An Improved Method to Calculate Generation Shift Keys · 2015-07-15 · 1 An Improved Method to Calculate Generation Shift Keys Kenneth Van den Bergha, Erik Delaruea, 2 3 aUniversity
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KULeuven Energy Institute
TME Branch
WP EN2015-12
An Improved Method to Calculate Generation Shift Keys
Kenneth Van den Bergh, and Erik Delarue
TME WORKING PAPER - Energy and Environment Last update: July 2015
An electronic version of the paper may be downloaded from the TME website:
http://www.mech.kuleuven.be/tme/research/
An Improved Method to Calculate Generation Shift Keys1
Kenneth Van den Bergha, Erik Delaruea,∗2
aUniversity of Leuven (KU Leuven) - Energy Institute & EnergyVille, Celestijnenlaan 300 box 2421, B-3001 Leuven,3
Belgium4
Abstract5
Transmission network constraints become increasingly relevant in generation scheduling models, given
the ongoing integration of market zones and the deployment of renewables in remote areas. However,
a full nodal network representation in generation models is often not possible due to computational
limitations. Therefore, reduced zonal network models are needed. A crucial step in the nodal-zonal
network reduction is the calculation of generation shift keys (GSKs). Generation shift keys denote
the nodal contribution to the zonal generation balance and are needed to compile different nodes into
one equivalent node. This paper discusses generation shift keys in detail and proposes an improved
method to calculate them. According to the improved method, the generation and load portfolio is
split up in different categories, and generation shift keys are determined separately for each category.
A case study of the central European electricity system indicates that the improved method is able to
approximate the nodal network without a considerable increase in computational cost.
Keywords: Network reduction, Generation Shift Key (GSK), Power Transfer Distribution Factor6
(PTDF).7
Nomenclature8
Sets9
I (index i) set of generation units
L (index l) set of lines in nodal network
K (index k) set of lines in zonal network
N (index n) set of nodes
T (index t) set of time steps
Y (index y) set of all categories
YG (index yG) subset of generation-related categories
10
∗Corresponding author: erik.delarue@mech.kuleuven.be (email address); +32 16 322 521 (phone number)
Preprint submitted to Electrical Power Systems Research June 1, 2015
YL (index yL) subset of load-related categories
Z (index z) set of zones11
Parameters12
AN,gen N-by-I matrix linking generation units to nodes
AZ,gen Z-by-I matrix linking generation units to zones
Bbus N-by-N bus impedance matrix
Bbus∗ (N-1)-by-(N-1) reduced bus impedance matrix
Bbranch L-by-N branch impedance matrix
Bbranch∗ L-by-(N-1) reduced branch impedance matrix
DN N-by-1 vector with nodal load
DZ Z-by-1 vector with zonal load
FN
L-by-1 vector with line capacities in nodal network
FZ
K-by-1 vector with line capacities in zonal network
FZ,0 K-by-1 vector with base case flows in zonal network
G I-by-1 vector with maximum power output of the generation units
GSK N-by-Z matrix with generation shift keys
GSKY N-by-Z matrix with generation shift keys of category Y
ISF L-by-N matrix with injection shift factors
ISF∗ L-by-(N-1) matrix with reduced injection shift factors
MC I-by-1 vector with the marginal generation costs
PTDFN L-by-N matrix with nodal power transfer distribution factors
PTDFN∗ K-by-N matrix with node-to-link power transfer distribution factors
PTDFZ K-by-Z matrix with zonal power transfer distribution factors
PTDFZ,Y K-by-Z matrix zonal power transfer distribution factors of category Y
13
Variables14
θ N-by-1 vector with nodal voltage angles
FN L-by-1 vector with line flows in nodal network
FZ K-by-1 vector with line flows in zonal network
G I-by-1 vector with power output of the generation units
PN N-by-1 vector with nodal power injections
15
2
PN,Y N-by-1 vector with nodal power injections of category Y
PZ Z-by-1 vector with zonal power injections
PZ,Y Z-by-1 vector with zonal power injections of category Y
16
1. Introduction17
A proper representation of transmission networks in electricity system models is becoming increasingly18
important. Given the integration of different market zones and the deployment of renewables in19
sometimes remote areas in the network, transmission constraints become more relevant and should20
hence be taken into account in operational and planning models of the electricity generation sector [1].21
A full implementation of the network in this kind of models is not always feasible due to the large size22
of real-life networks and the concomitant high computational cost. Therefore, reduced network models23
are needed, representing as good as possible the characteristics of the full network model without24
jeopardizing the computational tractability.25
Reduced network models can also be relevant for congestion management purposes in electricity mar-26
kets [2]. Policy makers might prefer a reduced network model to a full network implementation for,27
besides computational reasons, historical and socio-political reasons. For instance, current electricity28
markets in Europe are based on a strongly reduced model of the electricity network.29
A standard network reduction technique is equivalencing of the external network by computing impedances30
and eliminating unnecessary elements [3][4][5][6][7][8]. Equivalent networks have been used for short31
circuit analysis as they can reproduce the voltages and currents of the remaining buses. However,32
equivalent networks are not able to approximate flows of the eliminated branches. Therefore, the us-33
age of equivalent networks is limited in power flow analysis [9]. Another network reduction technique34
consists of grouping nodes in a limited number of zones, hereby reducing the number of nodes in35
the network [10][11]. A zone is assumed to be a copper plate, meaning that transmission constraints36
within a zone can be neglected. The remaining transmission lines between zones can be grouped in37
inter-zonal links. This second network reduction technique is referred to as the nodal-zonal reduction38
in this paper.39
The nodal-zonal reduction technique starts from a full nodal description and derives a simplified40
zonal network in three sequential steps. First, nodes with similar electric characteristics are clustered41
in zones [12][13][14][15][16]. Second, nodes within a zone are replaced by an equivalent node with42
- approximately - the same relationship between power injections in the network and power flows43
3
through the remaining inter-zonal lines [17][18][9][19]. Third, the remaining inter-zonal lines between44
two zones can be replaced by one equivalent inter-zonal link [12][20].45
A full non-linear AC power flow would be the most accurate network representation in electricity46
generation models [21]. However, due to the high computational cost of an AC power flow, a DC47
power flow is often preferred [22]. The DC power flow gives a linear relation between power injections48
and power flows by means of Power Transfer Distribution Factors (PTDF).49
This paper deals with the second step in the nodal-zonal reduction, i.e., grouping nodes in an equivalent50
node. A commonly used grouping approach is based on Generation Shift Keys (GSKs) [23]. GSKs are51
a mathematical expression of the spatial distribution of electricity generation (and load) within a zone.52
Each node within the zone contributes to the equivalent node in accordance with its GSK. The GSKs53
are an important parameter, influencing the nodal-zonal reduction to a great extent. Nevertheless,54
GSKs are underexposed in the literature. Typically, it is assumed that all nodes contribute equally to55
the equivalent node [9]. However, GSKs vary widely for different nodes in real-life electricity networks.56
Moreover, GSKs are time-dependent, as the spatial distribution of generation (and load) changes in57
time. This paper discusses the complexity of calculating GSKs and presents an improved method to58
determine them.59
The paper proceeds as follows. Section II presents in detail the nodal-zonal network reduction tech-60
nique. Section III discusses the challenges related to generation shift keys and proposes an improved61
method to calculate them. This improved method is evaluated in section IV based on a case study of62
the central European electricity network. Section V concludes.63
2. Nodal-zonal network reduction64
The nodal-zonal network reduction starts from a full nodal PTDF-matrix and ends up with a simplified65
zonal PTDF-matrix. The nodal-zonal reduction method is illustrated by a simple example (see Fig.66
1).67
2.1. Nodal PTDF-matrix68
The DC power flow is a linearization of the AC power flow equations, based on three assumptions:
(i) line resistances are negligible relative to line reactances, (ii) the voltage profile is flat and (iii) the
voltage angle differences between neighboring nodes are small. Given these assumptions, the static
4
(a) Nodal network.
(b) Zonal network.
Figure 1: Simple electricity network to illustrate the nodal-zonal network reduction. Generation unitsare located at node 1 (base load unit), node 2 (peak load unit) and node 4 (base load unit). Load islocated at nodes 2 and 3. Zone 1 consists of nodes 1 and 2, zone 2 of nodes 3 and 4.
AC power flow equation for active power injections simplifies to:
PN = Bbus θ (1)
The active power flow through a transmission line can be written as:
FN = Bbranch θ (2)
Substituting the voltage angles θ from Eqs. (1)-(2) gives the DC power flow equation:
FN = ISF PN (3)
or in scalar format:
FNl,t = ISFl,n PN
n,t ∀ l,∀ t (4)
with
ISF =[
0 ISF∗]
(5)
5
ISF∗ = Bbranch∗ (Bbus∗)−1 (6)
Since Bbus is a rank-deficient matrix, Eq. (6) can only be solved after removing the reference node.69
In this example, node 1 is denoted as reference node. The full LxN-dimension of the Injection Shift70
Factor matrix (ISF) can be restored by inserting a zero column in the reduced ISF-matrix (see Eq.71
(5)).72
An element in the ISF-matrix gives the sensitivity of the active power flow through line l with respect
to an additional power injection in node n and with the reference node as sink. Given the properties
of linearity and superposition, the sensitivity of line flows to power injections in node n1 with node
n2 as sink can be written as a linear combination of the ISF-elements with the reference node as sink
[24]:
PTDFNl,n1−n2
= ISFl,n1 − ISFl,n2 (7)
An element in the nodal PTDF-matrix (PTDFN) gives then the sensitivity of the active power flow73
through line l with respect to an additional power injection in node n1 and withdrawal at node n2. The74
nodal PTDF-matrix hence doesn’t depend on the chosen reference node, unlike the ISF-matrix. The75
nodal PTDF-matrix depends on the network topology but not on the operating point of the system76
[25].77
The before mentioned equations are valid for every time step. An additional equation is added to the
DC power flow equation to ensure a unique solution after removing the reference node from Eq. (6).
This equation imposes the sum of all power injections to be zero:
∑n
PNn,t = 0 ∀ t (8)
Consider the simple 4-node network in Fig. 1a. Assuming the same line susceptance for all lines (0.578
p.u.), defining positive flow directions as denoted in Fig. 1(a), and taking node 1 as the reference node,79
the ISF-matrix becomes:80
ISF =
0 −0.25 −0.75 −0.5
0 −0.25 0.25 −0.5
0 −0.25 0.25 0.5
0 0.75 0.25 0.5
(9)
6
The ISF-matrix indicates that a power injection of 1 MW in node 3 (see third column in ISF-matrix)81
with off-take in the reference node (i.e., node 1) results in a line flow of -0.75 MW in line A and 0.2582
MW in lines B, C and D.83
2.2. Clustering nodes into zones84
The first step in reducing the nodal to a zonal PTDF-matrix is defining the zones. Nodes should be85
clustered such that no congestion occurs within a zone, and that nodes within the same zone have86
a similar impact on the inter-zonal links. A well-known clustering approach is based on Locational87
Marginal Prices (LMP, also referred to as nodal prices). No congestion occurs between nodes with88
the same LMPs and hence these nodes can be grouped in one zone. Another clustering principle89
is based on nodal PTDFs. Nodes with similar nodal PTDFs are grouped in a zone. The before90
mentioned clustering approaches are based on electric characteristics of the network. However, in91
real-life examples, nodes are often clustered into zones based on administrative regions (e.g., one zone92
per country or province).93
Clustering algorithms are not part of this paper’s research question and it is assumed that the zones94
are already defined.95
In the simple example (see Fig. 1), nodes 1 and 2 are grouped in zone 1 and nodes 3 and 4 in zone96
2. The intra-zonal lines B and D can be removed from the ISF-matrix, resulting in the node-to-link97
PTDF-matrix:98
PTDFN∗ =
0 −0.25 −0.75 −0.5
0 −0.25 0.25 0.5
(10)
2.3. Zonal PTDF-matrix99
In a second step, the nodes within a zone are replaced by one equivalent node and the zonal PTDF-100
matrix is determined. The zonal PTDF-matrix gives the linear relation between the flow on inter-zonal101
lines and zonal power injections.102
The zonal PTDFs are derived from the node-to-link PTDF-matrix by means of Generation Shift Keys103
(GSKs). GSKs indicate the nodal contribution to (a change in) the zonal balance. As such, GSKs104
contain information about the spatial distribution of generation within a zone.105
In this paper, the GSKs are calculated as the nodal power injection divided by the zonal generation106
7
balance:107
GSKn,z,t =PNn,t∑
n∈z PNn,t
∀ z,∀n ∈ z,∀ t
GSKn,z,t = 0 ∀ z,∀n 6∈ z,∀ t(11)
The nodal power injection PNn,t can be positive (injection in the network) or negative (off-takes from108
the network). The sum of each column in the GSK-matrix is one. The GSK can not be determined in109
case of balanced zones (i.e.,∑
n∈z PNn,t = 0 ).110
GSKs have to be known a priori to derive a zonal network, whereas the zonal network is needed to111
determine the power injections and hence the GSKs. Therefore, GSKs are determined a priori based112
on expected nodal power injections. In this paper, the GSKs are based on a nodal simulation. In113
real-life applications, GSKs are determined based on simplified simulations and/or the expertise of114
system operators. However, the use of the nodal simulations to calculate the GSKs is justified since115
the aim of this paper is to compare an improved GSK-calculation method with the standard method,116
which can be done as long as both methods are based on the same assumptions.117
The zonal PTDF-matrix follows from the matrix multiplication of the node-to-link PTDF-matrix with118
the GSK-matrix. This multiplication indicates that the columns of the zonal PTDF-matrix are a119
weighted sum of the columns of the node-to-link PTDF-matrix, based on the spatial distribution of120
generation and load within a zone.121
PTDFZ = PTDFN∗ GSK (12)
Consider the simple example of Fig. 1 and assume that zone 1 exports 100 MW to zone 2 and nodal122
power injections are as follows: P1 = 120 MW, P2 = -20 MW, P3 = -150 MW, P4 = 50 MW. The123
GSK-matrix and the zonal PTDF-matrix for the simple example are, respectively:124
GSK =
1.2 0
−0.2 0
0 1.5
0 −0.5
(13)
PTDFZ =
0.050 −0.875
0.050 0.125
(14)
8
GSKs can correspond to a change of the system state relative to a so-called base case. One speaks125
then of an incremental GSK-matrix. A base case is, for instance, a system state in which each zone is126
balanced (i.e., no net export or import). An incremental GSK-matrix indicates the nodal contribution127
to a change in the zonal balance. A GSK-matrix can also correspond to a single system state (i.e.,128
absolute GSK-matrix), indicate the nodal contribution to the zonal balance. The zonal PTDF-matrix129
following from incremental GSKs is an incremental PTDF-matrix and only valid for the considered130
base case. The base case causes transmission flows, both within the zones and between the zones.131
Therefore, a term is added to the zonal DC power flow, corresponding to the inter-zonal flows in the132
base case. In case of absolute GSKs, the base-case flows are zero.133
The zonal DC power flow equation becomes:134
FZ = PTDFZ PZ + FZ,0 (15)
or in scalar format:
FZk,t =
∑z
PTDFZk,z P
Zz,t + FZ,0
z ∀k,∀ t (16)
The remainder of this paper focusses on absolute GSKs, but an analogous reasoning is applicable to135
incremental GSKs.136
3. Improved method to calculate GSKs137
The calculation of the GSK-matrix is a crucial but not straightforward step in the nodal-zonal network138
reduction. This section gives an overview of one of the main difficulties with GSK calculation, i.e.,139
time-averaging, and proposes an improved method to solve this issue.140
3.1. Time-averaging of GSKs141
The nodal-zonal network reduction is typically performed for multiple time steps at once (e.g., on a142
daily basis). In order to derive one zonal network which is valid for the whole considered time frame,143
time-independent GSKs are needed. However, GSKs can change from time step to time step as the144
spatial distribution of generation and load within a zone changes in time. Time-averaging of GSK145
results in a loss of accuracy of the zonal network.146
Fig. 2 shows for the considered case study of the central European electricity system (see section 4.1)147
the daily average GSKs and the standard deviation for the daily average GSKs. The daily average148
9
GSKs can be positive or negative; the absolute values of the daily GSKs are shown in Fig. 2. The149
standard deviation is a measure for time-variability. Fig. 2 indicates that certain nodes have a highly150
time-variable GSK. Both the average GSK as the standard deviation are expressed in percentage.151
0 50 100 150 200 250 3000
100
200
300
400
500
600
700
800
Daily average GSK [%]
Sta
ndar
d de
viat
ion
daily
GS
K [%
]
Figure 2: Daily average GSK (absolute value) and standard deviation for the simple zonal model (casestudy of Central Europe). The standard deviation is a measure for the variability in time of the GSKs.Certain nodes have a highly time-variable GSK.
Time-variability of GSKs is caused by changing nodal or zonal balances, and is enforced if the zonal152
balance is small compared to the nodal balance (see Eq. (11)). These effects are illustrated in Fig. 3153
for one particular node in the network. The nodal balance is constant, but the zonal balance varies154
during the day. Moreover, the zonal balance is smaller than the nodal balance and fluctuates around155
zero (i.e., the zone is a net exporter during certain hours of the day and a net importer during others).156
The result is a strongly fluctuating GSK.157
5 10 15 20−2000
−1500
−1000
−500
0
500
1000
1500
2000
2500
time step [h]
nodal balance [MW]zonal balance [MW]GSK [%]
Figure 3: The GSK of a particular node can vary strongly during a day if the zonal balance fluctuatesaround zero and is small compared to the nodal balance. The daily average GSK of the considerednode for the considered day is -230%, with a standard deviation of 520%.
10
Clearly, using one fixed GSK per day is a rather strong approximation for nodes with a highly vari-158
able GSK. Therefore, another method is required which results in less time-variable GSKs. Such an159
improved method is proposed in the following section.160
3.2. Improved GSKs161
The accuracy of the GSKs can be improved by increasing the number of zones or by shortening the162
time frame for which the zonal network has to be valid. Both improvements are mostly not possible163
as the number of zones and the time frame are set fixed (e.g., flow-based market coupling framework164
in the Central Western European region [26]).165
The GSK-matrix can also be improved by working with different GSK-matrices for different categories166
of generation and load units. Categories can refer to, for example, power injections from base load167
units, power injections from peak load units, load off-takes from industrial consumers and load off-takes168
from residential consumers.169
The spatial distribution of one category varies less in time than the spatial distribution of the whole170
generation and load portfolio together. Consider for instance base load units. The aggregated gen-171
eration from base load units is rather constant in time and the contribution of one base load unit to172
the aggregated base load generation is likely to be rather constant in time as well. Therefore, the173
GSKs for the category of base load generation vary little in time. As a result, less accuracy is lost174
with time-averaging a GSK-matrix which is specific for a certain category. When certain categories are175
predominant in some sub-regions, the category-specific GSK method is to a certain extent equivalent176
to dividing the system in smaller zones.177
The GSK for category Y can be calculated as follows:
GSKYn,z,t =
PN,Yn,t∑
n∈z PN,Yn,t
∀ z,∀n ∈ z,∀ t
GSKYn,z,t = 0 ∀ z,∀n 6∈ z,∀ t
(17)
with PN,Yn,t the nodal power injection of category Y. PN,Y
n,t is positive for injections in the network and178
negative for off-takes. Like in the basic GSK-method (see Eq. (11)), the nodal power injections are179
estimated a priori.180
The zonal PTDF-matrix is now different for the various categories:
PTDFZ,Y = PTDFN∗ GSKY (18)
11
The resulting DC power flow equation becomes:
FZ =∑Y
PTDFZ,Y PZ,Y + FZ,0 (19)
or in scalar format:
FZk,t =
∑z,y
PTDFZ,Yk,z PZ,Y
z,t + FZ,0z ∀ k, ∀ t (20)
The zonal injection of category Y, PZ,Yz,t , is positive for generation-related categories and negative for
load-related categories. The zonal injection per category is limited by the generation or load of that
category:
0 ≤ PZ,Yz,t ≤
∑n∈z
PN,Yn,t ∀ z,∀ y ∈ yG,∀ t
0 ≥ PZ,Yz,t ≥
∑n∈z
PN,Yn,t ∀ z,∀ y ∈ yL,∀ t
(21)
The net zonal injection is the sum of all zonal injection per category
PZz,t =
∑y
PZ,Yz,t ∀ z,∀ t (22)
The improved method is illustrated on the basis of the simple network in Fig. 1. Three different181
categories are considered: injections from base load generation (i.e., unit at node 1 and unit at node182
4), injections from peak load generation (i.e., unit at node 2), and load off-takes (i.e., nodes 2 and 3).183
The GSKs per category for this example are:184
GSKbase =
1 0
0 0
0 0
0 1
(23)
GSKpeak =
0 0
1 0
0 0
0 0
(24)
12
GSKload =
0 0
1 0
0 1
0 0
(25)
The zonal PTDF-matrices for the different categories then become:185
PTDFZ,base =
0 −0.50
0 −0.50
(26)
PTDFZ,peak =
−0.25 0
−0.25 0
(27)
PTDFZ,load =
−0.25 −0.75
−0.25 0.25
(28)
The DC power flow equation for the example in Fig. 1 becomes:
FZ = PTDFZ,base PZ,base + PTDFZ,peak PpeakZ
+PTDFZ,load PZ,load + FZ,0
(29)
4. Evaluation of improved GSK186
The central European electricity system is studied to evaluate the improved GSK-method.187
4.1. Case study - Central European electricity network188
The case study considers the interconnected high-voltage electricity system of 10 Central European189
countries: Austria, Belgium, Switzerland, Czech Republic, Germany, Denmark, France, Luxembourg,190
the Netherlands and Poland.191
The nodal network model contains 2,339 nodes, 3,367 line elements (transmission lines and transform-192
ers) [27]. The generation portfolio consists of 511 generation units with a total installed capacity of 205193
GW (82 GW nuclear units, 73 GW coal-fired units, 40 GW gas-fired units and 10 GW internal com-194
bustion engines). Residual load time series with an hourly time resolution are studied for two specific195
days: a high-load day with peak load of 200 GW (i.e., based on a working day in the winter time of196
13
2012) and a low-load day with peak load of 155 GW (i.e., based on a weekend day in the summer time197
of 2012) [28]. The residual load is the original electricity load minus generation from non-dispatchable198
renewable and cogeneration units. It is assumed that the residual load is fully inelastic.199
The optimal operational generation schedule for this electricity system is determined for three different200
network models:201
(1) a nodal network model as reference case;202
(2) a simple zonal network model with 40 zones;203
(3) an improved zonal network model with the same 40 zones as in the simple zonal network, but204
different GSK-matrices for 2 categories: generation-related and load-related injections.205
The nodes are clustered in 40 zones by a k-means algorithm with the nodal PTDF-values of the lines206
which are congested in the nodal simulation as observations. The resulting zonal network contains 315207
inter-zonal lines.208
The optimal generation schedule follows from from a deterministic generation dispatch model formu-
lated as a linear program in GAMS 24.2 and solved by CPLEX 12.6. The objective function is to
minimize the total operational system cost for the simulated days:
min∑i,t
MCi Gi,t (30)
The objective function is subject to the market clearing constraint (see Eq. (31) for a nodal and Eq.209
(32) for a zonal network model), the power plant generation limits (see Eq. (33)), the transmission210
line limits (see Eq. (34) for a nodal and Eq. (35) for a zonal network model), and the DC power flow211
equations (see Eqs. (4) and (36) for a nodal network, Eqs. (16) and (37) for a simple zonal network212
and Eqs. (20) and (37) for an improved zonal network). For the improved zonal network, Eqs. (21)213
and (22) are taken into account as well.214
AN,genn,i Gi,t = DN
n,t + PNn,t ∀n, ∀ t (31)
AZ,genz,i Gi,t = DZ
z,t + PZz,t ∀ z,∀ t (32)
14
0 ≤ Gi,t ≤ Gi ∀ i,∀ t (33)
−FN
l ≤ FNl,t ≤ F
N
l ∀ l,∀ t (34)
−FZ
k ≤ FZk,t ≤ F
Z
k ∀ k,∀ t (35)
∑n
PNn,t = 0 ∀ t (36)
∑z
PZz,t = 0 ∀ t (37)
4.2. Time variability of the GSKs215
The GSK-matrices for the case study are determined with Eq. (11) and Eq. (17), respectively for the216
simple zonal network and the improved zonal network. The results of the nodal simulation are used217
as estimation of the nodal power injections and off-takes.218
The zonal network model is determined on a daily basis. This implies that the GSKs are averaged219
over 24 hours, in order to derive a zonal network which is valid for a whole day. As mentioned before,220
this entails a loss in accuracy. The larger the variability in time of the GSKs, the larger the loss in221
accuracy due to time-averaging.222
Fig. 2 and Fig. 4 show the daily average GSK and the standard deviation in time for the simple223
zonal model and the improved zonal model, respectively. It is clear that the time-variability of the224
GSKs reduces drastically with the improved zonal method. The average GSK also becomes smaller in225
the improved method. In the simple method, GSK are calculated as the nodal injections divided by226
the zonal balance. The zonal balance can become small for almost balanced zones, resulting in high227
(average) GSKs. By splitting the zonal balance in zonal load and zonal generation in the improved228
method, the (average) GSK becomes smaller as the zonal generation or load is always at least as large229
as the nodal generation or load. The standard deviation of the GSKs in the improved method is230
maximal for average GSKs around 50%. For average GSK close to 0 (i.e., the node does not contribute231
to the zonal balance) or close to 1 (i.e., the node is the only contribution to the zonal balance), the232
standard deviation becomes zero.233
15
In short, the improved method thus results in less time-variable GSKs. This implies that less accuracy234
is lost by time-averaging the GSK-matrix.235
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
90
100
Daily average GSK [%]
Sta
ndar
d de
viat
ion
daily
GS
K [%
]
Figure 4: Daily average GSKs and standard deviation for the improved zonal network with 2 differentcategories. The standard deviation reduces drastically in the improved GSK-method, compared toFig. 2 (note the different scale of the axes).
4.3. Flows through the network236
The power flows through the network follow from the generation dispatch model. Depending on the237
network model, different flows are obtained. Table 1 shows the mean absolute error between the inter-238
zonal flows from a zonal and a nodal simulation. Table 1 indicates that the improved zonal network239
approximates the inter-zonal flows better than the simple zonal network.240
Working with GSKs for different categories (i.e., splitting generation-related injections from load-241
related off-takes) results in a more accurate time-averaging of GSKs, as seen in the previous subsection.242
However, introducing categories leads to additional freedom for the generation scheduling algorithm.243
With multiple categories, the algorithm has the freedom to choose which category exports to other244
regions (in case of generation-related injections) or imports from other regions (in case of load-related245
injections). For instance, consider a zone with a positive zonal generation balance of 100 MW, an246
aggregated zonal generation of 350 MW and a zonal load of 250 MW. In the simple zonal network,247
the zonal power injection PZz,t is 100 MW, whereas in an improved zonal network with generation-248
related and load-related injections, the zonal generation-related injection can vary from 0 to 350 MW249
and the zonal load-related off-take from 0 to 250 MW (as long as the net zonal balance remains 100250
MW). Depending on the specific PTDF-matrices for both categories, the algorithm can choose to251
minimize zonal import/export and cover the zonal load with zonal generation, or to maximize zonal252
16
import/export and import all zonal load and export all zonal generation, or any situation in between.253
Both situations will result in different flows through the network.254
Inter-zonal flows can deviate from the optimal flows as long as the line capacities are respected.255
However, this is not always the case. Table 2 shows how often a line overloading occurs if the zonal256
generation schedules are imposed to the nodal network. Line overloadings occur more often during the257
high-load day as the network is more used at these moments. Line overloadings occur also more often258
on inter-zonal lines than on intra-zonal lines. This can be explained by the fact that the zones were259
clustered such that intra-zonal congestion occured as little as possible. The improved zonal network260
model results in fewer line overloadings than the simple zonal network.261
MAE inter-zonal flow [MW] low-load day high-load daySimple zonal network 1,058 470Improved zonal network 266 316
Table 1: Mean absolute error (MAE) of inter-zonal flows, comparing the zonal with the nodal simula-tion results. A low-load and a high-load day are considered, each with an hourly time resolution.
Overloading penetration low-load day high-load dayrate [%] inter intra inter intraSimple zonal network 4.7 0.4 42.2 22.6Improved zonal network 3.0 0.1 30.1 14.1
Table 2: The overloading penetration rate is defined as the sum of overloaded lines over all time stepsdivided by the number of lines and the number of time steps (inter: inter-zonal lines; intra: intra-zonallines).
4.4. Zonal generation balances262
Different zonal import/export balances are obtained if different network models are used in the gener-263
ation scheduling model. Table 3 shows the mean absolute error between the hourly zonal generation264
balances from a zonal and a nodal network model. Again, the improved zonal network model outper-265
forms the simple zonal network model.266
The zonal balance error is larger in a low-load day than in a high-load day. In a high-load day, almost267
all generation units are used to fulfill load. Therefore, the generation scheduling algorithm has less268
options to optimize the generation schedule, leading to fewer deviations between the different network269
models.270
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MAE zonal balance [MW] low-load day high-load daySimple zonal network 777 440Improved zonal network 347 335
Table 3: Mean absolute error (MAE) of hourly zonal balance, comparing the zonal with the nodalsimulation result. A low-load and a high-load day are considered, each with an hourly time resolution.
4.5. Computational tractability271
The previous subsections have focussed on the accuracy of the zonal network models. This subsection272
deals with the computational cost (see table 4). Simulations were run on an Intel(R) Core(TM)273
i7-2620M CPU@2.7 GHz, 8 GB RAM.274
The size of the problem, i.e., the number of equations and variables, is an order of magnitude larger275
with the nodal network than with the various zonal networks. The improved zonal network results in a276
slightly larger optimization problem compared to the simple zonal network model. The run time with277
a nodal network is two orders of magnitudes larger than the run time with a zonal network model.278
The run times for the zonal network models increase by adding categories, but stay within the same279
order of magnitude.280
In this paper, an economic dispatch model, formulated as a linear program, is used. However, if one281
wants to include the dynamic constraints of the generation portfolio (e.g., minimum operation point,282
minimum down times), a mixed-integer program is needed, introducing binary variables in the op-283
timization problem. This increases the computational cost of the optimization problem drastically,284
which makes it more important to limit the size and the computational cost of the network representa-285
tion. Moreover, a rather short time frame is considered in this paper, i.e., one day. The computational286
cost increases exponentially if longer time frames are considered, again underlining the importance of287
computational tractable network models.288
Equations Variables Run timeNodal network 51.4·105 28.7·105 206minSimple zonal network 5.1·105 3.0·105 23sImproved zonal network 5.8·105 3.3·105 38s
Table 4: Computational cost of the different network models (generation scheduling optimization, 24time steps, 511 generation units, nodal network with 2,339 nodes, zonal network with 40 nodes).
5. Conclusion289
Reduced zonal network models are needed to take network constraints into account in generation290
scheduling models without losing computational tractability. A full nodal network can be reduced to291
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a zonal network model by clustering nodes into zones and replacing the nodes in the same zone by one292
equivalent node.293
Generation Shift Keys (GSK) are needed to group nodes into zones and determine the zonal Power294
Transfer Distribution Factors (PTDF). GSKs give the nodal contribution to the zonal balance, are295
estimated a priori and often averaged over multiple time steps in order to derive a zonal network296
model which is valid for multiple time steps.297
This paper presents an improved method to calculate GSK. According to this improved method,298
the injections and off-takes from the network are split up in different categories and the GSKs are299
determined separately for the different categories. A category can refer to, for example, power injections300
from base load generation or power off-takes from industrial consumers. The method is based on the301
insight that the GSKs from one category vary less in time than the GSK of the whole system. The302
improved method thus results in less time-variable generation shift keys and a lower loss in accuracy303
by time-averaging the GSK-matrix.304
The improved method is evaluated by means of a case study of the Central European electricity net-305
work. The study shows that an improved zonal network model, with generation-related injections306
considered separately from load-related off-takes, is able to better approach the flows and zonal gener-307
ation balances than a simple zonal network model with the same number of zones, and this with only308
a modest increase in computational cost.309
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