Spreading of Disturbances in Realistic Models of Transmission Grids: Dependence on Topology, Inertia and Heterogeneity

Kosisochukwu P. Nnoli Jacobs University Bremen Stefan Kettemann (  [email protected] ) Jacobs University Bremen

Research Article

Keywords: Disturbances, Transmission, RER, Inertia

Posted Date: August 4th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-769886/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Spreading of Disturbances in Realistic Models of Transmission Grids: Dependence on Topology, Inertia and Heterogeneity

Kosisochukwu P. Nnoli1 and Stefan Kettemann1,2*

1Department of Computer Science and Electrical Engineering, Jacobs University Bremen, 28759 Bremen, Germany 2Department of Physics, Jacobs University Bremen, 28759 Bremen, Germany and Division of Advanced Materials Science, Pohang 790-784, South Korea *[email protected]

ABSTRACT

The energy transition towards more renewable energy resources (RER) profoundly affects the frequency dynamics and stability of electrical power networks. Here, we investigate systematically the effect of reduced grid inertia, due to an increase in the magnitude of RER, its heterogeneous distribution and the grid topology on the propagation of disturbances in realistic power grid models. These studies are conducted with the DigSILENT PowerFactory software. By changing the power generation at one central bus in each grid at a specific time, we record the resulting frequency transients at all buses. Plotting the time of arrival (ToA) of the disturbance at each bus versus the distance from the disturbance, we analyse its propagation throughout the grid. While the ToAs are found to be distributed, we confirm a tendency that the ToA increases with geodesic distance linearly. Thereby, we can measure an average velocity of propagation by fitting the data with a ballistic equation. This velocity is found to decay with increasing inertia. Characterising each grid by its meshedness coefficient, we find that the distribution of the ToAs depends in more meshed grids less strongly on the grid inertia. In order to take into account the inhomogeneous distribution of inertia, we introduce an effective distance reff, which is weighted with a factor which strongly depends on local inertia. We find that this effective distance is more strongly correlated with the ToAs, for all grids. This is confirmed quantitatively by obtaining a larger Pearson correlation coefficient between ToA and reff than with r. Remarkably, a ballistic equation for the ToA with a velocity, as derived from the swing equation, provides a strict lower bound for all effective distances reff in all power grids. thereby yielding a reliable estimate for the smallest time a disturbance needs to propagate that distance as function of system parameters, in particular inertia. We thereby conclude that in the analysis of contingencies of power grids it may be advisable that system designers and operators use the effective distance reff, taking into account inhomogeneous distribution of intrtia as introduced in Eq. (11), to locate a disturbance. Moreover, our results provide evidence for the importance of the network topology as quantified by the meshedness coefficient β.

Introduction The inertia of rotating masses of synchronized generators automatically stores kinetic energy which is immediately available in case of disturbances to stabilise the system. As a consequence of the energy transition to RER the inertia will decrease continuously. This may severely weaken the resilience of the grids against disturbances. The current energy market design does not provide sufficient incentives for such important system stabilising technologies1. The reason behind that is historical, since conventional power plants provide part of the necessary grid service automatically whenever power is generated. Therefore, it is necessary to find out how the resilience of the system can be guaranteed, in future. This necessitates a better understanding of future power systems, which we aim to contribute to, by studying the response of the transmission grid to disturbances under these new conditions. The power system response to a disturbance is understood to unfold in four stages2: 1. electromagnetic stage, 2. inertial stage, 3. governor-response and 4. Automatic Generation Control (AGC) stage. The first stage begins the transient process at the time of disturbance. The response of the power system to this transient depends on the second and third stages describing the inertia and energy contributed by the spinning mass of synchronous generators as well as the primary response through the turbine-governor droop. The aftermath of these stages determines if the response is going to require the AGC stage response. This means, that in order to predict the time scales within which control measures would have to be in place after a contingency, it is crucial to understand the response to disturbances within the first two stages, and to get a better understanding of the time which disturbances need to spread a certain distance in a power grid in the presence of inertia. Moreover, as has been pointed out in Refs.3, 4, there are multiple delays in the power system control, which can lead to power instabilities. Therefore, the proper control action requires an understanding of all time scales in the power grid and its dependence on the location of the contingency. One way to achieve this is to extract the time which a disturbance needs to propagate in real power grids from direct frequency observations of power grids, as obtained from phasor unit measurements. This has been proposed by the GridEye project as outlined in Ref.5 and by the gridradar project6. Such Wide-area monitoring systems (WAMS), using the real-time, global positioning system (GPS) time-synchronized measurements will be crucial to understand and control future power grids, and are currently being set up in an increasing number of locations. The other way to get a better understanding is, to study the spreading of disturbances in realistic models of dynamic power grids. This is the goal of the present article. So far, this has been done systematically only in simplified network models based 7, 8 9 on second order swing equations , . There, as a measure of the disturbance spreading, the time of arrival (ToA) t −t0 has been considered. The ToA with t > t0 is defined to be the time, which a disturbance needs to arrive at distance r from the position of 7 8 the disturbance which occurred at time t0. It was found in Ref. analytically and in numerical simulations that for meshed grids and sufficiently large inertia, while the ToAs at a distance r can be distributed, a definite lower bound for the ToA at distance r is provided by a ballistic propagation formula, as given by t −t0 = r/v. For homogeneous regular grids, the velocity v has been derived7, 8 and its functional dependence on the power capacity of transmission lines, the inertia and the nominal grid frequency is known there, as outlined in section . For treelike grid topologies it was found in8 that the spreading is slowed down, so that the lower bound for the arrival times depends quadratically on distance r. Inhomogeneities in the power grid typically result in anisotropic propagation and a wider distribution of ToAs. Moroever, disturbances may become localised in inhomogeneous grids7. The patterns of spreading in models of inhomogeneous power grids has been studied in Ref.10. Beyond these studies, the time scale of spreading of disturbances has been studied only for an example of a small 3 machine- 6 bus system in Ref.11. Therefore, the aim of the present article is to fill this gap and to study the spreading of disturbances in realistic transmission grid models systematically as function of system parameters, such as inertia, and in dependence on grid topology, by exploring a wide variety of grid models. In particular we consider the Nigerian 330 kV transmission network, the Ghanaian 161 kV transmission network as well as the IEEE 118 bus test- 138 kV grid. In addition, we study model grids with a Cayley-tree topology, and a Square-Grid topology, where the substations are modeled with the same parameters as the Nigerian 330 kV transmission network. In the next section we introduce the five case study grids and its properties. In section we review the dynamic models of the power grids and control devices. Load flow calculations and stability analysis are reviewed in section . In Section we define the disturbance and analyse the electromechanical transient dynamics caused by it. The results of the simulations for the time of arrivals (ToA) of the disturbance at all buses are presented in section , and analysed by fitting with a ballistic equation, extracting an averaged velocity. In Section , we introduce a weighted effective distances which takes into account the nonuniform distribution of inertia and is found to be strongly correlated with the ToAs. We conclude in section with a summary of the main results and their relevance for the control design of future power grids.

Description, Modeling and Simulation of Electrical Power Networks Let us start with the basic description of the ﬁve different network models used in the frequency dynamics investigations.

Nigerian Transmission Grid Model The Nigerian 330 kV transmission network comprises a total of 71 buses (substations/power terminals), 81 overhead transmission lines made from an alloy of aluminium and steel (with a limiting current of 1320 A) of which some consist of 2 or 3 circuit pairs. There are 107 less-decommissioned Unit generators which form the existing 29 power plant stations with a total power capacity of about 13,208MW. The other voltage infrastructures include 132 kV, 66 kV (speciﬁcally for industrial utilities) and 33 kV sub-transmission networks. There are also 11 kV and 0.415 kV 3-phase distribution networks (for home and ofﬁce utility areas). This network is of ring topology where many nodes are connected to one another in such a way that they form a closed loop prior to step-down12, as seen in the single line diagram shown in Fig.1. All 330 kV voltage level buses, generators, controllers, and shunt compensators are represented13. For the purpose of simplifying the grid with respect to the number of generating units and to allow a comparison with other grid topologies without loosing their technical details and implications, we use in the following a reduced Nigerian 330 kV transmission network comprising of the 29 power plant stations with NS = 71 buses with their individual units integrated, as shown in Fig. 2 and listed and enumerated in the supplementary material14.

2/14 275.7 60.1 KAINJI L DAMATURU L MAIDUGURI L 15.516 B'KEBBI L Kainji G.S L KADUNA C Kadun.. KANO L 28.5 KANO C KADUNA L 18.3 SG 80.9 ~ GOMBE LGOMBE C 32.7 Afam I-III Pla.. 4.383 64.3 KAINJI T.S/KAINJI T.S 44 Damatu.. kaduna.. Damat.. 94.3 -13.0 51.1 29.8 3.662 KANO T.S/KANO T.S.. SG B'KEBBI T.S/B'KEBBI T.S 17 ~ MAIDUGURI T.S/MAIDUGURI T.S 48 Trans-Amadi .. 78.6

Kainji G.S-Kainji T.S SG ~ Gbaran Plant 8 29.0 63.4 KADUNA T.S./KADUNA T.S.L JOS 42 Jos-G.. AFAM IV-V B'Kebbi.. 60.4 20.2 TRANS-AMA.. DAMATURU T.S/DAMATURU T.S 21 143.7 77.1 8.186 GBARAIN G.S Jos-Ma.. MAKURDI C Jalingo..

6.1 MAKURDI L 35.4

B'Kebbi C C JOS 17.9 SG 43 G.S KAINJI/KAINJI ~ Jebba-K.. Jebba-.. Gwags-.. GOMBE T.S/GOMBE T.S 30 Gombe.. Rivers IPP Pl.. 77.0 31.0 69.6 9.9

JOS T.S/JOS T.S 41 T.S T.S/JOS JOS 29.9 JEBBA T.S/JEBBA T.S 40 Kaduna.. Ganmo L YOLA T.S/YOLA T.S.. RIVERS IPP.. Jebba G.. Aliade .. ALIADE UGW.. SHIRORO G.S/SHIRORO G.S. 68 44.7 29.9 PH MAIN T.S KATAMPE C MAKURDI T.S/MAKURDI T.S 49 Yenagoa T.S Jebba.. Inactive 353.5 131.9 14.522 NCC Osogbo 11.6 GWAGWALAD.. Asco Ajaokuta TX KATAMPE L LOKOJA C Aliade.. Out of Calculation 53.9 Gwags.. LOKOJA L SG 53.1 De-energised ~ Omoku Unit P.. 17.5 67.1 AJAOKUTA C

SHIRORO L JALINGO T.S/JALINGO T.S 38 T.S T.S/JALINGO JALINGO 7.1 L JALINGO Gwags-.. Voltages / Loading GANMO T.S/GANMO T.S 27 39.4 AJAOKUTA L LOKOJA T.S/LOKOJA T.S.. ALIADE UGWUAJI T.S/ALIADE UGWUAJI .. OMOKU G.S N/Have.. AHODA T.S 16.3 GWAG T.S/GWAGWALADA T.S 31 T.S T.S/GWAGWALADA GWAG JEBBA G.S C G.S JEBBA KATAMPEKatampe-Shiroro T.S./KATAMPE T.S. 46 46.2 YOLA C Lower Voltage Range Jebba-O.. NCC OSOGB.. Ihovbo.. 1. p.u. JEBBA L YOLA L 54.7 ASCO G.S/ASCO G.S.. ... 19.6 49.9 N/HAVEN T.S 1/N/HAVEN T.S 51 Created with DIgSILENT PowerFactory Thesis Licence

JEBBA G.S/JEBBA G.S.. G.S/JEBBA JEBBA 0.95 p.u. Ajaoku.. OSOGBO T.S./OSOGBO NC.. Ganmo.. AJAOKUTA T.S/AJAOKUTA T.S 7 AZURA IPP(2)/AZURA IPP.. BENIN NORT.. ... 399.1 151.1 N/HAVEN SOUTH T.S/N/HAVEN SOUTH T.. Ayede.. Benin N.. 16.423 121.8 88.2 5.787 40.3 0.9 p.u. IHOVBOR NIPP/IHOVBOR NIPP 33 12.6 EGBEMA

9.8 Ajaoku.. SG Upper Voltage Range Owerri 132KV ~ Alaoji NIPP P.. SG Ajaoku.. L N/HAVEN Okpai-.. 72.5 ~ N/Have.. Ibom Power P.. 15.8 Benin-.. 9.2 1. p.u. IBOM G.S 77.6 48.7 518.8

Benin-.. 27.8 ... 178.1 Papala.. 20.107 26.7 7.7 1.05 p.u. 383.0 BENIN NORTH T.S/BENIN NORTH T... 97.3 OMO PH1/NIPP/OMOTOSHO PH1/NIPP 57 14.487 SG ~ Papala.. L GEREGU N/HAVEN S.. ... ALAOJI G.S Afam IV-VII P.. 71.7 PAPALANTO L PAPALANTO 37.2 1.1 p.u. SG EYAEN T.S/EYAEN T.S 26 ~ IKOT-ABASI.. PAPALANTO/OLORU NIPP/PH1 55 CALABAR T.S/CALABAR T.S 20 Okpai Plant 15 Benin .. OKPAI P.S/OKPAI P.S 54 69.9 Loading Range OWERRI T.S 11.2 6.1 330.4 Ikeja .. Ikeja 149.1 Ikot E.. 80. % 13.143 ADIABOR T.S 24.6

368.7 AFAM VI -23.8 13.544 Azura-.. 15.9 ... SG ~ ONITSHA T.S./ONITSHA T.S. 58 Geregu Gas P.. 74.4 21.4 AYEDE L AYEDE ALAOJI G.S/ALAOJI G.S.. 100. % SG OKPAI P.S ~ CALABAR T.S Ikeja .. Ikeja Alaoji-.. Geregu NIPP .. ALAOJI T.S GERE G.S/NIPP/GEREGU G.S/ NIP.. G.S/ G.S/NIPP/GEREGU GERE 78.0

28.8 230.9 51.3 IKOT-EKPEN.. OMOTOSHO L 13.008 Ayede-.. Egbin Plant 7 AYEDE T.S./AYEDE T.S. 15 T.S. T.S./AYEDE AYEDE ODUKPANI .. 79.2 EYAEN L OKPAI P.S L ~ 5.2 BENIN T.S/BENIN T.S 19 SG IKEJA WEST T.S./IKEJA WEST T.S.. SG 498.8 293.5 ~ 29.056

18.4 AES Plant 1 854.7 263.1 64.4 363.0 -22.5 GERE G.S/N.. Benin-Onitsha 33.509 14.522 Calaba..

27.5 ONITSHA T.S. SG AES G.S ~

33.6 .. Alaoji Odukpani Pla..

14.6 ALAGBON T.. SG 78.7 Sakete.. SAKETE L Benin-.. ~ Omotosho Ga.. Alaoji-.. CALABAR L 63.1 N/HAVEN T... 42.9 392.7 28.2 -41.6 SAPELE NIPP/SAPELE NIPP 66 ALAOJI T.S/ALAOJI T.S 12 21.716 11.5 Ikeja .. N/HAVEN S.. Onitsh.. AJA T.S.

8.2 SG 29.2 ~ EYAEN T.S IKOT-EKPENE T.S/IKOT-EKPENE T... Omotosho NI..

3.7 63.2 IKEJA WEST C Benin-.. Benin-I.. BENIN NOR.. ONITSHA L OMO PH1/N.. Erunka.. 12.0 37.5 ONITSHA C EGBIN G.S. Benin-D.. 15.6 Alaoji-.. 58.3 IKORODU T.S 5.6 PARAS ENE.. AKANGBA T.S/AKANGBA T.S 8 IKEJA WEST L 3.219 13.8 AJAOKUTA .. 9.4 BENIN T.S

BENIN L ASCO G.S Egbin-I.. SG Benin-Sapele .. Benin-Sapele ~ ALIADE UG.. Paras-Energy.. BENIN C 73.2 SAPELE G.S 450.9 24.2 OKEARO T.S. 162.7 23.749 AKANGBA L ERUNKAN T.S/ERUNKAN T.S 25 Aladja.. OWERRI T.S/OWERRI T.S.. DELTA G.S ERUNKAN T.. 87.8 AZURA IPP(2) 0.4 SG 4.407 ~ IKEJA WEST.. DELTA G.S/DELTA G.S 22 Transcorp Del.. 582.7 Okearo.. 72.1 150.6 21.760

SAKETE T.S./SAKETE T.S. 65 T.S. T.S./SAKETE SAKETE Sapele G.S Load SG

4.4 4.7 ~

57.2 Asco Plant 5 67.8 SG ~ OKEARO T.S./OKEARO T.S. 53 15.2 SAPELE G.S/SAPELE ST..

Sapele Steam.. ALADJA T.S LOKOJA75.2 T.S Egbin-.. SAKETE T.S. Odukpani-Iko.. ALAOJI C MAKURDI T.S Okearo.. IBT 330/132 Owerri ALAOJI L OSOGBO T.S. OKEARO L OKEARO 19.4 OWERRI L 371.5 AKANGBA T.. -89.1 14.704 ERUNKAN L 23.7 Owerri 132KV/Owerri 132KV Bus 61 IBOM G.S/IBOM G.S.. Aladja-.. SAPELE NIPP PAPALANTO Owerri 132kv-Ahoda AYEDE T.S. SG ~

Azura IPP Pla.. 51.6 263.1 74.9 -101.4 ODUKPANI G.S/ODUKPANI G.S..IKOT-EKPEN.. 14.158 EGBIN G.S./EGBIN G.S. 24 SAPELE L GANMO T.S OKEARO C OKEARO

AJA L 11.8 607.0 SG GWAG T.S 60.2 ~ 33.540 392.7 Transcorp Delt.. 122.3 67.3 22.619 TRANS-AMADI/TRANS-AMADI G.S 69 Egbema.. SG IHOVBOR N.. ~ JOS T.S SG Olorunsogo N.. AHODA T.S/AHODA T.S 5 ~ 73.2 TransA.. ALADJA T.S/ALADJA T.S 9 Sapele IPP G.. JEBBA T.S Aja-Egbin Gbaran-.. 65.8 5.9 7.1 263.1 5.7 176.7 38.7 52.0 17.274 Ikot Ab.. Ikot 52.0 53.8 SHIRORO G.. 46.6 Afam VI.. Afam

36.0 KADUNA T.S. Adiabo.. 2.4 SG ~ YOLA T.S AJA T.S./AJA T.S. 6 Ahoada-.. Afam I.. Yenagoa T.S/Yenagoa T.S 71 Papal/Olorun.. GOMBE T.S 75.7 Ibom-Ik.. 14.8 KATAMPE T... EGBIN L IBOM POWER L JALINGO T.S

Egbin-AES Interbus TX Interbus Egbin-AES AFAM VI L 392.7 178.0 16.340 JEBBA G.S ALAGBON L TX 132KV-Egbin Ikorodu IKOT-ABASI L ODUKAPI L AES G.S/AES G.S 2 ADIABOR T.S/ADIABOR T.S 1 SG KAINJI T.S ~ Ihovbor NIPP .. ALADJA L 69.0 20.2

Aja-Al.. IKORODU T.S/IKORODU T.S 35 PH MAIN T.S/PORT HARCOURT 63 478.8 82.3 KANO T.S Paras-.. 17.532 DAMATURU .. 32.1 EGBEMA/EGBEMA.. SG IKOT-ABASI T.S/IKOT-ABASI T.S 36 ~ Shiroro Plant.. Rivers .. AFAM VI/AFAM VI 4 68.8 10.6 AFAM IV-V/AFAM IV-V 3 B'KEBBI T.S

461.2 223.3 Adiabor L 18.274

31.0 KAINJI ALAGBON T.S./ALAGBON T.S... PARAS ENERGY G.S/PARAS ENERGY G.. OMOKU L 28 G.S UBIE G.S/GBARAN GBARAIN

Egbem.. MAIDUGURI.. SG ~ Jebba Plant 12 75.3 19.4 Afam V.. EGBEMA L Omoku-.. 13.7 Topology of Nigerian Power Transmission Grid Project: CoNDyNet-2 RIVERS IPP G.S/RIVERS IPP G.S.. JU Bremen Dynamic Implementation Project: CoNDyNet-2 by Graphic: Ring Topology Port Hacourt K. P. Nnoli & K. P. Nnoli & S. Kettemann Date: 30/06/2021 of Nigerian Transmission Network Graphic: 330KV Grid 622.7 131.3 PowerFactory 2021 SP3 S. Kettemann 22.964 Annex: CoNDyNet-1 OMOKU G.S/OMOKU G.S .. AFAM IV-V L for Investigation on Date: 20/06/2021 SG ~ Kainji Plant13 PowerFactory 2021 SP3 Power System Stability Annex: CoNDyNet-1 79.6

Figure 1. Single-Line Diagram of Nigerian 330 kV Figure 2. The Topology of the Reduced Nigerian 330 kV Transmission Network Transmission Network with NS = 71 buses.

Ghana Transmission Grid Model Building on the results of15, with recent developments in the Volta River Authority that maintains the power systems in Ghana, the Ghana transmission grid comprises mostly of 161 kV high voltage transmission lines with only a few 330 kV and 225 kV line injections from neighbouring countries. Currently, its total power capacity is approximately 4,888MW with about 90 overhead transmission lines connecting 74 buses16. The Ghana transmission network comprises about 7 hydro-powered plants and 8 gas-powered plants. The single line diagram and the topology of Ghana transmission grid are shown in Figs. 3 and 4, respectively.

71/Wa 71 69/Tumu 69 22/Bolga 22 48/Navrongo 48 23/Bolga 34.5kV 23 27 Inactive Bolga 34.5(1) TX

Out of Calculation 0.3 Bolga 34.5 - Navrongo Sawla-Wa Wa - Tumu Bolga-Tumu De-energised 8.1 8.1 9.6 0.2 Shunt/Filter(4) Bolga Load Bolga 27/Dapaong(Togo) 27 Voltage Violations / Overloading 36/Kintampo 36 19 Bolga-Zebilla 74/Zebilla 74 18/Bawku 18 Terminals violating their max. voltage limit Bolga-Zebilla.. Terminals violating their min. voltage limit 24/Bui Plant st.. 0.3 Terminal Zebilla - Bawku Bawku34.5 - Dapaong 0.2 0.0 Edge elements violating their max. loading limit Bui - Kintampo 0.1 161.5 33.5 1.00 Voltages / Loading 57/Sawla 57 18 -10.2 Bawku 34.5 XT 0.1 66/Techiman 66 Lower Voltage Range 25/Buipe 25 Bui - Sawla Techiman - Kintampo 19/Bawku34.5kV 19 1. p.u. 19.1 15.3 64/Tamale 64 Tamale - Bolga ...

Sawla Load Sawla Buipe - Tamale 3.8 Shunt/Filter(3) 0.95 p.u. 25.4 Akosombo-Nkawkaw 38/Kpandu 38 74 ... 59.3 0.9 p.u. 28.4 26.6

Akosombo Load Akosombo Upper Voltage Range Akoso-Tafo

1. p.u.

Tamale - Yendi 57.9 7 7/Akosombo Shunt/Filter(1) Bui - Techiman - Bui ... Kintampo - Buipe - Kintampo 3.5 62/Tafo 62

67/Tow26 bus.. 61/Sunyani 61 1.05 p.u. Tamale Load Tamale 0.6 0.5 20.5

Techiman Load Techiman ...

Yendi Load Yendi 1.1 p.u.

Tafo_Load Sunyani Load Sunyani Bui - Sunyani - Bui 0.9 ... 35/Kenyasi 35 39/KpeveKpandu - Kpeve 39 Techiman - Sunyani - Techiman 20/Berekum 20 Created with DIgSILENT PowerFactory Thesis Licence Shunt/Filter

69 Tow26 - Kenyasi - Tow26 43/Kumasi 43 11/Anwomaso 11 73/Yendi 73 Asiekpe Load Sunyani - Berekum Ho - Kpeve 71 1.4 0.7 48 22 23 Anwomaso - Kumasi Nkawkaw - Anwomaso 33 2.4 1.3 36.2 37.2 Kumasi - Techiman 3.3 6.1 Line Tow26 - Kumasi 47/Mim 47 Sunyani - Kenyasi 54/Obuasi 54 49/New Aberim 49 57 32/Ho 32 20 4.0 Nkawkaw - New Aberim 47 1.3 42/Kpong SS 42 12 12/Asawinso 12 Obuasi - Tow26 64

17/Ayanfuri 17 37/Konongo 37 Akoso-Achimota 17.3 48.7 29.0

7.5

33/Juabeso 33 Ayanfuri - Asawinso Volta Load 73

Kumasi - Konongo Konongo - Nkawkaw Akoso-Volta Akosombo - Kpong SS 53/Nkawkaw 53 53/Nkawkaw Asawinso - Juabeso Load Obuasi Shunt/Filter(5) 4.8 Akwatia - Tafo

Nkawkaw - Tafo - Nkawkaw 16.8 13/Asiekpe 13 6.6 23.3 47.8 61

Shunt/Filter(8) 25 24 Konongo Load Konongo

Juabeso - Mim 1.3 Aswinso load Aswinso 17 5.1 Load Nkawkaw 5.1 Ku_Load 41/Kpong GS 41 35 Akosombo - Kpone 10 Shunt/Filter(6) 23.9 36

28/Dunkwa 28 12.8 Ho - KV 69 Asiekpe

Dunkwa - Ayanfuri - Dunkwa 70/Volta 70 17.0 39.6 New Obuasi - Kumasi 66 11.5 9/Akwatia 9 4.2

Bogoso - Akyempin Obuasi New - Dunkwa Shunt/Filter(9)

1.2 New Obuasi - Akwatia 28 21 Kpong SS - Volta - SS Kpong 40/Kpone 40 29 15.8 67

New Obuasi - Obuasi - Obuasi New

Kpone GS - Kpone Load Kpong 14/Asiekpe 69KV 14 69KV 14/Asiekpe 56 16.9 51.7 50/New Obuasi 50 20.0

Dunkwa Load TX 69KV Asiekpe-Asiekpe 43 Asiekpe-Aflao Load Akwatia 15.3 16.9 10/Akyempin 10 54 Bogoso - Dunkwa - Bogoso New Obuasi Load Obuasi New 6/Aflao 6 21/Bogoso 21 26/Cape Coast 26 46/Mallam 46 5/Achimota 5 50 55 60.3 volta-Lome TX volta-Lome Achimota - Mallam Kpone - Volta Lome - Asiekpe 35.8 23.6 16/Volta Bus 16 0.4 6.8 51

Prestia 1 - Bogoso Mallam - Cape Coast 31.4 Volta-Achimota Volta - Asogli 43.1 12.0 23.9 55.0 49.1 Tower1- Volta Tower1-

55/Prestia 1Bus 55 East Volta-Accra Mallam load Mallam 37 11 16.9

3/Abobo 3 Load Coast cape Volta - Smelter 15/Asogli 15 Prestia 1 - Obuasi - 1 Prestia

Prestia2.. Tema New - Volta 56/Prestia_225Bus 56 Sogakope - 69KV Asiekpe 65 Bogoso Load Winneba- Mallam 6.3 Lome - Aflao 8.4 21.7 Smelter1 Load

60/Sogakope 60 65/Tarkwa 65

2-Winding Tr.. Load.. General

63 52/New Tema 52 49 45.9 4/Accra East 4 31

45/Lome(Togo) 161K..

Prestia 1 Load 1 Prestia 53 29/Elubo 29 Load Prestia_225

Tarkwa -Prestia 29.7 62 31.3 Abobo load Abobo Elubo - .. Line(3) Tarkwa Load Tarkwa 52.6 load Winneba 5.7 7.9 Shunt/Filter(2) 72/Winneba 72 2 72 Tema Load Asogli-Tower1 Accra East - Achimota - East Accra Takoradi - Tarkwa 6 26 46 30.9 58/Smelter1 bus 58 Lome Load 7 1/Aboaze 330kv 1 5 1 68/Tower1 bus 68 13 Aboaze Load 45

28.2 8/Aksa 8

50.8 42

3.1 60 15.1 41 16 4 Takoradi load Takoradi 46.9

Aboaze - Tarkwa - Aboaze 34/Karpower 34 Cape Coast - Aboaze - Coast Cape 14 51/New Tarkwa 51 0.1 44/Lome 330KV 44 70 winneba - Aboaze - winneba Smelter2-Smelter3 49.6

59/Smelter2 bus 59Smelter2-Aksa

44 New Tarkwa - Prestia - Tarkwa New 40 15 9.5 68

32 Tarkwa - New Tarkwa New - Tarkwa 2-Winding Tr.. 2-Winding Essiama Load Essiama 48.4 58 New Tarkwa Load Tarkwa New 0.1 0.3 63/Takoradi 63 63/Takoradi Takoradi - Essiama 52 31/Essiama 31 31/Essiama smelter2 Load 4.6 30/Enclave 30 39 Aboaze - Takoradi - Aboaze Volta - Lome 330KV Smelter-Enclave

Karpower-Smelter2 6.7

2/Aboaze Plant 2 38

59 K. P. Nnoli & Dynamic implementation of Project: CoNDyNet-2 Graphic: Ghana Grid 8 S. Kettemann Ghana Transmission Grid volta_A.. Topology of Ghana Power Transmission Grid Project: CoNDyNet-2 Date: 30/06/2021 JU Bremen 9.2 Graphic: Ghana Grid(10) PowerFactory 2021 SP3 by 30 Annex: CoNDyNet-1 K. P. Nnoli & S. Kettemann Date: 30/06/2021 34 PowerFactory 2021 SP3 Annex: CoNDyNet-1

Figure 3. Single-Line Diagram of Ghana Transmission Figure 4. Ghana Transmission Network topology with Network NS = 75.

The IEEE 118 Bus Test-Grid Model and Further Grids We also consider the IEEE 118-bus test-grid which depicts an approximation of the American Electric Power system in the U.S. Midwest as of December 196214. In order to enable a comparative study with power grids of different topology, we also study a Cayley-tree topology and

3/14 Table 1. Grid Parameter Values

Grid Data Nigerian Square Cayley Ghana IEEE Total Active PT (MW) 13208 13208 13208 4888 4374 Total Rated ST (MVA) 15450 15450 15450 5525 5112 Grid Voltage V (KV) 330 330 330 161 138 Number of Lines NL 81 112 52 90 169 Number of Links NLK 85 112 52 95 179 Meshedness Coefﬁcient β 0.11 0.40 0.00 0.15 0.27 Avg. Line-Length a (Km) 92 100 100 59 46 Number of PV Buses NPV 25 22 21 7 19 Number of Gen Stations NG 29 29 29 14 19 Number of Substations/buses NS 71 64 53 74 118

Total Number of Effective Buses Nreff 25 22 21 7 54 Avg.Line-Inductance L per length (mH/Km) 1.076 1.054 1.054 1.332 1.292

a Square grid topology, where the substations are modeled with the same parameters as the Nigerian 330 kV transmission network and we choose the number of substations to be comparable, see the supplementary material14 for details. As a quantitative measure of the topology of the different power grids, we introduce the meshedness coefﬁcient β. It is deﬁned by the ratio of the bounded faces of the buses using Euler characteristics and the maximum possible bounded faces in a given electrical network as17, N − N + 1 β = LK S , (1) 2NS − 5 where NLK and NS are the total number of overhead transmission lines plus inter-bus transformer links and buses, respectively, as listed in Table 1. Note that β takes values between 0 and 1. It is weakly dependent on grid size Ns, but strongly dependent on grid topology: Tree grids are characterized by β = 0, while β = 1 indicates a completely meshed grid. We use the following transmission line parameters in the modeling of these power grid: the line connections (i.e. from terminal i to terminal j), number of circuits, lengths (in km), positive and zero sequence resistance, R1 and R0, reactances X1 and X0, capacitances C1 and C0, susceptances B1 and B0 and rated current Irated. Included are also their respective synchronous machines and/ condensers describing their active P, reactive Q, and apparent S powers and power factor (pf)12, 13, 15, 16,18. All data of the various grids used in this analysis are comparatively summarised in Table 1.

Dynamic Model for Power Grids and Control Devices

The swing equation describes the balance between the mechanical turbine torque Tt in N.m. of the plant’s individual turbines 19–21 and the electromagnetic torque Te in N.m. of their synchronous generators, as governed by the differential equation ,

dωR J i + D ωR = T − T , (2) i dt wi i t e where Dwi accounts for rotational loss due to windings for the generator at bus i in N.m.s. Here i denotes the index of power 2 generators in the grid, t time in seconds, Ji = 2HiSi/ω0 is the combined moment of inertia of the turbine and generator in 2 kg m with Hi the generator inertia constant in seconds and Si the generator apparent power in volt-amperes (VA). Also, R ωi is the angular velocity of the rotor in electrical rad/s and ω0 is its rated synchronous value in electrical rad/s. We assume that any change in the rotor’s angular velocity is a derivative of its angular position δ (in electrical radians with respect to its R − R synchronously rotating setpoint δ0), given by ωi ω0 = dδi/dt. Taking the derivative of ωi with respect to time, we get, since ω0 is constant, dωR d2δ i = . (3) dt dt2 Here, ω0 = 2πvo with the nominal grid frequency vo which is 50 or 60Hz depending on a country’s grid code. Substituting Eq.(3) into Eq.(2), we ﬁnd d2δ dδ J i + D i = T − T . (4) i dt2 wi dt m e

4/14 Here, the net mechanical shaft torque Tm = Tt − Dwiω0, which is the turbine torque minus the rotational losses at ω0 is 21 introduced . Multiplying both sides of Eq.(4) by the rated speed ω0 we get, since power is a product of torque and angular velocity,

d2δ dδ J ω i + D ω i = P − P , (5) i 0 dt2 wi 0 dt m e where Pm is the mechanical power from the turbine and Pe is the electrical power from the generator’s air-gap. If we denote the 2Hi angular momentum of the rotor at rated speed with Mi (i.e. Mi = Jiω0 = Si) and also denote the damping coefﬁcient at rated ω0 22–24 synchronous speed with Di (i.e. Di = Dwiω0), the swing equations can be written as ,

2 NS 2Hi d δi dδi − Si 2 + Di = Pi + ∑ Ki j sin(δ j δi). (6) ωo dt dt j=1 with the number of nodes NS, the power in node i, Pi and the transmission line power capacity between i and j, Ki j. The devices used to control the outputs of the synchronous generating-units in all model grids include, but are not limited to: the automatic voltage regulator (AVR), responsible for voltage control of the generator through its excitation control; the speed governor used in the model is the turbine governor (TGOV-model) of the generator’s prime mover which is responsible for keeping the generator’s rotor frequency synchronous to the grid frequency, thereby maintaining the frequency operational limits when there are contingencies. The control devices for the power plants also comprise the power system stabilizer (PSS), tasked with reducing the oscillatory instability and enhancing the damping of the power system oscillations through excitation control. The controlled signal is a derivative of the generator rotor speed injected to the AVR through the excitation system to cancel out the phase lag between exciter’s voltage set reference and the generator’s windings torque25. In a basic power plant frame, it is also important to add over-excitation limiter (OEL) and the under-excitation limiter (UEL) for the AVR device. Here, we carry out stability studies without the OEL and UEL devices, as many power plants stations may not have them. The choice of the synchronous machine models are based on IEEE guide for synchronous generator modeling practice and applications in power system stability analysis26. The turbine governor models used in the design simulations were chosen according to IEEE recommendations for dynamic models of turbine-governor in power system studies27. The following two models were used for the Nigerian, Cayley-tree, Square and Ghanaian power networks and the synchronous condensers part of IEEE 118-bus network: 1. The HYGOV model was used for the hydro power plants. 2. The simplified TGOV1 model was used for the thermal plants. The operations and basic signals are the same in both models. The excitation voltages of the synchronous generators in the networks are controlled by the simplified excitation (SEX-AVR) model and serve as their excitation and voltage control system except for the part of the synchronous generators in the IEEE 118 bus test-grid where the primary IEEET1-AVR model was used. The choice of the automatic voltage regulator for synchronous generators was influenced by ENTSO-E28. The PSS2A model was used for all power plants in the networks. Each was tuned to ensure that no destabilising signal is fed back into the excitation system voltage summation point. Power system stabilizers remain the most cost efficient way of enhancing the damping of power system oscillations. The PSS2A standard model used in this design was influenced by IEEE recommended practice for excitation system models for power system stability studies29.

Load Flow and Electromechanical Transient Stability Analysis in PowerFactory Load ﬂow calculation in complex power grids is used to determine voltage magnitude V and angle θ of the substations/buses, as well as active P and reactive Q power19. Here, DigSILENT PowerFactory is used to implement the Newton-Raphson (power equation, classical) method as its non-linear equation solver because of its fast-convergence. This method is used to obtain the initial conditions during the top of the maximum power ﬂows in all power grids. The currents at steady state are calculated ∗ − from Ii = Si /Vi =(Pi jQi)/Vi, where i = 1,2,...,NG and j the imaginary unit with NG the total number of generators in the generating stations. The currents at the substations are given in terms of the admittance matrix by Ibus = YbusVbus. All dynamic components of the plants (AVR, PSS, Turbine / GOV models) and all system parameters are modeled and simulated. The active power balance is ensured by direct coupling of the grid frequency to synchronous generators, ensuring inertia activation and nodal relaxations. The phase coupling between generators is described by the swing equations with frequency-dependent load damping. The effect of increasing the injection of renewable energy resources on the stability of the electrical power networks 30 is studied by varying the aggregated inertia constant Hgrid of the entire grid by ,

NS ∑i=1 HiSi Hgrid = , (7) ST

5/14 NS th with total rated apparent power in the grid ST = ∑i=1 Si, where Si and Hi are rated power and inertia constant of the i bus, respectively. Note that Hi = 0 on non-generator buses. The voltage profiles from stationary power flow were examined at no transient to ensure that the buses maintain the voltage tolerance according to the Nigerian grid code which specifies 0.85 p.u. minimum (i.e., 280.5 kV) and 1.05 p.u. maximum (i.e., 346.5 kV) tolerances. Voltage profiles of the Nigerian, Square, Cayley-tree, Ghana and IEEE 118 test power networks at steady operating conditions (undisturbed operation) were captured and represented in bar-charts in the supplementary material14. The results show that all buses were within the specified voltage tolerance. As overloading of power generators is defined to be any loading above 80% we choose this as our loading thresholds. The grid code specifies frequencies of 0.995 p.u. (min) and 1.005 (max) tolerance for network elements. Frequencies of 0.99 p.u. (min) and 1.01 p.u. (max) are used in actual daily operational procedure31.

[Hz] Time = 5 s 50 Hz Y = 50.001 Hz 50 Y = 49.999 Hz 5.013 s + -

49.99 Time of Arrival (ToA) +ve and -ve thresholds Created with DIgSILENT PowerFactory Thesis Licence 49.98 23.65167 s Local Max. = 49.97035 Hz Local Max. = 49.96621 Hz 49.97

49.96 Final Settling Time = 46.322 s Local Min. = 49.96614 Hz 49.966 Hz

49.95

49.94 9.912 s Local Min. = 49.929 Hz

49.93

0 10 20 30 40 50 60 70 80 [s] 90

EGBIN G.S.\EGBIN G.S. 24: Electrical Frequency

Figure 5. Green Line: Frequency as a function of time at a bus where a Synchronous Machine (SM) Event occurs at t0 = 5s. Blue line: Frequency deviation threshold at 50.001 Hz. Red Line: Deviation threshold at 49.999 Hz. Relevant times scales are indicated by thin lines: the Time of Arrival (ToA), the time when ﬁrst local minimum is reached, the time when ﬁrst local maximum is reached and the frequency settling time.

Transient Dynamic Analysis

The magnitudes of the injected perturbations Pinj for the synchronous machine (SM) events are chosen in such a way to make the transient dynamic analysis in the various power grids comparable. Therefore, we choose the disturbance to be N Pinj = 0.5PT /NPV, with PT = ∑i=1 Pi the total active power in the grid and NPV the total number of generator buses participating in the power injections as given in Table 1. Here, a PV bus is a bus from which generator(s) are connected to the power system. A comparison chart of the power in MW that participated in the SM-Event is shown in Table 2.

Table 2. Injected-Perturbations in the Grids

Perturbation Pinj Nigerian Square Cayley-Tree Ghana IEEE 118 SM-Event (MW) 264 300 314 349 115

Outage contingencies were simulated at those power plants substations which have the minimum geodesic distances to the rest of the substations for each grid. A geodesic distance between two nodes in a network is deﬁned to be the minimal number of edges which connect these two nodes in a given network. That way they are most centrally located and the spread of disturbances throughout the grid can be monitored. As shown in Fig. 5 the transient is characterised by several times scales, indicated by thin lines. We choose for the analysis the Time of Arrival (ToA), which measures the time, the frequency deviation ﬁrst surpasses a threshold. We place the threshold

6/14 meter at very small values of δν = ± 0.001 Hz, in order to record the propagation of the disturbances most directly. The synchronous machine event is chosen to occur at exactly t0 = 5s in the 90s transient electromechanical stability simulation time frame. The switching at the event takes a ﬁnite time of ∆t = 200 ms. The IEEE 118 Bus test-grid has a nominal ν0 = 60 Hz frequency. All other grids have ν0 = 50 Hz as nominal frequency. In order to be able to focus the investigation of the transient on the ﬁrst two stages, and in particular on the contribution of grid inertia on the disturbance spreading pattern and speed, we choose to turn off any controller in these numerical experiments.

Disturbance Propagation Analysis

In the following, we show results of the measurement of the ToA at each bus after the event at time t0, as obtained from the PowerFactory simulation. In Fig. 6, the ToAs for all buses of the Nigerian grid are shown as function of their geodesic distance r from the event location. Results are obtained for different aggregated grid inertia of Hgrid = 2s,4s,6s,8s, as varied by multiplying all local inertia Hi by a common factor. We see that, while the ToAs at nodes with same geodesic distance r are distributed over some range, they tentatively increase with r from the event location, as expected. We observe that the width of the ToA distribution increases with inertia Hgrid in the Nigerian grid. If the propagation would be ballistic, the ToAs would increase linearly with distance r as

t = b0 + r/v, (8) and one can identify the velocity v with which the perturbation spreads. We therefore ﬁt all ToAs as function of r to the ballistic 14 Eq. (8), magenta lines in Fig. 6, see supplementary material for further documentation . The parameter b0 is typically larger than t0 = 5.0s due to the fact that the synchronous machine contingency may have a small delay and the switching takes a small time of order δt = 200ms. In order to better understand these results, let us review the solution of the swing equation

5.03 5.03 5.05 H = 2s H = 4s Hgrid = 2s Hgrid = 4s grid grid 5.05

5.04 5.04 5.02 5.02 (secs) (secs) (secs) (secs)

t t t 5.03 t 5.03

5.02 5.02 5.01 5.01 Time of Arrival, of Time Arrival, of Time Arrival, of Time Arrival, of Time 5.01 5.01

5.00 5.00 5.00 5.00 0 2 4 6 8 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Geometric Distance, r (unweighted) Geometric Distance, r (unweighted) Geometric Distance, r (unweighted) Geometric Distance, r (unweighted) 5.03 5.03 5.08 Hgrid = 8s H = 8s Hgrid = 6s H = 6s grid 5.07 grid 5.08 5.07 5.06 5.02 5.02 5.06

(secs) (secs) (secs) 5.05 (secs) t t t t 5.05 5.04 5.04 5.03 5.01 5.01 5.03 5.02

Time of Arrival, of Time Arrival, of Time Arrival, of Time Arrival, of Time 5.02

5.01 5.01

Figure 6. ToA versus geodesic distance r for Nigerian Figure 7. ToA versus r for the Cayley-tree Power grid as power grid from transient dynamic simulations for inertia obtained for inertia Hgrid. Green (Blue) dots: ToA at bus(es) Hgrid. Green (Blue) dots: ToA at bus(es) with at least one with at least one (no) inertia injection. Magenta line: ﬁt to (no) inertia injection. Magenta line: ﬁt to ballistic Eq. (8). ballistic Eq. (8). Red line: ballistic Eq. (8) with velocity Eq. Red line: ballistic Eq. (8) with velocity Eq. (9). (9).

Eq.(6) for homogeneous parameters, where it can be solved analytically. For meshed grids and sufﬁciently large inertia one ﬁnds that a disturbance spreads ballistically according to Eq. (8) with velocity v = K/(ω0J)a, where a is the length of a transmission line7, 8. Here, J is the inertia of a synchronous generator or motor, respectively. In real power grids all parameters p are inhomogeneous: The power capacity Ki j and the length ai j are different for every transmission line between nodes i and j and the inertia is different for every generator, Ji at node i, with some of the nodes having no inertia. Let us therefore introduce

7/14 as a measure of the velocity in real grids, the effective velocity

K¯ω0 veff = a¯, (9) s2HgridSB

¯ which depends on the following averaged parameters: K = ∑i< j Ki j/NL, is the average power capacity of the NL transmission 2 lines, where Ki j = V /(ω0Li j) is the power capacity of a purely inductive transmission line between nodes i and j, V the voltage and Li j the line positive sequence inductance. a¯ is the average length of the transmission lines. Hgrid is the aggregated inertia constant, Eq. (7). SB = ST /NS is the total rated grid power in units of MVA divided by the total number of buses NS. With these definitions we are now ready to compare with the simulation results in realistic power grids with inhomogeneous parameters, plotting Eq.(8) (red line) with the velocity Eq. (9) in Fig. 6. We observe for the Nigerian power grid in Fig. 6 that the ballistic Eq. (8) with the velocity calculated according to Eq. (9) (red line) provides a lower bound for the ToAs at all nodes and for all inertia Hgrid = 2s,4s,6s,8s. The measured ToAs are found to be distributed in an interval b0 + r/veff < t < b0 − r/vmin, whose width is found to increase with inertia. We observe that the minimal velocity vmin decreases with increasing inertia. In order to understand the reason for this distribution, we plot the ToAs at buses with no inertia in Fig. 6 in blue, while the green dots indicate that at least one of the buses at this distance r and with this arrival time has finite inertia. From this we can conclude that one reason that the ToAs are widely distributed for same geodesic distance r could be the fact that the propagation is delayed in nodes with inertia while in nodes without inertia there is no delay. This can clearly be seen in Fig. 6, in particular for the example of the bus at distance r = 9, where the disturbance arrives at the same time as at the node at distance r = 8, which has no inertia. For the Cayley-tree power grid we observe that the ToAs are more widely distributed, Fig. 7. Moreover, the ballistic Eq. (8) with velocity Eq. (9) no longer provides a lower bound for the ToAs at all buses. Rather, there are outliers, which are the ToAs falling below the red line. In the Square grid we observe that the ballistic Eq. (8) with the velocity calculated according to Eq. (9) does provide a lower bound for the ToA at all buses for the smallest inertia Hgrid = 2s, while there are outliers at higher 14 Hgrid, as documented in Ref. . Fig. 8 shows the ToA as function of r in the Ghana grid. The ballistic Eq. (8) with velocity calculated according to Eq. (9) (red line) provides a lower bound for the ToAs at small and large distance, while there are outliers for all Hgrid at intermediate distances r = 2,...,5. In addition, we observe that there is a wide distribution of ToAs at intermediate distance r whose width increases with an increase in Hgrid. In the IEEE 118-bus test grid shown in Fig. 9, we observe that the ballistic equation with velocity Eq. (9) provides a lower bound for ToAs (red line) only for lowest inertia Hgrid = 2s. For higher inertia, there are outliers at intermediate r which are located at the nodes with no inertia. The distribution of ToAs is narrower than in the other grids. This may be because of the high degree of meshedness of this grid similar to what we observed in the square grid in Ref.14. In Fig. 10 we plot the fitted (magenta) and analytical (red) velocity versus inertia H for the Nigerian (left) and the Ghana (right) power grid. We observe that the fitted velocity decays with the inertia faster than the analytically predicted power law v ∼ H−1/2. We find for the IEEE 118-bus test grid as well as for the square grid that the fitted velocity decays with the inertia more slowly than the analytical velocity which decays with the power law ∼ H−1/2, as documented in Ref.14. In order to understand the effect of grid topology on these results better, let us compare the degree of meshedness Eq. (1) as listed in Table I for the various grids. It is vanishing β = 0 for the Cayley tree grid, and it is small, β = 0.11, for both the Nigerian and for the Ghanaian grid, where it is β = 0.15. We note that both the Nigerian and Ghana grid have a ring like topology at the center of the grid, while there are many tree like connections to outer buses, resulting overall in a small meshedness coefficient β. It is substantially larger in the IEEE test grid β = 0.27, and largest (in the considered grids) in the square grid β = 0.4. From the results for the ToAs versus distance r we notice that the more meshed a power grid is, as quantified by a higher meshedness coefficient β, the weaker does the distribution of ToAs depend on the grid inertia Hgrid. This can also be seen from a weaker dependence of the fitted ballistic velocity on the inertia in both Square and IEEE 118-bus test grid. Finally, we observe that in tree like grids with small β there is a wide spread of ToA. In particular, there are buses where the disturbance arrives much later than expected in a meshed grid. This observation is in accordance with Ref.8, where it was found that a disturbance tends to be more easily localised in a tree like grid, preventing or at least slowing down its spreading. Thus, the response to the contingency is in some buses slower in less meshed grids, like the Cayley-tree and the Ghana grids, as compared to the more meshed grids like the Square grid and the IEEE grid14.

8/14 5.03 5.03 5.03 H = 2s H = 4s grid grid Hgrid = 2s Hgrid = 4s 5.04

5.02 5.03 5.02 5.02 (secs) (secs) (secs) (secs) t t t t

5.02

5.01 5.01 5.01

5.01 Time of Arrival, of Time Arrival, of Time Arrival, of Time Arrival, of Time

5.00 5.00 5.00 5.00 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Geometric Distance, r (unweighted) Geometric Distance, r (unweighted) Geometric Distance, r (unweighted) Geometric Distance, r (unweighted) 5.03 5.03 H = 8s H = 6s Hgrid = 8s 5.07 Hgrid = 6s grid grid

5.06 5.08 5.05 5.07 5.02 5.02 (secs) (secs) (secs) (secs) t t t t 5.06 5.04 5.05 5.03 5.04 5.01 5.01 5.02 5.03

Time of Arrival, of Time Arrival, of Time 5.02 Arrival, of Time Arrival, of Time 5.01 5.01

Figure 8. ToA versus r for Ghana Power grid as obtained Figure 9. ToA versus r for IEEE Power grid as obtained from from transient dynamic simulations for inertia Hgrid. Green transient dynamic simulations for inertia Hgrid. Green (Blue) (Blue) dots: ToA at bus(es) with at least one (no) inertia dots: ToA at bus(es) with at least one (no) inertia injection. injection. Magenta line: ﬁt to ballistic Eq. (8). Red line: Magenta line: ﬁt to ballistic Eq. (8). Red line: ballistic Eq. (8) ballistic Eq. (8) with velocity Eq. (9). with velocity Eq. (9).

Weighted Effective Distance

In order to account for the variation of inertia in the buses, let us introduce a properly weighted effective distance reff. We begin by extending the analytical result for the velocity as derived from the swing equations to a velocity vi j of the disturbance when propagating between nodes i and j for inhomogeneous parameters,

Ki j v = min( a ,c ), (10) i j J ω i j i j r i where ai j is the length of this transmission line. In a transmission line which has no or low Ohmic losses, the velocity ci j along the transmission line between nodes i and j can be derived from the telegraph equation as a function of its inductance and 32 8 capacitance as ci j = 1/ Li jCi j, yielding a velocity which is close to the velocity of light c = 3x10 m/s for high voltage transmission lines. However, when there is inertia Ji at node i the transmission is delayed slowing the velocity down to a value p vi j = Ki j/(Jiω)ai j. Only if there is no inertia in node i, Ji = 0, the disturbance spreads to the next node faster, with velocity ci j, close to the speed of light. The inhomogeneous distribution of inertia is thus one of the reasons that the observed ToAs are distributedp widely, and that there are nodes which are reached by the disturbance faster than expected in a homogeneous grid. According to Eq. (10) we expect that nodes which are connected with the location of the disturbance through nodes with low or no inertia are reached faster by the disturbance. Let us therefore take this fact into account. Using Eq. (10) a more accurate estimate for the ToA at a node l when the disturbance occurs at node k is given by − tkl b0 = min(∑i j along Path from k to l ai j/vi j), where the sum is over all pairs of nodes i, j along the path between nodes k and l which has the minimal arrival time. Accordingly, we deﬁne an effective distance between nodes k and l which is weighted with the inverse of the velocity of the disturbance between nodes i and j, Eq. (10),

kl veff ai j reff = min( ∑ ), (11) a¯ i, j along Path from k to l vi j where veff is the effective velocity deﬁned in Eq. (9) and a¯ the average transmission line length. We note that, since vi j strongly depends on the presence and magnitude of inertia at node i, this effective distance is weighted with a local factor which takes into account the amount of inertia present on the geodesic path between nodes k and l. Now, let us plot the measured ToAs versus this effective distance reff, in order to see if we thereby get a less scattered plot and fewer outliers. We consider in particular those grids where we observed many outliers, namely the Nigerian, Cayley-tree

9/14 (v H) Relation at Bus 24 (v H) Analytical Relation

3 6 × 10 ) ) s s / / 3 3 4 × 10 10 Km Km ( ( v v

3 3 × 10 Velocity, Velocity,

3 2 × 10

(v H) Relation at Bus 7 (v H) Analytical Relation 2 10 0 0 0 0 1 0 0 0 0 1 2 × 10 3 × 10 4 × 10 6 × 10 10 2 × 10 3 × 10 4 × 10 6 × 10 10 Inertia Constant, H/secs Inertia Constant, H(s)

Figure 10. Fitted velocity v as function of inertia H (Magenta Line) and velocity Eq. (9) (red line) for Nigerian Power Grid Fig. 6 with contingency at bus 24 (left) and for Ghana Power Grid, Fig. 8 with contingency at bus 7 (right) in double logarithmic plot.

and Ghana power grid. In Fig. 11 we plot the times of arrival versus reff for the Nigerian grid. First of all, we observe that the ballistic motion with the analytical velocity Eq. (9) provides now a strict lower bound for all effective distances (red line). Secondly, the ToAs are less widely distributed when compared to the plot versus geodesic distances in Fig. 6. For the Cayley tree grid we also ﬁnd that the ballistic motion with the analytical velocity Eq. (9) provides now a strict lower bound for all ToAs when plotted as function of effective distance14, but its ToAs remain distributed without a strong correlation with the effective distance reff.

(v H) Relation at Bus 24 (v H) Analytical Relation

4 10

3 10 ) ) s / s / Km Km ( ( v v

Velocity, 2 Velocity, 10

3 10

(v H) Relation at Bus 7 (v H) Analytical Relation 1 10 0 0 0 0 1 0 0 0 0 1 2 × 10 3 × 10 4 × 10 6 × 10 10 2 × 10 3 × 10 4 × 10 6 × 10 10 Inertia Constant, H(s) Inertia Constant, H(s)

Figure 13. Fitted velocity v (Magenta) and velocity Eq. (9) (red) when the ToA is plotted versus effective distance for the Nigerian (left) and the Ghana Power Grid as function of inertia parameter H in double logarithmic plot.

Fig. 12 shows the ToAs as function of effective distance in the Ghana grid. There are no outliers and in comparison with Fig. 8, the width of the distribution of ToAs decreased strongly, the ToAs monotonuosly increase with reff with few excpetions. Fig. 13 shows a plot of the ﬁtted ballistic velocity (magenta) and the analytical velocity (red) versus inertia constant Hgrid for

10/14 5.03 5.03 5.03 H = 4s H = 2s Hgrid = 2s grid grid 5.04 Hgrid = 4s

5.03 5.02 5.02 5.02 (secs) (secs) (secs) (secs) t t t t

5.02

5.01 5.01 5.01 5.01 Time of Arrival, of Time Arrival, of Time Arrival, of Time Arrival, of Time

5.00 5.00 5.00 5.00 0 1 2 3 4 5 6 7 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Effective Distance, reff (weighted) Effective Distance, reff (weighted) Effective Distance, reff (weighted) Effective Distance, reff (weighted) 5.03 5.03 5.07 H = 8s Hgrid = 6s grid Hgrid = 6s Hgrid = 8s 5.06 5.08 5.05 5.07 5.02 5.02 (secs) (secs) (secs) (secs)

t t t t 5.06 5.04 5.05 5.03 5.04

5.01 5.01 5.02 5.03

Time of Arrival, of Time Arrival, of Time Arrival, of Time Arrival, of Time 5.02 5.01 5.01

5.00 5.00 5.00 5.00 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Effective Distance, reff (weighted) Effective Distance, reff (weighted) Effective Distance, reff (weighted) Effective Distance, reff (weighted)

Figure 11. ToA versus effective distance for the Nigerian Figure 12. ToA versus effective distance for the Ghana Power Power grid according to Eq. (11) for different grid parameter grid according to Eq.(11) for different grid inertia Hgrid. Hgrid. Magenta line: ﬁt to ballistic Eq. (8). Red line: ballistic Magenta line: ﬁt to ballistic Eq. (8). Red line: ballistic Eq. (8) Eq. (8) with velocity Eq. (10). with velocity Eq. (10).

the Nigerian power grid (left) and the Ghana grid (right) as obtained by ﬁtting all ToAs as function of effective distance reff. It monotonously decays as a power law with inertia in both grids, faster than the analytical results tv ∼ H−1/2. To statistically quantify the correlation of the frequency ToAs with the geodesic distance from the contingency r and effective distance reff for the various power grids, we calculated the Pearson correlation coefﬁcient Cr between all ToAs and r as given by

NS − − ∑i (ToAi ToA)(ri r) Cr = (12) NS − 2 NS − 2 ∑i (ToAi ToA) ∑i (ri r) q and accordingly the Pearson correlation coefﬁcient Creff between all ToAs and their respective reff as given by

NS − − ∑i (ToAi ToA)(reffi reff) Creff = , (13) NS − 2 NS − 2 ∑i (ToAi ToA) ∑i (reffi reff) q where ToAi is the measured time of arrival at node i at geodesic distance ri and effective distance reffi. ToA, r and reff are the respective mean values. The Cs for each grid inertia (i.e., Hgrid) are shown in Table 3. We observe that Creff is larger than Cr

Table 3. Pearson correlation coefﬁcients

Grid C ToAH=2s ToAH=4s ToAH=6s ToAH=8s Nigerian Cr 0.712 0.692 0.740 0.741 Creff 0.768 0.767 0.793 0.769 Ghana Cr 0.810 0.812 0.820 0.829 Creff 0.885 0.896 0.899 0.899 Cayley-tree Cr 0.211 0.213 0.198 0.184 Creff 0.218 0.295 0.303 0.311 for all power grids and all inertia values Hgrid. Thus, the effective distance reff correlates more strongly with the ToAs than the geodesic distance r, conﬁrming our observations above when plotting ToA versus r and reff. Also, we observe that the smaller the meshedness coefﬁcient β of the power grid is, Table 1, (Ghana β = 0.15, Nigerian β = 0.11, Cayley tree β = 0),

11/14 the smaller both correlation coefﬁcients are. While Cr depends for all power grids rather strongly on the inertia value Hgrid, the correlation with reff, Creff depends only weakly on it in the Nigerian and Ghanaian power grid. We note that another inertia weighted distance was introduced previously in Ref.33 which was used in Ref.34 to suggest that electrically-interfaced control resources enabled with damping should be placed at locations further away from the center of inertia (COI). That inertia weighted electrical distance is thereby rather different than the one we introduced here, reff, Eq. (11), which not only takes into account the inhomogeneous inertia, but also the inhomogeneous transmission line length ai j and power capacity Ki j.

Summary and Conclusions The spread of a disturbance in transmission power grids was studied on a large variety of realistic network models by means of the DigSILENT PowerFactory software. We measured the time of arrival (ToA) of a disturbance and analysed it systematically as function of the distance of the buses to the contingency. By plotting the measured ToA versus the geodesic distance r from the location of the perturbation, we find only a weak correlation between these measures. Lowering the inertia in the grid is found to reduce the ToA irrespective of the grid topology. Thus, a disturbance travels faster with less grid inertia, or equivalently, with more RES in the grid. Characterising each grid by its meshedness coefficient, we find that the distribution of the ToAs is in more meshed grids only weakly affected by a change in grid inertia. Moreover, we observe that in tree like grids, like the Cayley tree grid there is a wider distribution of ToAs, compared to grids with large meshedness, like the Square and IEEE 118 Bus test power grids. Plotting the ToA versus an effective distance reff, which we introduced here in order to take into account the inhomogeneous distribution of inertia, we find a much stronger correlation with the ToA than with geodesic distance. The increase of correlation is found to be particularly strong in tree like grids. This is confirmed quantitatively by obtaining a larger Pearson correlation coefficient between ToA and reff than with r. Remarkably, a ballistic equation for the ToA with a velocity, as derived from the swing equation, provides a strict lower bound for all effective distances reff in all power grids. thereby yielding a reliable estimate for the smallest time a disturbance needs to propagate that distance as function of system parameters, in particular inertia. We thereby conclude that in the analysis of contingencies of power grids it may be advisable that system designers and operators use the effective distance reff, taking into account inhomogeneous distribution of inertia as introduced in Eq. (11), to locate a disturbance. Moreover, our results provide evidence for the importance of the network topology as quantified by the meshedness coefficient β. Further studies will be necessary to find the optimal magnitude and precise location of inertia and other control measures in real power grids to keep power grids resilient beyond the energy transition. Being aware of the results presented here, which show that a decrease of the magnitude of grid inertia increases the velocity of disturbances and reduces the time a disturbance needs to spread throughout the grid, may help to optimise the design and positioning of control devices in future power grids.

ACKNOWLEDGMENT We acknowledge the support of Bundesministerium für Bildung und Forschung (BMBF) CoNDyNet-2, FK. 03EK3055D.

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Author contributions Research design and supervision were carried out by S.K. with K.P.N. performing the numerical experiments and analytical calculations.

13/14 Additional information Supplementary information accompanies this paper at https://dx.doi.org/10.21227/pjpt-nk47. Competing interests: The authors declare no competing interests.

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