Appendix B B
Modelling the dynamics and fragmentation of electric power systems
M´rk Szenes, Z´n´ Farkas, G´bor Papp a e o a
Collegium Budapest, H1014 Budapest, Szenth´roms´g u. 2. a a E¨tv¨s Lor´nd University, H1117 Budapest, P´zm´ny P. stny. 1/A o o a a a
December 2009
We have analysed the regular and breakdown dynamics of power grids. In the MANMADE project, we have spent considerable time on formulating a model for this purpose. Such a model should be simple, yet capable of capturing the
Modelling the dynamics and fragmentation of electric power systems
M´ark Szenes, Z´en´o Farkas, G´abor Papp
Collegium Budapest, H1014 Budapest, Szenth´aroms´ag u. 2.E¨otv¨os Lor´and University, H1117 Budapest, P´azm´any P. stny. 1/A
December 2009
We have analysed the regular and breakdown dynamics of power grids.In the MANMADE project, we have spent considerable time on formulating a model for this purpose. Such a model should be simple, yet capableof capturing the most fundamental properties of power ﬂow. For the betterpart of the project, we used the DC load ﬂow model [1]. However, this modelturned out to be incapable of taking into account the maximum transmissioncapacity of power lines correctly. At that point switched to a linear programming model, which proved to be an excellent tool for our investigations. Thedetails of the model is described below in detail.
1 Optimal power ﬂow
We analyzed the load ﬂow problem and studied the cascading breakdownphenomena with linear programming method. The objective function represents the generation and transmission costs that has to be minimized. Theset of equality constraints represents the inﬂow–outﬂow balances, and the setof inequality constraints represents the generating capacities of power plantsand the loadability limits of transmission lines. We can formulate the above1
B
Appendix B
mentioned linear programming problem:min: (
{
P
i

i
∈
V
g
}
,
{
F
ij
}
)
−→
i
∈
V
g
K
gi
·
P
i
·
∆
t
+
ij
K
tij
·
F
ij
·
∆
t
(1)
P
i
=
j
∈{
ij
}
F
ij
(2)
P
i
≤
C
gi
(
i
∈
V
g
) (3)

F
ij
 ≤
C
tij
(4)1.
P
i
denotes the actual power produced/consumed by the power plant/consumer
i
, measured in MW.2.
F
ij
is the power ﬂows from
i
to
j
, measured in MW.3.
K
gi
is the generation cost of the power plant
i
(which is element of theset of the power plants
V
g
), measured in EUR/MWh.4.
K
tij
is the transmission cost. For the sake of simplicity we choose it todepend only the transmission line length
l
ij
, so
K
tij
=
K
t
·
l
ij
(the unitof
K
t
is EUR/MWh
·
km).5.
C
gi
is the nominal capacity of power plant
i
, measured in MW.6.
C
tij
is the line loadability, measured in MW.The advantage of the application of linear programming technique described above is that it takes into account economic considerations undergiven physical constraints. Thus this method fairly reproduced the operation of the power transmission system operator that is responsible for themost eﬀective and reliable distribution of power.
2 Parameterizing the model
Our electricity network database [4] contains the information about:
ã
the network topology – the set of edges
{
ij
}
and nodes
{
i
}
(withcountry information)
ã
the length (
l
ij
) and the voltage level (
U
ij
) of transmission lines
ã
the fuel type (nuclear, coal, natural gas, fuel oil, lignite, wind, biomass,hydro, etc.) and the nominal capacity of power plants (
C
gi
)2
ã
the population belongs to the nearest substationHourly consumption data of European countries available from the page of the European Network of Transmission System Operators [2]. Combinedwith the population information we assigned the consumption values (
P
i
) tothe consumer nodes in the ratio of the corresponding populations.The determination of the generation and transmission costs (
K
g
and
K
t
)is based only on expert estimation which doesn’t take into account politicaland geographical and such speciﬁc factors that can aﬀect the real costs. Inour model the generation cost depends only on the fuel type of the powerplant.
3 Line loadability
For the purpose of determining the line loadability, we apply the methodsrcinally proposed by St. Clair [6], and which later on analytically derivedby Dunlop et al. [5]. Although the method has some limitations, and isbased on several assumptions (like the neglect of resistance, the terminalsystem impedance and the eﬀect of series or shunt compensation etc.), it is agood approximation for quickly estimating the line loading limit. The paperscited above showed that the loadability characteristics for uncompensatedhigh voltage transmission lines is universal, the maximal power in units of surge impedance loading (SIL) is independent of voltage levels, and dependsonly the line length. Three factors inﬂuence the maximal power that can betransmitted, these are:1. the thermal limitation;2. the linevoltagedrop limitation;3. the steadystatestability limitation.The thermal limitation is relevant only for lines shorter than 80 km, andwithin this range the maximal power is approximately 3
·
SIL. The maximum allowable voltage drop along the line is 5%, and relevant in the 80–320 km region. The steadystatestability limitation is important for lineslonger than 320 km. The steadystatestability margin is deﬁned as 100%
·
(
P
max
−
P
limit
)
/P
max
and it is assumed to be 35% (corresponds to
δ
= 40
.
5
◦
power angle). Fig. 1 shows the loadability curve, which we used to determinethe power transmission capability of the transmission lines in our load ﬂowsimulation. This is the so called St. Clair curve which gives the load carryingcapability in the units of SIL. We applied the following typical SIL values(Tab. 6.1 in [3]):3
02004006008001000
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5
Transmission line length (km)
P
m a x
S I L
35% steady−state−stability margin5% voltage−drop marginSt. Clair curve
Figure 1: Loadability curve for uncompensated overhead transmission linesVoltage levels 230 kV 345 kV 500 kV 765 kV 1100 kVSIL 140 MW 420 MW 1000 MW 2280 MW 5260 MWFor other voltages we interpolated the SIL values correspond to the nearestvoltage levels.
4 Breakdown process
A failure in the power network can cause successive failures which can propagate to the whole system. For example, when a transmission line (power4