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Variable step size P&O MPPT controller to improve static and dynamic PV system performances

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In this paper, a variable step size P&O algorithm is used in order to improve the performance of a photovoltaic system in both dynamic and static plans. The e ciency of the proposed algorithm has been investigated successfully using the BP SX150S solar module connected to the DC-DC derived by a P&O MPPT algorithm.

❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❱❛r✐❛❜❧❡ ❙t❡♣ ❙✐③❡ P✫❖ ▼PP❚ ❈♦♥tr♦❧❧❡r t♦ ■♠♣r♦✈❡ ❙t❛t✐❝ ❛♥❞ ❉②♥❛♠✐❝ P❱ ❙②st❡♠ P❡r❢♦r♠❛♥❝❡s 2,∗ ❍❛♠③❛ ❇❆❍❘■ ✱ ❆❜❞❡❧❣❤❛♥✐ ❍❆❘❘❆● ❊❧❡tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❉❡♣❛rt♠❡♥t✱ ▼♦❤❛♠❡❞ ❇♦✉❞✐❛❢ ❯♥✐✈❡rs✐t②✱ ✷✽✵✵✵ ▼s✐❧❛✱ ❆❧❣❡r✐❛ ❖♣t✐❝s ❛♥❞ Pr❡❝✐s✐♦♥ ▼❡❝❤❛♥✐❝s ■♥st✐t✉t❡✱ ❋❡r❤❛t ❆❜❜❛s ❯♥✐✈❡rs✐t② ❙❡t✐❢ ✶✱ ✶✾✵✵✵ ❙❡t✐❢✱ ❆❧❣❡r✐❛ ✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❆❜❞❡❧❣❤❛♥✐ ❍❆❘❘❆● ✭❡♠❛✐❧✿ ❛✳❜✳❤❛rr❛❣❅❣♠❛✐❧✳❝♦♠✮ ✭❘❡❝❡✐✈❡❞✿ ✵✾✲▼❛②✲✷✵✶✽❀ ❛❝❝❡♣t❡❞✿ ✵✽✲❏✉❧②✲✷✵✶✽❀ ♣✉❜❧✐s❤❡❞✿ ✷✵✲❏✉❧②✲✷✵✶✽✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✽✷✷✳✾✹ ❆❜str❛❝t✳ ■♥ t❤✐s ♣❛♣❡r✱ ❛ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ❛❧❣♦r✐t❤♠ ✐s ✉s❡❞ ✐♥ ♦r❞❡r t♦ ✐♠♣r♦✈❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ ♣❤♦t♦✈♦❧t❛✐❝ s②st❡♠ ✐♥ ❜♦t❤ ❞②✲ ♥❛♠✐❝ ❛♥❞ st❛t✐❝ ♣❧❛♥s✳ ❚❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❤❛s ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ s✉❝❝❡ss✲ ❢✉❧❧② ✉s✐♥❣ t❤❡ ❇P ❙❳✶✺✵❙ s♦❧❛r ♠♦❞✉❧❡ ❝♦♥✲ ♥❡❝t❡❞ t♦ t❤❡ ❉❈✲❉❈ ❞❡r✐✈❡❞ ❜② ❛ P✫❖ ▼PP❚ ❛❧❣♦r✐t❤♠✳ ❚❤❡ ❝♦♠♣❛r❛t✐✈❡ st✉❞② r❡s✉❧ts ♦❢ ❜♦t❤ ❝♦♥✈❡♥t✐♦♥❛❧ ✜①❡❞ st❡♣ s✐③❡ ❛♥❞ t❤❡ ♣r♦✲ ♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ❛❧❣♦r✐t❤♠s ♣r♦✈❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❝♦♠✲ ♣❛r❡❞ t♦ t❤❡ st❛♥❞❛r❞ ✜①❡❞ st❡♣ s✐③❡ P❖ ▼PP❚✳ ❚❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ r❡❞✉❝❡s r❡s♣♦♥s❡ t✐♠❡ ❜❡t✇❡❡♥ ✶✸✳✽✻✪ ❛♥❞ ✹✺✳✷✽✪ ❛♥❞ t❤❡ st❡❛❞② st❛t❡ ♦s❝✐❧❧❛t✐♦♥ ❜❡t✇❡❡♥ ✽✸✳✸✸✪ ❛♥❞ ✶✵✵✪ ❧❡❛❞✐♥❣ t♦ ❧❡ss ♣♦✇❡r ❧♦ss ❡s♣❡❝✐❛❧❧② ✐♥ ❝❛s❡ ♦❢ ❢❛st ❝❤❛♥❣✐♥❣ ❛t♠♦s♣❤❡r✐❝ ❝♦♥❞✐t✐♦♥s✳ ❑❡②✇♦r❞s P❤♦t♦✈♦❧t❛✐❝✱ ▼PP❚✱ P✫❖✱ ❋✐①❡❞ st❡♣ s✐③❡✱ ❱❛r✐❛❜❧❡ st❡♣ s✐③❡✱ ❉❈✲❉❈ ❝♦♥✲ ✈❡rt❡r✳ ✉s❡❞ ❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❡♥❡r❣② s♦✉r❝❡ ✐♥ ♣❧❛❝❡ ♦❢ ❡❧❡❝tr✐❝✐t② ❣❡♥❡r❛t❡❞ ❢r♦♠ ❝♦♥✈❡♥t✐♦♥❛❧ ❢♦ss✐❧ ❢✉✲ ❡❧s✳ P❤♦t♦✈♦❧t❛✐❝✱ ❛❧s♦ ❝❛❧❧❡❞ s♦❧❛r ❝❡❧❧s✱ ❛r❡ ❡❧❡❝✲ tr♦♥✐❝ ❞❡✈✐❝❡s t❤❛t ❝♦♥✈❡rt s✉♥❧✐❣❤t ❞✐r❡❝t❧② ✐♥t♦ ❡❧❡❝tr✐❝✐t②✳ ❚❤✐s ❡❧❡❝tr✐❝❛❧ ❡♥❡r❣② ❝❛♥ ❢❡❡❞s ♠❛♥② s②st❡♠s ❧✐❦❡✿ ✇❛t❡r ♣✉♠♣✐♥❣ s②st❡♠s ✉s❡❞ ❢♦r ✐r✲ r✐❣❛t✐♦♥✱ ❤②❞r♦❣❡♥ ♣r♦❞✉❝t✐♦♥✱ ❡❧❡❝tr✐❝❛❧ ♣♦✇❡r s②st❡♠s✱ ✳ ✳ ✳ ❡t❝ ❬✶❪✲❬✹❪✳ ❚❤❡ ■✲❱ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛ P❱ ♠♦❞✉❧❡ ✇✐❧❧ ✈❛r② ✇✐t❤ s♦❧❛r ✐♥s♦❧❛t✐♦♥ ❛♥❞ ❛t♠♦s♣❤❡r✐❝ t❡♠✲ ♣❡r❛t✉r❡✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♣♦✐♥t ♦♥ t❤❡ ■✲❱ ♦r P✲❱ ❝❤❛r❛❝t❡r✐st✐❝s✱ ❝❛❧❧❡❞ t❤❡ ▼❛①✲ ✐♠✉♠ P♦✇❡r P♦✐♥t ✭▼PP✮✱ ❛t ✇❤✐❝❤ t❤❡ ❡♥t✐r❡ P❱ s②st❡♠ ✭❛rr❛②✱ ❝♦♥✈❡rt❡r✱ ❡t❝✳ ✳ ✳ ✮ ♦♣❡r❛t❡s ✇✐t❤ ♠❛①✐♠✉♠ ❡✣❝✐❡♥❝② ❛♥❞ ♣r♦❞✉❝❡s ✐ts ♠❛①✲ ✐♠✉♠ ♦✉t♣✉t ♣♦✇❡r ❬✺❪✳ ❚♦ ❞❡t❡r♠✐♥❡ t❤❡ ▼PP ♦❢ P❱ s②st❡♠s✱ ♠❛♥② ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ s✉❝❤ ❛s s❧✐❞✐♥❣ ♠♦❞❡ ❛❧❣♦r✐t❤♠ ❬✻❪✲❬✾❪✱ ❢✉③③② ❧♦❣✐❝ ❛❧❣♦r✐t❤♠ ❬✶✵❪✲❬✶✷❪✱ ■♥❝r❡♠❡♥t❛❧ ❛♥❞ ❈♦♥✲ ❞✉❝t❛♥❝❡ ✭■❈✮ ❛❧❣♦r✐t❤♠ ❬✶✸❪✲❬✶✺❪ ❛♥❞ P❡rt✉r❜❛✲ t✐♦♥ ❛♥❞ ❖❜s❡r✈❛t✐♦♥ ✭P✫❖✮ ❛❧❣♦r✐t❤♠ ❬✶✻❪✲❬✶✽❪✳ ■♥ t❤✐s ✇♦r❦✱ ❛ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P❡rt✉r❜❛✲ t✐♦♥ ❛♥❞ ❖❜s❡r✈❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s ♣r♦♣♦s❡❞ t♦ ✐♠♣r♦✈❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡s ✐♥ ❜♦t❤ ❞②♥❛♠✐❝ ❛♥❞ st❛t✐❝ r❡❣✐♠❡s✳ ❚❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❤❛s ❜❡❡♥ st✉❞✐❡❞ s✉❝❝❡ss❢✉❧❧② ✉s✐♥❣ ▼❛t❧❛❜✴❙✐♠✉❧✐♥❦ ❡♥✈✐r♦♥♠❡♥t ✇❤❡r❡ t❤❡ ✇❤♦❧❡ P❤♦t♦✈♦❧t❛✐❝ s②st❡♠s ♣r♦✈✐❞❡ ❣r❡❡♥ r❡♥❡✇❛❜❧❡ s②st❡♠ ✐♥❝❧✉❞✐♥❣ P❱ ♠♦❞✉❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ❉❈✲ ♣♦✇❡r ❜② ❡①♣❧♦✐t✐♥❣ s♦❧❛r ❡♥❡r❣②✳ ❚❤❡② ❝❛♥ ❜❡ ❉❈ ❜♦♦st ❝♦♥✈❡rt❡r ❞❡r✐✈❡❞ ❜② t❤❡ ♣r♦♣♦s❡❞ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✽✻ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ Vd ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ▼PP❚ ❛❧❣♦r✐t❤♠ ✐s ♠♦❞✲ Id = Is (exp − 1) ✭✹✮ AVt ❡❧❡❞ ❛♥❞ ✐♥✈❡st✐❣❛t❡❞✳ ❚❤❡ r❡s✉❧ts ♦❢ ❛ ❝♦♠♣❛r✲ ❛t✐✈❡ st✉❞② ❜❡t✇❡❡♥ t❤❡ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡ st❡♣ ❚❤❡ ❞✐♦❞❡ s❛t✉r❛t✐♦♥ ❝✉rr❡♥t ✐s ❣✐✈❡♥ ❜②✿ s✐③❡ P✫❖ ❛❧❣♦r✐t❤♠s ❝♦♥✜r♠s t❤❛t t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❝❛♥ ❡✛❡❝t✐✈❡❧② ❛♥❞ s✐♠✉❧t❛♥❡♦✉s❧② ✐♠✲ qEg ( Tref − T1 ) T exp , Is = Is,ref ♣r♦✈❡✿ t❤❡ ❛❝❝✉r❛❝②✱ t❤❡ r❛♣✐❞✐t②✱ t❤❡ r✐♣♣❧❡ ❛♥❞ Tref kA t❤❡ ♦✈❡rs❤♦♦t ✐♥ ❝❛s❡ ♦❢ ❢❛st ❛♥❞ ❝❤❛♥❣✐♥❣ ❛t♠♦✲ ✭✺✮ s♣❤❡r✐❝ ❝♦♥❞✐t✐♦♥ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ✜①❡❞ st❡♣ s✐③❡ ❛❧❣♦r✐t❤♠✳ ✇✐t❤ ❚❤❡ r❡st ♦❢ t❤✐s ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✿ Isc,ref Is,ref = ✭✻✮ ❙❡❝t✐♦♥ ✷ ❞❡s❝r✐❜❡s t❤❡ P❱ s②st❡♠ ♠♦❞❡❧✐♥❣✳ V −1 exp oc,ref ❚❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ▼PP❚ ✐s AVt ❞❡t❛✐❧❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳ ❲❤✐❧❡✱ ❙❡❝t✐♦♥ ✹ s❤♦✇ t❤❡ s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❛♥❞ t❤❡✐r ❞✐s❝✉ss✐♦♥✳ ❋✐♥❛❧❧②✱ ❛♥❞ ❙❡❝t✐♦♥ ✺ ❝♦♥❝❧✉❞❡s t❤❡ ♣❛♣❡r ❜② ❞r❡ss✐♥❣ t❤❡ kT Vt = ✭✼✮ ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥s ❛♥❞ ❣✐✈✐♥❣ t❤❡ ♣❡rs♣❡❝t✐✈❡s q ♦❢ s♦♠❡ ❢✉t✉r❡ ✇♦r❦s✳ ❋r♦♠ ❡q✉❛t✐♦♥s ✭✶✮ t♦ ✭✼✮✱ Ipv ✐s ❞❡✜♥❡❞ ❜②✿ ✷✳ P❱ ❙❨❙❚❊▼ ▼❖❉❊▲■◆● Ipv = Iph − Is (exp [λ (Vpv + Ipv Rs )] − 1) Vpv + Ipv Rs , ✭✽✮ − Rsh ❆ P❱ ❝❡❧❧ ❡q✉✐✈❛❧❡♥t ❝✐r❝✉✐t ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳ ❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❝❛♥ ❜❡ s✐♠♣❧② ❡①✲ ♣r❡ss❡❞ ❛s ❬✶✾❪✿ ❋✐❣✳ ✶✿ ✭✶✮ ✇❤❡r❡ Iph ✐s t❤❡ ♣❤♦t♦✲❝✉rr❡♥t❀ IRp ✐s t❤❡ s❤✉♥t ❝✉rr❡♥t❀ Id ✐s t❤❡ ❞✐♦❞❡ ❝✉rr❡♥t✳ Iph = ζ ζref [Isc,ref + ϑIsc (T − Tref )] IRp = Vpv + Ipv Rs Rp λ= q = AVt AkT ✭✾✮ ❚❤❡ ❇P ❙❳✶✺✵❙ ♠♦❞✉❧❡ ♣❛r❛♠❡t❡rs ✉s❡❞ t♦ ❞r❛✇ t❤❡ P❱ ❝❤❛r❛❝t❡r✐st✐❝s ❛r❡ ❧✐st❡❞ ✐♥ ❚❛✲ ❜❧❡ ✶✳ ❋✐❣✉r❡s ✷ ❛♥❞ ✸ s❤♦✇ t❤❡ t②♣✐❝❛❧ ♦✉t✲ ♣✉t ❝❤❛r❛❝t❡r✐st✐❝s ✭■✲❱✮ ❛♥❞ ✭P✲❱✮ ♦❢ P❱ ❝❡❧❧✱ ✇❤✐❝❤ ❛r❡ s✐♠✉❧❛t❡❞ ✉♥❞❡r ✈❛r✐❛❜❧❡ s♦❧❛r ✐rr❛✲ ❞✐❛t✐♦♥ ✭ζ = 600, 700, 800, 900 ❛♥❞ 1000 ❲✴♠✷ ✮ ❛♥❞ ❝♦♥st❛♥t t❡♠♣❡r❛t✉r❡ ✭T = 25♦ ❈✮❀ ✇❤✐❧❡ ❋✐❣✉r❡s ✹ ❛♥❞ ✺ s❤♦✇ t❤❡ t②♣✐❝❛❧ ♦✉t♣✉t ❝❤❛r❛❝✲ t❡r✐st✐❝s ✭■✲❱✮ ❛♥❞ ✭P✲❱✮ ♦❢ P❱ ❝❡❧❧ ✉♥❞❡r ✈❛r✐✲ ❛❜❧❡ t❡♠♣❡r❛t✉r❡ ✭T = 25, 50 ❛♥❞ 75♦ ❈✮ ✇✐t❤ ❝♦♥st❛♥t ✐rr❛❞✐❛♥❝❡ ✭ζ = 1000 ❲✴♠✷ ✮✳ ❖♥❡ ❞✐♦❞❡ ♠♦❞❡❧✳ Ipv = Iph − IRp − Id , ✇❤❡r❡ ✭✷✮ ✭✸✮ ✸✳ P❘❖P❖❙❊❉ ❱❆❘■❆❇▲❊ ❙❚❊P ❙■❩❊ P✫❖ ▼PP❚ ❋r♦♠ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❛♥❞ ❋✐❣✉r❡s ✷ t♦ ✺✱ ✐t ✐s ❝❧❡❛r t❤❛t s♦❧❛r P❱ ♣r❡s❡♥ts ♥♦♥❧✐♥❡❛r ❝❤❛r❛❝✲ t❡r✐st✐❝s ✈❛r②✐♥❣ ✇✐t❤ s♦❧❛r ✐rr❛❞✐❛t✐♦♥ ❛♥❞ t❡♠✲ ♣❡r❛t✉r❡✳ ❈♦♥s❡q✉❡♥t❧②✱ ✐t ✐s ♠❛♥❞❛t♦r② t♦ ✐♥t❡✲ ❣r❛t❡ ♠❛①✐♠✉♠ ♣♦✇❡r st❛❣❡ t♦ ❛❞❛♣t ❛♥❞ ❣✉❛r✲ ❛♥t❡❡ t❤❡ ♠❛①✐♠✉♠ ♣♦✇❡r tr❛♥s❢❡r t♦ t❤❡ ❧♦❛❞ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✽✼ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✶✳ ❖♥ ❞✐♦❞❡ P❱ ❝❡❧❧ ♠♦❞❡❧✳ P❛r❛♠❡t❡r ❱❛❧✉❡ ▼❛①✐♠✉♠ P♦✇❡r ✭Pmax ✮ 150 ❲ ❱♦❧t❛❣❡ ❛t Pmax ✭Vmpp ✮ 34.5 ❱ ❈✉rr❡♥t ❛t Pmax ✭Impp ✮ 4.35 ❆ ❲❛rr❛♥t❡❞ ♠✐♥✐♠✉♠ Pmax 140 ❲ ❙❤♦rt ❝✐r❝✉✐t ❝✉rr❡♥t ✭ISC ✮ 4.75 ❆ ❖♣❡♥✲❝✐r❝✉✐t ✈♦❧t❛❣❡ ✭VOC ✮ 43.5 ❱ ▼❛①✐♠✉♠ ❙②st❡♠ ❱♦❧t❛❣❡ 600 ❱ ❚❡♠♣✳ ❈♦❡✣❝✐❡♥t ♦❢ ISC (0.065 ± 0.015) %/♦ ❈ ❚❡♠♣✳ ❈♦❡✣❝✐❡♥t ♦❢ VOC (160 ± 20) ♠❱/♦ ❈ ❚❡♠♣✳ ❈♦❡✣❝✐❡♥t ♦❢ P♦✇❡r (0.5 ± 0.05) %/♦ ❈ ◆❖❈❚ 47 ± 2♦ ❈ ❋✐❣✳ ✷✿ ■✲❱ ❝❤❛r❛❝t❡r✐st✐❝s ✉♥❞❡r ✈❛r✐❛❜❧❡ ✐rr❛❞✐❛t✐♦♥✳ ❋✐❣✳ ✹✿ ■✲❱ ❝❤❛r❛❝t❡r✐st✐❝s ✉♥❞❡r ✈❛r✐❛❜❧❡ t❡♠♣❡r❛t✉r❡✳ ❋✐❣✳ ✸✿ P✲❱ ❝❤❛r❛❝t❡r✐st✐❝s ✉♥❞❡r ✈❛r✐❛❜❧❡ ✐rr❛❞✐❛t✐♦♥✳ ❋✐❣✳ ✺✿ P✲❱ ❈❤❛r❛❝t❡r✐st✐❝s ✉♥❞❡r ✈❛r✐❛❜❧❡ t❡♠♣❡r❛t✉r❡✳ ✐♥ ❝❛s❡ ♦❢ ❛❧t❡r❛t✐♦♥ ♦❢ ❛t♠♦s♣❤❡r✐❝ ❝♦♥❞✐t✐♦♥s✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✈❛r✐♦✉s ▼PP❚ ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❛♠♦♥❣ t❤❡♠ t❤❡ P❡rt✉r❜❛t✐♦♥ ❛♥❞ ❖❜s❡r✈❛t✐♦♥ ♠❡t❤♦❞ ✐s t❤❡ ♠♦st ❝♦♠♠♦♥❧② ❛❧❣♦r✐t❤♠ ✉s❡❞ ✐♥ ♣❤♦t♦✈♦❧t❛✐❝ s②st❡♠s ❞✉❡ t♦ ✐ts s✐♠♣❧✐❝✐t② ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✽✽ ❛♥❞ ❢❡✇ ❝♦♥tr♦❧ ♣❛r❛♠❡t❡rs r❡q✉✐r❡❞ ❬✶✺✱ ✶✻❪✳ ❚❤❡ P✫❖ ✢♦✇❝❤❛rt ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✻✳ ❚❤❡ ❛✐♠ ✐s t♦ ❣❡t t❤❡ ♠❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t ❜② ❛❞❥✉st✐♥❣ ♣❡rt✉r❜❛t✐♦♥ ✐♥ t❤❡ ❞✉t② ❝②❝❧❡ ❜② ❝♦♥t✐♥✉❛❧❧② ♠❡❛s✉r✐♥❣ ✈♦❧t❛❣❡✴❝✉rr❡♥t ❢r♦♠ P❱ ♣❛♥❡❧✱ t❤❡ ♣♦✇❡r ♦✉t♣✉t ✐s ♠❡❛s✉r❡❞ t❤❡♥✿ ✐❢ t❤❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ ✈❛❧✉❡ ♦❢ ♣♦✇❡r ♦✉t♣✉t Pk > Pk−1 t❤❡♥ ♣❡rt✉r✲ ❜❛t✐♦♥ ✐s ✉♥✐♥t❡rr✉♣t❡❞ ✐♥ t❤❡ s❛♠❡ s❡♥s❡✳ ■❢ t❤❡ ♥❡✇ ✈❛❧✉❡ ♦❢ ♣♦✇❡r ♦✉t♣✉t Pk < Pk−1 t❤❡♥ ♣❡r✲ t✉r❜❛t✐♦♥ ✐s ❛♣♣❧✐❡❞ ✐♥ t❤❡ ♦♣♣♦s✐t❡ s❡♥s❡✳ ❚❤✐s ♦♣❡r❛t✐♦♥ ♦♣❡r❛t❡s ✇❤✐❧❡ ♠❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t ✐s ♥♦t r❡❛❝❤❡❞✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ P✫❖ ❛❧❣♦r✐t❤♠ ✇✐t❤ ✜①❡❞ st❡♣ s✐③❡ ❤❛s t✇♦ ♠❛❥♦r ❞r❛✇❜❛❝❦s ❬✶✼✱ ✶✽❪ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ r❡❧❛t✐♦♥✲ s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ♦s❝✐❧❧❛t✐♦♥ ❛♥❞ r❡s♣♦♥s❡ s♣❡❡❞✳ ❲✐t❤ ❛ s♠❛❧❧ ✜①❡❞ st❡♣ s✐③❡✱ ✇❡ ❣❡t ❧♦✇ ♦s❝✐❧✲ ❧❛t✐♦♥ ❛t t❤❡ ❡①♣❡♥s❡ ♦❢ r❡s♣♦♥s❡ s♣❡❡❞ ❛♥❞ ✐♥ ❝❛s❡ ♦❢ ❧❛r❣❡ st❡♣ s✐③❡ ✇❡ ❤❛✈❡ ❛ ❣♦♦❞ s♣❡❡❞ ✇✐t❤ ♦✈❡r str♦♥❣ ♦s❝✐❧❧❛t✐♦♥s✳ ■♥ ♦r❞❡r t♦ ♦✈❡r❝♦♠❡ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❋✐❣✉r❡ ✼ s❤♦✇s t❤❡ ♦✉t♣✉t ♣♦✇❡r ❢♦r ❜♦t❤ ❝♦♥✈❡♥t✐♦♥❛❧ ✜①❡❞ st❡♣ s✐③❡ ❛♥❞ ♣r♦♣♦s❡❞ ✈❛r✐✲ ❛❜❧❡ st❡♣ s✐③❡ P✫❖ ♠❡t❤♦❞ ✇❤❡♥ t❤❡ ✐rr❛❞✐✲ ❛♥❝❡ ❝❤❛♥❣❡s✳ ❋r♦♠ ❋✐❣✳ ✼✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ❜♦t❤ ❛❧❣♦r✐t❤♠s tr❛❝❦ t❤❡ ♠❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t ✐♥ ❝❛s❡ ♦❢ ✈❛r✐❛❜❧❡ ✐rr❛❞✐❛t✐♦♥ ✇✐t❤ ❜❡st ♣❡r❢♦r✲ ♠❛♥❝❡s ❢♦r t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✜①❡❞ st❡♣ s✐③❡ ❛❧❣♦r✐t❤♠ ✐♥ t❡r♠ ♦❢ r❛♣✐❞✐t② ❛♥❞ ♦✈❡rs❤♦♦t✳ ❆✮ ❉②♥❛♠✐❝ P❡r❢♦r♠❛♥❝❡s✳ ❋✐❣✳ ✼✿ ❖✉t♣✉t ♣♦✇❡r✳ ❋✐❣✉r❡s ✽ t♦ ✶✵ s❤♦✇ t❤❡ ③♦♦♠✲✐♥ ♦❢ t❤❡ ♣♦✐♥ts ✻✵✵✲✶✱ ✼✺✵✲✶ ❛♥❞ ✶✵✵✵✲✶ ❣✐✈✐♥❣ t❤❡ ❞②♥❛♠✐❝ t❤❡ ❞r❛✇❜❛❝❦s ♦❢ t❤❡ ✜①❡❞ st❡♣ s✐③❡ ❛❧❣♦r✐t❤♠✱ ❛ r❡s♣♦♥s❡ ♦❢ ❜♦t❤ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✐rr❛❞✐❛t✐♦♥ ❧❡✈❡❧ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ❛❧❣♦r✐t❤♠ ♦♣❡r❛t❡s ✇✐t❤ P❖ ▼PP❚s ✷ ✱ 750 ❲✴♠✷ ❛♥❞ 1000 ❲✴♠✷ ✱ r❡s♣❡❝✲ 600 ❲✴♠ ❛ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡✱ ✇❤❡r❡ t❤❡ ❞✉t② ❝②❝❧❡ ✐s ❛❞✲ ❥✉st❡❞ ❞✐r❡❝t❧② ✇✐t❤ ❧❛r❣❡ st❡♣ s✐③❡ ✇❤❡♥ ❢❛r ❢r♦♠ t✐✈❡❧②✳ ■♥ t❤✐s ❝❛s❡ ✇❡ ❝♦♠♣❛r❡ t❤❡ r❡s♣♦♥s❡ t❤❡ ▼PP❚ ♣♦✐♥t ❛♥❞ ✇✐t❤ s♠❛❧❧ st❡♣ s✐③❡ ❛r♦✉♥❞ t✐♠❡ ❢♦r t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ ▼PP❚ t❤❡ ▼PP❚ ♣♦✐♥t✳ ❚❤❡ ♥♦✈❡❧ st❡♣ s✐③❡ ✐s ❣✐✈❡♥ ❜②✿ ✭❚r✲❱❙❙✮ t♦ t❤❡ r❡s♣♦♥s❡ t✐♠❡ ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ ✜①❡❞ st❡♣ s✐③❡ ▼PP❚ ✭❚r✲❋❙❙✮✳ ❋r♦♠ ❋✐❣s✳ ✽ Dk = Dk−1 ± ∆D ∗ SF, ✭✶✵✮ ❋✐❣✳ ✻✿ ❋❧♦✇❝❤❛rt ♦❢ ❝♦♥✈❡♥t✐♦♥❛❧ P✫❖ ❛❧❣♦r✐t❤♠✳ ✇❤❡r❡ SF ✐s t❤❡ s❝❛❧✐♥❣ ❢❛❝t♦r ❡q✉❛❧ t♦ |dP/dV |❀ dP ❛♥❞ dV ❛r❡ t❤❡ ❝❤❛♥❣❡s ✐♥ ♣♦✇❡r ❛♥❞ ✈♦❧t❛❣❡✳ ✹✳ ❘❊❙❯▲❚❙ ❆◆❉ ❉■❙❈❯❙❙■❖◆❙ ❚♦ ✈❡r✐❢② t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐✲ ❛❜❧❡ st❡♣ s✐③❡ P✫❖ ▼PP❚ ❛❧❣♦r✐t❤♠✱ t❤❡ ✇❤♦❧❡ s②st❡♠ ✐♥❝❧✉❞✐♥❣ P❱ ♠♦❞✉❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ❉❈✲ ❉❈ ❜♦♦st ❝♦♥✈❡rt❡r ❞r✐✈❡❞ ❜② t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐✲ ❛❜❧❡ st❡♣ s✐③❡ P✫❖ ▼PP❚ ❛❧❣♦r✐t❤♠ ✐s ♠♦❞✲ ❋✐❣✳ ✽✿ ❖✉t♣✉t ♣♦✇❡r✿ ❩♦♦♠ ✐♥ P♦✐♥t ✻✵✵✲✶✳ ❡❧❡❞ ❛♥❞ ✐♥✈❡st✐❣❛t❡❞ ✉s✐♥❣ ▼❛t❧❛❜✴❙✐♠✉❧✐♥❦ ❡♥✲ ✈✐r♦♥♠❡♥t✳ t♦ ✶✵✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✽✾ ❱❖▲❯▼❊✿ ✷ ❋✐❣✳ ✾✿ ❖✉t♣✉t ♣♦✇❡r✿ ❩♦♦♠ ✐♥ P♦✐♥t ✼✺✵✲✶✳ ❋✐❣✳ ✶✵✿ ❖✉t♣✉t ♣♦✇❡r✿ ❩♦♦♠ ✐♥ P♦✐♥t ✶✵✵✵✲✶✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❋✐❣✳ ✶✶✿ ❖✉t♣✉t ♣♦✇❡r✿ ❩♦♦♠ ✐♥ P♦✐♥t ✻✵✵✲✷✳ ❋✐❣✳ ✶✷✿ ❖✉t♣✉t ♣♦✇❡r✿ ❩♦♦♠ ✐♥ P♦✐♥t ✼✺✵✲✷✳ ❋✐❣✳ ✶✸✿ ❖✉t♣✉t ♣♦✇❡r✿ ❩♦♦♠ ✐♥ P♦✐♥t ✶✵✵✵✲✷✳ s✐③❡ P❖ ▼PP❚ ♦✉t♣❡r❢♦r♠s t❤❡ st❛♥❞❛r❞ ✜①❡❞ st❡♣ s✐③❡ P❖ ▼PP❚ ✐♥ ❝❛s❡ ♦❢ ❝❤❛♥❣✐♥❣ ✐rr❛❞✐❛✲ t✐♦♥ ❧❡✈❡❧ ❢r♦♠ 600 ❲✴♠✷ t♦ 750 ❲✴♠✷ ❛♥❞ ❢r♦♠ 750 ❲✴♠✷ t♦ 1000 ❲✴♠✷ ✳ ❚❤❡ r❡s♣♦♥s❡ t✐♠❡ r❡✲ ❞✉❝t✐♦♥ r❛t✐♦ ✉s✐♥❣ t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P❖ ▼PP❚ ✐s ❜❡t✇❡❡♥ 13.86% ❛♥❞ 45.28%✳ ❇✮ ❙t❡❛❞② ❙t❛t❡ P❡r❢♦r♠❛♥❝❡s✳ ❋✐❣✉r❡s ✶✶ t♦ ✶✸ s❤♦✇ t❤❡ ③♦♦♠✲✐♥ ♦❢ t❤❡ ♣♦✐♥ts ✻✵✵✲✷✱ ✼✺✵✲✷ ❛♥❞ ✶✵✵✵✲✷ ❣✐✈✐♥❣ t❤❡ st❡❛❞② st❛t❡ r❡s♣♦♥s❡ ♦❢ ❜♦t❤ ✜①❡❞ ❛♥❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P❖ ▼PP❚s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✐rr❛❞✐❛t✐♦♥ ❧❡✈❡❧ 600 ❲✴♠✷ , 750 ❲✴♠✷ ❛♥❞ 1000 ❲✴♠✷ ✱ r❡✲ s♣❡❝t✐✈❡❧②✳ ■♥ t❤✐s ❝❛s❡ ✇❡ ❝♦♠♣❛r❡ t❤❡ ♦s❝✐❧✲ ❧❛t✐♦♥ ❛r♦✉♥❞ t❤❡ ▼PP ♣♦✐♥t ❢♦r t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ ▼PP❚ ✭❖s❝✲❱❙❙✮ t♦ t❤❡ ♦s❝✐❧✲ ❧❛t✐♦♥ ❛r♦✉♥❞ t❤❡ ▼PP ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ ✜①❡❞ st❡♣ s✐③❡ ▼PP❚ ✭❖s❝✲❋❙❙✮✳ ❋r♦♠ ❋✐❣✉r❡s ✶✶ t♦ ✶✸✱ ✐t ✐s ❝❧❡❛r t❤❛t t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P❖ ▼PP❚ ♣❡r❢♦r♠s ❜❡tt❡r t❤❛♥ t❤❡ st❛♥❞❛r❞ ✜①❡❞ st❡♣ s✐③❡ P❖ ▼PP❚ ✐♥ ❝❛s❡ ♦❢ ❝❤❛♥❣✐♥❣ ✾✵ ✐rr❛❞✐❛t✐♦♥ ❧❡✈❡❧ ❢r♦♠ 600 ❲✴♠✷ t♦ 750 ❲✴♠✷ ❛♥❞ ❢r♦♠ 750 ❲✴♠✷ t♦ 1000 ❲✴♠✷ ✳ ❚❤❡ st❡❛❞② st❛t❡ ♦s❝✐❧❧❛t✐♦♥ r❡❞✉❝t✐♦♥ r❛t✐♦ ✉s✐♥❣ t❤❡ ♣r♦✲ ♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P❖ ▼PP❚ ✐s ❜❡t✇❡❡♥ ✽✸✳✸✸✪ ❛♥❞ ✶✵✵✪✳ ❈✮ ❈♦✉rs❡ ❆r♦✉♥❞ ▼PP✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✷✳ ❘❡s♣♦♥s❡ t✐♠❡ r❡❞✉❝t✐♦♥✳ ❚r✲❋❙❙ ✭s✮ ❚r✲❱❙❙ ✭s✮ ❘❡❞ r❛t✐♦ ✭✪✮ P♦✐♥t ✻✵✵✲✶ ✵✳✷✺✼ ✵✳✶✻✸ ✸✻✳✺✽ P♦✐♥t ✼✺✵✲✶ ✵✳✶✵✶ ✵✳✵✽✼ ✶✸✳✽✻ P♦✐♥t ✶✵✵✵✲✶ ✵✳✶✵✻ ✵✳✵✺✽ ✹✺✳✷✽ ❚❤❡ ▼PP ♣♦✐♥t tr❛❝❦✐♥❣ ✉s✐♥❣ ❜♦t❤ P❖ ▼PP❚s ✐♥ ❝❛s❡ ♦❢ ❝❤❛♥❣✐♥❣ ✐rr❛❞✐❛t✐♦♥ ❧❡✈❡❧ ❢r♦♠ 600 ❲✴♠✷ t♦ 750 ❲✴♠✷ ❛♥❞ ❢r♦♠ 750 ❲✴♠✷ t♦ 1000 ❲✴♠✷ ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✳ ✶✹✳ ❲❤✐❧❡ t❤❡ ③♦♦♠✲ ✐♥ ❛r♦✉♥❞ t❤❡ ▼PP ♣♦✐♥t ✐s ❣✐✈❡♥ ✐♥ ❋✐❣s✳ ✶✺ ❛♥❞ ✶✻✱ r❡s♣❡❝t✐✈❡❧②✳ ■♥ t❤✐s ❝❛s❡ ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ❝♦✉rs❡ ♦❢ t❤❡ ▼PP ♣♦✐♥t ✭❖s❝✲▼PP✮ ❢♦r ❜♦t❤ ❛❧❣♦r✐t❤♠s t❤❡ ❝❧❛ss✐❝❛❧ ✜①❡❞ st❡♣ s✐③❡ ❛♥❞ t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ ♦♥❡✳ ❋r♦♠ ❋✐❣s✳ ❋✐❣✳ ✶✻✿ ❋✐❣✳ ✶✹✿ P✲❱ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❋✐❣✳ ✶✼✿ ❋✐❣✳ ✶✺✿ P✲❱ ❝❤❛r❛❝t❡r✐st✐❝s✿ ❩♦♦♠ ✐♥ P♦✐♥t ▼PP ✭♦s✲ ❝✐❧❧❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ✜①❡❞ st❡♣ s✐③❡ ▼PP❚✮✳ ❱❛r✐❛❜❧❡ st❡♣ s✐③❡ ❉✉t② ❈②❝❧❡✳ t❤❡ ✜①❡❞ st❡♣ s✐③❡ ❛❧❣♦r✐t❤♠ ✭❛r♦✉♥❞ ✵✳✵✵✵✶ ❲ ❢♦r t❤❡ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P❖ ▼PP❚ ✐♥st❡❛❞ ♦❢ ✵✳✵✷✷✺ ❲ ❢♦r t❤❡ ✜①❡❞ st❡♣ s✐③❡ P❖ ▼PP❚✮✳ ❋✐❣✲ P✲❱ ❝❤❛r❛❝t❡r✐st✐❝s✿ ❩♦♦♠ ✐♥ P♦✐♥t ▼PP ✭♦s✲ ✉r❡ ✶✼ ❣✐✈❡s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✉t② ❝②❝❧❡ ❝♦♠✲ ❝✐❧❧❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ ♣✉t❡❞ ❜② ❜♦t❤ ❛❧❣♦r✐t❤♠s✳ ▼PP❚✮✳ ❚❛❜❧❡s ✷ ❛♥❞ ✸ s✉♠♠❛r✐③❡ t❤❡ ♠❛✐♥ ❝♦♥tr✐❜✉✲ ✶✺ ❛♥❞ ✶✻✱ ✇❡ ❝❛♥ s❡❡ ❝❧❡❛r❧② t❤❛t ✇❡ ❤❛✈❡ ❧❡ss t✐♦♥s ♦❢ t❤✐s st✉❞② r❡❣❛r❞✐♥❣ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ♦s❝✐❧❧❛t✐♦♥ ❛r♦✉♥❞ t❤❡ ▼PP ♣♦✐♥t ✇✐t❤ t❤❡ ♣r♦✲ r❡s♣♦♥s❡ t✐♠❡ ❛s ✇❡❧❧ ❛s t❤❡ ♦s❝✐❧❧❛t✐♦♥ ❛r♦✉♥❞ ♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ ❛❧❣♦r✐t❤♠ ❝♦♠♣❛r❡❞ t♦ t❤❡ ▼PP✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✾✶ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✸✳ ❙t❡❛❞② st❛t❡ ♦s❝✐❧❧❛t✐♦♥ r❡❞✉❝t✐♦♥✳ ❖s❝✲❋❙❙ ✭❲✮ ❖s❝✲❱❙❙ ✭❲✮ ❘❡❞ r❛t✐♦✭✪✮ P♦✐♥t ✻✵✵✲✷ ✵✳✶✻ ✵✳✵✵ ✶✵✵ P♦✐♥t ✼✺✵✲✷ ✵✳✶✵ ✵✳✵✵ ✶✵✵ P♦✐♥t ✶✵✵✵✲✷ ✵✳✻✵ ✵✳✶✵ ✽✸✳✸✸ ✺✳ ❈❖◆❈▲❯❙■❖◆❙ ■♥ t❤✐s ♣❛♣❡r✱ ❛ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ♠❛①✐✲ ♠✉♠ ♣♦✇❡r ♣♦✐♥t ❝♦♥tr♦❧❧❡r ❢♦r P❱ s②st❡♠s ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❛♥❞ ✐♥✈❡st✐❣❛t❡❞ ✈✐❛ t❤❡ ♠♦❞❡❧✲ ✐♥❣ ♦❢ t❤❡ ✇❤♦❧❡ s②st❡♠ ✐♥❝❧✉❞✐♥❣ t❤❡ P❱ ❣❡♥✲ ❡r❛t♦r ❛s ✇❡❧❧ ❛s t❤❡ ❉❈✲❉❈ ❜♦♦st ❝♦♥✈❡rt❡r ❞❡r✐✈❡❞ ✉s✐♥❣ t❤❡ P✫❖ ▼PP❚ ❝♦♥tr♦❧❧❡r ✉♥✲ ❞❡r ▼❛t❧❛❜✴❙✐♠✉❧✐♥❦ ❡♥✈✐r♦♥♠❡♥t✳ ❚❤❡ ❝♦♠✲ ♣❛r❛t✐✈❡ st✉❞② ♦❢ r❡s✉❧ts ❢♦r ❜♦t❤ ❝♦♥✈❡♥t✐♦♥❛❧ ✜①❡❞ st❡♣ s✐③❡ ❛♥❞ t❤❡ ♣r♦♣♦s❡❞ ✈❛r✐❛❜❧❡ st❡♣ s✐③❡ P✫❖ ❛❧❣♦r✐t❤♠s ♣r♦✈❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ❝♦♠♣❛r❡❞ t♦ t❤❡ st❛♥✲ ❞❛r❞ ✜①❡❞ st❡♣ s✐③❡ P❖ ▼PP❚✳ ❚❤❡ ♣r♦♣♦s❡❞ ❛❧✲ ❣♦r✐t❤♠ r❡❞✉❝❡s r❡s♣♦♥s❡ t✐♠❡ ❜❡t✇❡❡♥ ✶✸✳✽✻✪ ❛♥❞ ✹✺✳✷✽✪ ❛♥❞ t❤❡ st❡❛❞② st❛t❡ ♦s❝✐❧❧❛t✐♦♥ ❜❡✲ t✇❡❡♥ ✽✸✳✸✸✪ ❛♥❞ ✶✵✵✪ ❧❡❛❞✐♥❣ t♦ ❧❡ss ♣♦✇❡r ❧♦ss ❡s♣❡❝✐❛❧❧② ✐♥ ❝❛s❡ ♦❢ ❢❛st ❝❤❛♥❣✐♥❣ ❛t♠♦✲ s♣❤❡r✐❝ ❝♦♥❞✐t✐♦♥s✳ ❆s ❢✉t✉r❡ ✇♦r❦s✱ ✇❡ ♣❧❛♥ t♦ ✈❛❧✐❞❛t❡ ❡①♣❡r✐♠❡♥t❛❧❧② t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ✉s✐♥❣ ❛ ❤❛r❞✇❛r❡ ✐♥ t❤❡ ❧♦♦♣ ♣❧❛t❢♦r♠✳ ❣r✐❞✲❝♦♥♥❡❝t❡❞ P❱ ✐♥st❛❧❧❛t✐♦♥ ❢♦r ❧♦✇ ❝♦st ❡❧❡❝tr✐❝✐t② ♣r♦❞✉❝t✐♦♥✳ ❊♥❡r❣② ❛♥❞ ❇✉✐❧❞✲ ✐♥❣s✱ ✸✹✭✼✮✱ ✼✺✸✲✼✺✽✳ ❬✺❪ ●♦♠❛t❤②✱ ❙✳ ❙✳ ❚✳ ❙✳✱ ❙❛r❛✈❛♥❛♥✱ ❙✳✱ ✫ ❚❤❛♥❣❛✈❡❧✱ ❙✳ ✭✷✵✶✷✮✳ ❉❡s✐❣♥ ❛♥❞ ✐♠♣❧❡✲ ♠❡♥t❛t✐♦♥ ♦❢ ♠❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t tr❛❝❦✲ ✐♥❣ ✭▼PP❚✮ ❛❧❣♦r✐t❤♠ ❢♦r ❛ st❛♥❞❛❧♦♥❡ P❱ s②st❡♠✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥t✐✜❝ ✫ ❊♥❣✐♥❡❡r✐♥❣ ❘❡s❡❛r❝❤✱ ✸✭✸✮✱ ✶✲✼✳ ❬✻❪ ●❛rr❛♦✉✐✱ ❘✳✱ ❍❛♠❡❞✱ ▼✳ ❇✳✱ ✫ ❙❜✐t❛✱ ▲✳ ✭✷✵✶✺✮✳ ❆ r♦❜✉st ♦♣t✐♠✐③❛t✐♦♥ t❡❝❤♥✐q✉❡ ❜❛s❡❞ ♦♥ ✜rst ♦r❞❡r s❧✐❞✐♥❣ ♠♦❞❡ ❛♣♣r♦❛❝❤ ❢♦r ♣❤♦t♦✈♦❧t❛✐❝ ♣♦✇❡r s②st❡♠s✳ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❆✉t♦♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✲ ✐♥❣✱ ✶✷✭✻✮✱ ✻✷✵✲✻✷✾✳ ❬✼❪ ❆rt❡❛❣❛ ❖r♦③❝♦✱ ▼✳■✳✱ ❱á③q✉❡③✱ ❏✳ ❘✳✱ ❙❛❧♠❡ró♥✱ P✳✱ ▲✐trá♥✱ ❙✳P✳✱ ❆❧❝á♥t❛r❛✱ ❋✳ ❏✳ ✭✷✵✵✾✮✳ ▼❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t tr❛❝❦❡r ♦❢ ❛ ♣❤♦t♦✈♦❧t❛✐❝ s②st❡♠ ✉s✐♥❣ s❧✐❞✐♥❣ ♠♦❞❡ ❝♦♥✲ tr♦❧✳ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❘❡♥❡✇✲ ❛❜❧❡ ❊♥❡r❣✐❡s ❛♥❞ P♦✇❡r ◗✉❛❧✐t② ❱❛❧❡♥❝✐❛ ✭❙♣❛✐♥✮✱ ✶✳ ❘❡❢❡r❡♥❝❡s ❬✶❪ ▼♦t❛❧❧❡❜✱ ▼✳✱ Ð✉❦✐➣✱ ❆✳✱ ✫ ❋✐r❛❦✱ ▼✳ ✭✷✵✶✺✮✳ ❙♦❧❛r ❤②❞r♦❣❡♥ ♣♦✇❡r s②st❡♠ ❢♦r ✐s♦❧❛t❡❞ ♣❛ss✐✈❡ ❤♦✉s❡✳ ■♥t❡r♥❛t✐♦♥❛❧ ❥♦✉r♥❛❧ ♦❢ ❤②❞r♦❣❡♥ ❡♥❡r❣②✱ ✹✵✭✹✻✮✱ ✶✻✵✵✶✲✶✻✵✵✾✳ ❬✷❪ ❈❛♠♣❛♥❛✱ P✳ ❊✳✱ ▲✐✱ ❍✳✱ ❩❤❛♥❣✱ ❏✳✱ ❩❤❛♥❣✱ ❘✳✱ ▲✐✉✱ ❏✳✱ ✫ ❨❛♥✱ ❏✳ ✭✷✵✶✺✮✳ ❊❝♦♥♦♠✐❝ ♦♣✲ t✐♠✐③❛t✐♦♥ ♦❢ ♣❤♦t♦✈♦❧t❛✐❝ ✇❛t❡r ♣✉♠♣✐♥❣ s②st❡♠s ❢♦r ✐rr✐❣❛t✐♦♥✳ ❊♥❡r❣② ❈♦♥✈❡rs✐♦♥ ❛♥❞ ▼❛♥❛❣❡♠❡♥t✱ ✾✺✱ ✸✷✲✹✶✳ ❬✸❪ ❆❦②✉③✱ ❊✳✱ ❈♦s❦✉♥✱ ❈✳✱ ❖❦t❛②✱ ❩✳✱ ✫ ❉✐♥✲ ❝❡r✱ ■✳ ✭✷✵✶✶✮✳ ❍②❞r♦❣❡♥ ♣r♦❞✉❝t✐♦♥ ♣r♦❜✲ ❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s ❢♦r ❛ P❱✲❡❧❡❝tr♦❧②s❡r s②st❡♠✳ ✐♥t❡r♥❛t✐♦♥❛❧ ❥♦✉r♥❛❧ ♦❢ ❤②❞r♦❣❡♥ ❡♥❡r❣②✱ ✸✻✭✶✼✮✱ ✶✶✷✾✷✲✶✶✷✾✾✳ ❬✽❪ ●❛rr❛♦✉✐✱ ❘✳✱ ❊❧ ❆r♦✉❞✐✱ ❆✳✱ ❍❛♠❡❞✱ ▼✳ ❇✳✱ ❙❜✐t❛✱ ▲✳✱ ✫ ❆❧✲❍♦s❛♥✐✱ ❑✳ ✭✷✵✶✻✮✳ ❆ ❈♦♠✲ ♣❛r❛t✐✈❡ ❙t✉❞② ❇❡t✇❡❡♥ ❚✇♦ ▼PP❚ ❈♦♥✲ tr♦❧❧❡rs ❇❛s❡❞ ♦♥ t❤❡ Pr✐♥❝✐♣❡ ♦❢ ❙❧✐❞✐♥❣✲ ▼♦❞❡ ❈♦♥tr♦❧ ❚❤❡♦r② ❛♥❞ ■♥t❡❧❧✐❣❡♥t ❈♦♥✲ tr♦❧ ❚❡❝❤♥✐q✉❡ ✐♥ P❤♦t♦✈♦❧t❛✐❝ ❙②st❡♠s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ▼❡❞✐t❡rr❛♥❡❛♥ ❈♦♥✲ ❢❡r❡♥❝❡ ♦♥ ■♥❢♦r♠❛t✐♦♥ ✫ ❈♦♠♠✉♥✐❝❛t✐♦♥ ❚❡❝❤♥♦❧♦❣✐❡s ✷✵✶✺ ✭♣♣✳ ✺✵✺✲✺✶✺✮✳ ❙♣r✐♥❣❡r✱ ❈❤❛♠✳ ❬✾❪ ●❛rr❛♦✉✐✱ ❘✳✱ ❍❛♠❡❞✱ ▼✳ ❇✳✱ ✫ ❙❜✐t❛✱ ▲✳ ✭✷✵✶✼✮✳ ▼PP❚ ❈♦♥tr♦❧❧❡rs ❇❛s❡❞ ♦♥ ❙❧✐❞✐♥❣✲▼♦❞❡ ❈♦♥tr♦❧ ❚❤❡♦r② ❛♥❞ ❋✉③③② ▲♦❣✐❝ ✐♥ P❤♦t♦✈♦❧t❛✐❝ P♦✇❡r ❙②st❡♠s✿ ❆ ❈♦♠♣❛r❛t✐✈❡ ❙t✉❞②✳ ■♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❙❧✐❞✲ ✐♥❣ ▼♦❞❡ ❈♦♥tr♦❧ ✭♣♣✳ ✷✶✺✲✷✸✶✮✳ ❙♣r✐♥❣❡r✱ ❙✐♥❣❛♣♦r❡✳ ❬✹❪ ❇❛❦♦s✱ ●✳ ❈✳✱ ✫ ❙♦✉rs♦s✱ ▼✳ ✭✷✵✵✷✮✳ ❚❡❝❤✲ ❬✶✵❪ ▼❛❤❛♠✉❞✉❧✱ ❍✳✱ ❙❛❛❞✱ ▼✳✱ ✫ ■❜r❛❤✐♠ ♥✐❝❛❧ ❢❡❛s✐❜✐❧✐t② ❛♥❞ ❡❝♦♥♦♠✐❝ ✈✐❛❜✐❧✐t② ♦❢ ❛ ❍❡♥❦✱ ▼✳ ✭✷✵✶✸✮✳ P❤♦t♦✈♦❧t❛✐❝ s②st❡♠ ♠♦❞✲ ✾✷ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ ❡❧✐♥❣ ✇✐t❤ ❢✉③③② ❧♦❣✐❝ ❜❛s❡❞ ♠❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t tr❛❝❦✐♥❣ ❛❧❣♦r✐t❤♠✳ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ P❤♦t♦❡♥❡r❣②✱ ✷✵✶✸✳ ❬✶✶❪ ▼❛❦❤❧♦✉✜✱ ▼✳ ❚✳✱ ❑❤✐r❡❞❞✐♥❡✱ ▼✳ ❙✳✱ ❆❜✲ ❞❡ss❡♠❡❞✱ ❨✳✱ ✫ ❇♦✉t❛r❢❛✱ ❆✳ ✭✷✵✶✹✮✳ ▼❛①✲ ✐♠✉♠ P♦✇❡r P♦✐♥t ❚r❛❝❦✐♥❣ ♦❢ ❛ P❤♦t♦✲ ✈♦❧t❛✐❝ ❙②st❡♠ ✉s✐♥❣ ❛ ❋✉③③② ▲♦❣✐❝ ❈♦♥✲ tr♦❧❧❡r ♦♥ ❉❈✴❉❈ ❇♦♦st ❈♦♥✈❡rt❡r✳ ■♥t❡r✲ ♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ■s✲ s✉❡s ✭■❏❈❙■✮✱ ✶✶✭✸✮✱ ✶✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♥tr♦❧ ❆♣♣❧✐✲ ❝❛t✐♦♥s ✭■❈❊❊❈❆✬✷✵✶✼✮✳ ❬✶✾❪ ❑❛r❛♠✐✱ ◆✳✱ ▼♦✉❜❛②❡❞✱ ◆✳✱ ✫ ❖✉t❜✐❜✱ ❘✳ ✭✷✵✶✹✮✳ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t ❢♦r ❛ P❊▼❋❈✕P❱ ❤②❜r✐❞ s②st❡♠✳ ❊♥❡r❣② ❈♦♥✲ ✈❡rs✐♦♥ ❛♥❞ ▼❛♥❛❣❡♠❡♥t✱ ✽✷✱ ✶✺✹✲✶✻✽✳ ❆❜♦✉t ❆✉t❤♦rs ❬✶✷❪ ❍❛rr❛❣✱ ❆✳✱ ▼❡ss❛❧t✐✱ ❙✳ ✭✷✵✶✽✮✳ ■♠♣r♦✈✲ ▼r✳ ❍❛♠③❛ ❇❆❍❘■ ✇❛s ❜♦r♥ ✐♥ ▼s✐❧❛✱ ✐♥❣ P❱ P❡r❢♦r♠❛♥❝❡s ✉s✐♥❣ ❋✉③③②✲❜❛s❡❞ ❆❧❣❡r✐❛✳ ❍❡ r❡❝❡✐✈❡❞ t❤❡ ❇❙❝ ❛♥❞ ▼❛st❡r ❉❡✲ ▼PP❚✱ ■♥ ❜♦♦❦✿ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ✐♥ ❣r❡❡s ✐♥ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❢r♦♠ ▼♦❤❛♠❡❞ ❇♦✉❞✐❛❢ ❯♥✐✈❡rs✐t②✱ ▼s✐❧❛✱ ❆❧❣❡r✐❛✱ ✐♥ ✷✵✵✾ ❛♥❞ ❘❡♥❡✇❛❜❧❡ ❊♥❡r❣❡t✐❝ ❙②st❡♠s✱ ✷✸✻✲✷✹✹✳ ✷✵✶✹✱ r❡s♣❡❝t✐✈❡❧②✳ ❈✉rr❡♥❧② ❤❡ ♣r❡♣❛r❡s t❤❡ ❬✶✸❪ ❙❡❧✈❛♥✱ ❉✳ ❙✳ ✭✷✵✶✸✮✳ ▼♦❞❡❧✐♥❣ ❛♥❞ s✐♠✉✲ P❤❉ ❞❡❣r❡❡ ❛t ▼♦❤❛♠❡❞ ❇♦✉❞✐❛❢ ❯♥✐✈❡rs✐t②✱ ❧❛t✐♦♥ ♦❢ ✐♥❝r❡♠❡♥t❛❧ ❝♦♥❞✉❝t❛♥❝❡ ▼PP❚ ▼s✐❧❛✱ ❆❧❣❡r✐❛✱ ♦♥ ❤②❜r✐❞ s②st❡♠ ✐♥❝❧✉❞✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ♣❤♦t♦✈♦❧t❛✐❝ ❛♣♣❧✐❝❛t✐♦♥s✳ ■♥✲ ♣❤♦t♦✈♦❧t❛✐❝ ❛♥❞ ❤②❞r♦❣❡♥ ♣❛rts ❜❛s❡❞ ♦♥ P❊▼ t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥t✐✜❝ ❊♥❣✐♥❡❡r✲ ❢✉❡❧ ❝❡❧❧s✳ ❍✐s ✇♦r❦ ❝♦♥❝❡r♥s t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ❛♥❞ t❤❡ ♠❛♥❛❣❡♠❡♥t ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♣♦✇❡r ✐♥❣ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✷✭✼✮✱ ✻✽✶✲✻✽✺✳ s②st❡♠✳ ❬✶✹❪ ❑❤❛r❜✱ ❘✳ ❑✳✱ ❙❤✐♠✐✱ ❙✳ ▲✳✱ ❈❤❛tt❡r❥✐✱ ❙✳✱ ✫ ❆♥s❛r✐✱ ▼✳ ❋✳ ✭✷✵✶✹✮✳ ▼♦❞❡❧✐♥❣ ♦❢ s♦❧❛r P❱ ❉r✳ ❆❜❞❡❧❣❤❛♥✐ ❍❆❘❘❆● ✇❛s ❜♦r♥ ♠♦❞✉❧❡ ❛♥❞ ♠❛①✐♠✉♠ ♣♦✇❡r ♣♦✐♥t tr❛❝❦✐♥❣ ✐♥ ❙❡t✐❢✱ ❆❧❣❡r✐❛✳ ❍❡ r❡❝❡✐✈❡❞ ❇❙❝✱ ▼❛st❡r ❛♥❞ ✉s✐♥❣ ❆◆❋■❙✳ ❘❡♥❡✇❛❜❧❡ ❛♥❞ ❙✉st❛✐♥❛❜❧❡ P❤❉ ❉❡❣r❡❡s ✐♥ ❊❧❡❝tr♦♥✐❝s ❢r♦♠ ❋❡r❤❛t ❆❜❜❛s ❊♥❡r❣② ❘❡✈✐❡✇s✱ ✸✸✱ ✻✵✷✲✻✶✷✳ ❯♥✐✈❡rs✐t② ✭❯❋❆❙✮✱ ❙❡t✐❢✱ ❆❧❣❡r✐❛✱ ✐♥ ✶✾✾✺✱ ✶✾✾✽ ❛♥❞ ✷✵✶✶✱ r❡s♣❡❝t✐✈❡❧②✳ ■♥ ✷✵✶✷✱ ❤❡ r❡❝❡✐✈❡❞ t❤❡ ❬✶✺❪ ❍❛rr❛❣✱ ❋✳ ❆✳✱ ❚✐tr❛♦✉✐✱ ❙✳ ❆✳✱ ✫ ❇❛❤r✐✱ ❚✳ ❍❉❘ ❞❡❣r❡❡ ✐♥ ❡❧❡❝tr✐❝❛❧ ❡♥❣✐♥❡❡r✐♥❣ ❢r♦♠ ▼♦✲ ❍✳ ✭✷✵✶✼✱ ▼❛②✮✳ P✫❖ ♦r ■❈ ❢♦r P❱ ♣✉♠♣✲ ❤❛♠❡❞ ❑❤✐❞❡r ❯♥✐✈❡rs✐t②✱ ❇✐s❦r❛✱ ❆❧❣❡r✐❛✳ ❍❡ ✐♥❣ s②st❡♠✿ ❲❤❛t ▼PP❚ ❛❧❣♦r✐t❤♠ t♦ ✐♠✲ ✇♦r❦❡❞ ❛s Pr♦❥❡❝t ▼❛♥❛❣❡r ❞✉r✐♥❣ ♠♦r❡ t❤❛♥ ♣r♦✈❡ ♣❡r❢♦r♠❛♥❝❡s❄✳ ■♥ ❙②st❡♠s ❛♥❞ ❈♦♥✲ ✶✵ ②❡❛rs ✐♥ ❋r❛♥❝❡ ✇✐t❤ ❋r❡♥❝❤ ❛♥❞ ❆♠❡r✐❝❛♥ tr♦❧ ✭■❈❙❈✮✱ ✷✵✶✼ ✻t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r✲ ❙♦❝✐❡t✐❡s✳ ❍❡ ✐s t❤❡ ❝r❡❛t♦r ♦❢ t❤❡ st❛♥❞❛r❞ ❡♥❝❡ ♦♥ ✭♣♣✳ ✷✷✵✲✷✷✺✮✳ ■❊❊❊✳ ❆r❛❜✐❝ ❧❛♥❣✉❛❣❡ ♦♥ ❛❧❧ ♠♦❜✐❧❡ ❛♥❞ ✐♥t❡❧❧✐❣❡♥t s②st❡♠s s♦❧❞ ❜② ❆❧❝❛t❡❧ ▲✉❝❡♥t ❛❧❧ ♦✈❡r t❤❡ ❬✶✻❪ ❚❧✐❧✐✱ ◆✳✱ ◆❡✐❧②✱ ❇✳ ✫ ❇❡♥ ❙❛❧❡♠✱ ❋✳ ✭✷✵✶✹✮✳ ✇♦r❧❞ ✐♥❝❧✉❞✐♥❣ ❆r❛❜ ❝♦✉♥tr✐❡s ❢r♦♠ ❆t❧❛♥t✐❝ ❈♦✉♣❧✐♥❣ ❛ P❤♦t♦✈♦❧t❛✐❝ ●❡♥❡r❛t♦r✱ ❛ P❊▼ ♦❝❡❛♥ t♦ ❆r❛❜✐❝ ❣✉❧❢✳ ❍❡ t❛✉❣❤t ❛t ❯♥✐✈❡rs✐t② ❋✉❡❧ ❈❡❧❧ ❛♥❞ ❛♥ ❊❧❡❝tr♦❧②③❡r ✭P❛rt ■✮✱ P✐❡rr❡ ▼❡♥❞❡s ❋r❛♥❝❡ ❛♥❞ ❏♦s❡♣❤ ❋♦✉r✐❡r ■❊❊❊✱ ✶✲✼✳ ✶✾✾✾✲✷✵✵✵✱ ●r❡♥♦❜❧❡✱ ❋r❛♥❝❡✱ ❛t ❋❡r❤❛t ❆❜❜❛s ❬✶✼❪ P❛❧✐③❜❛♥✱ ❖✳✱ ✫ ▼❡❦❤✐❧❡❢✱ ❙✳ ✭✷✵✶✶✱ ◆♦✈❡♠✲ ❯♥✐✈❡rs✐t② ✶✾✾✻✲✶✾✾✾✱ ❙❡t✐❢✱ ❆❧❣❡r✐❛✳ ■♥ ✷✵✵✾✱ ❜❡r✮✳ ▼♦❞❡❧✐♥❣ ❛♥❞ ❝♦♥tr♦❧ ♦❢ ♣❤♦t♦✈♦❧t❛✐❝ ❤❡ ❥♦✐♥❡❞ ▼♦❤❛♠❡❞ ❇♦✉❞✐❛❢ ❯♥✐✈❡rs✐t②✱ ▼s✐❧❛✱ ♣❛♥❡❧s ❜❛s❡ ♣❡rt✉r❜❛t✐♦♥ ❛♥❞ ♦❜s❡r✈❛t✐♦♥ ❆❧❣❡r✐❛✳ ❈✉rr❡♥t❧②✱ ❤❡ ✇♦r❦s ❛s Pr♦❢❡ss♦r ❛t ▼PP❚ ♠❡t❤♦❞✳ ■♥ ❈♦♥tr♦❧ ❙②st❡♠✱ ❈♦♠✲ ❋❡r❤❛t ❆❜❜❛s ❯♥✐✈❡rs✐t②✳❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ♣✉t✐♥❣ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣ ✭■❈❈❙❈❊✮✱ ✷✵✶✶ ♠❛✐♥❧② ❝♦♥❝❡r♥❡❞ ✐♥t❡❧❧✐❣❡♥t ❝♦♥tr♦❧✱ r❡♥❡✇❛❜❧❡ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ✭♣♣✳ ✸✾✸✲ ❡♥❡r❣②✱ ❤❡✉r✐st✐❝ ❛♥❞ ❡✈♦❧✉t✐♦♥❛r② ♦♣t✐♠✐③❛t✐♦♥✱ ❡♠❜❡❞❞❡❞ s②st❡♠s ❛♥❞ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✳ ❍❡ ✐s ✸✾✽✮✳ ■❊❊❊✳ ♠❡♠❜❡r ♦❢ s❡✈❡r❛❧ r❡s❡❛r❝❤ ♣r♦❥❡❝ts ❛t ❯♥✐✈❡r✲ ❬✶✽❪ ❇❛❤r✐✱ ❍✳✱ ❍❛rr❛❣✱ ❆✳ ✭✷✵✶✼✮✳ ❱❛r✐❛❜❧❡ ❙t❡♣ s✐t② ♦❢ ▼s✐❧❛ ❛♥❞ ❈❈◆❙ ▲❛❜♦r❛t♦r② ❛t ❯❋❆❙ ❙✐③❡ P✫❖ ▼PP❚ ♦♥t♦❧❧❡r t♦ ■♠♣r♦✈❡ ❙t❛t✐❝ ❯♥✐✈❡rs✐t②✳ ❛♥❞ ❉②♥❛♠✐❝ P❱ ❙②st❡♠ P❡r❢♦r♠❛♥❝❡s✱ "This is an Open Access article distributed under the ❚❤❡ ❚❤✐r❞ ■♥t❡r♥♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ 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)." ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✾✸ ... ♦❢ s♦♠❡ ❢✉t✉r❡ ✇♦r❦s✳ ❋r♦♠ ❡q✉❛t✐♦♥s ✭✶✮ t♦ ✭✼✮✱ Ipv ✐s ❞❡✜♥❡❞ ❜②✿ ✷✳ P❱ ❙❨❙❚❊▼ ▼❖❉❊▲■◆● Ipv = Iph − Is (exp [λ (Vpv + Ipv Rs )] − 1) Vpv + Ipv Rs , ✭✽✮ − Rsh ❆ P❱ ❝❡❧❧ ❡q✉✐✈❛❧❡♥t ❝✐r❝✉✐t ✐s... t❤❡ s❤✉♥t ❝✉rr❡♥t❀ Id ✐s t❤❡ ❞✐♦❞❡ ❝✉rr❡♥t✳ Iph = ζ ζref [Isc,ref + ϑIsc (T − Tref )] IRp = Vpv + Ipv Rs Rp λ= q = AVt AkT ✭✾✮ ❚❤❡ ❇P ❙❳✶✺✵❙ ♠♦❞✉❧❡ ♣❛r❛♠❡t❡rs ✉s❡❞ t♦ ❞r❛✇ t❤❡ P❱ ❝❤❛r❛❝t❡r✐st✐❝s... t❡♠♣❡r❛t✉r❡ ✭T = 25, 50 ❛♥❞ 75♦ ❈✮ ✇✐t❤ ❝♦♥st❛♥t ✐rr❛❞✐❛♥❝❡ ✭ζ = 1000 ❲✴♠✷ ✮✳ ❖♥❡ ❞✐♦❞❡ ♠♦❞❡❧✳ Ipv = Iph − IRp − Id , ✇❤❡r❡ ✭✷✮ ✭✸✮ ✸✳ P❘❖P❖❙❊❉ ❱❆❘■❆❇▲❊ ❙❚❊P ❙■❩❊ P✫❖ ▼PP❚ ❋r♦♠ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧

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