Dynamic fault classification and location in distribution networks

This paper presents a method for detecting, classifying and localizing faults in MV distribution networks. This method is based on only two samples of current or voltage signals. The fault detection, faultclassi cation and fault localization are based on the maximum value of current and voltage as a function of time.

❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ❉②♥❛♠✐❝ ❋❛✉❧t ❈❧❛ss✐❢✐❝❛t✐♦♥ ❛♥❞ ▲♦❝❛t✐♦♥ ✐♥ ❉✐str✐❜✉t✐♦♥ ◆❡t✇♦r❦s ∗ ❆❜❞❡❧❤❛❦✐♠ ❇❖❯❘■❈❍❆ ✱ ❚❛❤❛r ❇❖❯❚❍■❇❆✱ ❙❛♠✐r❛ ❙❊●❍■❘✱ ❘❡❜✐❤❛ ❇❖❯❑❍❆❘■ P♦✇❡r ❙②st❡♠ ❖♣t✐♠✐③❛t✐♦♥ ▲❛❜♦r❛t♦r②✭▲❖❘❊✮ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡s ❛♥❞ ❚❡❝❤♥♦❧♦❣② ♦❢ ❖r❛♥ ▼♦❤❛♠♠❡❞ ❇♦✉❞✐❛❢✱ ❯❙❚❖ ❇✳P✳ ✶✺✵✺ ❊❧✲▼♥❛♦✉❛r✱ ❖r❛♥ ✸✶✵✵✵ ✲ ❆❧❣❡r✐❛ ✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❆❜❞❡❧❤❛❦✐♠ ❇❖❯❘■❈❍❆ ✭❡♠❛✐❧✿ ❛❜❞❡❧❤❛❦✐♠✳❜♦✉r✐❝❤❛❅✉♥✐✈✲✉st♦✳❞③✮ ✭❘❡❝❡✐✈❡❞✿ ✶✷✲▼❛r❝❤✲✷✵✶✽❀ ❛❝❝❡♣t❡❞✿ ✵✹✲❙❡♣t❡♠❜❡r✲✷✵✶✽❀ ♣✉❜❧✐s❤❡❞✿ ✸✶✲❖❝t♦❜❡r✲✷✵✶✽✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✽✷✸✳✶✶✹ ❆❜str❛❝t✳ ❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛ ♠❡t❤♦❞ ❢♦r ❞❡✲ t❡❝t✐♥❣✱ ❝❧❛ss✐❢②✐♥❣ ❛♥❞ ❧♦❝❛❧✐③✐♥❣ ❢❛✉❧ts ✐♥ ▼❱ ❞✐str✐❜✉t✐♦♥ ♥❡t✇♦r❦s✳ ❚❤✐s ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ ♦♥❧② t✇♦ s❛♠♣❧❡s ♦❢ ❝✉rr❡♥t ♦r ✈♦❧t❛❣❡ s✐❣♥❛❧s✳ ❚❤❡ ❢❛✉❧t ❞❡t❡❝t✐♦♥✱ ❢❛✉❧t❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❢❛✉❧t ❧♦❝❛❧✐③❛t✐♦♥ ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ ❝✉rr❡♥t ❛♥❞ ✈♦❧t❛❣❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❆ st✉❞② ✐s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ✇♦r❦ t♦ ❡✈❛❧✉❛t❡ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✳❆ ❝♦♠♣❛r❛t✐✈❡ st✉❞② ❜❡t✇❡❡♥ ❝✉rr❡♥t ❛♥❞ ✈♦❧t❛❣❡ ♠❡t❤♦❞ ❞❡t❡❝t✐♦♥ ❤❛s ❜❡❡♥ ❞♦♥❡ t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ✐s t❤❡ ❢❛st❡st✳ ■♥ ❛❞❞✐✲ t✐♦♥✱ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ❢❛✉❧ts ✇❡r❡ ♠❛❞❡ ❜② t❤❡ s❛♠❡ ♠❡t❤♦❞ ✉s✐♥❣ t✇♦ s❛♠✲ ♣❧❡s s✐❣♥❛❧✳ ❙✐♠✉❧❛t✐♦♥ ✇✐t❤ r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ ▼❆❚▲❆❇ ✴ ❙✐♠✉❧✐♥❦ s♦❢t✲ ✇❛r❡✳ ❘❡s✉❧ts ❛r❡ r❡♣♦rt❡❞ ❛♥❞ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❞r♦✇♥✳ ♦♠②✳ ❲✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ ✇♦r❧❞ ♣♦♣✉❧❛t✐♦♥✱ ❛♥❞ ❞❡✈❡❧♦♣♠❡♥t ✐♥ ❛❧❧ ❛r❡❛s✱ t❤❡ ❞❡♠❛♥❞ ❢♦r ❡❧❡❝tr✐❝ ♣♦✇❡r ✐s ❣r♦✇✐♥❣ r❛♣✐❞❧②✳ ▼❡❞✐✉♠✲❱♦❧t❛❣❡ ❡❧❡❝tr✐❝❛❧ ♣♦✇❡r ❞✐str✐❜✉t✐♦♥ ❧✐♥❡s ❛r❡ ❛♥ ❡ss❡♥t✐❛❧ ♣❛rt ♦❢ ❛♥ ❡❧❡❝tr✐❝❛❧ ♣♦✇❡r ❣r✐❞ t❤❛t ♠✉st ❡♥s✉r❡ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ♣♦✇❡r s✉♣♣❧② t♦ ▼❡❞✐✉♠ ❱♦❧t❛❣❡ ✭▼❱✮ ❛♥❞ ▲♦✇ ❱♦❧t✲ ❛❣❡ ✭▲❱✮ ❝♦♥s✉♠❡rs✳ ❚❤❛t ✐s ♥♦t ❛❧✇❛②s t❤❡ ❝❛s❡✱ ❚❤❡s❡ ❧✐♥❡s ❡①♣❡r✐❡♥❝❡ ❢❛✉❧ts ✇❤✐❝❤ ❛r❡ ❝❛✉s❡❞ ❜② st♦r♠s✱ ❧✐❣❤t♥✐♥❣✱ s♥♦✇✱ ❢r❡❡③✐♥❣ r❛✐♥✱ ✐♥s✉✲ ❧❛t✐♦♥ ❜r❡❛❦❞♦✇♥ ❛♥❞✱ s❤♦rt ❝✐r❝✉✐ts ❝❛✉s❡❞ ❜② ❜✐r❞s ❛♥❞ ♦t❤❡r ❡①t❡r♥❛❧ ♦❜❥❡❝ts ❬✶❪✳ ❚❤❡s❡ ❢❛✉❧ts ♠✉st ❜❡ ❞❡t❡❝t❡❞✱ ❝❧❛ss✐✜❡❞ ❛♥❞ ❧♦❝❛❧✐③❡❞ q✉✐❝❦❧② ❛♥❞ ❝♦rr❡❝t❧② s♦ t❤❛t ♦✉r s②st❡♠ r❡♠❛✐♥s st❛❜❧❡✳ ❲❤❡♥ ❛ ❢❛✉❧t ♦❝❝✉rs ✐♥ ❛ ❞✐str✐❜✉t✐♦♥ ♥❡t✲ ✇♦r❦s✱ t❤❡ ❢❛✉❧t ❝✉rr❡♥t ✐s ❛❧✇❛②s ❣r❡❛t❡r t❤❛♥ t❤❡ r❛t❡❞ ❧♦❛❞ ❝✉rr❡♥t ❛♥❞ t❤❡ ❢❛✉❧t ✈♦❧t❛❣❡ ✇✐❧❧ ❜❡ s♠❛❧❧❡r t❤❛♥ t❤❡ ♥♦♠✐♥❛❧ ♥❡t✇♦r❦ ✈♦❧t❛❣❡✳ ❑❡②✇♦r❞s ❚❤❡ ❞❡t❡❝t✐♦♥ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ❢❛✉❧ts ✐♥ ❡❧❡❝✲ tr✐❝❛❧ ♥❡t✇♦r❦s ♣❧❛②s ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ t❤❡ ❉✐str✐❜✉t✐♦♥ ◆❡t✇♦r❦✱ ❋❛✉❧t ❉❡t❡❝t✐♦♥✱ ❋❛✉❧t ❈❧❛ss✐✜❝❛t✐♦♥✱ ❋❛✉❧t ▲♦❝❛❧✐③❛t✐♦♥✳ ❝♦rr❡❝t ♦♣❡r❛t✐♦♥ ♦❢ ♣r♦t❡❝t✐✈❡ r❡❧❛②s✳ ❋❛✉❧t ❞❡t❡❝t✐♦♥ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥ ❝♦♥✈❡♥t✐♦♥❛❧ ♠❡t❤♦❞s ❢♦r ❞✐str✐❜✉t✐♦♥ ❧✐♥❡s ❛r❡ ❜r♦❛❞❧② ❝❧❛ss✐✲ ✜❡❞ ❛s ✐♠♣❡❞❛♥❝❡ ❜❛s❡❞ ♠❡t❤♦❞ ✇❤✐❝❤ ✉s❡s t❤❡ st❡❛❞② st❛t❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ ✈♦❧t✲ ✶✳ ❛❣❡ ❛♥❞ ❝✉rr❡♥t ✈❛❧✉❡s ❬✷❪✲❬✻❪✳ ❲❛✈❡❧❡t ♠❡t❤♦❞ ■◆❚❘❖❉❯❈❚■❖◆ ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ ❧♦✇ ♣❛ss ✜❧t❡rs ❛♥❞ ❤✐❣❤ ♣❛ss ✜❧t❡rs ❬✼❪✲❬✾❪✱ ❛♥❞ ❦♥♦✇❧❡❞❣❡ ❜❛s❡❞ ♠❡t❤♦❞ ❊❧❡❝tr✐❝ ♣♦✇❡r s②st❡♠s ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ r❛♣✐❞❧② ✇❤✐❝❤ ✉s❡s ❛rt✐✜❝✐❛❧ ♥❡✉r❛❧ ♥❡t✇♦r❦ ❛♥❞✴♦r ♣❛t✲ ✐♥ r❡❝❡♥t ②❡❛rs ❛♥❞ t❤❡s❡ s②st❡♠s ❤❛✈❡ ❜❡❝♦♠❡ t❡r♥ r❡❝♦❣♥✐t✐♦♥ t❡❝❤♥✐q✉❡s ❬✶✵❪✲❬✶✷❪✳ ✐♠♣♦rt❛♥t ✐♥ ❛❧❧ ❜r❛♥❝❤❡s ♦❢ t❤❡ ♠♦❞❡r♥ ❡❝♦♥✲ ✶✽✽ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ ❉✐❣✐t❛❧ r❡❧❛②s t❤❛t ✉s❡ t❤❡ ✇❛✈❡❧❡t ♠❡t❤♦❞ ❛♥❞ ♠❡t❤♦❞s ❜❛s❡❞ ♦♥ ❛rt✐✜❝✐❛❧ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❢♦r ❞❡t❡❝t✐♥❣ ❛♥❞ ❧♦❝❛t✐♥❣ ❢❛✉❧ts ❤❛✈❡ ❛ ✇❡❛❦♥❡ss ❚❤❡r❡❢♦r❡✱ ❊q✳ ✭✺✮ ❜❡❝♦♠❡s✿ vk+1 = Vmax ∗ sin (w0 ∗ tk ) ∗ cos (w0 ∗ ∆t) + Vmax ∗ cos (w0 ∗ tk ) ∗ sin (w0 ∗ ∆t) ❜❡❝❛✉s❡ t❤❡② ❤❛✈❡ ❜❡❡♥ ❞❡s✐❣♥❡❞ ❢♦r s♣❡❝✐✜❝ ♥❡t✲ ✇♦r❦s ✉♥❧✐❦❡ t❤❡ ❞✐❣✐t❛❧ r❡❧❛② ❜❛s❡❞ ♦♥ ❝♦♥✈❡♥✲ t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s t❤❛t ❛r❡ ❞❡s✐❣♥❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ ❝✉rr❡♥t ♦r ✈♦❧t❛❣❡ ❛♠♣❧✐t✉❞❡ ♠❡❛s✉r❡♠❡♥ts | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ✭✼✮ ❲❡ r❡♣❧❛❝❡ ❊q✳ ✭✷✮ ✐♥ ❊q✳ ✭✼✮ ❛♥❞ ✇❡ ❣❡t✿ vk+1 = vk ∗ cos (w0 ∗ ∆t) ■♥❝r❡❛s❡ ♦❢ ❝✉rr❡♥t ♠❛❣♥✐t✉❞❡ ♦r ❞❡❝r❡❛s❡ ♦❢ + Vmax ∗ cos (w0 ∗ tk ) ∗ sin (w0 ∗ ∆t) ✭✽✮ ✈♦❧t❛❣❡ ♠❛❣♥✐t✉❞❡ ❝♦✉❧❞ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ♠❡❛✲ s✉r❡ t♦ ❞❡t❡❝t ❛♥❞❝❧❛ss✐❢②❛ s②st❡♠ ✐♥ ❢❛✉❧t✳ ❚❤❡ ❙♦ ♠❡❛s✉r❡ ♦❢ r❡❛❝t❛♥❝❡ ♦r ✐♠♣❡❞❛♥❝❡ ♦❢ t❤❡ ❧✐♥❡ ✐s Vmax ∗ cos (w0 ∗ tk ) = ❝♦♥s✐❞❡r❡❞ t♦ ❧♦❝❛t❡ t❤❡ ❢❛✉❧t✳ vk+1 − vk ∗ cos (w0 ∗ ∆t) sin (w0 ∗ ∆t) ■♥ ❬✶✸❪ t❤❡ ❛✉t❤♦rs ✉s❡ t✇♦ ♠❡t❤♦❞s t♦ ❧♦❝❛❧✐s❡ ✭✾✮ t❤❡ ❢❛✉❧t ✐♥ tr❛♥s♠✐ss✐♦♥ ❧✐♥❡✳ ❚❤❡ ✜rst ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❝✐r❝✉✐t ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ s❡❝♦♥❞ ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ❝✐r❝✉✐t ❡q✉❛t✐♦♥✳ ■♥ t❤✐s ♣❛♣❡r✱ ❛♥ ❛❧❣♦r✐t❤♠ ✐s ♣r♦♣♦s❡❞ t♦ ❞❡✲ Vmaxk = vk + vk+1 − ∗ vk ∗ vk+1 ∗ cos (w0 ∗ ∆t) (sin (w0 ∗ ∆t)) t❡❝t✱ ❝❧❛ss✐❢② ❛♥❞ ❧♦❝❛t❡ ❢❛✉❧ts ♦♥ ❞✐str✐❜✉t✐♦♥ ♥❡t✇♦r❦ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ❲✐t❤ ✭❊q✳✭✷✮ ✮ ✰✭❊q✳✭✾✮ ✮ ✱ ✇❡ ❣✐✈❡✿ ❚❤❡ ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥❧② ♦♥ t✇♦ s❛♠♣❧❡s ♦❢ s✐❣♥❛❧ ❝✉rr❡♥t ♦r ✈♦❧t❛❣❡✳ ✭✶✵✮ ❲❡ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ❢♦r t❤❡ ❝✉rr❡♥t ✧i✧✱ t❤❡ r❡s✉❧t ✐s✿ ✷✳ Imax k = ❯❙❊❉ ▼❊❚❍❖❉ ik + ik+1 − ∗ ik ∗ ik+1 ∗ cos (w0 ∗ ∆t) (sin (w0 ∗ ∆t)) ❋♦r t❤❡ ❞❡t❡❝t✐♦♥ ♦❢ ❡❧❡❝tr✐❝❛❧ ❢❛✉❧ts ✐♥ ❛♥② ♥❡t✲ ✭✶✶✮ ✇♦r❦ t❤❡r❡ ❛r❡ s❡✈❡r❛❧ ♠❡t❤♦❞s✳ ▼♦st ♠❡t❤♦❞s ✉s❡ t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡s ♦❢ t❤❡ ✈♦❧t❛❣❡ ♦r ❝✉r✲ r❡♥t ✭❱♠❛① ✱ ■♠❛① ✮ ❝♦♠♣❛r✐♥❣ t❤❡♠ t♦ ❛ t❤r❡s❤✲ ❚❤❡ ❝✉rr❡♥t ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s✿ ♦❧❞ ✈❛❧✉❡✱ ✐♥ t❤✐s ♣❛♣❡r ✇❡ ✇✐❧❧ ✉s❡ ❛ ♥❡✇ ♠❡t❤♦❞ ik = Imax ∗ sin (w0 ∗ tk + θk ) ❜❛s❡❞ ♦♥ t✇♦ s❛♠♣❧❡s ✇❤✐❝❤ ✐s ❛s ❢♦❧❧♦✇s✿ ✭✶✷✮ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✈♦❧t❛❣❡ ✐s ❛s ❢♦❧❧♦✇s✿ ik = Imax ∗ sin (w0 ∗ tk ) ∗ cos θk v = Vmax ∗ sin (w0 ∗ t) ❲❡ ❤❛✈❡ t❤❡ ✈♦❧t❛❣❡ ❛t t❤❡ ♠♦♠❡♥t ✭✶✮ ✭✶✸✮ ik+1 = Imax ∗ sin (w0 ∗ tk+1 + θk ) ✭✶✹✮ ik+1 = Imax ∗ sin (w0 ∗ (tk + ∆t) + θk ) ✭✶✺✮ k✿ vk = Vmax ∗ sin (w0 ∗ tk ) ❚❤❡ ✈♦❧t❛❣❡ ❛t t❤❡ ♠♦♠❡♥t + Imax ∗ cos (w0 ∗ tk ) ∗ sin (θk ) ✭✷✮ k + 1✿ vk = Vmax ∗ sin (w0 ∗ tk ) ✭✸✮ vk+1 = Vmax ∗ sin (w0 ∗ (tk + ∆t)) ✭✹✮ ik+1 = ✭✺✮ Imax ∗ vk+1 = Vmax ∗ sin ((w0 ∗ tk ) + (w0 ∗ ∆t)) ∆t = tk+1 − tk = 0.001 s❡❝ + cos (w0 ∗ tk ) ∗ sin (w0 ∗ ∆t) + Imax ∗ ❲❡ ❦♥♦✇ t❤❛t✿ sin (A + B) = sin A ∗ cos B + cos A ∗ sin B sin (w0 ∗ tk ) ∗ cos (w0 ∗ ∆t) ✭✻✮ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ∗ cos (θk ) cos (w0 ∗ tk ) ∗ cos (w0 ∗ ∆t) − sin (w0 ∗ tk ) ∗ sin (w0 ∗ ∆t) ∗ sin (θk ) ✭✶✻✮ ✶✽✾ ❱❖▲❯▼❊✿ ✷ ✷✳✶✳ ❋r♦♠ ❊q✳ ✭✷✮ ✇❡ ❤❛✈❡✿ vk sin (w0 ∗ tk ) = Vmax | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ❋❛✉❧t ❞❡t❡❝t✐♦♥ ❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥ ✭✶✼✮ ❋✐❣✳ ✶ ♣r❡s❡♥ts t❤❡ ✢♦✇❝❤❛rt ♦❢ t❤❡ ♠❡t❤♦❞ t♦ ❋r♦♠ ❊q✳ ✭✼✮ ✇❡ ❤❛✈❡✿ ❞❡t❡❝t ❛♥❞ ❝❧❛ss✐❢② t❤❡ ❢❛✉❧t✳ vk+1 − vk ∗ cos (w0 ∗ ∆t) cos (w0 ∗ tk ) = Vmax ∗ sin (w0 ∗ ∆t) ✭✶✽✮ ❯s✐♥❣ ❊q✳ ✭✶✸✮ ❛♥❞ ❊q✳ ✭✶✻✮ ✇❡ ❝❛♥ ♦❜t❛✐♥✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ θk ❯s✐♥❣ ❊q✳ ✭✶✽✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ✜♥❛❧ ✈❛❧✉❡ ♦❢ E1 E2 θk = −cos−1 ✭✶✼✮ ❛♥❞ ❊q✳ θk ✿ , ✭✶✾✮ ✇❤❡r❡ E1 = ik ∗ vk + ik+1 ∗ vk+1 − (ik ∗ vk+1 + ik+1 ∗ vk ) ∗ cos (w0 ∗ ∆t) E2 = Imaxk ∗ Vmaxk ∗ (sin (w0 ∗ ∆t)) ❯s✐♥❣ ❊q✳ ✭✷✮ ❛♥❞ ❊q✳ ✭✶✷✮✱ t❤❡ ❢❛✉❧t ✐♠♣❡❞❛♥❝❡ Zk ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❛s✿ Zk = vk Vmaxk ∗ sin (w0 ∗ tk ) = ik Imaxk ∗ sin (w0 ∗ tk + θk ) Zk = ✭✷✵✮ Vmax k −jθk e Imax k ✭✷✶✮ ❲❡ ♥♦t❡✿ θz k = −θk ❚♦ ❧♦❝❛❧✐③❡ t❤❡ ❢❛✉❧t✱ t❤❡ ❢❛✉❧t ✐♠♣❡❞❛♥❝❡ Zk ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜②✿ Zk = Vmaxk ∗ cos θz k + j ∗ sin θz k Imaxk ✭✷✷✮ ✷✳✷✳ ❲✐t❤✿ −1 θz k = cos E3 E4 ❚❤❡ ✢♦✇❝❤❛rt ♦❢ t❤❡ ♠❡t❤♦❞❢♦r ❢❛✉❧t ❞❡t❡❝t✐♦♥ ❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✳ ❋✐❣✳ ✶✿ , ✭✷✸✮ ❋❛✉❧t ❧♦❝❛❧✐③❛t✐♦♥ ❚❤❡ ❛♣♣❛r❡♥t ♣♦s✐t✐✈❡✲s❡q✉❡♥❝❡ ❢❛✉❧t ✐♠♣❡❞❛♥❝❡ ♠❡❛s✉r❡❞ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❢❛✉❧t ❞✐st❛♥❝❡✱ ✇❤❡r❡ ✇❤✐❝❤ ❝❛♥ ❜❡ ❡st✐♠❛t❡ ❢♦r ❡❛❝❤ ❢❛✉❧t t②♣❡ ❬✶✹✱ ✶✺❪ E3 = ik ∗ vk + ik+1 ∗ vk+1 ❛s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✶✳ − (ik ∗ vk+1 + ik+1 ∗ vk ) ∗ cos(w0 ∗ ∆t) ❲❤❡r❡ E4 = Imaxk ∗ V maxk ∗ (sin(w0 ∗ ∆t)) Zk = Rk + jXk Rk = Xk = ✶✾✵ a, b ✭✷✹✮ Vmaxk ∗ cos θz k Imaxk ✭✷✺✮ Vmax k ∗ sin θz k Imax k ✭✷✻✮ g ❛♥❞ c ✐♥❞✐❝❛t❡s ❢❛✉❧t② ♣❤❛s❡s✳ ✐♥❞✐❝❛t❡s ❣r♦✉♥❞ ❢❛✉❧t✳ Va ✱ Vb Ia , Ib ❛♥❞ ❛♥❞ Vc Ic ✐♥❞✐❝❛t❡ ✈♦❧t❛❣❡ ♣❤❛s♦rs✳ ✐♥❞✐❝❛t❡ ❝✉rr❡♥t ♣❤❛s♦rs✳ k= ZOL − ZdL 3ZdL ✭✷✼✮ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ ZOL ✐s t❤❡ ③❡r♦✲s❡q✉❡♥❝❡ ❧✐♥❡ ✐♠♣❡❞❛♥❝❡✳ ✐s t❤❡ ♣♦s✐t✐✈❡✲s❡q✉❡♥❝❡ ❧✐♥❡ ✐♠♣❡❞❛♥❝❡✳ IR I0 r❡❛❝t✐✈❡ ♣♦✇❡r ♦❢ ✷✵✵ ❦✈❛r✳ ❋✐❣✳ ZdL | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ✸ s❤♦✇s t❤❡ st❡♣s ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❞✐❣✲ ✐t❛❧ r❡❧❛② ❢♦r ❢❛✉❧t ❞❡t❡❝t✐♦♥✱ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥✳ ✐s t❤❡ r❡s✐❞✉❛❧ ❝✉rr❡♥t (3I0 )✳ ✐s t❤❡ ③❡r♦✲ s❡q✉❡♥❝❡ ❝✉rr❡♥t✳ ❚❤❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ✭m✮ ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ✉s✐♥❣ ✐♠♣❡❞❛♥❝❡ Zk ♦r t❤❡ r❡❛❝t❛♥❝❡ Xk ✳ ❯s✐♥❣ ❋✐❣✳ ✸✿ t❤❡ r❡❛❝t❛♥❝❡✱ t❤❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ✭m✮ ✐s✿ m= Xd ✐s t❤❡ ✭Ω✴❦♠✮✳ ♣♦s✐t✐✈❡ Xk Xd ✭✷✽✮ s❡q✉❡♥❝❡ ❧✐♥❡ r❡❛❝t❛♥❝❡ ❚❤❡ ❝✉rr❡♥ts ❛♥❞ ✈♦❧t❛❣❡ s✐❣♥❛❧s ❛r❡ ✜❧t❡r❡❞ ✉s✐♥❣ t❤❡ ❛♥t✐❛❧✐❛s✐♥❣ ✜❧t❡r ✭❇✉tt❡r✇♦rt❤ ❧♦✇✲ ♣❛ss✮ ❛♥❞ s❛♠♣❧❡❞ ❛t ✶ ❦❍③✳ ✹✳ ✸✳ ❚❤❡ st❡♣s ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❞✐❣✐t❛❧ r❡❧❛② ❢♦r ❢❛✉❧t ❞❡t❡❝t✐♦♥✱ ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥✳ P❖❲❊❘ ❙❨❙❚❊▼ ❙■▼❯▲❆❚■❖◆❙❆◆❉ ❘❊❙❯▲❚❙ ▼❖❉❊▲ ❲❤❡♥ ❛ ❢❛✉❧t ❛♣♣❡❛rs ✐♥ ❞✐str✐❜✉t✐♦♥ ❧✐♥❡✱ t❤❡ ❋✐❣✳ ✷ s❤♦✇s t❤❡ ❜❧♦❝❦ ❙✐♠✉❧✐♥❦ ♦❢ ♦✉r ✷✺ ❦❱✱ ✺✵ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ ❝✉rr❡♥t ✐♥❝r❡❛s❡s ❛♥❞ t❤❡ ❍③ ♥❡t✇♦r❦ ✉♥❞❡r t❤❡ s♦❢t✇❛r❡ ▼❆❚▲❆❇✳ ❚❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ ✈♦❧t❛❣❡ ❞❡❝r❡❛s❡s✳ ❇② ❝♦♠✲ ♣❛r✐♥❣ ✇✐t❤ ❛ t❤r❡s❤♦❧❞ ❛t ❡❛❝❤ s❛♠♣❧❡ ✏❦✑ ✇❡ ❝❛♥ ❞❡t❡❝t ❛♥❞ ❝❧❛ss✐❢② t❤❡ ❢❛✉❧t✳ ✹✳✶✳ ❋❛✉❧t ❞❡t❡❝t✐♦♥ ❚♦ ❞❡t❡❝t ❢❛✉❧t ✐♥ ❞✐str✐❜✉t✐♦♥ ❧✐♥❡✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ t❤❡ ❝✉rr❡♥t ♦r ✈♦❧t❛❣❡ s✐❣✲ ♥❛❧✳ ❯s✐♥❣ t❤❡ ♥❡t✇♦r❦ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳ ❋✐❣✳ ✷✿ ✷✱ ❛ s✐♥❣❧❡✲♣❤❛s❡ t♦ ❣r♦✉♥❞ ❢❛✉❧t ✭❛✲❣✮ ✇❛s ❛♣♣❧✐❡❞ P♦✇❡r s②st❡♠ ♠♦❞❡❧✳ ❛t t❤❡ ✐♥st❛♥t ✻✵ ♠s ✇✐t❤ ❛ ❞✐st❛♥❝❡ ♦❢ ✺ ❦♠ ✉s✲ ✐♥❣ ♥❡✉tr❛❧ r❡❣✐♠❡ ❝♦♥♥❡❝t❡❞ ❞✐r❡❝t❧② t♦ ❣r♦✉♥❞ ❞✐str✐❜✉t✐♦♥ ❧✐♥❡ ♣❛r❛♠❡t❡rs ❛r❡ ❛s ❢♦❧❧♦✇s✿ P♦s✐t✐✈❡ ❙❡q✉❡♥❝❡ ❘❡s✐st❛♥❝❡✿ Rd = 0.2236 Ω✴❦♠✳ Rf ault = Ω✳ ❚❤❡ ❢❛✉❧t ✐s ❞❡t❡❝t❡❞ ❜② ❜♦t❤ ♠❛①✐♠✉♠ ✈♦❧t❛❣❡ ❛♥❞ ❝✉rr❡♥t✈❛❧✉❡s✳ R0 = 0.368 Ω✴❦♠✳ ■♥❞✉❝t❛♥❝❡✿ Ld = 1.11 ❩❡r♦ ❙❡q✉❡♥❝❡ ❘❡s✐st❛♥❝❡✿ P♦s✐t✐✈❡ ❛♥❞ ❛ ③❡r♦✲❢❛✉❧t r❡s✐st❛♥❝❡ ❙❡q✉❡♥❝❡ ♠❍✴❦♠✳ ♥❛❧ ✐♥ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ L0 = 5.05 ♠❍✴❦♠✳ ❈❛♣❛❝✐t❛♥❝❡✿ Cd = 11.13 ❩❡r♦ ❙❡q✉❡♥❝❡ ■♥❞✉❝t❛♥❝❡✿ P♦s✐t✐✈❡ ❙❡q✉❡♥❝❡ ❋✐❣✳ ✹ s❤♦✇s t❤❡ ❝✉rr❡♥t s✐❣♥❛❧✱ t❤❡ ♠❛①✐♠✉♠ ❝✉rr❡♥t ✈❛❧✉❡ ❛♥❞ t❤❡ ♦✉t♣✉t ❢❛✉❧t ❞❡t❡❝t♦r s✐❣✲ ♥❋✴❦♠✳ ❩❡r♦ ❙❡q✉❡♥❝❡ ❈❛♣❛❝✐t❛♥❝❡✿ C0 = ♥❋✴❦♠✳ ❚❤❡ ❧✐♥❡ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ✺ ♣❛rts ♦❢ ✺ ❦♠❀ ❛t t❤❡ ❋✐❣✳ ✺ s❤♦✇s t❤❡ ✈♦❧t❛❣❡ s✐❣♥❛❧✱ t❤❡ ♠❛①✐♠✉♠ ✈♦❧t❛❣❡ ✈❛❧✉❡ ❛♥❞ t❤❡ ♦✉t♣✉t ❢❛✉❧t ❞❡t❡❝t♦r s✐❣✲ ♥❛❧ ✐♥ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳ ■♥ ❋✐❣✳ ✹✭❛✮ t❤❡ ❜❧❛❝❦ ❞♦ts r❡♣r❡s❡♥t t❤❡ ♠❛①✲ ❡♥❞ ♦❢ ❡❛❝❤ ♣❛rt✱ ✇❡ ❤❛✈❡ ❛ ❧♦❛❞✳ ✐♠✉♠ ❝✉rr❡♥t ✈❛❧✉❡ ❝❛❧❝✉❧❛t❡❞ ❛t ❡❛❝❤ ✐♥st❛♥t✳ ❆❧❧ ❧♦❛❞s ❤❛✈❡ ❛♥ ❛❝t✐✈❡ ♣♦✇❡r ♦❢ ✺✵✵ ❦❲ ❛♥❞ ❛ ■♥ ❋✐❣✳ ✹ ✭❜✮ ✇❡ ❝❛♥ s❡❡✱ t❤❡ ✜rst ❞♦t t❤❛t ✐s ❞✐❢✲ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✾✶ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ❚❛❜❧❡ ✶✳ ❙✐♥❣❧❡ ❢❛✉❧t ✐♠♣❡❞❛♥❝❡ ❡q✉❛t✐♦♥ ❢♦r ♥❡❣❧✐❣✐❜❧❡ ❢❛✉❧t r❡s✐st❛♥❝❡✳ ❋❛✉❧t t②♣❡ ❛✲❣ ❜✲❣ ❝✲❣ ❛✲❜ ♦r ❛✲❜✲❣ ❜✲❝ ♦r ❜✲❝✲❣ ❝✲❛ ♦r ❝✲❛✲❣ ❛✲❜✲❝ ♦r ❛✲❜✲❝✲❣ ❋✐❣✳ ✹✿ Zk Va (Ia +kIR ) Vb (Ib +kIR ) Vc (Ic +kIR ) Va −Vb (Ia −Ib ) Vb −Vc (Ib −Ic ) Vc −Va (Ic −Ia ) Vb −Vc Va −Vb Vc −Va (Ia −Ib ) ♦r (Ib −Ic ) ♦r (Ic −Ia ) ❋❛✉❧t ✐♠♣❡❞❛♥❝❡ ✭❛✮ ✭❛✮ ✭❜✮ ✭❜✮ ❋❛✉❧t ❞❡t❡❝t♦r ♦✉t♣✉t ✉s✐♥❣ t❤❡ ♠❛①✐♠✉♠ ❝✉r✲ r❡♥t ✈❛❧✉❡✳ ❋✐❣✳ ✺✿ ❋❛✉❧t ❞❡t❡❝t♦r ♦✉t♣✉t ✉s✐♥❣ t❤❡ ♠❛①✐♠✉♠ ✈♦❧t✲ ❛❣❡ ✈❛❧✉❡✳ ❢❡r❡♥t ❢r♦♠ ③❡r♦ ✐s ❛t t❤❡ ✐♥st❛♥t ✵✳✵✻✷ s❡❝✳ ✭✻✷ ❜② t❤❡ ❞❡✈❡❧♦♣❡❞ ♠❡t❤♦❞ ✉s✐♥❣ t❤❡ ♠❛①✐♠✉♠ ♠s✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❢❛✉❧t ✐s ❞❡t❡❝t❡❞ ✷ ♠s ❧❛t❡✳ ✈♦❧t❛❣❡ ✈❛❧✉❡ ✐s ❢❛st❡r t❤❛♥ t❤❡ ❞❡t❡❝t✐♦♥ ❜② t❤❡ ■♥ ❋✐❣✳ ✺ ✭❛✮ t❤❡ ❜❧❛❝❦ ❞♦ts r❡♣r❡s❡♥t t❤❡ ♠❛①✲ ♠❛①✐♠✉♠ ❝✉rr❡♥t ✈❛❧✉❡✳ ✐♠✉♠ ✈♦❧t❛❣❡ ✈❛❧✉❡ ❝❛❧❝✉❧❛t❡❞ ❛t ❡❛❝❤ ✐♥st❛♥t✳ ■♥ ❋✐❣✳ ✺ ✭❜✮ ✇❡ ❝❛♥ s❡❡✱ t❤❡ ✜rst ❞♦t t❤❛t ✐s ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦ ✐s ❛t t❤❡ ✐♥st❛♥t ✵✳✵✻✶ s❡❝✳ ✭✻✶ ♠s✮✳ ❋❛✉❧t ❝❧❛ss✐✜❝❛t✐♦♥ ❚❤❡r❡❢♦r❡✱ t❤❡ ❢❛✉❧t ✐s ❞❡t❡❝t❡❞ ✶ ♠s ❜❡❤✐♥❞✳ ❚♦ ❝❧❛ss✐❢② t❤❡ ❢❛✉❧t ❜② t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✱ ❚❤❡r❡❢♦r❡ ✐t ✐s ❝♦♥❝❧✉❞❡❞ t❤❛t t❤❡ ❞❡t❡❝t✐♦♥ ✶✾✷ ✹✳✷✳ t❤❡ ♠❛①✐♠✉♠ ✈♦❧t❛❣❡ ✈❛❧✉❡s ❛r❡ ✉s❡❞ ❛♥❞ t❤❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r s❛♠❡ st❡♣s ❛s t❤❡ ❞❡t❡❝t✐♦♥ ❢♦r ❡❛❝❤ ♣❤❛s❡ ❛r❡ ❢♦❧❧♦✇❡❞✳ ❚♦ ❝❧❛ss✐❢② t❤❡ ❣r♦✉♥❞ ❢❛✉❧t✱ ✇❡ ✉s❡ t❤❡ ③❡r♦ s❡q✉❡♥❝❡ ✈♦❧t❛❣❡ s✐❣♥❛❧✳ ❲❡ ♣r♦❣r❛♠♠❡❞ ❛ ❢❛✉❧t ❝❧❛ss✐✜❡r ❛❧❣♦r✐t❤♠ ❛♥❞ ✇❡ ❝r❡❛t❡❞ s❡✈❡r❛❧ t②♣❡s ♦❢ ❢❛✉❧ts✳ ❚❤❡ r❡✲ s✉❧ts ❛r❡ ❛s ❢♦❧❧♦✇s✿ ❋✐❣✳ ✻ r❡♣r❡s❡♥ts t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ❢♦r ❛ s✐♥❣❧❡✲♣❤❛s❡ t♦ ❣r♦✉♥❞ ❢❛✉❧t ✭❛✲❣✮✳ ❋✐❣✳ ✼✿ ❋✐❣✳ ✻✿ ❋❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❢♦r t❤❡ s✐♥❣❧❡✲♣❤❛s❡ t♦ ❣r♦✉♥❞ ❢❛✉❧t ✐♥ t❤❡ ♣❤❛s❡ ✧❛✧✳ ❋❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❢♦r t❤❡ s✐♥❣❧❡✲♣❤❛s❡ t♦ ❣r♦✉♥❞ ❢❛✉❧t ✐♥ t❤❡ ♣❤❛s❡ ✧❛✧✳ ❚❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ♣❤❛s❡s ✧❜✧ ❛♥❞ ✧❝✧ ❛r❡ ❛❧✇❛②s ③❡r♦✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t ✐t ✐s ❛ s✐♥❣❧❡✲♣❤❛s❡ ❢❛✉❧t ✭❛✲❣✮✳ ❚❤❡ ❢❛✉❧t ✐s ❝❧❛s✲ s✐✜❡❞ ♦♥ ♣❤❛s❡ ✧❛✧ ❛t t✐♠❡ ✵✳✵✻✷ s❡❝✳ ❛♥❞ ♦♥ t❤❡ ❣r♦✉♥❞ ❛t t✐♠❡ ✵✳✵✻✸ s❡❝✱ s♦ t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❝❛✲ ❋✐❣✳ ✽✿ ❋❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❢♦rt❤❡ ❞♦✉❜❧❡✲♣❤❛s❡s ❢❛✉❧t ✇✐t❤ ❣r♦✉♥❞ ✐♥ t❤❡ ♣❤❛s❡s ✬❛✬✱ ✬❝✬ ❛♥❞ t❤❡ ❣r♦✉♥❞✳ t✐♦♥ t✐♠❡ ✐s ❡q✉❛❧ t♦ ✵✳✵✻✸ s❡❝✱ ✐t✬s ❧❛t❡ ❜② ✸ ♠s✳ ❋✐❣✳ ✼ r❡♣r❡s❡♥ts t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ❢♦r ❛ ❞♦✉❜❧❡✲♣❤❛s❡ ❢❛✉❧t ✇✐t❤✲ ♣❤❛s❡t♦ ❣r♦✉♥❞ ❢❛✉❧t ✭❛✲❝✲❣✮✳ ❚❤❡ ❢❛✉❧t ✐s ❝❧❛s✲ ♦✉t ❣r♦✉♥❞ ✭❛✲❜✮✳ s✐✜❡❞ ♦♥ ♣❤❛s❡ ✧❛✧ ❛t t✐♠❡ ✵✳✵✻✸ s❡❝✳ ❚❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ♣❤❛s❡ ✧❝✧ ❛♥❞ t❤❡ ❣r♦✉♥❞ ❛r❡ ❛❧✇❛②s ③❡r♦✱ ✇❤✐❝❤ ✐♠✲ ♣❧✐❡s t❤❛t ✐t ✐s ❛ ❞♦✉❜❧❡✲♣❤❛s❡❢❛✉❧t ✭❛✲❜✮✳ ❚❤❡ ❛♥❞ ♦♥ ♣❤❛s❡ ✧❝✧ ❛♥❞ t❤❡ ❣r♦✉♥❞ ❛t t✐♠❡ ✵✳✵✻✷ s❡❝✳ s♦ t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❝❛t✐♦♥ t✐♠❡ ✐s ❡q✉❛❧ t♦ ✵✳✵✻✸ s❡❝✱ ✐t✬s ❧❛t❡ ❜② ✸ ♠s✳ ❢❛✉❧t ✐s ❝❧❛ss✐✜❡❞ ♦♥ ♣❤❛s❡ ✧❛✧ ❛t t✐♠❡ ✵✳✵✻✶ s❡❝✳ ❋✐❣✳ ✾ r❡♣r❡s❡♥ts t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❛s ❛♥❞ ♦♥ ♣❤❛s❡ ✧❜✧ ❛t t✐♠❡ ✵✳✵✻✷ s❡❝✳ s♦ t❤❡ ❢❛✉❧t ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ❢♦r ❛ t❤r❡❡✲♣❤❛s❡ ❢❛✉❧t ✭❛✲❜✲❝✮✳ ❝❧❛ss✐✜❝❛t✐♦♥ t✐♠❡ ✐s ❡q✉❛❧ t♦ ✵✳✵✻✷ s❡❝✱ ✐t✬s ❧❛t❡ ❜② ✷ ♠s✳ ❋✐❣✳ ✽ r❡♣r❡s❡♥ts t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ❢♦r ❛ ❞♦✉❜❧❡✲♣❤❛s❡ ❢❛✉❧t ✇✐t❤✲ ❣r♦✉♥❞ ✭❛✲❝✲❣✮✳ ❚❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ♣❤❛s❡s ✧❛✧✱ ✧❜✧ ❛♥❞ ✧❝✧ ✈❛r② ❢r♦♠ ✧✵✧ t♦ ✧✶✧ s♦ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t ✐t ✐s ❛ t❤r❡❡✲♣❤❛s❡ ❢❛✉❧t✳ ❚❤❡ ❢❛✉❧t ✐s ❝❧❛ss✐✜❡❞ ♦♥ t❤❡ ♣❤❛s❡ ✧❜✧ ❛t t❤❡ ✐♥st❛♥t ✵✳✵✻✶ ❚❤❡ ❢❛✉❧t ❝❧❛ss✐✜❡r ✐♥❞✐❝❛t❡s t❤❛t ♣❤❛s❡ ✧❜✧ ✐s s❡❝✳ ❛♥❞ t❤❡ ♣❤❛s❡s ✧❛✧ ❛♥❞ ✧❝✧ ❛t t❤❡ ✐♥st❛♥ts ❛❧✇❛②s ③❡r♦✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t ✐t ✐s ❛ ❞♦✉❜❧❡✲ ✵✳✵✻✷ s❡❝✳ s♦ t❤❡ ❢❛✉❧t ❝❧❛ss✐✜❝❛t✐♦♥ t✐♠❡ ✐s ❡q✉❛❧ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✾✸ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ❋r♦♠ ❋✐❣✳ ✶✵ ✇❡ ❝❛♥ s❡❡ ❛ st❛❜✐❧✐t② ✐♥ t❤❡ r❡✲ s♣♦♥s❡ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ✐s ❞❡t❡❝t❡❞ r❛♣✐❞❧②✱ ✐t ✐s ❝❧❡❛r t❤❛t t❤❡ ✜♥❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ❢❛✉❧t ❧♦❝❛t♦r ✐s t❤❡ s❛♠❡ ✈❛❧✉❡ ♦❢ t❤❡ s✉♣♣♦s❡❞ ❢❛✉❧t ❞✐st❛♥❝❡✳ ❋✐❣✉r❡ ✶✶ s❤♦✇s t❤❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝✲ t✐♦♥ ♦❢ t✐♠❡ ✉s✐♥❣ t❤❡ r❡❛❝t❛♥❝❡ ❢♦r ❞♦✉❜❧❡✲♣❤❛s❡ ❢❛✉❧t ✇✐t❤ ❣r♦✉♥❞✳ ❋✐❣✳ ✾✿ ❋❛✉❧t ❝❧❛ss✐✜❡r ♦✉t♣✉t ❢♦rt❤❡ t❤r❡❡✲♣❤❛s❡ ❢❛✉❧t✬❛✬✱ ✬❜✬✱ ❛♥❞ ✬❝✬✳ t♦ ✵✳✵✻✷ s❡❝✱ ✐t✬s ❧❛t❡ ❜② ✷ ♠s✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ t❡sts st✉❞✐❡❞ ✇❡ ♥♦t❡ t❤❛t ❢❛✉❧ts ✇✐t❤♦✉t ❣r♦✉♥❞ ❛r❡ ❝❧❛ss✐✜❡❞ ❢❛st❡r t❤❛♥ ❢❛✉❧ts ✇✐t❤ ❣r♦✉♥❞✳ ❋✐❣✳ ✶✶✿ ✹✳✸✳ ❋❛✉❧t ❧♦❝❛❧✐③❛t✐♦♥ ❋❛✉❧t ❧♦❝❛t✐♦♥ ❛s ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✉s✐♥❣ t❤❡ r❡✲ ❛❝t❛♥❝❡ ❢♦r ❞♦✉❜❧❡✲♣❤❛s❡ ❢❛✉❧t ✇✐t❤ ❣r♦✉♥❞✳ ❋✐❣✉r❡ ✶✷ s❤♦✇s t❤❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝✲ ❚❤❡ ❢❛✉❧t ✐s s✉♣♣♦s❡❞ ❛♣♣❡❛rs ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ t✐♦♥ ♦❢ t✐♠❡ ✉s✐♥❣ t❤❡ r❡❛❝t❛♥❝❡ ❢♦r ❞♦✉❜❧❡✲♣❤❛s❡ s❡❝t✐♦♥ t❤❛t ✐s t♦ s❛② ❛t ✺ ❦♠✱ ✶✵ ❦♠✱ ✶✺ ❦♠ ❛♥❞ ❢❛✉❧t ✇✐t❤♦✉t ❣r♦✉♥❞✳ ✷✵ ❦♠ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❧✐♥❡✳ ❋✐❣✳ ✶✵ s❤♦✇s t❤❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✉s✐♥❣ t❤❡ r❡❛❝t❛♥❝❡ ❢♦r s✐♥❣❧❡✲♣❤❛s❡ t♦ ❣r♦✉♥❞✳ ❋✐❣✳ ✶✷✿ ❋❛✉❧t ❧♦❝❛t✐♦♥ ❛s ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✉s✐♥❣ t❤❡ r❡✲ ❛❝t❛♥❝❡ ❢♦r ❞♦✉❜❧❡✲♣❤❛s❡ ❢❛✉❧t ✇✐t❤♦✉t ❣r♦✉♥❞✳ ❋r♦♠ ❋✐❣✳ ✶✶ ❛♥❞ ❋✐❣✳ ✶✷✱ ✇❡ ❝❛♥ s❡❡ ❛♥ ✐♥✲ ❋✐❣✳ ✶✵✿ ✶✾✹ ❋❛✉❧t ❧♦❝❛t✐♦♥ ❛s ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ✉s✐♥❣ t❤❡ r❡✲ ❛❝t❛♥❝❡ ❢♦r s✐♥❣❧❡✲♣❤❛s❡ t♦ ❣r♦✉♥❞✳ st❛❜✐❧✐t② ✐♥ t❤❡ r❡s♣♦♥s❡ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ✐s ♥♦t ❞❡t❡❝t❡❞ r❛♣✐❞❧②✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ✜♥❛❧ ✈❛❧✉❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ♦s❝✐❧❧❛t❡s ❛r♦✉♥❞ t❤❡ ✜♥❛❧ ✈❛❧✉❡ ♦❢ t❤❡ s✉♣♣♦s❡❞ ♦❢ ❙❝✐❡♥❝❡✱ ❢❛✉❧t ❞✐st❛♥❝❡✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧✱ ❈♦♠✲ ◆♦t❡✿ t❤❡ t❤r❡❡✲♣❤❛s❡ ❢❛✉❧t ❣✐✈❡s ✉s t❤❡ s❛♠❡ r❡s✉❧ts ❛s t❤❡ ❞♦✉❜❧❡✲♣❤❛s❡ ❢❛✉❧t ✇✐t❤ ❣r♦✉♥❞✳ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ♣✉t❡r✱ ❊♥❡r❣❡t✐❝✱ ❊❧❡❝tr♦♥✐❝ ❛♥❞ ❈♦♠♠✉✲ ♥✐❝❛t✐♦♥ ❊♥❣✐♥❡❡r✐♥❣✱ ✶✶✭✺✮✱ ✺✸✹✲✺✸✽✳ ❍♦✇❡✈❡r✱ t❤❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ❡st✐♠❛t✐✈❡ ✐s ❛❢✲ ❢❡❝t❡❞ ❜② ♠❛♥② ♣❛r❛♠❡t❡rs✱ ✐♥❝❧✉❞✐♥❣ ❢❛✉❧t r❡s✐s✲ ❬✹❪ ▼♦r❛✈❡❥✱ ❩✳✱ ❍❛❥❤♦ss❛♥✐✱ ❖✳✱ ✫ P❛③♦❦✐✱ ▼✳ t❛♥❝❡ ❘❋✱ ✇❤✐❝❤ ♠❛② ❜❡ ❤✐❣❤ ❢♦r ❣r♦✉♥❞ ❢❛✉❧ts✳ ✭✷✵✶✼✮✳ ❋❛✉❧t ❧♦❝❛t✐♦♥ ✐♥ ❞✐str✐❜✉t✐♦♥ s②s✲ ■♥ t❤✐s st✉❞② ✇❡ ❤❛✈❡ ♥♦t❡❞ t❤❛t t❤❡ ♠❛①✐♠✉♠ t❡♠s ✇✐t❤ ❉● ❜❛s❡❞ ♦♥ s✐♠✐❧❛r✐t② ♦❢ ❢❛✉❧t ✈❛❧✉❡ ♦❢ ❢❛✉❧t r❡s✐st❛♥❝❡ t❤❛t ❝❛♥ ❜❡ ❛❝❝❡♣t❡❞ ❜② ✐♠♣❡❞❛♥❝❡✳ ❚✉r❦✐s❤ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧ t❤❡ ♣r♦♣♦s❡❞ t❡❝❤♥✐q✉❡ ✐s ✽ Ω ❢♦r ❛❧❧ ❢❛✉❧t t②♣❡ ❊♥❣✐♥❡❡r✐♥❣ ✫ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡s✱ ✷✺✭✺✮✱ ✸✽✺✹✲✸✽✻✼✳ ❛♥❞ ❛t ❡❛❝❤ s❡❝t✐♦♥✳ ❬✺❪ ❏❛②✱ P✳ ❑✳✱ ❍❛r♣❛❧ ❚✳ ✭✷✵✶✼✮✳ ❋❛✉❧t ❧♦❝❛✲ ✺✳ t✐♦♥ ✐♥ ♦✈❡r❤❡❛❞ tr❛♥s♠✐ss✐♦♥ ❧✐♥❡ ✇✐t❤♦✉t ❈❖◆❈▲❯❙■❖◆❙ ✉s✐♥❣ ❧✐♥❡ ♣❛r❛♠❡t❡r✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ❖❢ ❊❧❡❝tr✐❝❛❧✱ ❊❧❡❝tr♦♥✐❝s ❆♥❞ ❉❛t❛ ❈♦♠✲ ❆ ♠❡t❤♦❞ ♦❢ t✇♦ s❛♠♣❧❡s ✇❛s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ♣❛♣❡r t♦ ❞❡t❡❝t✱ ❝❧❛ss✐❢② ❛♥❞ ❧♦❝❛❧✐③❡ t❤❡ ❢❛✉❧t ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♥❡t✇♦r❦✇✐t❤ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡✳❚❤❡ ♠❡t❤♦❞ ❝❛♥ ❜❡ ✉s❡❞ ❜② ♥✉♠❡r✐❝❛❧ r❡❧❛②✳❚❤❡ ❢❛✉❧t ❞❡t❡❝t✐♦♥ ❜② t❤❡ ✈♦❧t❛❣❡ ❣✐✈❡s ❛ ❢❛st❡r r❡s♣♦♥s❡ ✐♥st❡❛❞ ♦❢ t❤❡ ❝✉rr❡♥t✳ ■♥ ❛❞✲ ❞✐t✐♦♥✱ t❤❛t ❢❛✉❧ts ✇✐t❤♦✉t ❣r♦✉♥❞ ❛r❡ ❝❧❛ss✐✜❡❞ ❢❛st❡r t❤❛♥ ❢❛✉❧ts ✇✐t❤ ❣r♦✉♥❞✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ❢❛✉❧t ❧♦❝❛t♦r✱ ❢♦r s✐♥❣❧❡ ♣❤❛s❡ t♦ ❣r♦✉♥❞ ❢❛✉❧t✱ t❤❡r❡ ✐s st❛❜✐❧✐t② ✐♥ t❤❡ r❡s♣♦♥s❡ ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ✐s ❞❡t❡❝t❡❞ r❛♣✐❞❧②✳ ❚❤❡ ❞✐st❛♥❝❡ ✐s ❞❡t❡r♠✐♥❡❞ ❛❢t❡r ✶✵✵ ♠s✳ ❋♦r ♠✉❧t✐♣❤❛s❡ ❢❛✉❧t✱ t❤❡ ❢❛✉❧t ❧♦✲ ❝❛t♦r t❛❦❡s s♦♠❡ t✐♠❡ t♦ t❤❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✜♥❛❧ ✈❛❧✉❡✱ t❤❡ r❡s♣♦♥s❡ ✐s✉♥st❛❜❧❡ ❛♥❞ t❤❡ ❞✐s✲ t❛♥❝❡ ✐s ♥♦t ❞❡t❡❝t❡❞ r❛♣✐❞❧②✱ ❜✉t t❤❡ ❞✐st❛♥❝❡ ✐s ❞❡t❡r♠✐♥❡❞ ❛❢t❡r ✹✵✵ ♠s✳ ♠✉♥✐❝❛t✐♦♥✱ ✺✭✺✮✱ ✺✲✾✳ ❬✻❪ ❘✉✐✱ ▲✳✱ ◆❛♥✱ P✳✱ ❩❤✐✱ ❨✳✱ ✫ ❩❛r❡✱ ❋✳ ✭✷✵✶✽✮✳ ❆ ♥♦✈❡❧ s✐♥❣❧❡✲♣❤❛s❡✲t♦✲❡❛rt❤ ❢❛✉❧t ❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❞✐str✐❜✉t✐♦♥ ♥❡t✇♦r❦ ❜❛s❡❞ ♦♥ ③❡r♦✲s❡q✉❡♥❝❡ ❝♦♠♣♦♥❡♥ts ❞✐str✐✲ ❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐st✐❝s✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r✲ ♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧ P♦✇❡r ✫ ❊♥❡r❣② ❙②st❡♠s✱ ✶✵✷✱ ✶✶✲✷✷✳ ❬✼❪ ❙❛✐♥✐✱ ▼✳✱ ❩✐♥✱ ❆✳ ▼✳✱ ▼✉st❛❢❛✱ ▼✳ ❲✳✱ ❙✉❧✲ t❛♥✱ ❆✳ ❘✳✱ ✫ ◆✉r✱ ❘✳ ✭✷✵✶✽✮✳ ❆❧❣♦r✐t❤♠ ❢♦r ❋❛✉❧t ▲♦❝❛t✐♦♥ ❛♥❞ ❈❧❛ss✐✜❝❛t✐♦♥ ♦♥ P❛r❛❧✲ ❧❡❧ ❚r❛♥s♠✐ss✐♦♥ ▲✐♥❡ ✉s✐♥❣ ❲❛✈❡❧❡t ❜❛s❡❞ ♦♥ ❈❧❛r❦❡✬s ❚r❛♥s❢♦r♠❛t✐♦♥✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐✲ ♥❡❡r✐♥❣ ✭■❏❊❈❊✮✱ ✽✭✷✮✱ ✻✾✾✲✼✶✵✳ ❬✽❪ Pr❡❡t✐✱ ●✳✱ ▼❛❤❛♥t②✱ ❘✳ ◆✳ ✭✷✵✶✽✮✳ ❈♦♠♣❛r✲ ❘❡❢❡r❡♥❝❡s ❛t✐✈❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ ❲❆❱❊▲❊❚ ❛♥❞ ❆◆◆ ❜❛s❡❞ ♠❡t❤♦❞s ❢♦r ❢❛✉❧t ❧♦❝❛t✐♦♥ ♦❢ tr❛♥s✲ ❬✶❪ ❙❛❤❛✱ ▼✳ ▼✳✱ ❉❛s✱ ❘✳✱ ❱❡r❤♦✱ P✳✱ ✫ ◆♦✈♦s❡❧✱ ❉✳ ✭✷✵✵✷✮✳ ❘❡✈✐❡✇ ♦❢ ❢❛✉❧t ❧♦❝❛t✐♦♥ t❡❝❤✲ ♥✐q✉❡s ❢♦r ❞✐str✐❜✉t✐♦♥ s②st❡♠s✳ P♦✇❡r ❙②s✲ t❡♠s ❛♥❞ ❈♦♠♠✉♥✐❝❛t✐♦♥s ■♥❢r❛str✉❝t✉r❡s ❢♦r t❤❡ ❢✉t✉r❡✱ ❇❡✐❥✐♥❣✳ ♠✐ss✐♦♥ ❧✐♥❡s✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✶✶✽✭✶✾✮✱ ✷✺✽✼✲ ✷✻✶✶✳ ❬✾❪ ●✉♦✱ ▼✳ ❋✳✱ ❨❛♥❣✱ ◆✳ ❈✳✱ ✫ ❨♦✉✱ ▲✳ ❳✳ ✭✷✵✶✽✮✳ ❲❛✈❡❧❡t✲tr❛♥s❢♦r♠ ❜❛s❡❞ ❡❛r❧② ❞❡✲ ❬✷❪ ●❛♥✐②✉✱ ❆✳ ❆✳✱ ❙❡❣✉♥✱ ❖✳ ❙✳ ✭✷✵✶✻✮✳ ❆♥ t❡❝t✐♦♥ ♠❡t❤♦❞ ❢♦r s❤♦rt✲❝✐r❝✉✐t ❢❛✉❧ts ✐♥ ♦✈❡r✈✐❡✇ ♦❢ ✐♠♣❡❞❛♥❝❡✲❜❛s❡❞ ❢❛✉❧t ❧♦❝❛t✐♦♥ ♣♦✇❡r ❞✐str✐❜✉t✐♦♥ ♥❡t✇♦r❦s✳ ■♥t❡r♥❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ✐♥ ❡❧❡❝tr✐❝❛❧ ♣♦✇❡r tr❛♥s♠✐ss✐♦♥ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧ P♦✇❡r ✫ ❊♥❡r❣② ❙②s✲ ♥❡t✇♦r❦✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ t❡♠s✱ ✾✾✱ ✼✵✻✲✼✷✶✳ ❊♥❣✐♥❡❡r✐♥❣ ❘❡s❡❛r❝❤ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✷ ✭✸✮✱ ✶✷✸✲✶✸✵✳ ❬✶✵❪ ●✉r✉r❛❥❛♣❛t❤②✱ ❙✳ ❙✳✱ ▼♦❦❤❧✐s✱ ❍✳✱ ■❧❧✐❛s✱ ❍✳ ❆✳ ❇✳✱ ❆❜✉ ❇❛❦❛r✱ ❆✳ ❍✳✱ ✫ ❆✇❛❧✐♥✱ ▲✳ ❬✸❪ ❉r❛❣♦♠✐r✱ ▼✳✱ ✫ ❉r❛❣♦♠✐r✱ ❆✳ ✭✷✵✶✼✮✳ ■♥✲ ❏✳ ✭✷✵✶✽✮✳ ❋❛✉❧t ❧♦❝❛t✐♦♥ ✐♥ ❛♥ ✉♥❜❛❧❛♥❝❡❞ ✢✉❡♥❝❡ ♦❢ t❤❡ ▲✐♥❡ P❛r❛♠❡t❡rs ✐♥ ❚r❛♥s♠✐s✲ ❞✐str✐❜✉t✐♦♥ s②st❡♠ ✉s✐♥❣ s✉♣♣♦rt ✈❡❝t♦r s✐♦♥ ▲✐♥❡ ❋❛✉❧t ▲♦❝❛t✐♦♥✳ ❲♦r❧❞ ❆❝❛❞❡♠② ❝❧❛ss✐✜❝❛t✐♦♥ ❛♥❞ r❡❣r❡ss✐♦♥ ❛♥❛❧②s✐s✳ ■❊❊❏ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✾✺ ❱❖▲❯▼❊✿ ✷ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐♥❡❡r✐♥❣✱ ✶✸✭✷✮✱ ✷✸✼✲✷✹✺✳ ❬✶✶❪ Pr❛s❛❞✱ ❆✳✱ ❊❞✇❛r❞✱ ❏✳ ❇✳✱ ✫ ❘❛✈✐✱ ❑✳ ✭✷✵✶✼✮✳ ❆ r❡✈✐❡✇ ♦♥ ❢❛✉❧t ❝❧❛ss✐✜❝❛t✐♦♥ ♠❡t❤♦❞♦❧♦❣✐❡s ✐♥ ♣♦✇❡r tr❛♥s♠✐ss✐♦♥ s②s✲ t❡♠s✿ P❛rt✖■✳ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧ ❙②s✲ t❡♠s ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ❚❡❝❤♥♦❧♦❣②✱ ✺✭✶✮✱ ✹✽✲ ✻✵✳ | ■❙❙❯❊✿ ✸ | ✷✵✶✽ | ❙❡♣t❡♠❜❡r ❙❛♠✐r❛ ❙❊●❍■❘ r❡❝❡✐✈❡❞ t❤❡ ▼✳❙❝✳ ❞❡❣r❡❡ ❢r♦♠ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤✲ ♥♦❧♦❣② ♦❢ ❖r❛♥ ❝✐t②✱ ❆❧❣❡r✐❛ ✐♥ ✷✵✶✺✳ ❙❤❡ ✐s ❝✉rr❡♥t❧② ❛ P❤✳❉✳ st✉❞❡♥t ❛t t❤❡ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✐♥ t❤❡ s❛♠❡ ✉♥✐✈❡rs✐t②✳ ❙❤❡ ✐s ❛ ♠❡♠❜❡r ♦❢ P♦✇❡r ❙②st❡♠ ❖♣t✐♠✐③❛t✐♦♥ ▲❛❜♦r❛t♦r②✳ ❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ❢❛✉❧t ❧♦❝❛t✐♦♥ ✐♥ tr❛♥s♠✐ss✐♦♥ ❧✐♥❡✱ ❞②♥❛♠✐❝ ❛r❝ ❢❛✉❧t ❬✶✷❪ ❊❣❡♦❧✉✱ ■✳✱ ❖♥❛❤✱ ❏✳ ✭✷✵✶✽✮✳ ❙✐♥❣❧❡ ♣❤❛s❡ t♦ ❣r♦✉♥❞ ❢❛✉❧t ❧♦❝❛t✐♦♥ ♦♥ ✹✶✺✈ ❞✐str✐❜✉✲ t✐♦♥ ❧✐♥❡s ✉s✐♥❣ ❛rt✐✜❝✐❛❧ ♥❡✉r❛❧ ♥❡t✇♦r❦ ❛❧✲ ❣♦r✐t❤♠✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ■♥♥♦✈❛✲ t✐✈❡ ❙❝✐❡♥❝❡ ❛♥❞ ❘❡s❡❛r❝❤ ❚❡❝❤♥♦❧♦❣②✱ ✸✭✶✮✱ ✸✼✶✲✸✽✺✳ s✐♠✉❧❛t✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ r❡❧❛② ❢♦r tr❛♥s♠✐ss✐♦♥ ❧✐♥❡ ♣r♦t❡❝t✐♦♥✳ ❘❡❜✐❤❛ ❇❖❯❑❍❆❘■ r❡❝❡✐✈❡❞ t❤❡ ▼✳❙❝✳ ❞❡❣r❡❡ ❢r♦♠ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤✲ ♥♦❧♦❣② ♦❢ ❖r❛♥ ❝✐t②✱ ❆❧❣❡r✐❛ ✐♥ ✷✵✶✶✳ ❙❤❡ ✐s ❝✉rr❡♥t❧② ❛ P❤✳❉✳ st✉❞❡♥t ❛t t❤❡ ❋❛❝✉❧t② ♦❢ ❙✳✱ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✐♥ t❤❡ s❛♠❡ ✉♥✐✈❡rs✐t②✳ ✭✷✵✶✽✮✳ ❙❤❡ ✐s ❛ ♠❡♠❜❡r ♦❢ P♦✇❡r ❙②st❡♠ ❖♣t✐♠✐③❛t✐♦♥ ❋❛✉❧t ▲♦❝❛t✐♦♥ ✐♥ ❍✐❣❤ ❱♦❧t❛❣❡ ❚r❛♥s♠✐s✲ ▲❛❜♦r❛t♦r②✳ ❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ❢❛✉❧t s✐♦♥ ▲✐♥❡s ❯s✐♥❣ ❘❡s✐st❛♥❝❡✱ ❘❡❛❝t❛♥❝❡ ❛♥❞ ❧♦❝❛t✐♦♥ ✐♥ ❝♦♠♣❡♥s❛t❡❞ tr❛♥s♠✐ss✐♦♥ ❧✐♥❡ ❛♥❞ ■♠♣❡❞❛♥❝❡✳ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✲ ♥✉♠❡r✐❝❛❧ ✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✷✭✷✮✱ ✼✽✲✽✺✳ ❙❡r✐❡s✲❈♦♠♣❡♥s❛t❡❞ ❬✶✸❪ ❙❡❣❤✐r✱ ❙✳✱ ❇♦✉❦❤❛r✐✱ ❇♦✉t❤✐❜❛✱ ❘✳✱ ✫ ❚✳✱ ❇♦✉r✐❝❤❛✱ ❉❛❞❞❛✱ ❆✳ r❡❧❛② ✐♥ ❉✐st❛♥❝❡ Pr♦t❡❝t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ ▲✐♥❡ ❢♦r ✉s✐♥❣ ◆❡✉r♦✲❋✉③③② ❚❡❝❤♥✐q✉❡ ❆◆❋■❙✳ ❬✶✹❪ ❋✐❧♦♠❡♥❛✱ ❆✳ ❉✳✱ ❙❛❧✐♠✱ ❘✳ ❍✳✱ ❘❡s❡♥❡r✱ ▼✳✱ ✫ ❇r❡t❛s✱ ❆✳ ❙✳ ✭✷✵✵✼✱ ❏✉♥❡✮✳ ❋❛✉❧t r❡s✐s✲ t❛♥❝❡ ✐♥✢✉❡♥❝❡ ♦♥ ❢❛✉❧t❡❞ ♣♦✇❡r s②st❡♠s ❚❛❤❛r ❇❖❯❚❍■❇❆ r❡❝❡✐✈❡❞ ❚❤❡ P❤✳❉✳ ❞❡✲ ❣r❡❡ ✐♥ P♦✇❡r ❙②st❡♠ ✐♥ ✷✵✵✹✳ ❍❡ ✐s ❝✉rr❡♥t❧② ❛ ✇✐t❤ ❞✐str✐❜✉t❡❞ ❣❡♥❡r❛t✐♦♥✳ ■♥ Pr♦❝❡❡❞✐♥❣s Pr♦❢❡ss♦r ♦❢ ❡❧❡❝tr✐❝❛❧ ❡♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❛ ❧❡❝t✉r❡r ♦❢ t❤❡ s❡✈❡♥t❤ ✐♥t❡r♥❛t✐♦♥❛❧ ❝♦♥❢❡r❡♥❝❡ ♦♥ ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ♦❢ ♣♦✇❡r s②st❡♠s tr❛♥s✐❡♥ts✱ ▲②♦♥✱ ❋r❛♥❝❡✳ ❖r❛♥ ❝✐t② ❆❧❣❡r✐❛✳ ❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ❬✶✺❪ ❈♦✉r②✱ ❉✳ ❱✳✱ ❖❧❡s❦♦✈✐❝③✱ ▼✳✱ ✫ ❙♦✉③❛✱ ❙✳ ❆✳ ✭✷✵✶✶✮✳ ●❡♥❡t✐❝ ❛❧❣♦r✐t❤♠s ❛♣♣❧✐❡❞ t♦ ❛ ❢❛st❡r ❞✐st❛♥❝❡ ♣r♦t❡❝t✐♦♥ ♦❢ tr❛♥s✲ ♠✐ss✐♦♥ ❧✐♥❡s✳ ❙❜❛✿ ❈♦♥tr♦❧❡ tỗó rsr tt tr r ❛♥❞ ❝♦♥tr♦❧ s✇✐t❝❤✐♥❣ ✉s✐♥❣ ❞✐❣✐t❛❧ t❡❝❤♥✐q✉❡s ❛♥❞ ❛rt✐✜❝✐❛❧ ✐♥t❡❧❧✐❣❡♥❝❡✳ "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0)." ❆❜♦✉t ❆✉t❤♦rs ❆❜❞❡❧❤❛❦✐♠ ❇❖❯❘■❈❍❆ ▼✳❙❝✳ r❡❝❡✐✈❡❞ t❤❡ ❞❡❣r❡❡ ❢r♦♠ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ♦❢ ❖r❛♥ ❝✐t②✱ ❆❧❣❡r✐❛ ✐♥ ✷✵✶✻✳ ❍❡ ✐s ❝✉rr❡♥t❧② ❛ P❤✳❉✳ st✉❞❡♥t ❛t t❤❡ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✐♥ t❤❡ s❛♠❡ ✉♥✐✈❡rs✐t②✳ ❍❡ ✐s ❛ ♠❡♠❜❡r ♦❢ P♦✇❡r ❙②st❡♠ ❖♣t✐♠✐③❛t✐♦♥ ▲❛❜♦r❛t♦r②✳ ❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ❉❡✲ t❡❝t✐♦♥ ❛♥❞ ▲♦❝❛t✐♦♥ ♦❢ ❍✐❣❤ ■♠♣❡❞❛♥❝❡ ❋❛✉❧ts ✐♥ ▼❡❞✐✉♠ ❱♦❧t❛❣❡ ❉✐str✐❜✉t✐♦♥ ◆❡t✇♦r❦s ✉s✐♥❣ ◆❡✉r♦✲❋✉③③② ❚❡❝❤♥✐q✉❡ ❆◆❋■❙ ❛♥❞ st❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ❛r❝ ❢❛✉❧t s✐♠✉❧❛t✐♦♥✳ ✶✾✻ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ... sin (w0 ∗ tk+1 + θk ) ✭✶✹✮ ik+1 = Imax ∗ sin (w0 ∗ (tk + ∆t) + θk ) ✭✶✺✮ k✿ vk = Vmax ∗ sin (w0 ∗ tk ) ❚❤❡ ✈♦❧t❛❣❡ ❛t t❤❡ ♠♦♠❡♥t + Imax ∗ cos (w0 ∗ tk ) ∗ sin (θk ) ✭✷✮ k + 1✿ vk = Vmax ∗ sin... Vmax ∗ sin (w0 ∗ (tk + ∆t)) ✭✹✮ ik+1 = ✭✺✮ Imax ∗ vk+1 = Vmax ∗ sin ((w0 ∗ tk ) + (w0 ∗ ∆t)) ∆t = tk+1 − tk = 0.001 s❡❝ + cos (w0 ∗ tk ) ∗ sin (w0 ∗ ∆t) + Imax ∗ ❲❡ ❦♥♦✇ t❤❛t✿ sin (A + B) = sin A... A ∗ sin B sin (w0 ∗ tk ) ∗ cos (w0 ∗ ∆t) ✭✻✮ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ∗ cos (θk ) cos (w0 ∗ tk ) ∗ cos (w0 ∗ ∆t) − sin (w0 ∗ tk ) ∗ sin (w0 ∗ ∆t) ∗ sin (θk