The investigated model is a mathematical model in which operating power, installed power, power ows between nodes in electric power system (EPS) are optimized for the last year of the calculation period. The model is static, multinodal and it is represented as a large dimension linear programming problem. The aim of this study is analysis of the relationships between dual variables as the nodal and line prices.
To describe the model we need rst to describe sets, parameters, variables, objective function and constraints.
Sets:
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In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
J | set of nodes;
I | set of types of stations;
S | set of seasons (winter, spring, summer, autumn);
T | set of hours in days (
Node parameters: yj0i and yji | initial and maximum installed powers of station of type i2I in node j2J ;
0 xjist and xjist | minimum and maximum allowable operating powers of station of type i2I in node j2J in hour t2T of season s2S; vji | unit variable costs of station of type i2I in node j2J ; kji and bji | relative capital investments and unit xed costs of station of type i2I in node j2J ; Djst | consumer load in node j2J in hour t2T of season s2S.
Line parameters:
jj0 and jj0 | relative capital investments and unit xed costs for new and developing lines between nodes j and j0; ajj0 | maximum power line capacity between nodes j2J and j02J ;
jj0 | unit line losses between nodes j2J and j02J .
Other parameters:
| power reserve ratio (i.e. this is a power reserve coe cient for stations repair, emergency situations and etc.); sw ( sh) | equivalent number of working days (holidays) in season s2S (i.e. such a number of days which when multiplied by season maximum load results in electrical energy consumption equal to an accepted season value); f | capital recovery factor (CRF), f = (1+(1+)M)M1 ; where | discount rate, M | number of years, in which the capital is returned.
Capital recovery factor is calculated on the condition of capital recovery in equal parts G during M years with discount rate .
Node variables: yji | installed power of station of type i2I in node j2J ; xjwist(xjhist) | operating power of station of type i2I in node j2J in hour t2T of season s2S on working days (holidays).
Line variables: ajj0 | power line capacity between nodes j2J and j02J ; ujwj0st(ujw0jst) | operating power ow in the set from node j2J to node j02J (from node j02J to node j2J ) in hour t2T of season s2S on working days; ujhj0st(ujh0jst) | operating power ow in the set from node j2J to node j02J (from node j02J to node j2J ) in hour t2T of season s2S on holidays; uejj0st(uej0jst) | "emergency" power ow in the set from node j2J to node j02J (from node j02J to node j2J ) in hour t2Qs of season s2R.
The objective function: The objective function is a function of the total costs of the whole EPS and it has the form of: +f X X On operating power balance of power stations on working days and holidays (it is based on Kirchho 's rst law):
w i2I
h i2I
X
w ujj0st+
X
w uj0jst(1 j02J j06=j X h ujj0st+ j02J j06=j
X ujh0jst(1
On installed power in peak load hours:
The mathematical model is a linear programming problem. One needs to nd a
minimum of the objective function (
We derive the dual problem using technique from [7] based on the Lagrange function
for problem (
Dual function consists of several parts. The dual part corresponding to constraints on
electric power station development, left part of constraints (
X X
jwist:
X X X X
j2J i2I s2S t2T
jwist(xj0ist
xjwist) =
(14)
(15)
(16)
(18)
= X X X X h
The dual part corresponding to minimum power loading on holidays, left part of
constraints (
X X X X jhist(xj0ist
xjhist) = j2J i2I s2S t2T = X X X X h jhisti xjhist + X X X X The dual part corresponding to constraints on ows in power lines on working days, left part of constraints (8), dual variables jwj0st:
X X X X h The dual part corresponding to constraints on ows in power lines on holidays, left part of constraints (9), dual variables jhj0st:
X X X X h The dual part corresponding to constraints on "emergency" ows in power lines, left part of constraints (10), dual variables
jj0st: The dual part corresponding to constraints on "emergency" ows in power lines, right part of constraints (10), dual variables jj0st: j2J j02J s2R t2Qs
j06=j = X X X X
jj0stuejj0st j2J j02J s2R t2Qs j06=j
X X X X j2J j02J s2R t2Qs j06=j
jj0stajj0 : 0 0 The dual part corresponding to constraints on operating power balance of power stations on working days (11), dual variables jwst: The dual part corresponding to constraints on operating power balance of power stations on holidays (12), dual variables jhst:
X X X = X X X j0j) h jj0st w jj0st+ (30) (31) (32) + jwj0st)ujwj0st + X X X X( jhst j2J j02J s2S t2T
j06=j + X X X
X ( jst j2J j02J s2R t2Qs j06=j jh0st(1 j0j) jj0st + jhj0st)ujhj0st+ h
) j0st(1 j0j)
jj0st + jj0st)uejj0st ; j2J s2S t2T + X X X jstDjst(1 + ) + X X
X X X X
X X X X jst jist + jwist = 0; j2J; i2I; s2S; t2T ; w jist + jhist = 0; j2J; i2I; s2S; t2T ; h X X
ji + ji = 0; j2J; i2I; s2R t2Qs (35) (36) (37)
X X s2R t2Qs w jst jst j2J; j2J; j2J; jst ji To give an interpretation of the above approach we consider an illustrative example. The network consists of three nodes: J = f"V olga"; "Center"; "South"g. There are two periods of time: T = f0; 1g. Period 0 is a peak period. Assume that f =0.02, =0.025, =0.1. There are connections between the lines ("Volga" { "Center", "Center" { "South", "South" { "Volga"). The node and lines parameters of the model are presented in Tab. 1{2. The values of the primal and dual variables are presented in Tab. 3{6. It is shown in [3, 4] that the dual variables corresponding to every constraint in the primal problem have a certain conceptual meaning. Let us consider some of the most
xj0
prices vary in time. The concept of nodal prices is well studied. In the very beginning
the nodal prices were proposed and developed by Schweppe and his collaborators in [6].
For this example the equations (35){(36) are wrote in the following form:
Substitute the value of primal and the dual variables in the equation (48):
Node "Volga" at 0 and 1 hours respectively:
vj
jt
jt + jt = 0 :
(48)
Node "Center" at 0 and 1 hours respectively:
Node "South" at 0 and 1 hours respectively:
640
960
The prices in the node "Volga" at 0 and 1 hours are higher than unit variable costs.
Producers get producer surplus: 857.60 at 0 hour and 412.80 at 1 hour per unit. Also
producers get producer surplus in the node "South" at 0 hour and it equals 85.56 per
unit. In the node "Center" at 0 and 1 hours, in the node "South" at 1 hour there is no
producer surplus, the electricity hour price equals unit variable costs.
Now we consider equations (39){(40). They are wrote in the following form:
jt
j0t(
0:025) + 94:60 = 0 : We can see that the price in the node where power come is higher than the price in another node for the value of line losses. Moreover, if the line is used on maximum capacity the consumer pays for the line. When transferring power from the node "South" to the node "Center" the consumers pay 132.04 at 0 hour and 152 at 1 hour per unit for the use of line. 5
In conclusion it is possible to state the following: