OF RUSSIA
A. V. Kolosnitsyn
Melentiev Energy Systems Institute SB RAS 130, Lermontov Str., Irkutsk, 664033, Russian Federation E-mail: ankolos25@mail.ru
We suggest an equilibrium search methodology under uncertainty conditions using the example of gas transportation model of Russia. This model includes gas producers and consumers that are joined by the network. Two-stage approach to finding the network equilibrium is described in details. For the first stage we investigate the method of demand and supply functions forming for gas consumers and producers that let us to find the equilibrium price as well as production and consumption volumes of gas. On the second stage we formulate a problem offinding the optimal plan of gas transportation with the network constraints. Then we add to our model the case of demand uncertainty and state a problem of finding the stochastic equilibrium in the gas transportation model. Gas production volume and price which provide demand satisfaction with specified probability is determined in our model. Described method of finding the stochastic equilibrium is applied to the gas transportation model of Russia. Results of numerical calculations are also given in this paper.
Keywords: stochastic equilibrium, network model, chance constraints convex optimization problem.
Вестник СибГАУ Том 17, № 1. С. 56-61
ПОИСК ТРАНСПОРТНОГО СТОХАСТИЧЕСКОГО РАВНОВЕСИЯ С ПРИЛОЖЕНИЯМИ В ГАЗОТРАНСПОРТНОЙ МОДЕЛИ РОССИИ
А. В. Колосницын
Институт систем энергетики им. Л. А. Мелентьева СО РАН Российская Федерация, 664033, г. Иркутск, ул. Лермонтова, 130 E-mail: ankolos25@mail.ru
Рассматривается поиск равновесия в условиях неопределенности на примере сетевой газотранспортной модели России. Модель включает в себя поставщиков и потребителей природного газа, объединенных сетью транспортных линий. Подробно описывается двухэтапный подход к поиску сетевого равновесия. Для первого этапа разбирается способ формирования функций спроса и предложения для потребителей и поставщиков газа, а также поиск равновесной цены, объемов производства и потребления газа. На втором этапе ставится задача определения оптимального плана транспортировок газа с учетом сетевых ограничений. Рассматривается случай неопределенности спроса, и ставится задача поиска стохастического равновесия в газотранспортной модели. Определяются объемы производства и цена на газ, которые обеспечивают удовлетворение спроса с заданной вероятностью. Описанная методика поиска стохастического равновесия применяется к газотранспортной модели России. Приводятся результаты численных расчетов.
Ключевые слова: стохастическое равновесие, сетевая модель, задача выпуклой оптимизации с ограничениями по вероятности.
Introduction. We consider a gas network system in conditions of deregulated operation. This means that we have to take into account different (not necessarily antagonistic) aims of the participants of the system. The below suggested model is based on game-theoretical approach (see also [1]). The key point is consideration of stochastic gas demand uncertainty. In general, uncertainty is one of the main problems in energy system modelling [2]. We assume that uncertainty has stochastic nature, i. e. all uncertain parameters are random variables with known
distributions. Different approaches are used for studying games with stochastic uncertainty. In [3] authors base their investigations on Markov chains and stochastic dynamic programming. Random variational inequalities are used for finding the Wardrop traffic equilibria in [4]. Among other interesting and relevant topics the following can be mentioned: stochastic differential equations [5], stochastic Stackelberg games [6], generalization of the maxmin strategy [7; 8], stochastic Cournot models [9], two-stage stochastic programming model [10]. In our
approach we use standard stochastic programming technique [11; 12] and basic knowledge of probability theory [13]. Recent investigation concerning the paper topic can be found in [14] and [15].
The standard transportation model that joins gas producers and consumers and takes into account network constraints is considered. Minimizing total network costs, we obtain optimal gas transportation plan as a solution of the following mathematical programming problem:
min jz Cj (Xj ),
Ax = b, - d < x < d !
(1)
Here A is an m*n incidence matrix, which defines relations between m nodes and n transportation lines; xj -volume of the gas, transporting on the line j; cj(xj) - transportation costs; bi - volume of the gas production or consumption in the node i; dj - capacity of the line j.
Two stage approach to finding the network equilibrium. We assume that the considering model reflects operation in the market conditions. It means that supply and demand functions are given in every production and consumption node respectively. Having that data, we can build total demand and supply functions that let us to find an equilibrium price, production and consumption volumes of gas in corresponding nodes.
Let Ip be node set of producers, Ic - node set of consumers, I0 - set of the branching nodes. Production costs have nonlinear structure in general and for our purposes we represent it as the quadratic functions that practically appropriate:
) = «b + pA +li, i eIp
(2)
where ai > 0, Pi > 0, yi > 0, i e Ip - specified constants. Consumption linearly depends on market price p:
bi (p ) = ki - riP, j e Ic ,
where ki > 0, ri > 0, i e Ic - specified constants. The
following explanation can be given here to make justification to introduced functions representations. Production cost form is our assumption as the model designer, whereas consumption function form is standard assumption that reflects inverse dependency of the consumption volume from the price level.
In the branch nodes we have bj = 0, i e I0. One can naturally suppose that producers maximize their profit:
)Pbi -aibi -Pibi -"fi ^ maX U < a < A, i e T ,
(3)
wherep - market price on the product. Then using (2), (3) we can build supply function of every producer:
b* (P ) =
A, p < 2aiA_ p-P
2af
2aibi +рг. < p < 2aA + p;, (4)
bi, P ^ 2aibi +Pi, i e Tp•
Total supply equals to
^ (p) = Z b*( p). (5)
ielp
Total demand function equals to
D(p) = Zb = Z (k - rtp) = Zk - p z r.
ieIc ieIc ieIc ieIc
Intersection point of total supply and demand functions defines an equilibrium price p , which is set on the market of gas production and consumption:
P : ^(/) = D()•
(6)
We consider (6) as the key condition in defining the state of the network equilibrium in the represented model. Situation on the market of gas production and consumption in which total volume of production equals to the total volume of consumption taking into account network constraints will be understood to be the network equilibrium.
Note that considering market conditions it is quite reasonable to divide searching the network equilibrium into two stages. On the first stage we find equilibrium market price and also equilibrium volumes of gas production and consumption. To do this we need to implement next sequence of steps:
1. Considering gas production costs build supply function of each gas producer in the form of (4).
2. Build total supply function S(p).
3. Build total demand function D(p).
4. Using the relation (6) find the equilibrium pricep .
5. Determine equilibrium volumes of gas production for each provider:
b* = A
( P* ), i e Tp •
6. Determine equilibrium volumes of gas demand for each consumer:
b* = ki -r•P*, i e Ic.
On the second stage we optimize gas transportation volumes considering throughput of each transport line and demand satisfaction in every consumption node. We can represent mathematical model of this problem in the next form:
c(x) ^ min,
i z.*
ax = A ,
i u* ax = -A
i e I
p' i e L
(7)
a'x = 0, i e I, -d < x < d,
0'
where b*, i = 1, ...,m - equilibrium volumes of gas production and consumption that were obtained on the first stage. Solution of the problem (7) is optimal plan of gas transportation in the market conditions with network constraints.
Network equilibrium with demand uncertainty.
Sometimes we can come across the difficulty in determination of consumers demand functions. In this case we can consider its uncertainty.
In our model demand functions have the form bi = ki - rip, i e Ic. Now let the parameters ki and ri be random variables with normal distribution and known expectations and variances. Hence, bt (p), i e Ic and D(p) become random variables as well with normal distribution. This representation of demand function includes a particular case, when parameters r = 0, i e Ic. Then demand value of each consumer will be determined by the random variable ki, i e Ic with normal distribution.
In the uncertainty conditions we can set the problem of searching such price and corresponding gas production volumes that provide satisfaction gas consumption with the probability not less than given positive constant 5. It means that we move to finding stochastic equilibrium in our model. We can write this condition formally in the next form:
P{S(p) -D(p) > 0} >8.
(8)
®( x)-
1
•v/2K 0
e z /2dz.
® (S ( p)) = ®
( S(p) - M[D(p)] ^ VA[ D( P )]
>8,
(9)
transform the expression (9):
S(p) - M[D(p)] > S^A[D(p)], or S(p) > M[D(p)] + S&4A[D(p)],
where S8 : F(S8 ) = 8.
Total demand will be entirely satisfied with the probability not less then 5 with the price
p : S(p) = M[D(p)] + S^A[D(p)] .
Following the two stage approach of finding the stochastic equilibrium in describing model, now we need to solve gas transportation problem, using the obtained price and gas production volume. For this purpose it is quite convenient to use statement of the convex programming problem with chance constraints [11; 12]:
Z(S8- S8f )2 ^ min
ielc
ax = b*(p ), i e Ip,
P{-dx > bt (p)| > 8i, i e Ic, 8i e (0,1),
a'x = 0, i e I0, - y - d < x < y + d, y > 0. Here on the
(10)
one
hand,
expressions
We can rewrite (8) as F(S(p)) > 8, where F(S(p)) -is the distribution function of the random variable D(p) with fixed price.
It is well known that the probability of random variable realization from the interval (0, x) can be found using Laplace function [13]:
To use Laplace function we need to normalize the random variable D(p), that is count expression
(S (p) - M [ D( p)])/V A[ D( p)], where M [ D( p)] and A[ D( p)] - are expectation and variance of the random variable D(p) accordingly.
Using Laplace function we have
Py-a'x > bi (p)j>8i, i e Ic - are chance constraints,
reflecting the requirement of demand satisfaction in each consumption node with the determine probabilities 8j, i e Ic and with fixed price p . On the other hand, this constraints may be considered as distribution functions of the random variables bi (p), i e Ic.
It is important to note that for each consumption node the probability of demand satisfaction differs from the probability which is set on the first stage problem. It is related with weaker condition of demand satisfaction for whole model in general in comparison with the condition of demand satisfaction for each consumption node with the same probability. In order to find probabilities of demand satisfaction in every consumption node, we consider 8j, i e Ic to be the variables, that we try to approximate to the probability of demand satisfaction of the first stage problem as close as possible. Note that throughput constraints may prevent us from obtaining the solution of the problem (10), that is why we add variables yj , j = 1, ...,n , which give us the possibility to broaden
throughput of transport lines if it is necessary.
After determination of the values 8t, i e Ic, we can use the Laplace function for moving to the determine equivalent in the next form:
Z cj(xj) + Z hjyj ^ ^
j=1 j=1
ax = b*( p ), i e Ip, -a'x > bj, i e Ic , a'x = 0,
(11)
. i e I0, - y - d < x < y + d,
y > 0,
where bi = M\jbt (p)] + S^A[b; (p)] , i e Ic. Constants
hj , j = 1, ...,n - are capital costs for broaden transport
line j. Solution of the problem (11) is the optimal plan of gas transportation in the market conditions with the demand uncertainty.
Stochastic network equilibrium in the gas transportation model of Russia. Described approach to finding the stochastic equilibrium was applied to gas transportation model of Russia, which includes 70 transport lines and 51 nodes. There are 9 of these nodes that contain gas producers, 41 nodes with consumers and 4 branch nodes (fig. 1).
Fig. 1. Gas transportation system of Russia
Basic data of the model let us to set the production cost in the form (2) for every gas producer and build supply functions and total supply function in the form (5).
The demand functions were set for each consumer in two variants. In the first variant we set the deterministic parameters of the demand functions and then we obtained and solved deterministic problem of the finding the network equilibrium. In this case demand functions were formed automatically with using special technique described in [16] by means of solving next mathematical programming problem:
a'x = A i e I p
i p
a'x = A i e Ic a'x = A i e I, -d < x < d•
(12)
0'
bi (P) = ki - ri (p) = bi, = dbi ( p) p dp bi ( p)
i e L
(13)
For instance we need to deliver 27 units of gas to Tyumen, elasticity of demand equals -0.07 and shadow price obtained from problem (12) equals 44.3 conventional units. Use the system (13):
bj(44.3) = k - 44.3r = 27,
-0.07 =-
"1P
ki - ri P
We can define dual variables for equalities a'x = bi, i = 1, ..., m from the solution of the problem (12). These variables or shadow prices reflect the resource value for producers and consumers and equal for them out of the problem properties (matrix A is the incidence matrix). Denote this price as p. We can assume that the demand elasticity e = -0.07 is known (almost inelastic demand). Moreover we know the desirable consumption level from basic data of the model, denote it as bi, i e Ic. To set the consumption functions in the form bi (p ) = kt - rtp , i e Ic, we need to find parameters k and r , i e Ic from the following equations system:
Finally we get k1 = 28.89 and r = 0.043 . Consumption function in Tyumen equals to b1(p) = 28.89 - 0.043p.
Second variant of our gas system modeling included stochastic uncertainty of the demand functions that led to another solution of the stochastic network equilibrium problem. Both variants of problems were solved using program complex GAMS [17], the results are demonstrated in the fig. 2, where it is showed how demand uncertainty influence the main parameters of our model.
We obtained the following results for deterministic model: S(p) = D(p) = 354.3 billions m3, price for billion m3 of gas equals to 33.9 conventional units. For stochastic model we set the level of demand satisfaction with the probability equals to 90 % and we got S (p) = 370.4 billions
m3
and price equals to 36.7 conventional units. As we can see, considering demand uncertainty increase the price, supply of the gas and the network load in the transport model. We received 4.5 % of the price growth, 8.3 % of the supply volume growth and 37.5 % of the network load growth under uncertainty conditions with the 90 % of demand satisfaction probability (fig. 2).
Fig. 2. How demand uncertainty influence the main parameters of the model
Conclusion. We represented the methodology of the finding of stochastic network equilibrium with specified probability level for natural gas demand volumes. Our approach consists of two stages. On the first stage we defined the equilibrium volumes of natural gas consumption and supply and equilibrium price. On the second stage we obtained the optimal plan of gas transportation. To consider the gas consumption uncertainty we modified our two-stage approach to finding the network equilibrium and obtained problem statement for finding the stochastic network equilibrium that included convex programming problem with chance constraints. This problem was solved by program complex GAMS. Our method of finding the stochastic network equilibrium was successfully applied to the gas transportation model of Russia.
Acknowledgments. This work was supported by the grant of RFBR № 15-07-08986.
Благодарности. Работа поддержана грантом РФФИ № 15-07-08986.
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© Kolosnitsyn A. V., 2016
