—The energy transition towards more renewable en- ergy resources (RER) together with the reduction of conventional energy sources profoundly affects the frequency dynamics and electromechanical stability of electrical power networks. Here, we investigate the effect of different levels of grid inertia and its heterogeneous distribution on propagation patterns of distur- bances in realistic models of power grids, including the Nigerian transmission network, the Ghana transmission network and the IEEE 118 bus test-grid as well as a Cayley-tree network, as a model grid without cycles, and the Square-grid power network as a strongly meshed model grid. These studies are conducted with the DigSILENT PowerFactory software and compared with results obtained from swing equations. Results show that the times of arrival of frequency deviations as function of geodesic distance from the contingency are more clustered and decay faster in more meshed grids than in tree- like or weakly meshed grids. In order to take into account the inhomogenous distribution of inertia, we deﬁne an effective distance reff and ﬁnd it to be more strongly correlated with the arrival time than the geodesic distance. A ballistic equation for the arrival time, with a velocity derived from the swing equation, provides then a strict lower bound for all effective distances. The ﬁtted velocity decays with a power law with grid inertia. Decreasing the magnitude of grid inertia by increasing the magnitude of RER is thereby found to increase the travel velocity of disturbances and reduce the time a disturbance needs to spread throughout the grid. Index Terms —Grid disturbance spread, power grid topology, transient stability, linear response theory, frequency dynamics, heterogeneous inertia, effective distance. I. I NTRODUCTION THE inertia of the rotating masses of synchronised genera- tors automatically store kinetic energy. This stored energy is available in the case of disturbances immediately to stabilise the system. As a consequence of the energy transition to RER the inertia will decrease continuously and may weaken thereby the resilience of the grids against disturbances severely. The current energy market design does not provide sufﬁicient sustainable incentives for such important system stabilising technologies [1]. The reason behind that is historical, since the conventional power plants did provide part of the neces- sary grid service automatically whenever it generated power. Kosisochukwu P. Nnoli is with the Department of Computer Science and Electrical Engineering, Jacobs University Bremen, 28759 Bremen, Germany (e-mail:k.nnoli@jacobs-university.de). Stefan Kettemann is with the Department of Physics and Earth Science, and the Department of Computer Science, Jacobs University Bremen, 28759 Bremen, Germany, and also with the Division of Advanced Materials Science, Pohang 790-784, South Korea (e-mail:s.kettemann@jacobs-university.de).Therefore, it is necessary to ﬁnd out how the resilience of the system can be guaranteed, in the future. Since DC-AC converters of wind- and photovoltaic- generators feed their power directly into the grid and do not provide inertia of rotating masses to the grid, they have to be enabled technically to provide momentary reserve, as can be achieved by installing virtual synchronous machines. Therefore, we aim to study the response of the transmission grid to disturbances under these new conditions. To this end we investigate the spreading pattern of various kinds of disturbances in realistic transmission grid models. In particular we consider the Nigerian 330 kV transmission network, the Ghanaian 161 kV transmission network as well as the well known IEEE 118 bus test- 138kV 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. This investigation takes into account the topology of these grids with emphasis on their degree of meshedness, the power capacity of the transmission lines [2], momentary reserve power (in units of MW ) as well as different grid inertia H(in units of seconds) in order to investigate quantitatively their inﬂuence on frequency dynamics of the grid. The spatial propagation of disturbances is investigated by measuring the time of arrival of a disturbance ( t), as measured from the time t0when the disturbance occurred ﬁrst, from which we would trace the travel pattern and subsequently the travel velocity of the disturbance. II. D ESCRIPTION , 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 for the purpose of understanding the inﬂuence and interplay of system parameters and their contributions to the stability of the electrical power network. A. The Nigerian Transmission Network Model The Nigerian 330 kV transmission network comprises a total of 71buses (substations/power terminals), 81overhead 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 107less-decommissioned Unit generators which form the existing 29power plant stations with a total power capacity of about 13;208MW . JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 2 The other voltage infrastructures include 132 kV , 66 kV (speciﬁcally for industrial utilities) and 33kV sub- transmission networks. There are also 11kV and 0:415 kV 3-phase distribution networks (for home and ofﬁce utility areas). The Nigerian network is of ring topology where all the nodes are connected to one another in such a way that they form a closed loop prior to step-down [3], as seen in the single or one-line diagram of the Nigerian transmission network shown in Fig.1. All 330 kV voltage level buses, generators, controllers, and shunt compensators are represented [4]. Fig. 1: Single-Line Diagram of the Nigerian 330 kV Trans- mission Network Fig. 2: The Topology of the Reduced Nigerian 330 kV Transmission Network with NS= 71 buses. For the purpose of simplifying this network with respect to the number of generating units and to allow a comparison with other network topologies without loosing their technical detailing and implications, we use in the following a reduced Nigerian 330 kV transmission network comprising of the 29power plant stations with NS= 71 buses with their individual units integrated, as shown in Fig. 2 and listed and enumerated in Appendix A and in Table IV. B. The Cayley-Tree and Square Grid Transmission Model In order to enable a comparative study with power grids of different topology, we use 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 and we choose the number of substations to be comparable. We use a Cayley-tree transmission network of branching number b= 3 andM= 3 generations, resulting in a total of NS=1+ 4 + (4 x 3) + (4 x 3 x 3) =53 buses [2]. Note that M= 4generations would give NS= 90 , exceeding the number of substations in the Nigerian power grid withNS= 71 . The other buses, loads, network shunt compensators and power injections were assigned starting from the beginning. See Appendix A and Table IV for details of these mappings. The power capacity, network loading, number of generating stations and other network parameters of the Cayley-tree network is chosen the same as in the Nigerian transmission network model. The single-line diagram of the Cayley-tree transmission network is shown in Appendix B in Fig. 21 while its tree-like topology is shown in Fig. 22. The Square power grid is modeled with NS=8 x 8 = 64 buses with the remaining loads, shunt compensators and power injections added, recounting from the beginning as in the case of Cayley-tree network. This process enables the comparison of the Square power network model with the Nigerian model ensuring frequency and voltage tolerances according to [5]. The single line diagram and the topology of Square transmission grid are shown in Appendix B in Fig. 23 and Fig. 24 respectively. See also Appendix A and Table IV for details of these mappings. C. The Ghana Power Transmission Model Building on the results of [6] and with the recent developments in the V olta River Authority that maintains the power systems in Ghana, the Ghana transmission network comprises mostly of 161 kV high voltage transmission lines with only a few 330 kV and 225 kV line injections from neighbouring countries. At the moment, the total power capacity of the Ghana-grid is approximately 4;888MW with about 90overhead transmission lines connecting 74buses [7]. The Ghana transmission network comprises about 7 hydro-powered plants and 8gas-powered plants. The single line diagram and the topology of Ghana transmission grid are shown in Figs. 3 and 4 respectively. JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 3 Fig. 3: Ghana Transmission Network Single-Line Diagram Fig. 4: Ghana Transmission Network topology with NS= 75 buses. D. The IEEE 118 Bus Test-Grid Model The IEEE 118-bus test-grid used depicts an approximation of the American Electric Power system (in the U.S. Midwest) as of December 1962. The IEEE 118-bus system is made up of 19synchronous generators, 35synchronous condensers, 169 transmission lines of which few consist of double circuits, 9inter-bus transformers, and 91loads [8] [9]. The total power capacity of both the power plants and the synchronous condensers in this network is about 4882MW with a reserve of about 1069:33MW . The one-line diagram and the topology of IEEE 118-bus test-grid are shown in Fig. 5 and Fig. 6 respectively. The following transmission line parameters are used in the modeling of the Nigerian, Cayley tree, Square, Ghana and IEEE 118 Bus test power grid: the line connections (i.e. from terminal i to terminal j), number of circuits (pair number), lengths (in km), positive and zero sequence resistance ( R1andR0), reactance ( X1andX0), capacitance ( C1andC0), susceptance ( B1andB0) 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) [3] [4] [6] [7] [8]. Fig. 5: Single-Line Diagram of IEEE 118-Bus Test-Grid E. The Swing Equation and the Control Devices for the Network Power Plants The devices used to control the outputs of the synchronous generating-units in all the grids include, but are not limited to, the automatic voltage regulator (A VR), 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 frequency of the grid, thereby maintaining the frequency operational limits when there are contingencies. The swing equation describes the balance between the mechanical torque Tmin N.m. of the plant’s individual turbines and the electromagnetic torque Tein N.m. of their synchronous generators, as governed by the differential equation [10] [11] [12], Jid!R i dt+Dwi!R i=TmTe; (1) whereDwiaccounts for the rotational loss due to windings for the generator at bus iin N.m.s. Here idenotes the index of power generators in the grid, ttime in seconds, Ji=2Hi !2 0Siis the combined moment of inertia of the turbine and generator in kg.m2withHithe generator inertia constant in seconds and Sithe generator apparent power in volt-amperes (V A). Also, !R iis the angular velocity of the rotor in electrical rad/s and !0is its rated synchronous value in electrical rad/s. We assume that any change in the rotor’s angular velocity !R i!0is a derivative of its angular position (in electrical radians with respect to its synchronously rotating setpoint 0), given by !R i!0=di dt: (2) JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 4 Fig. 6: The Topology of IEEE 118-Bus Test-Grid Taking the derivative of !R iwith respect to time, we get, d!R i dt=d dtdi dt +d!0 dt: (3) Since!0is a constant rated synchronous value, Eq.(3) becomes d!R i dt=d2 dt2: (4) In reality,!0is related to the grid frequency voby 2vo, wherevocould be 50or60Hz depending on a country’s grid code. Substituting Eq.(4) into Eq.(1), we then have, Jid2i dt2+Dwidi dt =TmTe: (5) The assumption here is that the perturbation in the network effected on the rotors from the initial position is small. Multiplying both sides of Eq.(5) by the rated speed !0in order to ensure that we maintain a uniform or synchronous speed throughout the system, balancing the power, we have Ji!0d2i dt2+Dwi!0di dt =Tm!0Te!0: (6) Since power is a product of torque and angular velocity, the right side of Eq.(6) can be rewritten as Ji!0d2i dt2+Dwi!0di dt =PmPe; (7) wherePmis the mechanical power from the turbine and Peis the electrical power from the generator’s air-gap. If we denote the angular momentum of the rotor at rated speed with Mi(i.e. Mi=Ji!0=2Hi !0Si) and also denote the damping coefﬁcient at rated synchronous speed with Di(i.e.Di=Dwi!0), the swing equation can be written as Mid2i dt2+Didi dt =PmPe; (8)and also as [13] [14] [15], 2Hi !oSid2i dt2+Didi dt=Pi+NSX j=1Kijsin(ji): (9) NSis the number of nodes/buses, Piis the power in the grid’s ithnode.Kijis the power capacity of the transmission lines which depends on the voltage. (ji)are phase differences. The control devices for the power plants also comprise the power system stabilizer (PSS) which is tasked with reducing the oscillatory instability and enhancing the damping of the entire power system’s oscillations through excitation control. The controlled signal is a derivative of generator’s rotor speed injected to the A VR through the excitation system to cancel out the phase lag between exciter’s voltage set reference and the generator’s windings torque [16]. In a basic power plant frame, it is also important to add over-excitation limiter (OEL) and the under-excitation limiter (UEL) for the A VR device, but we carry out the stability studies of the networks 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 analysis [17]. The turbine governor models used in the design simulations were chosen according to IEEE recommendations for dynamic models of turbine-governor in power system studies [18]. 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-A VR) model and serve as their excitation and voltage control system except for JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 5 the part of the synchronous generators in the IEEE 118 bus test-grid where the primary IEEET1-A VR model was used. The choice of the automatic voltage regulator for syn- chronous generators was inﬂuenced by ENTSO-E [19]. 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 studies [20]. III. L OAD FLOW AND ELECTROMECHANICAL TRANSIENT STABILITY ANALYSIS IN POWER FACTORY Load ﬂow calculation in complex power grids is used to determine voltage magnitude Vand angle ()of the substations/buses, as well as the active (P) and reactive (Q) power [10]. In this design, 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 condi- tions during the top of the maximum power ﬂows in the entire power grids. The currents at steady state are calculated from Ii=S i Vi=Pi+jQi Vi; (10) wherei= 1;2;:::;N Gandjthe imaginary unit. Here, NGis the total number of generators in the generating stations. The currents at the substations can be represented by the equation in admittance matrix form as, Ibus=YbusVbus: (11) All dynamic components of the plants (A VR, PSS, Turbine / GOV models) and all system parameters were modeled and simulated. The active power balance is ensured by direct coupling of the grid frequency 50=60Hzto the synchronous generators, ensuring inertia activation and nodal relaxations. The phase coupling between the generators is described by the classical swing equation model with frequency-dependent load damping. The effect of increasing the injection of renewable energy resources on the stability of the electrical power networks based on the classical swing equation is studied on these power grids by varying the aggregated inertia constant Hgridof the entire grid by [21], Hgrid=PNS i=1HiSi ST: (12) whereST=PNS i=1Siis the total rated apparent power in the grid,SiandHiare the rated power and the inertia constant of theithnode/bus respectively. Note that Hion non-generator buses are set to zero. The data of the various grids used in this analysis are summarised in Table I.A. Transient Dynamic Analysis The voltage proﬁles from the stationary power ﬂow of the grids were examined at no transient to investigate whether the buses actually maintained their realistic voltage tolerance according to the Nigerian grid code. The Nigerian grid code speciﬁes voltages of 0.85 p.u. minimum (i.e. 280.5 kV) and 1.05 p.u. maximum (i.e. 346.5 kV) tolerances for the buses. For the frequencies, the grid code speciﬁes frequencies of 0.995 p.u (min) and 1.005 (max) tolerance for network elements where frequencies of 0.99 p.u. (min) and 1.01 p.u. (max) are used in actual daily operational procedure [5]. The voltage proﬁles of the Nigerian, Square, Cayley- tree, Ghana and IEEE 118 test power networks at steady operating conditions (undisturbed operation) were captured and respectively represented in bar-charts in Appendix C. The results show that the buses were all within the speciﬁed voltage tolerance. Overloading of power generators is deﬁned to be any loading above 80 %which we therefore choose as our maximum loading thresholds. The magnitude of the injected perturbations Pinjfor the synchronous machine (SM) events are chosen such that the studied power grids are comparable. We choose it to be Pinj= 0:5PT NPV; (13) wherePT=PN i=1Piis the total active power in the grid and NPVthe total number of generator buses participating in the power injections, as given in Table I. A PV bus is a bus from which generator(s) are connected to the power system. TABLE II: Injected-Perturbations in the Grids tylePerturbation Pinj Nigerian Square Cayley-Tree Ghana IEEE 118 SM-Event (MW) 264 300 314 349 115 A comparison chart showing the amount of power in MW that participated in the SM-Event is shown in Table II. Local minima arrival time Local maxima arrival time Final settling time Fig. 7: Green Line: Frequency as a function of time at a bus where a Synchronous Machine (SM) Event occurs at t= 5s. Blue line: Frequency deviation threshold at 50:001 Hz. Red Line: Deviation threshold at 49:999Hz. These outage contingencies were simulated at the power plants substations which were chosen to have the minimum JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 6 TABLE I: Grid Parameter Values Grid Data Nigerian Square Cayley Ghana IEEE Total Active PT(MW) 13208 13208 13208 4888 4374 Total Rated ST(MV A) 15450 15450 15450 5525 5112 Grid V oltage,V (KV) 330 330 330 161 138 Number of Lines NL 81 112 52 90 169 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 Nreff25 22 21 7 54 Avg.Line-Inductance L per length (mH/Km) 1.076 1.054 1.054 1.332 1.292 geodesic distances to the rest of the substations for each grid so that they are most centrally located and the spread of disturbances throughout the grid can be monitored. As shown in Fig.7, the threshold meter used to monitor the arrival of frequency deviations was placed at very small values of0:001Hz, in order to record the direct spread of the disturbances. The synchronous machine event occurred at exactlyt0= 5sin the 90stransient electromechanical stability simulation time frame. The switching at the event took a ﬁnite time of t= 200 ms. The IEEE 118 Bus test-grid has a nominal 0= 60 Hz frequency. All other grids have0= 50 Hz as nominal frequency. In the simulations we choose not to include any controller at the power plants, in order to be able to focus the investigation on the contribution of grid inertia on the disturbance spreading pattern and speed. In Figs. 8 to 14, we show the results of the measurement of the arrival time of the disturbance at each bus in the grids, after the event at time t0;as obtained in the simulation with PowerFactory. That arrival time tis deﬁned as the time, when the frequency ﬁrst deviates by more than the threshold deviation=:001Hz from the nominal frequency 0. In Fig. 8, the arrival times are shown as function of the geodesic distance rfrom the event location for different aggregated grid inertia of Hgrid= 2s;4s;6s;8s. The grid inertia was varied by multiplying all Hiby the same factor. First of all, we see that the arrival times tentatively increase with the geodesic distance rfrom the event location. Secondly, we ﬁnd that the arrival times are spread over a wider range as the inertia Hgrid is increased. If the propagation would be ballistic, that is the perturbation spreads with a deﬁnite velocityv, the arrival time tafter the disturbance starting from the time of disturbance t0would be linearly related to the distance ras, t=b0+r v: (14) The Table III shows the result for the linear regression parameterb0which is used to ﬁt the arrival time as function of geodesic distance in the various power grids to the ballistic equation Eq. (14), shown as magenta lines in Figs. 8 to 14. TheGrid Hgrid(s) b0(s) Nigerian 2 5.0130 4 5.0133 6 5.0132 8 5.0133 Cayley-Tree 2 5.0150 4 5.0149 6 5.0149 8 5.0148 Square 2 5.0134 4 5.0137 6 5.0137 8 5.0137 Ghana 2 5.0012 4 5.0012 6 5.0012 8 5.0007 IEEE 118 Bus 2 5.0130 4 5.0133 6 5.0134 8 5.0135 TABLE III: The Regression values for the Frequency Dynam- ics Curve-Fit parameterb0is found to be typically larger than the actual time t0= 5:0swhen the disturbance occurred, which can be due to the fact that the synchronous machine contingency may have a small delay, and the switching takes a small time t= 200ms. In order to better understand these results, let us compare them with results obtained by the solution of the swing equa- tion Eq.(9). For homogeneous parameters and assuming purely inductive transmission lines, which is a good assumption for high voltage transmission lines, one can solve the swing equations analytically. For meshed grids and sufﬁciently large inertia one ﬁnds in linear response theory that a disturbance spreads ballistically according to Eq. (14) with a velocity v=p K=(!0J)a;whereais the length of a transmission line [2], [22]. Here Jis the inertia of a synchronous generator or motor, respectively, where it is assumed that the inertia is the same at every node. In real power grids all parameters are inhomogeneous, that is the power capacity Kijand the length aijare different for every transmission line between node i andj, and the inertia Jis different for every generator Ji, with some of the nodes having no inertia. In an attempt to compare if the velocity in a real grid has a similar parametric JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 7 /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b 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 Hgrid=2s /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b 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 Hgrid=4s /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b 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 Hgrid=6s /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b 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 Hgrid=8s Fig. 8: Time of arrival of disturbance versus geodesic distance for the Nigerian Power grid from results of transient dynamic simulations for different inertia parameter Hgrid. Green dots: arrival time(s) at bus(es) with at least one inertia injection. Blue dots: arrival time(s) at bus(es) with no inertia. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. 15. dependence let us therefore deﬁne a velocity in terms of average parameters ve =s K!0 2HgridSBa; (15) where K=P i<jKij=NL;whereNLis the total number of transmission lines in the grid. Here, Kij=V2=(!0Lij) is the power capacity of a purely inductive transmission line between nodes iandj,Vis the voltage and Lijis the line positive sequence inductance. ais the average length of the transmission lines. Hgrid is the aggregated inertia constant in Eq. (12). SB=ST=NSis the total rated grid power in units ofMVA divided by the total number of nodes/buses NS. With these deﬁnitions we are now ready to compare with the simulation results by PowerFactory, which is based on swing equations in the realistic electricity networks with inhomogeneous parameters. We plot there also Eq.(14) (red line) using the analytical averaged velocity Eq. (15) in Figs. 8 to 15. The parameter b0is ﬁtted as given in Table III. We observe for the Nigerian power grid in Fig. 8 that the ballistic Eq. (14) with the velocity calculated according to Eq. 15 (red line) provides a lower bound for the time of arrival for all nodes and for all inertia Hgrid= 2s;4s;6s;8s. The actual measured arrival times tare found to be spread in an intervalr=veff<tb0<r=v min, where the width of this interval is found to increase with inertia. Thus, we ﬁnd that the minimal velocity of the disturbance vmindecreases with increasing inertia. This can be understood by the fact, that the parameters are inhomogenous. Let us therefore explore the /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /uni00000015/uni00000011/uni00000013 /uni00000015/uni00000011/uni00000018 /uni00000016/uni00000011/uni00000013 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 Hgrid=2s /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /uni00000015/uni00000011/uni00000013 /uni00000015/uni00000011/uni00000018 /uni00000016/uni00000011/uni00000013 /uni0000002a/uni00000048/uni00000052/uni00000050/uni00000048/uni00000057/uni00000055/uni0000004c/uni00000046/uni00000003/uni00000027/uni0000004c/uni00000056/uni00000057/uni00000044/uni00000051/uni00000046/uni00000048/uni0000000f/uni00000003r/uni00000003/uni0000000b/uni00000058/uni00000051/uni0000005a/uni00000048/uni0000004c/uni0000004a/uni0000004b/uni00000057/uni00000048/uni00000047/uni0000000c/uni00000018/uni00000011/uni00000013/uni00000013/uni00000018/uni00000011/uni00000013/uni00000014/uni00000018/uni00000011/uni00000013/uni00000015/uni00000018/uni00000011/uni00000013/uni00000016/uni00000018/uni00000011/uni00000013/uni00000017/uni00000018/uni00000011/uni00000013/uni00000018/uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni00000052/uni00000049/uni00000003/uni00000024/uni00000055/uni00000055/uni0000004c/uni00000059/uni00000044/uni0000004f/uni0000000f/uni00000003t/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni00000056/uni0000000c Hgrid=4s /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /uni00000015/uni00000011/uni00000013 /uni00000015/uni00000011/uni00000018 /uni00000016/uni00000011/uni00000013 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 Hgrid=6s /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /uni00000015/uni00000011/uni00000013 /uni00000015/uni00000011/uni00000018 /uni00000016/uni00000011/uni00000013 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 Hgrid=8sFig. 9: Time of arrival of disturbance versus geometrical dis- tance for the Cayley-Tree Power grid from transient dynamic simulations for different inertia parameter Hgrid. Green dots: arrival time(s) at bus(es) with at least one inertia injection. Blue dots: arrival time(s) at bus(es) with no inertia. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. 15. /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H/secs/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s)(vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni00000015/uni00000017 (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s)(vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni00000014 (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 Fig. 10: Fitted ballistic velocity vas function of inertia constantH(Magenta Line), and the velocity according to the analytical Eq. (15) (red line), for the Nigerian Power Grid Fig. 8 (left) with contingency at bus 24 and for the Cayley Tree Grid, Fig. 9 (right) with contingency at bus 1 in double logarithmic plot. effect of the inhomogeneous distribution of the inertia further. In Fig. 8 the arrival times at buses with no inertia are plotted in blue, while the green dots indicates that at least one of the buses at this distance and with this arrival time has inertia. The fact that the propagation of the disturbance is delayed in nodes with inertia while in nodes without inertia it propagates without delay, can be seen in Fig. 8, 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 in Fig. 9 that JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 8 /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=2s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=4s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=6s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=8s Fig. 11: Time of arrival of disturbance versus geodesic distance for the Square Power grid as obtained by transient dynamic simulations for different inertia parameter Hgrid. Green dots: arrival times at buses with at least one inertia injection. Blue dots: arrival times at buses with no inertia. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. 15. the ballistic Eq. (14) with the velocity calculated according to Eq. 15 no longer provide a lower bound for the time of arrival for all buses. The lower the inertia Hgrid, the more outliers there are, which are the data points which fall below the red line. This can again be explained by the lack of inertia in some of the buses, f.e. the fact that the bus at distance r= 2 with smallest time of arrival has no inertia, as indicated by the blue colour, explains that the disturbance arrived at almost the same time, at a bus at distance r= 3. In Fig. 10 we show the double logarithmic plot of the ﬁtted ballistic velocity (magenta) and the analytical velocity (red) versus the inertia constant Hfor the Nigerian power grid (left) and the Cayley tree grid (right). We observe that the ﬁtted velocity decays with the inertia in both power grids faster than the analytical velocity which decays with the power law H1=2: In the Square grid shown in Fig.11, we observe that the ballistic Eq. (14) with the velocity calculated according to Eq. 15 does provide a lower bound for the time of arrival for all buses, only for the smallest inertia Hgrid= 2, while there are outliers at higher Hgrid. Fig.12 shows the result for the time of arrivals as function of the geodesic distance in the Ghana grid. The ballistic Eq. (14) with the velocity calculated according to Eq. 15 (red line) does provide a lower bound for the time of arrival only at small and large distance, while there are outliers for all values ofHgridat intermediate distances r= 2;:::;5). In addition, we observe that there is a wide spreading of the time of arrivals at intermediate distance from the point of contingency which increases with corresponding increase in the Hgrid. We note /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013 /uni00000014/uni00000015 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 Hgrid=2s /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013 /uni00000014/uni00000015 /uni0000002a/uni00000048/uni00000052/uni00000050/uni00000048/uni00000057/uni00000055/uni0000004c/uni00000046/uni00000003/uni00000027/uni0000004c/uni00000056/uni00000057/uni00000044/uni00000051/uni00000046/uni00000048/uni0000000f/uni00000003r/uni00000003/uni0000000b/uni00000058/uni00000051/uni0000005a/uni00000048/uni0000004c/uni0000004a/uni0000004b/uni00000057/uni00000048/uni00000047/uni0000000c/uni00000018/uni00000011/uni00000013/uni00000013/uni00000018/uni00000011/uni00000013/uni00000014/uni00000018/uni00000011/uni00000013/uni00000015/uni00000018/uni00000011/uni00000013/uni00000016/uni00000018/uni00000011/uni00000013/uni00000017/uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni00000052/uni00000049/uni00000003/uni00000024/uni00000055/uni00000055/uni0000004c/uni00000059/uni00000044/uni0000004f/uni0000000f/uni00000003t/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni00000056/uni0000000c Hgrid=4s /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013 /uni00000014/uni00000015 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 Hgrid=6s /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013 /uni00000014/uni00000015 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 Hgrid=8sFig. 12: Time of arrival of disturbance versus geodesic distance for the Ghana Power grid as obtained from transient dynamic simulations for different inertia parameter Hgrid. Green dots: arrival times at buses with at least one inertia injection. Blue dots: arrival times at buses with no inertia. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. 15. /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s)(vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni00000015/uni0000001c (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s) (vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni0000001a (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 Fig. 13: Fitted ballistic velocity vas function of inertia constantH(Magenta Line), and the velocity according to the analytical Eq. (15) (red line), for the Square Power Grid Fig. 11 (left) with contingency at bus 29 and for the Ghana Power Grid, Fig. 12 (right) with contingency at bus 7 in double logarithmic plot. that the Ghana grid is highly connected with ring like topology at the center of the grid, while there are tree like connections to outer buses. This may explain the wide spreading of arrival times. In Fig. 13 we show the double logarithmic plot of the ﬁtted ballistic velocity (magenta) and the analytical velocity (red) versus the inertia constant Hfor the square power grid (left) and the Ghana grid (right). We observe that the ﬁtted velocity decays with the inertia in the square power grid more slowly than the analytical velocity which decays with the power law H1=2;while in the Ghana grid, the ﬁtted velocity decays JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 9 with the inertia as in the square power grid with a power law, but faster than the analytical velocity. In the IEEE 118-bus test grid shown in Fig.14, we observe thatvprovides a good lower bound for the disturbance dispersion in the ballistic path (red line) only for the lowest inertia value Hgrid= 2s. For higher inertia, there are outliers at intermediate geodesic distances. Here, the spreading of the time of arrivals are smaller 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 Fig. 11. /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=2s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=4s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b 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 Hgrid=6s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a /uni0000001b /uni0000002a/uni00000048/uni00000052/uni00000050/uni00000048/uni00000057/uni00000055/uni0000004c/uni00000046/uni00000003/uni00000027/uni0000004c/uni00000056/uni00000057/uni00000044/uni00000051/uni00000046/uni00000048/uni0000000f/uni00000003r/uni00000003/uni0000000b/uni00000058/uni00000051/uni0000005a/uni00000048/uni0000004c/uni0000004a/uni0000004b/uni00000057/uni00000048/uni00000047/uni0000000c/uni00000018/uni00000011/uni00000013/uni00000013/uni00000018/uni00000011/uni00000013/uni00000014/uni00000018/uni00000011/uni00000013/uni00000015/uni00000018/uni00000011/uni00000013/uni00000016/uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni00000052/uni00000049/uni00000003/uni00000024/uni00000055/uni00000055/uni0000004c/uni00000059/uni00000044/uni0000004f/uni0000000f/uni00000003t/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni00000056/uni0000000c Hgrid=8s Fig. 14: Time of arrival of disturbance versus geodesic distance for the IEEE 118 Bus Power grid as obtained from the transient dynamic simulations for different inertia parameter Hgrid. Green dots: arrival times at buses with at least one inertia injection. Blue dots: arrival times at buses with no inertia. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. 15. /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s)(vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni00000019/uni0000001c (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 Fig. 15: Fitted ballistic velocity vas function of inertia constantH(Magenta Line), and the velocity according to the analytical Eq. (15) (red line), for the IEEE Power Grid Fig. 14 with contingency at bus 69 in double logarithmic plot. In Fig. 15 we show the double logarithmic plot of the ﬁtted ballistic velocity (magenta) and the analytical velocity (red)versus the inertia constant Hfor the IEEE 118-bus test grid. We observe that the ﬁtted velocity decays with the inertia more slowly than the analytical velocity which decays with the power lawH1=2: In summary, we conclude that the more meshed a power grid is, that is the more loops it has, the weaker is the dependence of the spreading of the disturbance on the grid inertia Hgrid. This can be seen from the weaker dependence of the ballistic velocity on the inertia in both the square grid and the IEEE 118-bus test grid. Moreover we observe that in more tree like grids there is a wider spread of arrival times, and in particular there are buses where the disturbance arrives much later than expected in a meshed grid. This observation is in accordance with the result of Ref. [2], 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 the less meshed grid, like the Cayley-tree and the Ghana grids, as compared to the more meshed grids like the square grid Fig. 11 and the IEEE grid Fig. 14. In order to better account for the lack of inertia in some of the buses, one could introduce an effective geodesic distance re between the node i, where the disturbance started, to a nodejin the power grid along a path Pij=fi;k1;k2;::;jg, by counting only nodes k1;k2;:::which contribute inertia Hkn, n= 1;:::;rA e to the network inertia Hgrid(i.e. a generator or synchronous condenser injection node). Thus, rA e = Geodesic Distance d(i;k1;k2;:::;j )jHi;Hkn6=0:(16) However, the implementation of this formula, requires a re- duction of the graph to the nodes with inertia, and thereby a change of the topology of the power grid. Let us therefore rather introduce a properly weighted ef- fective distance re . Extending the analytical result for the velocity as derived from the swing equations to the swing equations with inhomogenous parameters, we arrive at the fol- lowing velocity or the disturbance when propagating between nodesiandj vij= min(r Kij Ji!aij;cij); (17) whereaijis the length of this transmission line, and cijis the velocity of propagation along the transmission line between nodesiandj. In a transmission line, which has no or low Ohmic losses, the velocity along the transmission line cijcan be derived from the effective wave equation of the transmission line, which has the form of the telegraph equation. One obtains it as a function of the inductance and capacitance of the transmission line [23] as cij= 1=p LijCij, yielding a result which is close to the velocity of light c= 3x108m=s for high voltage transmission lines. Thus, the velocity along the transmission line, cijis typically much larger than the velocity, which is caused by the delay due to the momentary reserve of a bus at nodeiwith inertia JiThus, when node ihas inertia, we getvij= (q Kij Ji!aij). If there is no inertia in node i,Ji= 0, however, the disturbance spreads to the next node faster, with velocitycij, close to the speed of light. Thus, this explains the fact that the observed arrival times are spread widely, and that JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 10 there are also nodes which are reached by the disturbance faster than expected in a homogeneous grid. According to Eq. (17) 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 try to take this fact into account. Using Eq. (17) we can give a more accurate estimate for the time of arrival at a node l when the disturbance occurs at node kby tklb0= min X Path from k to laij vij! ; (18) where the sum is over all pairs of nodes i;jalong that path between nodes kandlwhich has the minimal arrival time. The velocity vijis given by Eq. (17). Accordingly, we can now deﬁne an effective distance between nodes kandlby rkl eff=veff amin X Path from k to laij vij! ; (19) whereve is the effective velocity deﬁned in Eq. (15) and a the average transmission line length. /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019 /uni0000001a 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 Hgrid=2s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 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 Hgrid=4s /uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000028/uni00000049/uni00000049/uni00000048/uni00000046/uni00000057/uni0000004c/uni00000059/uni00000048/uni00000003/uni00000027/uni0000004c/uni00000056/uni00000057/uni00000044/uni00000051/uni00000046/uni00000048/uni0000000f/uni00000003reff/uni00000003/uni0000000b/uni0000005a/uni00000048/uni0000004c/uni0000004a/uni0000004b/uni00000057/uni00000048/uni00000047/uni0000000c/uni00000018/uni00000011/uni00000013/uni00000013/uni00000018/uni00000011/uni00000013/uni00000014/uni00000018/uni00000011/uni00000013/uni00000015/uni00000018/uni00000011/uni00000013/uni00000016/uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni00000052/uni00000049/uni00000003/uni00000024/uni00000055/uni00000055/uni0000004c/uni00000059/uni00000044/uni0000004f/uni0000000f/uni00000003t/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni00000056/uni0000000c Hgrid=6s /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /uni00000015/uni00000011/uni00000013 /uni00000015/uni00000011/uni00000018 /uni00000016/uni00000011/uni00000013 /uni00000016/uni00000011/uni00000018 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 Hgrid=8s Fig. 16: Time of arrival of disturbance versus effective distance for the Nigerian Power grid according to Eq. (19) for different inertia parameter Hgrid. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. (17). Now, let us check whether plotting the measured arrival time versus this effective distance re , we get a less scattered plot with fewer outliers. We consider in particular those grids where we observed many outliers, namely the Nigerian, Cayley-tree and Ghana power grid. In Fig. 16 we plot the times of arrival versus the re for the Nigerian grid. We observe that the ballistic motion with the analytical velocity Eq. (15) provides now a strict lower bound for all effective distances (red line). Also, the arrival times are more clustered as compared to the plot versus geodesic distances shown in Fig 8. /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000014/uni00000013/uni00000016/uni00000014/uni00000013/uni00000017/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s)(vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni00000015/uni00000017 (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s) (vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni00000014 (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 Fig. 17: Fitted ballistic velocity vas function of inertia constantH(Magenta Line), and the velocity according to the analytical Eq. (15) (red line), for the Nigerian Power Grid when the time of arrival is plotted against the effective distance, Fig. 16 (left) with contingency at bus 24 and for the Cayley Tree Power Grid, Fig. 18 (right) with contingency at bus 1 in double logarithmic plot. /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000015/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 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 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000015/uni00000013 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 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000017 /uni00000013/uni00000011/uni00000013/uni0000001b /uni00000013/uni00000011/uni00000014/uni00000015 /uni00000013/uni00000011/uni00000014/uni00000019 /uni00000013/uni00000011/uni00000015/uni00000013 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 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000017 /uni00000013/uni00000011/uni00000013/uni0000001b /uni00000013/uni00000011/uni00000014/uni00000015 /uni00000013/uni00000011/uni00000014/uni00000019 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 H = 2sgrid H = 4s H = 6s H = 8sgrid grid grid Fig. 18: Time of arrival of disturbance versus effective distance for the Cayley-Tree Power grid according to Eq. (19) for different inertia parameter Hgrid. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. (17). In Fig. 18, we plot the times of arrival versus the re for Cayley tree power grid. We ﬁnd also in this tree grid topology, that the ballistic motion with the analytical velocity Eq. (15) provides now a strict lower bound for all effective distances (red line). There remains however a wide distribution of the times of arrival without a strong correlation with the effective distancere . In Fig. 17 the double logarithmic plot of the ﬁtted ballistic velocity (magenta) and the analytical velocity (red) versus the inertia constant His shown for the Nigerian power grid (left) and the Cayley tree grid (right). We observe that the agreement of the ﬁtted velocity with a power law decay with inertia is in both power grids better as compared to Fig. 10, JOURNAL OF L ATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 11 which was extracted from the plot versus geodesic distance. The power law decay is found to be faster than the analytical velocity which decays with the power law vH1=2. /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018 /uni00000015/uni00000011/uni00000013 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 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000019 /uni00000013/uni00000011/uni0000001b /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000015 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 H = 2s H = 4s H = 6s H = 8s grid gridgrid grid Fig. 19: Time of arrival of disturbance versus effective distance for the Ghana Power grid according to Eq.(19) for different grid inertiaHgrid. Magenta line: ﬁt to ballistic Eq. (14). Red line: ballistic Eq. (14) with velocity calculated according to Eq. (17). /uni00000014/uni00000013/uni00000014/uni00000015/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000016/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000017/uni000000ee/uni00000014/uni00000013/uni00000013/uni00000019/uni000000ee/uni00000014/uni00000013/uni00000013 /uni0000002c/uni00000051/uni00000048/uni00000055/uni00000057/uni0000004c/uni00000044/uni00000003/uni00000026/uni00000052/uni00000051/uni00000056/uni00000057/uni00000044/uni00000051/uni00000057/uni0000000f/uni00000003H(s)/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016/uni00000039/uni00000048/uni0000004f/uni00000052/uni00000046/uni0000004c/uni00000057/uni0000005c/uni0000000fv(Km/s) (vH)/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000044/uni00000057/uni00000003/uni00000025/uni00000058/uni00000056/uni00000003/uni0000001a (vH)/uni00000003/uni00000024/uni00000051/uni00000044/uni0000004f/uni0000005c/uni00000057/uni0000004c/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000035/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 Fig. 20: Fitted ballistic velocity vas function of inertia constantH(Magenta Line), and the velocity according to the analytical Eq. (15) (red line), for the Ghana Power Grid Fig. 19 with contingency at bus 7 in double logarithmic plot. The Fig.19 shows the result for the times of arrival as a function of the effective distance in the Ghana grid. In comparison with Fig.12, we observe that the times of arrival become more clustered with no outliers. In Fig. 20 we show the double logarithmic plot of the ﬁtted ballistic velocity (magenta) and the analytical velocity (red) versus the inertia constant Hfor the Ghana grid as obtained by ﬁtting the times of arrival as a function of the effective distance shown in Fig. 19. We observe that the ﬁtted velocity decays with the inertia in the Ghana power grid with a powerlaw, but still faster than predicted by the analytical velocity vH1=2. In summary, we found that the plot of the arrival time versus effective distance re shows a stronger correlation than with geodesic distance. The ballistic equation with analytical velocity Eq. (15) is found to provide a strict lower bound for all effective distances. The ﬁtted velocity is found to decay with a power law, albeit with a faster decay than the one expected from the analytical velocity vH1=2. IV. S UMMARY AND CONCLUSIONS The spread of a disturbance in transmission power grids was studied on different realistic network topologies by means of the DigSILENT PowerFactory commercial power software and compared with analytical results obtained from the swing equations. We ﬁnd a strong dependence on the grid inertia and the degree of meshedness, which we have analysed systematically. By plotting the measured arrival time of a disturbance versus the geodesic distance from the location of the perturbation, we ﬁnd only a weak correlation between these parameters. Lowering the inertia in the grid is found to reduce the time of arrival irrespective of the grid topology. Thus, the disturbances is found to travel faster with less inertia, or equivalently, with more RES in the grid. Moreover, we observe that in tree like grids, like the Cayley tree grid and grids with small meshedness, like the Ghana grid, there is a wider spread of arrival times, while in more meshed grids, like the Square and IEEE 118 Bus test power grids, there is a stronger clustering of arrival times versus distance and the disturbance decays faster. Plotting instead the arrival time versus an effective distance reff, which we deﬁned here in order to take into account the inhomogenous distribution of inertia, we ﬁnd a much stronger correlation than with the geodesic distance. A ballistic equation for the arrival time with an analytical velocity, as derived from the swing equation, is then found to provide a strict lower bound for all effective distances. The ﬁtted velocity is found to decay with a power law with the grid inertia. These insights on the response of the transmission grid to disturbances under the new conditions when more wind- and photovoltaic- generators feed their power directly into the grid and do not provide inertia of rotating masses to the grid, may therefore help to better prepare the power grids for the energy transition. ACKNOWLEDGMENT We gratefully acknowledge the support of Bundesminis- terium für Bildung und Forschung (BMBF) CoNDyNet-2, FK. 03EK3055D.