Power Engineering II. Power System Transients

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1 Power System Transients

2 Oeroltage Maximum supply oltage U m the maximum effectie alue of line oltage that can occur at any time or place under normal operating conditions Rated oltage (k) Maximum supply oltage (k) 7, , Oeroltage any line or phase oltage that exceeds the amplitude of the respectie maximum supply oltage (u > 2U m ) Power System Transients 2

3 Types of Oeroltage Diided by Magnitude Duration Cause of occurrence The magnitude of an oeroltage is expressed by an oeroltage coefficient that is determined from either relatie or absolute alues: k f = u fm 2U 3 Where U is the maximum supply oltage and u fm is the maximum line-to-ground oeroltage By duration Permanent oeroltage power frequency, constant effectie alue Temporary oeroltage power frequency, 0.03 s s Transient oeroltage (slow, fast, ery fast) duration of units of ms or shorter, damped oscillating or impulse characteristic Combined oeroltage a combination of two types of oeroltage Power System Transients 3

4 Types of Oeroltage By cause of occurrence Internal oeroltage (operating) its magnitude can be defined as a multiple of the rated oltage of the network External oeroltage (lightning) its magnitude is independent on the rated oltage of the network Internal oeroltage Faults (ground fault, short-circuit) Switching operations (switching of capacitie and inductie currents, semiconductor deices) Electrical resonance (resonance and ferroresonance) Switching operations on power lines (Load switching, automatic reclosing) External oeroltage Direct lightning strike to the network Oeroltage induced by a near lightning strike Lightning-caused oeroltage in buildings Power System Transients 4

5 Switching off of Short-Circuit AC R u R ~ U C γ = R L L u L u C ω 0 2 = 1 LC d 2 u C dt 2 Ri + L di dt + u C = Ucos ωt i = C du C dt RC du C dt + LC d2 u C dt 2 + γ du C dt +ω2 u C = U cos ωt LC + u C = Ucos ωt We assume that the transient eent is periodic and damped (weak damping), i.e. γ < 2ω 0 ; therefore the general solution is: u C0 = e γ 2 t sin ω f t + φ, where ω f = And the particular solution is: u Cp = Bcos ωt + ψ ω 0 2 γ2 4 Power System Transients 5

6 Switching off of Short-Circuit AC The resulting oltage u C can be then expressed as: u C = u C0 + u Cp = e γ 2 t sin ω f t + φ + Bcos(ωt + ψ) By putting u c = u Cp into the differential equation for u C, we receie the following equation: Bω 2 cos ωt + ψ Bγω sin ωt + ψ + Bω 0 2 cos ωt + ψ = U LC cos(ωt) This expression can be isualized by a phasor diagram in the following manner: We can see from the figure that B a Ψ are: B(ω 0 2 ω 2 ) Bω 0 2 U LC = ω 0 2 U Bω 2 Bγω B = ψ = arctg Uω 0 2 ω 0 2 ω γ 2 ω 2 γω ω 0 2 ω 2 Power System Transients 6

Switching off of Short-Circuit AC During the switching off of short-circuit current, we assume that the oltage u c is equal to zero in the beginning of the process, i.e.: u c (0) = 0 By employing the preious equation as an initial condition, we can determine the only remaining unknown parameter φ.

7 Switching off of Short-Circuit AC During the switching off of short-circuit current, we assume that the oltage u c is equal to zero in the beginning of the process, i.e.: u c (0) = 0 By employing the preious equation as an initial condition, we can determine the only remaining unknown parameter φ. The resulting oltage trend of u c (on switch contacts) Time (s) Power System Transients 7

8 Switching off of Small Inductie Currents Inductie currents are relatiely small when compared to the rated current (e.g. no-load current of transformer, no-load current of a squirrel-cage rotor engine, reactor or Peterson coil current) The unstable character of an electric arc inside a switch creates oscillations with erratic amplitudes The current should be switched off when its instantaneous alue reaches zero. The oeroltage factor can reach alues of oer 2.5. L s L L L T i ~ u C s u s C T u T Power supply side Transformer side Power System Transients 8

Switching off of Small Inductie Currents i 1 i 1 2 3 4 Transient eents caused by unstable electric arc create oscillations that are superposed on power frequency.

9 Switching off of Small Inductie Currents i 1 i Transient eents caused by unstable electric arc create oscillations that are superposed on power frequency. Therefore, the resulting current reaches zero alue already in times 1-4 (see figure). If the transient recoery oltage between the switch contacts rises faster than the dielectric strength of the space between both contacts, the circuit reconnects and needs to be switched off again. Power System Transients 9

10 Switching off of Small Inductie Currents i u T t t The oltage between both contacts rises (drops) sharply whilst the distance between switch contacts increases insufficiently. This causes re-ignition of the arc after the first switch operation. Repetitie re-ignition creates saw tooth oltage on the transformer side (u T ) of the circuit, as seen in the figure. Power System Transients 10

11 Switching of Capacitie Currents Oeroltage is caused by switching on and off of capacitie currents (primarily switching of capacitor batteries that sere as reactie power compensators) Connecting an uncharged capacitor battery C B to a power source (u,l,c) ia switch S forms a current that is limited only by inductance L of the power source in the first few moments. Since C B is much larger than C, oltage and current oscillations emerge. Frequency of these oscillations is gien by parameters L and C B. Oeroltage during the switching reaches a maximum alue equal to the double of the source s amplitude. L ~ u C S C B Power System Transients 11

12 Switching off of Capacitie Currents With S switched on, the oltage on C B is higher than on the power source. When the switch is disconnected, the current ceases to flow when it passes zero alue. Subsequently, a oltage difference emerges between the switch contacts. The oltage of the source part of the circuit drops down to the network oltage ia a transient eent. The oltage of the battery part of the circuit remains constant. The oltage difference between contacts increases and an electric arc ignites at time 2. oltage on C B changes to u C ia a transient. The maximum oeroltage during the eent is equal to the triple of u C u c u s t i 1 2 t Power System Transients 12

13 Traelling Waes i(x,t) Ldx Rdx i(x+dx,t) u(x,t) Cdx Gdx u(x+dx,t) dx Circuit equations: i x + dx, t u x, t + Rdxi x + dx, t + Ldx + u x + dx, t = 0 t u x, t i x, t Gdxu x, t Cdx i x + dx, t = 0 t u x + dx, t u(x, t) = Ri(x + dx, t) L i dx t i x + dx, t i(x, t) u(x, t) = Gu(x, t) C dx t u x i x = Ri L i t = Gu C u t Power System Transients 13

14 Traelling Waes By current/oltage elimination and if the lossless line is assumed (R->0 and G->0), we can get the final wae equations in the form: 2 u t 2 = 1 2 u LC x 2 2 i t 2 = 1 LC i x, t = f 1 x t + f 2 (x + t) u After substitution in oltage eq. : x = L t f 1 x t f 2 (x + t) 2 i x 2 D Alambert s solution is any function with argument (x±t), where = 1 LC elocity of wae propagation. The solution for a current wae: u(x, t) = L f 1 x t f 2 (x + t) is the u(x, t) = L C f 1 x t f 2 (x + t) Power System Transients 14

15 Traelling Waes u i i u u i i u The oltage backward wae is in phase with the forward wae, the current backward wae is in opposite phase to the forward wae. Power System Transients 15

16 Boundary influence to traelling waes U d U r Z 01 Z 02 U p i, I i r, I r t, I t incident wae reflected wae transmitted wae Currents can be expressed as : I i = i Z 01 I r = r Z 01 I t = t Z 02 oltage and current equation for border: t = i + r I t = I i + I r Power System Transients 16

17 Boundary influence to traelling waes oltage reflection coefficient ρ = r i : i Z 01 r Z 01 = i Z 02 + r Z 02 r i = Z 02 Z 01 Z 01 Z 02 = Z 01 + Z 02 Z 01 Z 02 Z 02 Z 01 Z 01 + Z 02 Similarly we can get current reflection coefficient: ρ I = Z 01 Z 02 Z 01 + Z 02 and oltage and current transmission coefficients: τ = 2Z 02 Z 01 + Z 02 τ I = 2Z 01 Z 01 + Z 02 Power System Transients 17

18 Effect of line terminations Short-circuit line: Z 0 Open-circuit line: Z 0 =0 I=0 I I I I I I x x Power System Transients 18

19 Effects of line terminations Line termination in inductance: Z 0 L Line termination in capacitance: Z 0 C t t Power System Transients 19

20 Lightning discharge Causes the oeroltage in electric power systems The lightning discharge originates in lightning clouds (cumulonimbus), which are consequences of intensie ertical air flow Power System Transients 20

Charge distribution in clouds Charge separation occurs inside a cloud many theories of charge separation mechanism exists, the whole process is still not fully understood The upper

and positie charge in its upper part or induced positie charge on the ground can be neutralized by lightning discharge, so we can recognize: Lightning discharge between centers of

21 Charge distribution in clouds Charge separation occurs inside a cloud many theories of charge separation mechanism exists, the whole process is still not fully understood The upper part of a cloud consists of ice crystals with positie charge while the lower part of a cloud consists of negatiely charged water droplets Negatie charge in the lower part of clouds and positie charge in its upper part or induced positie charge on the ground can be neutralized by lightning discharge, so we can recognize: Lightning discharge between centers of negatie and positie cahrge inside the clouds Lightning discharge between clouds and Earth s surface Power System Transients 21

22 Lightning discharge propagation A discharge propagates from a cloud to the ground in steps (approx. 20 m) which are elongated towards the surface. This discharge is called stepped leader and the head of the leader is moing with the elocity in order of 10 5 m/s to ground. While approaching the ground, the upward discharge emerges. When both discharges connect, a highly conductie path is created and the potential difference between the cloud and the surface causes a current impulse and a return stroke. The return stroke (the main isible eent) follows the path of both preious discharges (with the speed approx m/s), flashing with a lasting approx. 0.1 s 20 ms 40 ms 1 ms 40 ms 1 ms 1 ms Power System Transients 22

23 Current of Lightning Discharge Lightning current waeform is important for an assessment of lightning effects to electric power systems The parameters of current waeforms are statistical alues determined on the base of obserations Negatie first partial discharge has higher amplitude than subsequent partial discharges Positie lightning has higher amplitude and lower steepness Total time of lightning duration is usually hundreds of milliseconds Peaks of currents (ka) Cumulatie frequency 95% 50% 5% Negatie first partial discharge Negatie subsequent partial discharge 4, Positie lightning 4, Power System Transients 23

24 Oeroltage caused by lightning discharge Oeroltage can be caused by: oltage drop across conductor with lightning current Electromagnetic field which arises as consequence of lightning discharge Oeroltage on oerhead lines and in buildings can be classified as: Direct lightning stroke to oerhead lines Induced by lightning discharge Caused by lightning discharge in buildings Power System Transients 24

25 Direct stroke to oerhead line Injected current impulse propagates to both sides of oerhead line -> current and oltage traelling waes oltage and current waes are reflected in locations where the change of surge impedance is present If the oer head line ground wire is struck, the reflections occurs in the place of joints with towers and in the place of grounding of towers The most dangerous oeroltage arises at direct stroke of lightning to phase conductor of line; assuming the peak current of lightning discharge I m = 30 ka and line surge impedance Z =300 Ω, then the peak alue of oeroltage is: m = Z 2 I m = = 4,5 M Power System Transients 25

26 Oeroltage induced to line Lightning stroke causes rapid change of electromagnetic field and creates induced oltages Dangerous oeroltage for an insulation system can arise if the lightning strikes somewhere in the distance of up to 5 km from an oerhead line Capacitie coupling Inductie coupling C m Z 01 i/2 i/2 Z 02 Z 0 Z 01 L m + E - Z 02 Z 0 Power System Transients 26

Transmission Line Transients

8 5 Transmission Line Transients CHAPTER OBJECTIES After reading this chapter, you should be able to: Proide an analysis of traelling waes on transmission lines Derie a wae equation Understand the effect