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

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: [email protected] (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 [email protected] 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

References310

[1] E. G. Kardakos, C. K. Simoglou, A. G. Bakirtzis, Optimal bidding strategy in transmission-311

constrained electricity markets, Electric Power Systems Research 109 (2014) 141–149.312

[2] A. Ehrenmann, Y. Smeers, Inefficiencies in European congestion management proposals, Utilities313

Policy 13 (2) (2005) 135–152.314

[3] J. B. Ward, Equivalent circuits for power-flow studies, Electrical Engineering 68 (9) (1949) 794–315

794.316

[4] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, O. Alsac, Numerical testing of power system317

load flow equivalents, IEEE Transactions on Power Apparatus and Systems 6 (1980) 2292–2300.318

[5] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, O. Alsac, Studies on power system load flow319

equivalencing, IEEE Transactions on Power Apparatus and Systems 6 (1980) 2301–2310.320

19

[6] E. Housos, G. Irisarri, R. Porter, A. Sasson, Steady state network equivalents for power system321

planning applications, IEEE Transactions on Power Apparatus and Systems 6 (1980) 2113–2120.322

[7] W. F. Tinney, J. M. Bright, Adaptive reductions for power flow equivalents, IEEE Transactions323

on Power Systems 2 (2) (1987) 351–359.324

[8] M. K. Enns, J. J. Quada, Sparsity-enhanced network reduction for fault studies, IEEE Transac-325

tions on Power Systems 6 (2) (1991) 613–621.326

[9] H. Oh, A new network reduction methodology for power system planning studies, IEEE Transac-327

tions on Power Systems 25 (2) (2010) 677–684.328

[10] J. B. Bart, M. Andreewsky, Network modelling for congestion management: zonal representation329

versus nodal representation, in: Proc. 15th Power Systems Computation Conference, 2005, pp.330

1–6.331

[11] A. Papaemmanouil, G. Andersson, On the reduction of large power system models for power332

market simulations, in: Proc. 17th Power Systems Computation Conference, 2011, pp. 1–7.333

[12] H. Oh, Aggregation of buses for a network reduction, IEEE Transactions on Power Systems 27 (2)334

(2012) 705–712.335

[13] M. Klos, K. Wawrzyniak, M. Jakubek, G. Orynczak, The scheme of a novel methodology for336

zonal division based on power transfer distribution factors, in: Conference of IEEE Industrial337

Electronics Society, 2014, pp. 1–7.338

[14] P. N. Biskas, A. Tsakoumis, A. G. Bakirtzis, A. Koronides, J. Kabouris, Transmission loss allo-339

cation through zonal aggregation, Electric Power Systems Research 81 (10) (2011) 1973–1985.340

[15] C. Kang, Q. Chen, W. Lin, Y. Hong, Q. Xia, Z. Chen, Y. Wu, J. Xin, Zonal marginal pric-341

ing approach based on sequential network partition and congestion contribution identification,342

International Journal of Electrical Power & Energy Systems 51 (2013) 321–328.343

[16] E. Shayesteh, B. F. Hobbs, L. Soder, M. Amelin, ATC-based system reduction for planning power344

systems with correlated wind and loads, IEEE Transactions on Power Systems.345

[17] X. Cheng, T. J. Overbye, PTDF-based power system equivalents, IEEE Transactions on Power346

Systems 20 (4) (2005) 1868–1876.347

20

[18] K. Purchala, E. Haesen, L. Meeus, R. Belmans, Zonal network model of european interconnected348

electricity network, in: in Proc. CIGRE/IEEE PES International Symposium, 2005, pp. 362–369.349

[19] M. Doquet, Zonal reduction of large power systems: Assessment of an optimal grid model ac-350

counting for loop flows, IEEE Transactions on Power Systems.351

[20] C. Duthaler, M. Emery, G. Andersson, M. Kurzidem, Analysis of the use of power transfer distribu-352

tion factors (PTDF) in the UCTE transmission grid, in: Proc. 16th Power Systems Computation353

Conference, 2008, pp. 1–6.354

[21] A. Kumar, S. Srivastava, S. Singh, A zonal congestion management approach using AC transmis-355

sion congestion distribution factors, Electric Power Systems Research 72 (1) (2004) 85–93.356

[22] B. Stott, J. Jardim, O. Alsac, DC power flow revisited, IEEE Transactions on Power Systems357

24 (3) (2009) 1290–1300.358

[23] M. Vukasovic, S. Skuletic, Implementation of different methods for PTDF matrix calculation in359

flow-based coordinated auction, in: International Conference on Power Engineering, Energy and360

Electrical Drives, POWERENG, IEEE, 2007, pp. 791–795.361

[24] T. Guler, G. Gross, M. Liu, Generalized line outage distribution factors, IEEE Transactions on362

Power Systems 22 (2) (2007) 879–881.363

[25] R. Baldick, Variation of distribution factors with loading, IEEE Transactions on Power Systems364

18 (4) (2003) 1316–1323.365

[26] Amprion, APX, Belpex, Creos, Elia, Transnet BW, EpexSpot, RTE, Tennet, CWE enhanced366

flow-based market clearing intuitiveness report, Available at www.elia.be/en/projects/market-367

integration/flow-based-marktkoppeling-centr-w-europaf (2013).368

[27] European Network of Transmission System Operators for Electricity (ENTSO-E), Study model369

2020 - Continental Europe, Available at request via www.entsoe.eu (2013).370

[28] European Network of Transmission System Operators for Electricity (ENTSO-E), Data portal -371

statistical database, Available at www.entsoe.eu/data (2013).372

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