—This paper presents the studies and investigations on the dynamics of momentary reserves in electrical power systems under contingency. Momentarily reserve through the machine’s inertia serves the purpose of primary frequency control and prevents voltage collapse in the case of reactive power reserves. A simulation was performed on a realistic Nigerian 330 kV transmission network in PowerFactory software to study and investigate the mechanism of these reserve functions on the network buses as an inertia active power control method. Moreover, we investigated the inﬂuence of geodesic increment of momentary reserve on the decay of disturbances. The results indicated that the momentary reserve by inertia alone reduces the frequency deviation from its nominal value, delays the trans- mission of disturbances and enhances the damping of oscillations by reducing the ﬁnal frequency settling time at the buses under contingency. This numerical experiment also suggests the optimal placement of the momentary reserves in the grid in order to improve system stability against power outage disturbances. Index Terms —momentary reserve, system disturbance, inertia control, oscillations damping, frequency stability. I. I NTRODUCTION The active and reactive power reserves of synchronous generators or battery energy storage systems (BESS) are the keys to a successful system control in power systems. For each power generator in the grid, power reserve represents the total amount of power remaining after the supply of system loads and losses. Of course, this deﬁnition does not extend to exceeding the power capability curve of the generators. These reserves are particularly referred to as spinning reserves in synchronous machines more than in other kinds of power generators, like the wind and solar generators using BESS [1]. These reserves can be used for both or either primary frequency control, secondary control and tertiary control [2] [3] [4]. Again, not all of these remaining power reserves from generators are assigned for primary control function alone, we classify the ones momentarily made available through Bundesministerium f ¨ur Bildung und Forschung (BMBF) CoNDyNet-2, FK. 03EK3055D.the generators’ droop functions speciﬁcally for few seconds primary frequency and active power control as momentarily reserves of the generators. A lot about system stability and reliability rest on the grid’s momentarily reserve. The control and response of generators to network dis- turbances or contingencies depends heavily on the kind and magnitude of the disturbance and amount of momentary reserves available for use in the primary frequency control or in voltage security. Since primary control requires fast control action to be taken within few seconds of contingency, momentary reserves and their placement play important roles in the dynamics of the disturbance from the event location to the rest of the electrical power network. In this paper, we investigate how the dynamics of a disturbance is inﬂuenced by momentary reserves ﬁrst at the contingent node, its neighbour nodes and other nodes located far away from the fault location. We would see whether a disturbance is damped as it travels across a grid and whether it could be contained on fewer nodes (i.e. localized) based on the function of the grid’s momentarily reserves. These investigations are carried out in the DigSILENT PowerFactory software [5], using the Nigerian 330kV grid as a case study. II. M ODELING AND SIMULATION OF THE NIGERIAN TRANSMISSION NETWORK MODEL To understand our case study system and its parameter interactions, we would describe the components that make up the network. The Nigerian 330 kV transmission grid consist of NS=71 substations/nodes, NL=81 over-head transmission lines (from an alloy of aluminium and steel) with an average length of 92 Km, each with a limiting current of 1320 A. The grid is comprised of 107 less-decommissioned units of generators, accounting for the present 29 power stations. The apparent power capacity of the Nigerian grid is about 13;208MW as of 2020 [6]. There are other lower voltage networks including the 132 kV and 33kV sub-transmission networks. For household utilities, there are 11kV and 0:415 kV 3-phase distribution networks. The Nigerian network op- erates at0=50 Hz frequency and can be described as a grid where most of the nodes are connected to one another in a ring form [7], as seen in the diagram shown in Fig.1. To control the voltage outputs of the generators through their excitation control, the simpliﬁed excitation automatic voltage regulator (A VR) model is used [8]. Other controllers also include power system stabilizers (PSS2A-model) tasked with enhancing the damping of the entire power system’s oscillations through excitation control. The input signal to the PSS2A controller is the derivative of generator’s rotor speed injected to the A VR through the excitation system. This injection works to terminate the phase-lag between the voltage reference and the windings’ torque of the generator [9]. For the synchronous machine model, the choice of the model is inﬂuenced by the IEEE guide in [10]. The speed governor in the model is the turbine governor (TGOV) model used to maintain the frequency operational limits according to swing equation [11]. Here, the swing equation describes the torque balance between the mechanical torque Ttin N.m. of each synchronous machine’s turbine and the electromagnetic torque Tein N.m. as governed by the differential equation given as [2] [3] [12], Jid! i dt+Dri! i=TtTeDri!0; (1) whereJi=2Hi !2 0Siis the combined moment of inertia of the generator and turbine in kg.m2withHithe generator inertia constant in seconds and Sithe generator apparent power in volt-amperes ( VA).Drirepresents the rotational loss due to generator rotor windings for each ithgenerator in N.m.s and idenotes the index of power generators in the grid. Here, tis time in seconds and ! iis the angular velocity of the rotor in electrical rad/s with !0as its rated synchronous value in electrical rad/s. If we assume that a change in the rotor’s angular velocity ( ! i!0) is a derivative of its angular position in electrical radians with respect to its rotating setpoint, 0 at t=0 given as ! i!0=di dt; (2) and with respect to time, the derivative of ! iwould give d! i dt=d dtdi dt +d!0 dt; (3) where!0is the constant rated synchronous value whose derivative with respect to time gives zero (i.e.d!0 dt=0 ), Eq.(3) becomes, d! i dt=d2 dt2: (4) In practice,!0is related to the grid frequency ( vo) by 2vo, wherevois50Hz in the Nigerian power grid. If we representthe net mechanical shaft torque at grid frequency to be Tm= TtDri!0, substituting Eq.(4) into Eq.(1), we then have, Jid2i dt2+Dridi dt =TmTe: (5) Here, we assumed that the network perturbation effected on the rotors from the fault location is small. Multiplying both sides of Eq.(5) by the rated speed ( !0) in order to ensure that we maintain a synchronous 50Hz revolution throughout the system, balancing the power, we have Ji!0d2i dt2+Dri!0di dt =Tm!0Te!0: (6) As powerP=T!, the right side of Eq.(6) can now be written as Ji!0d2i dt2+Dri!0di dt =PmPe; (7) wherePmis the turbine’s mechanical power and Peis the generator’s air-gap electrical power. If we represent the rotor angular momentum at rated speed with Mi(i.e.Mi=Ji!0= 2Hi !0Si) and also represent the damping coefﬁcient at rated synchronous speed with Di(i.e.Di=Dri!0), the swing equation can then be re-written in many forms as, Mid2i dt2+Didi dt =PmPe; (8) and also as [13] [14], 2Hi !oSid2i dt2+Didi dt=Pi+NSX j=1Wijsin(ji):(9) WherePiis the power in the grid’s ithnode,NSis the number of nodes/buses and Wijis the power capacity in Watt of the transmission lines and it is dependent on the network voltage with sin (ji)modeling the dependence of their phase differences which informs the direction of powerﬂow and the transmission of disturbances in the case of contingencies. We performed load-ﬂow calculations using the Newton-Raphson method and electromechanical simulations in DigSILENT PowerFactory software, as documented in Ref. [15]. Here, we report results applying these simulations to study the effect of momentary reserves on system dynamics and its contribution to the overall system stability. D I g S I L E N T S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ SG ~ S G ~ S G ~ SG ~ S G ~ S G ~ SG ~ S G ~ S G ~ S G ~ S G ~ S G ~ S G ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ SG ~ S a p e l e G a s . . J e b b a 2 G 1 E g b i n S T 1 w i t h 6 U n i t s S h i r o r o 4 1 1 G 1 S a p e l e S t e a . . P a p a l / O l o r u . . T r a n s c o r p D e l t . . T r a n s c o r p G T . . O m o t o s h o G . . A s c o P l a n t G e r e g u G T 1 R i v e r s I P P G . . I b o m P o w e r A f a m I V - V I . . A f a m I - I I I 1 O m o k u U n i t . . A z u r a I P P G T 1 I h o v b o r N I P . . O m o t o s h o N . . G e r e g u N I P P 1 T r a n s - A m a d i . . O l o r u n s o g o . . A l a o j i N I P P . . A E S G T 1 P l a n t A E S T X E g b i n - A E S I n . . T r a n s - A m a d i . . A f a m I V - V T X 1 O m o k u T X 1 R i v e r s I P P T X 1 G e r e g u T X 1 T r a n s c o r p D e . . A s c o T X 1 I b o m C a l a b a . . A s c o A j a o k u t . . A f a m I - I I I T X 1 T r a n s c o r p 2 . . O d u k p a n i T X 1 I h o v b o r N I P . . O m o t o s h o N . . P a r a s - E n e r g . . A z u r a I P P T X 1 G b a r a n T X G e r e g u N I P P . . O l o r u n s o g o . . A l a o j i T X 1 O m o t o s h o T X 1 P a p a l a n t o / O l . . A l a d j a - D e l t a C a l a b a r - I k o t . . A l a o j i - O w e r r i A f a m V I - I k o t . . G o m b e - Y o l a I k o t E k p e n e - . . A f a m I V _ V - P o . . T r a n s A m a d i - P . . J o s - M a k u r d i A l a d j a - S a p e l . . A l i a d e - N / H a v . . A j a o k u t a B e n i n A l a o j i - I k o t E . . O k p a i - O n i t s h a I k o t A b a s i - I k . . A j a o k u t a - G e . . B e n i n - O m o t o . . I b o m - I k o t - A b . . R i v e r s I P P - P o . . B e n i n - S a p e l . . B e n i n - D e l t a N / H a v e n - O n i . . D a m a t u r u - M a . . N / H a v e n - N / H . . B e n i n - B e n i n . . E g b e m a - O w e r . . A l a o j i - O n i t s h a A l a o j i G . S - A l a . . B e n i n - O s o g b o O m o k u - P o r t . . A l i a d e U g w u a . . A j a o k u t a - L o k . . I h o v b o r - O s o . . E g b e m a - O m . . A f a m V I - A l a o j i B e n i n N o r t h - . . J a l i n g o - Y o l a O w e r r i 1 3 2 k . . O d u k p a n i - I k o . . B e n i n N o r t h - . . B e n i n - I k e j a . . J o s - G o m b e A d i a b o r - O d u . . G b a r a n - Y e n e . . A z u r a - B e n i n E g b i n - E r u n k a n P a r a s - E n e r g y . . k a d u n a - J o s B e n i n - O n i t s h a O n i t s h a - O w e r r i A h o a d a - Y e n . . E r u n k a n - I k e j . . E g b i n S T 2 K a i n j i 1 G 1 1 K a i n j i 1 G 7 S h i r o r o 4 1 1 G 4 Paras-Energy.. S h i r o r o 4 1 1 G 2 K a i n j i 1 G 1 2 Gbaran Plant K a i n j i 1 G 1 0 K a i n j i 1 G 8 Odukpani GT1 A E S G T 2 P l a n t K a i n j i 1 G 9 K a i n j i 1 G 6 S h i r o r o 4 1 1 G 3 T r a n s c o r p 2 . . T r a n s c o r p 2 . . T r a n s c o r p 2 . . G e r e g u N I P P . . T r a n s c o r p 2 . . O d u k p a n i T X 4 O d u k p a n i T X 5 O m o t o s h o N . . O d u k p a n i T X 2 O d u k p a n i T X 3 P a p a l a n t o / O l . . I h o v b o r N I P . . I h o v b o r N I P . . P a p a l a n t o / O l . . A l a o j i T X 2 P a p a l a n t o / O l . . P a p a l a n t o / O l . . P a p a l a n t o / O l . . A z u r a I P P T X 3 I h o v b o r N I P . . P a p a l a n t o / O l . . A z u r a I P P T X 4 A z u r a I P P T X 2 A l a o j i T X 3 O m o t o s h o T X 2 O l o r u n s o g o . . O l o r u n s o g o . . P a p a l a n t o / O l . . O m o t o s h o N . . O m o t o s h o T X 3 O m o t o s h o N . . O l o r u n s o g o . . O l o r u n s o g o . . O m o t o s h o N . . S a p e l e N I P P . . S a p e l e N I P P . . A f a m I V - V T X 3 A f a m I V - V T X 2 A f a m I - I I I T X 3 A f a m I - I I I T X 2 S a p e l e N I P P . . S a p e l e N I P P . . A f a m I V - V T X 4 S a p e l e U n i t . . J e b b a T X 2 S a p e l e U n i t . . S a p e l e U n i t . . J e b b a T X 4 J e b b a T X 5 J e b b a T X 6 O m o k u T X 2 S a p e l e U n i t . . O k p a i T X 3 J e b b a T X 3 O k p a i T X 2 S a p e l e U n i t . . R i v e r s I P P T X 2 G e r e g u T X 2 G e r e g u T X 3 G e r e g u N I P P . . G e r e g u N I P P . . E g b i n G T 5 T X K a i n j i T X 5 K a i n j i T X 6 E g b i n G T 2 T X S h i r o r o T X 4 K a i n j i T X 7 Kainji TX8 S h i r o r o T X 3 K a i n j i T X 3 K a i n j i T X 4 E g b i n G T 6 T X S h i r o r o T X 2 E g b i n G T 5 T . . E g b i n G T 3 T X K a i n j i T X 2 A E S T X 2 K a i n j i 1 G 5 O k p a i G T 1 Jebba 2G4 Egbin ST3 Sapele Stea.. Sapele Stea.. Jebba 2G6 Jebba 2G5 Jebba 2G3 Jebba 2G2 Egbin ST6 Egbin ST4 Egbin ST5 Sapele Stea.. Sapele Stea.. A j a - A l a g b o n E g b i n - I k e j a . . B ' K e b b i - K a i n j i O k e a r o - E g b i n J e b b a - O s o g b . . J e b b a - K a i n j i . . A y e d e - I k e j a . . K a d u n a - S h i r . . S a k e t e - I k e j a . . I k e j a W e s t - O . . G a n m o - O s o . . I k e j a W e s t - A . . P a p a l a n t o - I k . . A y e d e - O s o g b o G w a g s - S h i r o r o G w a g s - L o k o j a G w a g s - K a t a . . J e b b a G . S - J e . . K a t a m p e - S h i . . J e b b a - S h i r o r o I k e j a W e s t - . . D a m a t u r u - G . . K a i n j i G . S - K a i . . O k e a r o - I k e j a . . K a d u n a - K a n o J e b b a - G a n m o P a p a l a n t o - A . . B e n i n - E g b i n A j a - E g b i n Azura IPP GT2 Olorunsogo .. Afam IV-VI .. Omotosho N.. Odukpani GT2 Papal/Oloru.. Ihovbor NIP.. Afam IV-VI .. Azura IPP GT4 Afam I-III 2 Omotosho N.. Sapele Gas .. Sapele Gas .. Azura IPP GT3 Papal/Oloru.. Papal/Oloru.. Odukpani GT3 Olorunsogo .. Papal/Oloru.. Transcorp GT.. Olorunsogo .. Afam I-III 3 Omotosho N.. Geregu NIPP 4 Omotosho N.. Alaoji NIPP .. Odukpani GT5 Sapele Gas .. Papal/Oloru.. Ihovbor NIP .. Papal/Oloru.. Olorunsogo .. Papal/Oloru.. Omotosho G.. Sapele Gas .. Transcorp GT.. Omotosho G.. Transcorp GT.. Transcorp GT.. Ihovbor NIP.. Alaoji NIPP .. Odukpani GT4 Omoku Unit .. Okpai GT3 Okpai GT2 Sapele Stea.. Geregu NIPP 2 Geregu GT3 Geregu NIPP 3 Afam IV-VI .. Rivers IPP G.. Geregu GT2 O k p a i T X 1 K a i n j i T X 1 S a p e l e N I P P . . S a p e l e U n i t . . E g b i n G T 1 T X J e b b a T X 1 S h i r o r o T X 1 I k o r o d u 1 3 2 K . . O w e r r i 3 3 0 / 1 . . B'KEBBI T .S IKOT-ABASI .. IBOM G.S ADIABOR T.S ODUKPANI .. GBARAIN G.S EGBEMA ALAOJI G.S ALIADE UGW.. AFAM IV-V RIVERS IPP.. ALADJA T .S TRANS-AM.. SAPELE NIPP DAMATURU .. MAIDUGURI.. PARAS ENE.. OKEARO T.S. ALAGBON T... AJA T .S. PAPALANTO SAKETE T.S. IHOVBOR N.. GWAG T.S KATAMPE T ... JEBBA G.S KAINJI G.S GANMO T.S KAINJI T .S JEBBA T.S AFAM VI ALAOJI T .S N/HAVEN T... N/HAVEN S.. GERE G.S/.. ASCO G.S AZURA IPP(2) YOLA T.S MAKURDI T .S GOMBE T.S ERUNKAN T.. AES G.S IKORODU T .S AYEDE T.S. KANO T .S JOS T .S KADUNA T.S. SHIRORO G.. OWERRI T .S OKPAI P.S JALINGO T.S PH MAIN T.S OMOKU G.S EYAEN T.S AJAOKUTA .. SAPELE G.S ONITSHA T .S. EGBIN G.S. DELTA G.S BENIN NOR.. BENIN T .S IKOT-EKPEN.. CALABAR T .S Yenagoa T .S OMO PH1/N.. AKANGBA T .S OSOGBO T.S. IKEJA WEST.. AHODA T .S Owerri 132KV LOKOJA T.S Load Flow Balanced Nodes Ul Line-Line Voltage, Magnitude [kV] u Voltage, Magnitude [p.u.] phiu Voltage, Angle [deg] Branches P Active Power [MW] Q Reactive Power [Mv I Current, Magnitude Inactive Out of Calculation De-energised Voltages / Loading Lower Voltage Range 1. p.u. ... 0.95 p.u. ... 0.9 p.u. Upper Voltage Range 1. p.u. ... 1.05 p.u. ... 1.1 p.u. Loading Range 80. % ... 100. %Fig. 1: The Nigerian 330 kV Electrical Power Grid 60.048.036.023.911.9-0.10[s]50.0149.9949.9749.9549.9349.91[Hz] X = 6.242 s49.917 Hz49.966 HzY = 49.966 Hz 5.319 s17.177 s27.829 s39.846 sX = 27.829 s49.966 Hz49.966 HzX = 18.612 s49.968 Hz49.966 Hz22.472 s49.968 Hz39.792 s49.966 Hz54.842 s49.966 Hz27.822 s49.966 HzLocal Max. = 49.968 Local Min. = 49.91750.00 Hz50.00 Hz Local Minima Magnitude and Arrival timeLocal Minima Magnitude and Arrival timeLocal Maxima MagnitudeFinal Frequency Settling MagnitudeLocal Maxima MagnitudeLocal Maxima Arrival timeLocal Maxima Arrival timeFinal Frequency Settling timeFinal Frequency Settling timeFinal Frequency Settling Magnitude Frequency deviationThreshold = 0.002 Hz Fig. 2: Green Line: Frequency as a function of time at a node where a Synchronous Machine (SM) Event occurred at t= 5s. Red line: Frequency magnitude at ﬁnal settling time. Black Arrows: are pointing to the local maxima and minima of the frequency magnitude and arrival times. III. D YNAMICS OF MOMENTARY RESERVE For a power system to be in a balanced state, the total power generated ( P) has to be equal to the sum of the total load and the transmission losses ( Px). This means that for the system to always remain balanced, any change in the loaddemand and/ losses would require a balancing change in the generated power P. Therefore, the condition P >> Px should always be fulﬁlled to ensure the security and integrity of the power grids. These dynamic changes result in a per- manently changing frequency according to the speed-droop characteristics of power turbines, given by [3] i 0=iPi Px i(10) where0is the nominal frequency in Hertz and ithe local droop of the generation characteristics at the ithgenerator node,i=Px i=Pi. Thus, any change in frequency resulting from a change in load demand or losses requests a generator response. In this way, if the node is a synchronous generator node, its inertia is expected to slow down the transmission of disturbances to other nodes connected to it, while attempting to damp the frequency oscillations with its momentary reserve according to its droop ifunction. However, its ability to do so depends on the amount of momentary reserve that the generator can supply, according to its dead zone speed-droop characteristics setting. Since it is not feasible to inject power equally at every node, we aim to understand the dynamics of these reserves to see how an available momentary reserve at one node inﬂuences the frequency dynamics at other nodes as function of their distance from the fault location. A better understanding of this inﬂuence in the realistic Nigerian grid model would be advantageous for its optimal placement and could improve primary frequency control in real power grids. To show the inﬂuence of geographical distribution of inertia with available momentary reserve, we chose the Nigerian transmission grid as the case study grid. As reported in Section II, we modelled this network in PowerFactory software and calculated the load ﬂow of the network. We considered the voltage dependency of grid loads, and the active power control according to the grid inertia. The effect of provision of momentary reserves by system inertia on the frequency stability of the electrical power networks is studied by varying the aggregated inertia constant Haggof the entire grid, as deﬁned by [16], Hagg=PNS i=1HiSi Sn; (11) whereSiandHiare the rated apparent power and inertia constant of the ithbus, respectively, Sn=PNS i=1Siis the total rated power in the grid. Note that Hion non-generator buses are set to zero. The Haggwas varied in the entire grid by multiplying the Siwith a common factor. Note that the Nigerian transmission network does not comprise tie-line connections and power transfer [6] [17]. The grid frequency is operated uniformly at 50Hz across the entire network. Here, we choose 11buses for the investigation. The fault location bus 24, two buses at the same geodesic distance (unweighted), r= 2 from fault location with no inertia (i.e. buses 8and10) and three buses with inertia (i.e. buses 22, 55, and 57). We again choose two buses at the same geodesic distance (unweighted), r= 7 with no inertia (i.e. buses 7and 30) and three buses with inertia (i.e. buses 3,28, and 69). The reason is that we want to investigate the effect of momentary reserve both in the vicinity of the reserve (i.e. near to the fault location ) and at far distances from the injection node. We also want to understand how momentary reserve tentatively contributes to damping of disturbances as they propagate along the network. A reference to the network buses numbers is in [18]. Keeping all system parameters, generations, losses and loads constant under undisturbed operations, a synchronous machine outage event (disturbance/contingency) is induced at bus 24at exactlyt= 5s of the 90s transient electromechanical stability simulation time frame with a 200ms switching, enabling the observations on the buses in PowerFactory power simulation software. The change in frequency propagated across the grid is related to the change of power by Eq. (10). From the nodal points described in 2, we observe at each study case node, the frequency’s time of arrival (ToA), as deﬁned when the frequency deviation ﬁrst reaches a small threshhold of= 0:0001 Hz, as deﬁned in more detail in Ref. [18]. Furthermore we record the time of the ﬁrst maximum (maxima t) of the transient and its magnitude (maxima mag), the time of the ﬁrst minimum (minima t) and its magnitude (minima mag). We also observe the ﬁnal frequency’s settling time (FS t), its magnitude (FS mag) and frequency deviation(Dev mag) from the nominal 50Hz, for each of these nodes. In Tables I-IV we show the observations in milliseconds for the case study nodes, with static network power ﬂow of the Nigerian transmission grid. TABLE I: Nodal observations with large disturbance and no reserve at fault location given that Hagg= 2s Bus r ToA minima t minima mag FSt FSmag Devmag (s) (s) (Hz) (s) (Hz) (Hz) 24 0 5.012 6.522 49.596 27.200 49.829 0.171 8 2 5.012 6.522 49.596 27.201 49.829 0.171 10 2 5.012 6.522 49.596 27.200 49.829 0.171 22 2 5.013 6.282 49.596 27.195 49.829 0.171 55 2 5.013 6.532 49.596 27.201 49.829 0.171 57 2 5.012 6.532 49.597 27.196 49.829 0.171 7 7 5.013 6.372 49.598 27.205 49.829 0.171 30 7 5.013 6.322 49.600 27.206 49.829 0.171 3 7 5.013 6.422 49.590 27.185 49.829 0.171 28 7 5.013 6.942 49.596 27.184 49.829 0.171 69 7 5.013 6.352 49.592 27.186 49.829 0.171 TABLE II: Nodal observations with large disturbance and large reserve at fault location given that Hagg= 2s Bus r ToA minima t minima mag FSt FSmag Devmag (s) (s) (Hz) (s) (Hz) (Hz) 24 0 5.014 6.132 49.920 25.700 49.967 0.033 8 2 5.014 6.112 49.920 25.691 49.967 0.033 10 2 5.014 6.102 49.920 25.690 49.967 0.033 22 2 5.015 6.282 49.919 25.687 49.967 0.033 55 2 5.014 6.082 49.920 25.691 49.967 0.033 57 2 5.015 6.192 49.919 25.688 49.967 0.033 7 7 5.016 6.292 49.920 25.696 49.967 0.033 30 7 5.016 6.372 49.920 25.698 49.967 0.033 3 7 5.017 6.252 49.918 25.679 49.967 0.033 28 7 5.017 6.142 49.916 25.678 49.967 0.033 69 7 5.017 6.182 49.918 25.680 49.967 0.033 TABLE III: Nodal observations with large disturbance and large reserve at fault location given that Hagg= 6s Bus r ToA minima t minima mag FSt FSmag Devmag (s) (s) (Hz) (s) (Hz) (Hz) 24 0 5.015 9.312 49.928 29.304 49.967 0.033 8 2 5.015 9.262 49.928 29.304 49.967 0.033 10 2 5.015 9.272 49.928 29.304 49.967 0.033 22 2 5.017 8.352 49.928 29.302 49.967 0.033 55 2 5.015 9.442 49.928 29.305 49.967 0.033 57 2 5.017 9.142 49.928 29.302 49.967 0.033 7 7 5.019 9.182 49.928 29.308 49.967 0.033 30 7 5.020 8.932 49.928 29.309 49.967 0.033 3 7 5.028 8.672 49.927 29.296 49.967 0.033 28 7 5.074 8.602 49.927 29.295 49.967 0.033 69 7 5.029 8.652 49.927 29.296 49.967 0.033 In Table I and with network Haggat 2s, a typical behaviour of high renewable energy source injected grids, we observe a high frequency deviation (i.e. 0:171Hz) induced by the large disturbance (of 1320MW power magnitude outage) at bus 24 fault location. We also observe the disturbance arrival at the fault location ﬁrst and at the same time as most of its nearest neighbours at r=2, but arrived a little later at the distant buses 7, 30, 28 and 69. The frequency dip described by the minima magshows the lowest magnitude of frequency deviation before the actions of generator governor-turbine. TABLE IV: Nodal observations with large disturbance and large reserve at fault location and an increased reserve at bus 22given thatHagg= 2s Bus r ToA minima t minima mag FSt FSmag Devmag (s) (s) (Hz) (s) (Hz) (Hz) 24 0 5.014 6.612 49.920 27.645 49.967 0.033 8 2 5.014 6.122 49.921 27.647 49.967 0.033 10 2 5.014 6.612 49.920 27.645 49.967 0.033 22 2 5.015 6.352 49.919 27.641 49.967 0.033 55 2 5.014 6.582 49.920 27.647 49.967 0.033 57 2 5.015 6.152 49.921 27.642 49.967 0.033 7 7 5.016 6.342 49.921 27.653 49.967 0.033 30 7 5.016 6.452 49.921 27.655 49.967 0.033 3 7 5.017 6.282 49.919 27.630 49.967 0.033 28 7 5.017 6.142 49.917 27.629 49.967 0.033 69 7 5.017 6.252 49.919 27.631 49.967 0.033 AtHagg =2s and in comparison with Table I, the Table II shows the observations when the fault location is injected with inertia and large reserve. We observe that there are delays in the frequency ToA at the buses, a reduction in minima twhich corresponds to a reduced frequency dip (i.e. minima mag), a shorter ﬁnal frequency settling time and smaller magnitude and hence, a decrease in the frequency deviation from the nominal value (0.033 Hz). In Table III, we kept the network parameters constant and only increased the grid inertia, Hagg=6s using Eq. (11) without changing the active power injections at any network node. We did not observe any further decrement in the frequency deviation or any increment in the frequency ﬁnal settling magnitude, rather we observe a delay in its time of arrival at the buses with a corresponding increase in its dip and ﬁnal settling time. This suggest that increasing the Haggwithout a corresponding increase in the reserve does not improve the frequency dynamics during contingencies, rather it increases the arrival time of the disturbance while reducing the frequency dip across the network (i.e. improved minima mag). This further delay in ToA also increases the frequency ﬁnal settling time. Table IV shows our nodal observations on the case study buses when we injected more power reserve prior to con- tingency at another bus apart from the fault location and keeping all other system operations parameters constant from observations in Table II. Here the generator’s power and reserve at the fault location are higher in magnitude than the injected ones. We observe that the frequency ﬁnal settling magnitude did not increase, the ToA at the buses remained the same but there is an observable reduction in the frequency dip at some nodes. In summary, we observe that an increase in the nodal momentary reserve generally delays the travel and arrival of disturbances in a power grid at contingencies. In particular, it improves the frequency magnitude at the local minima point and reduces its ﬁnal settling time. We found out that the optimal placement of momentary reserve is at the point of contingency as it contributes more in the damping of disturbance across the network more than in any other place.Since we may not always be able to predict a fault location, the optimal solution would be to place momentary reserve at all buses where resources allow. In this way, the power system could quickly recover most contingencies within few seconds after its occurrence. Also, we concluded that increasing the grid inertia without a corresponding increase in the magnitude of the reserve could only delay the travel and arrival of disturbances in electrical network but does not reduce the frequency deviation from the nominal value. This result would be important when injecting virtual inertia in the energy transition to renewable schemes. IV. CONCLUSION In this paper, we have studied the dynamics of power system reserves. By realistic numerical experiments we explored how they contribute to the damping of system oscillations at con- tingencies and thereby work to restore the grid frequency to its nominal value. We have shown that the optimal placement of momentary reserve would be at the fault location, particularly in the case of generator outage events at plant stations. Since this situation could not always be predicted, we suggest placement of reserves in all the nodes where resources permit as this would improve the overall ﬁnal frequency settling time, frequency dips, and reduce overall frequency deviation from the nominal value, thereby contributing to primary frequency control and reducing the amount of secondary control needed. Refer to our preprint here [18]. ACKNOWLEDGMENT We gratefully acknowledge the support of Bundesminis- terium f ¨ur bildung und forschung (BMBF) CoNDyNet-2, FK. 03EK3055D.