‫ ﻓﻲ ﻤﺤﺎﻜﺎﺓ ﺃﺴﺘﻘﺭﺍﺭﻴﺔ ﻤﻨﻅﻭﻤﺔ ﻗﺩﺭﺓ ﺤﺴﺏ‬MATLAB ‫ﺃﺴﺘﺨﺩﺍﻡ ﺒﺭﻨﺎﻤﺞ‬

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‫ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ‬
USING MATLAB PROGRAM IN SIMULATION OF POWER
SYSTEM STABILITY BY LYAPUNOV'S DIRECT METHOD

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‫ﻤﺠﻠﻲ ﻨﻌﻤﺔ ﺤﻭﺍﺱ ﺍﻝﺨﺯﺭﺠﻲ‬

:‫ﺍﻝﻤﺴﺘﺨﻠﺹ‬

– ‫ﻴﻌﺭﺽ ﺍﻝﺒﺤﺙ ﻭﺼﻑ ﻤﺨﺘﺼﺭ ﻝﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻭﺘﻌﺭﺽ ﻜﻔﺎﺌﺘﻬﺎ ﻤﻘﺎﺭﻨﺔ ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‬
‫ ﻭﺍﻝﻔﻭﺍﺌﺩ ﺍﻝﻌﻤﻠﻴﺔ ﻝﻬﺫﻩ‬،‫ﺨﻁﻭﺓ ﻏﻴﺭ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻤﻥ ﺍﺠل ﺤﺴﺎﺏ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ ﻝﻤﻨﻅﻭﻤﺔ ﻗﺩﺭﺓ ﻜﻬﺭﻭﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬
:‫ﺍﻝﻁﺭﻴﻘﺔ ﺍﻝﺘﺤﻠﻴﻠﻴﻪ‬
.‫ ﺴﺭﻋﺔ ﺍﻝﺤل‬.١
. ‫ ﺍﻤﻜﺎﻨﻴﺔ ﺤﺴﺎﺏ ﺩﺭﺠﺔ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﻪ‬.٢
‫ ﻤﻊ‬V ‫ ﻝﺤﺴﺎﺏ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ﻨﻜﺎﻤل ﻤﻌﺎﺩﻻﺕ ﺍﻝﻌﻁل ﺤﺘﻰ ﺘﺘﺴﺎﻭﻱ ﻗﻴﻤﺔ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‬.٣
.V1 ‫ﺍﻝﻘﻴﻤﺔ ﺍﻝﺤﺭﺠﺔ ﻝﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‬
‫ ﻭﺘﻡ ﺍﻋﺘﻤﺎﺩ ﻤﻨﻅﻭﻤﺔ ﺍﺨﺘﺒﺎﺭﻴﻪ ﻝﻌﺭﺽ ﻜﻔﺎﺀﺓ ﻁﺭﻴﻘﺔ‬.‫ﺘﻡ ﺘﻠﺨﻴﺹ ﺍﻝﻤﻔﺎﻫﻴﻡ ﺍﻻﺴﺎﺴﻴﻪ ﻭﺍﻝﺘﻤﺜﻴل ﺍﻝﺭﻴﺎﻀﻲ‬
.‫ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻤﻘﺎﺭﻨﺔ ﻤﻊ ﻜﻔﺎﺀﺓ ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ – ﺨﻁﻭﺓ ﻏﻴﺭ ﺍﻝﻤﺒﺎﺸﺭﺓ‬

Abstract:

This paper presents a brief description of Lyapunov's direct method, and

demonstrates its efficiency with respect to previous indirect step by step method,
in order to compute the stability of electromechanically power system. The
practical advantages of this direct method of analysis are:
(i) Rapid solution.
(ii) Ability to compute the degree of stability.
(iii) For computing the critical clearing time integrate the faulted equation,
until V=V1.
The basic concepts and the mathematical formulation of the method are
summarized. Practical test system are used to demonstrate the superior
efficiency of the method relative to step by step method.

:‫ﺍﻝﻤﻘﺩﻤﺔ‬

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٢٠٠٨/١٢/٢٤ : ‫ ﺘﺎﺭﻴﺦ ﻗﺒﻭل ﺍﻝﻨﺸﺭ‬، ٢٠٠٦/٤/٥ : ‫ﺘﺎﺭﻴﺦ ﺍﺴﺘﻼﻡ ﺍﻝﺒﺤﺙ‬
‫ ﻜﻠﻴﺔ ﺍﻝﺘﻘﻨﻴﺎﺕ ﺍﻝﻜﻬﺭﺒﺎﺌﻴﺔ ﻭﺍﻹﻝﻜﺘﺭﻭﻨﻴﺔ‬/ ‫*ﻤﺩﺭﺱ‬
‫ﺩﺭﺍﺴﺎﺕ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻤﻥ ﺍﻝﺩﺭﺍﺴﺎﺕ ﺒﺎﻝﻐﺔ ﺍﻷﻫﻤﻴﺔ ﻝﺘﺤﺩﻴﺩ ﺍﻝﺴﻠﻭﻜﻴﺔ ﺍﻝﺩﻴﻨﺎﻤﻴﻜﻴﺔ ﻝﻤﻨﻅﻭﻤﺔ ﺍﻝﻘـﺩﺭﺓ‬
‫ﺍﻝﻜﻬﺭﻭﻤﻴﻜﺎﻨﻴﻜﻴﺔ ﻭﺘﻬﺩﻑ ﺍﻝﺘﺤﻠﻴﻼﺕ ﺍﻝﺭﻴﺎﻀﻴﺔ ﻭﺍﻝﺩﺭﺍﺴﺎﺕ ﺍﻝﺘﺼﺤﻴﺤﻴﻪ ﻝﻸﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﺇﻝﻰ ﻤﻌﺭﻓﺔ ﺇﻤﻜﺎﻨﻴـﺔ‬
‫ﺒﻘﺎﺀ ﺍﺴﺘﻘﺭﺍﺭ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻋﻨﺩ ﺤﺩﻭﺙ ﺍﻀﻁﺭﺍﺒﺎﺕ ﺭﺌﻴﺴﻴﺔ ﻤﺜل ﺃﻋﻁﺎل ﺨﻁﻭﻁ ﻨﻘل ﺍﻝﻘـﺩﺭﺓ ﻭﺍﻝﺘﻐﻴﻴـﺭ ﺍﻝﻔﺠـﺎﺌﻲ‬
‫ﻝﻸﺤﻤﺎل ﺍﻝﻜﻬﺭﺒﺎﺌﻴﺔ ﻭﺍﻝﻔﻘﺩﺍﻥ ﺍﻝﻔﺠﺎﺌﻲ ﻝﻭﺤﺩﺍﺕ ﺘﻭﻝﻴﺩ ﺍﻝﻁﺎﻗﺔ ﺍﻝﻜﻬﺭﺒﺎﺌﻴﺔ‪.‬ﻭﺒﺴﺒﺏ ﻀﺨﺎﻤﺔ ﻤﻨﻅﻭﻤﺔ ﺍﻝﻘﺩﺭﺓ ﻭﺘﻭﺴـﻊ‬
‫ﺍﻝﺸﺒﻜﺎﺕ ﺍﻝﻤﺨﺘﻠﻔﺔ ﺍﻝﻤﺭﺘﺒﻁﺔ ﺒﻬﺎ ﻗﺩ ﺃﺩﻯ ﺇﻝﻰ ﻭﺠﻭﺏ ﺍﻝﺘﻭﺴﻴﻊ ﻓﻲ ﺩﺭﺍﺴﺎﺕ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻝﻐﺭﺽ ﺍﻝﺤﺼﻭل‬
‫ﻋﻠﻰ ﺩﺭﺠﺔ ﻋﺎﻝﻴﺔ ﺠﺩﹰﺍ ﻤﻥ ﺍﻝﻀﺒﻁ ﻭﺍﻝﺩﻗﺔ ﻓﻲ ﺍﻝﻨﺘﺎﺌﺞ ﺍﻝﻜﻔﻴﻠﺔ ﺒﺈﻋﻁﺎﺀ ﺍﻝﻘﺭﺍﺭ ﺍﻝﺼﺤﻴﺢ ﻓﻲ ﻜﻴﻔﻴﺔ ﺍﺴﺘﺠﺎﺒﺔ ﺍﻝﻤﻨﻅﻭﻤﺔ‬
‫ﻝﻼﻀﻁﺭﺍﺒﺎﺕ ﺍﻝﺘﻲ ﻴﺤﺘﻤل ﺤﺩﻭﺜﻬﺎ‪ .‬ﻭﻴﻜﻭﻥ ﻝﻠﻤﻜﺎﺌﻥ ﻋﺎﺩﺓ ﻤﻨﻅﻭﻤـﺎﺕ ﺇﺜـﺎﺭﺓ ﻭﻤﻨﻅﻭﻤـﺎﺕ ﺴـﻴﻁﺭﺓ‪) ،‬ﺍﻝـﺘﺤﻜﻡ‬
‫ﺍﻝﺘﻭﺭﺒﻴﻨﻲ( ﻴﺘﻭﺠﺏ ﻭﻀﻌﻬﺎ ﻓﻲ ﻨﻤﺎﺫﺝ ﺒﺸﻜل ﺩﻗﻴﻕ ﻋﻨﺩ ﺘﻤﺜﻴﻠﻬﺎ ﻓﻲ ﺍﻝﺤﺎﺴﺒﺔ ﻝﻜﻲ ﺘﻌﻜﺱ ﺍﻻﺴـﺘﺠﺎﺒﺔ ﺍﻝﺩﻴﻨﺎﻤﻴﻜﻴـﺔ‬
‫ﺍﻝﺼﺤﻴﺤﺔ ﻝﻤﻨﻅﻭﻤﺔ ﺍﻝﻘﺩﺭﺓ ﻓﻲ ﺤﺎﻝﺔ ﺍﻻﻀﻁﺭﺍﺒﺎﺕ ]‪. [2,1‬‬

‫ﻤﻌﺎﺩﻝﺔ ﺍﻝﺘﺄﺭﺠﺢ‪:‬‬

‫ﻤﻌﺎﺩﻝﺔ ﺃﻝﺘﺄﺭﺠﺢ ﺫﺍﺕ ﺃﻫﻤﻴﻪ ﺒﺎﻝﻐﺔ ﻓﻲ ﺩﺭﺍﺴﺎﺕ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ ﻭﻫﻲ ﻤﻌﺎﺩﻝﺔ ﺘﻔﺎﻀﻠﻴﺔ ﻤﻥ ﺍﻝﺩﺭﺠﺔ ﺍﻝﺜﺎﻨﻴﺔ‬
‫ﻭﺘﻜﺘﺏ ﺒﺎﻝﺼﻴﻐﺔ ﺍﻝﺘﺎﻝﻴﺔ ]‪. [3,2‬‬

‫‪2 H d 2δ‬‬
‫‪.‬‬ ‫‪= Pa = Pm − Pe‬‬ ‫)‪...(1‬‬
‫‪ω sm dt 2‬‬
‫ﻤﻘﺎﺴﻪ ﺒﺩﺭﺠﺎﺕ ﻨﺼﻑ ﻗﻁﺭﻴﺔ ﻝﻜل ﺜﺎﻨﻴﺔ‪.‬‬ ‫ﻭ ‪ω sm‬‬ ‫ﺇﺫ ﺃﻥ ‪ δ‬ﻤﻘﺎﺴﻪ ﺒﺩﺭﺠﺎﺕ ﻨﺼﻑ ﻗﻁﺭﻴﺔ‬
‫ﻭﺒﺎﻹﻤﻜﺎﻥ ﺇﻋﺎﺩﺓ ﻜﺘﺎﺒﺔ ﺍﻝﻤﻌﺎﺩﻝﺔ )‪ (1‬ﻋﻠﻰ ﺍﻝﻨﺤﻭ ﺍﻝﺘﺎﻝﻲ ﺒﺎﻝﻨﺴﺒﺔ ﻝﻤﻨﻅﻭﻤﺔ ﺘﻌﻤل ﺒﺎﻝﺘﺭﺩﺩ ﺍﻝﻜﻬﺭﺒﺎﺌﻲ ‪ f‬ﻫﻴﺭﺘﺯ‪.‬‬
‫‪H d 2δ‬‬
‫‪.‬‬ ‫‪= Pa = Pm − Pe‬‬ ‫)‪...(2‬‬
‫‪π f dt 2‬‬
‫ﺇﺫ ﺃﻥ ‪ δ‬ﻤﻘﺎﺴﻪ ﺒﺩﺭﺠﺎﺕ ﻨﺼﻑ ﻗﻁﺭﻴﻪ ﻜﻬﺭﺒﺎﺌﻴﺔ‪.‬‬
‫‪H d δ‬‬ ‫‪2‬‬
‫‪.‬‬ ‫‪= Pa = Pm − Pe‬‬ ‫)‪...(3‬‬
‫‪180 f dt 2‬‬
‫ﻋﻠﻰ ﺍﻋﺘﺒﺎﺭ ‪ δ‬ﻤﻘﺎﺴﻪ ﺒﺎﻝﺩﺭﺠﺎﺕ ﺍﻝﻜﻬﺭﺒﺎﺌﻴﺔ ‪.‬‬

‫ﺘﻁﺒﻴﻘﺎﺕ ﻤﻌﺎﺩﻝﺔ ﺍﻝﺘﺄﺭﺠﺢ‪:‬‬

‫ﻴﻤﻜﻥ ﻜﺘﺎﺒﺔ ﻤﻌﺎﺩﻝﺔ ﺍﻝﺘﺄﺭﺠﺢ ﺒﺸﻜل ﻤﻌﺎﺩﻝﺘﻴﻥ ﺘﻔﺎﻀﻠﻴﺘﻴﻥ ﻤﻥ ﺍﻝﺩﺭﺠﺔ ﺍﻷﻭﻝﻰ ﻭﻜﻤﺎ ﻴﺄﺘﻲ‪:‬‬
‫‪2 H dω‬‬
‫‪.‬‬ ‫‪= Pm − Pe‬‬ ‫)‪...(4‬‬
‫‪ω s dt‬‬
‫‪dδ‬‬
‫‪= ω − ωs‬‬ ‫)‪...(5‬‬
‫‪dt‬‬

‫ﻭﺘﻜﻤﻥ ﺍﻝﻐﺎﻴﺔ ﻤﻥ ﺤل ﻤﻌﺎﺩﻝﺔ ﺃﻝﺘﺄﺭﺠﺢ ﻓﻲ ﺇﻋﻁﺎﺀ ﺍﻝﻌﻼﻗﺔ ﺒﻴﻥ ‪ δ‬ﻭ ‪ t‬ﺤﻴﺙ ﺘﻜﻭﻥ ﺍﻝﺯﺍﻭﻴﺔ ‪ δ‬ﻋﻨﺩﻫﺎ‬
‫ﻋﺒﺎﺭﺓ ﻋﻥ ﺩﺍﻝﺔ ﺯﻤﻨﻴﺔ‪ .‬ﻭﻴﻁﻠﻕ ﻋﻠﻰ ﺍﻝﻤﺨﻁﻁ ﺍﻝﺒﻴﺎﻨﻲ ﻝﺯﺍﻭﻴﺔ ‪ δ‬ﺇﺯﺍﺀ ﺍﻝﺯﻤﻥ ‪ t‬ﺒﻤﻨﺤﻨﻰ ﺃﻝﺘﺄﺭﺠﺢ ﻝﻠﻤﺎﻜﻨﺔ ]‪[5,4‬‬
‫‪.‬‬
 ‫ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ ‪ -‬ﺨﻁﻭﺓ ﻝﻸﺴﺘﻘﺭﺍﺭﻴﺔ‪:‬‬

‫ﻫﻲ ﻁﺭﻴﻘﻪ ﻝﺤل ﻤﻌﺎﺩﻝﺔ ﺃﻝﺘﺄﺭﺠﺢ ﻭﺫﻝﻙ ﻝﻠﺤﺼﻭل ﻋﻠﻰ ﻤﻨﺤﻨﻲ ﺃﻝﺘﺄﺭﺠﺢ‪ .‬ﺃﻱ ﺍﻝﺤﺼﻭل ﻋﻠﻰ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ‬
‫‪ δ‬ﺒﻭﺼﻔﻬﺎ ﺩﺍﻝﺔ ﻝﻠﺯﻤﻥ ‪ t‬ﻷﺯﻤﻨﺔ ﺇﺯﺍﻝﺔ ﺤﺭﺠﺔ ﻤﻌﻴﻨﺔ‪ .‬ﻭﻴﺘﻡ ﺫﻝﻙ ﺒﺘﻁﺒﻴﻕ ﻋﻤﻠﻴﺎﺕ ﺤﺴﺎﺒﻴﻪ ﺘﻘﺭﻴﺒﻴﺔ ﻭﺘﻜﺭﺍﺭﻴﺔ‪.‬‬
‫ﺘﺴﺘﺨﺩﻡ ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ ﻓﻲ ﺍﻝﺤﺼﻭل ﻋﻠﻰ ﺍﻝﺤل ﺒﻭﺍﺴﻁﺔ ﺍﻝﺒﺭﻤﺠﺔ ﻓﻲ ﺍﻝﺤﺎﺴﺒﺔ ﻭﺨﺎﺼﺔ ﻋﻨﺩﻤﺎ‬
‫ﺘﻜﻭﻥ ﻫﺫﻩ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻜﺒﻴﺭﺓ ﻭﺤﺎﻭﻴﺔ ﻤﺠﻤﻭﻋﺔ ﻤﻥ ﺍﻝﻤﻜﺎﺌﻥ‪ ،‬ﻭﻴﺠﺭﻱ ﺍﻝﺤل ﺒﺈﻴﺠﺎﺩ ﻗﻴﻡ ﺍﻝﺯﺍﻭﻴﺔ ﺒﻭﺼﻔﻬﺎ ﺩﺍﻝﺔ ﻝﻠﺯﻤﻥ‬
‫‪ t‬ﻭﻷﻱ ﻤﻥ ﺍﻝﻤﻜﺎﺌﻥ ﻝﻔﺘﺭﺓ ﺯﻤﻨﻴﺔ ﻜﺎﻤﻠﺔ‪ ،‬ﻭﻋﻨﺩﻫﺎ ﻴﺠﺭﻱ ﺘﻤﺜﻴل ﺍﻝﻌﻼﻗﺔ ﻓﻴﻤﺎ ﺒﻴﻥ ‪ δ‬ﻭ‪ t‬ﺒﻤﻨﺤﻨﻰ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ‪/‬‬
‫ﺍﻝﺯﻤﻥ‪ ،‬ﻝﺘﺤﺩﻴﺩ ﻤﺎ ﺇﺫﺍ ﻜﺎﻨﺕ ‪ δ‬ﺴﺘﺯﺩﺍﺩ ﺤﺘﻰ ﺘﺼل ﺇﻝﻰ ﺤﺩ ﺃﻗﺼﻰ ﺜﻡ ﺘﺒﺩﺃ ﺒﺎﻝﺘﻨﺎﻗﺹ ﺃﻭ ﺘﺴﺘﻤﺭ ﺩﻭﻥ ﺤﺩ‪.‬‬
‫ﻭﺍﻝﻭﻀﻊ ﺍﻷﺨﻴﺭ ﻴﻌﻨﻲ ﻓﻘﺩﺍﻥ ﺍﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻤﻨﻅﻭﻤﺔ‪ .‬ﻭﺍﻥ ﻓﺘﺭﺓ ﺍﻹﺯﺍﻝﺔ ﺍﻝﻜﻠﻴﺔ )‪ (Overall Clearing Time‬ﻫﻲ‬
‫ﺘﻠﻙ ﺍﻝﻌﺎﺌﺩﺓ ﻝﻤﺠﻤﻭﻉ ﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻓﻲ ﺇﺼﺩﺍﺭ ﺍﻻﻓﻼﺕ ﻤﻥ ﻗﺒل ﺃﺠﻬﺯﺓ ﺍﻝﺤﻤﺎﻴﺔ ﻭﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻓﻲ ﻓﻌل‬
‫ﻗﺎﻁﻊ ﺍﻝﺩﻭﺭﺓ ﻝﻔﺼل ﺍﻝﺘﻭﺼﻴل ﺍﻝﻜﻬﺭﺒﺎﺌﻲ ]‪. [6,3‬‬
‫ﻫﻨﺎﻙ ﻋﺩﺩ ﻤﻥ ﺍﻝﻁﺭﻕ ﻝﺘﻁﺒﻴﻕ ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪-‬ﺨﻁﻭﺓ ﻝﺤل ﺍﻝﻤﻌﺎﺩﻻﺕ ﺍﻝﺘﻔﺎﻀﻠﻴﺔ ﻤﻥ ﺍﻝﺩﺭﺠﺔ ﺍﻝﺜﺎﻨﻴﺔ‪ ،‬ﻭﻫﻲ‬
‫ﺠﻤﻴﻌﻬﺎ ﺃﺴﺎﻝﻴﺏ ﻋﺩﺩﻴﺔ )‪ (Numerical Methods‬ﺘﺘﻌﻠﻕ ﺒﺈﻀﺎﻓﺔ ﺯﻴﺎﺩﺍﺕ ﺠﺯﺌﻴﺔ ﺼﻐﻴﺭﺓ ﺠﺩﺍ ﺇﻝﻰ ﺍﻝﺩﺍﻝﺔ ﺃﻭ‬
‫ﺘﺴﺘﺨﺩﻡ ﺍﻝﺤﺎﺴﺒﺔ ﻷﺠﺭﺍﺀ ﻫﺫﻩ ﺍﻝﻌﻤﻠﻴﺎﺕ ﺍﻝﺤﺴﺎﺒﻴﺔ‪.‬‬
‫ﻭﺘﻬﺩﻑ ﺍﻝﺤﺴﺎﺒﺎﺕ ﻓﻲ ﺠﻤﻴﻊ ﺍﻷﺤﻭﺍل ﺇﻝﻰ ﺃﻴﺠﺎﺩ ﺍﻝﺘﻐﻴﺭﺍﺕ ﻓﻲ ﺍﻝﻤﻭﻗﻊ ﺍﻝﺯﺍﻭﻱ ﻝﻠﺩﻭﺍﺭ ﺨﻼل ﻓﺘﺭﺍﺕ ﺯﻤﻨﻴﺔ ﻗﺼﻴﺭﺓ‬
‫ﺠﺩﹰﺍ‪-‬ﻤﺴﺘﻨﺩﺓ ﺇﻝﻰ ﺍﻝﻔﺭﻀﻴﺎﺕ ﺍﻵﺘﻴﺔ‪:‬‬

‫ﺃ‪ .‬ﺍﻋﺘﺒﺎﺭ ﺍﻝﻘﺩﺭﺓ ﺍﻝﺘﻌﺠﻴﻠﻴﺔ ‪ Pa‬ﺍﻝﻤﺤﺴﻭﺒﺔ ﻓﻲ ﺒﺩﺍﻴﺔ ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﻤﻨﺘﻬﻴﺔ )ﺍﻝﺴﺎﺒﻘﺔ( ﻭﻝﻐﺎﻴﺔ ﻤﻨﺘﺼﻑ ﺍﻝﻔﺘﺭﺓ‬
‫ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﺤﺎﻝﻴﺔ‪.‬‬
‫ﺏ‪ .‬ﺍﻋﺘﺒﺎﺭ ﺴﺭﻋﺔ ﺍﻝﺯﺍﻭﻴﺔ ﺜﺎﺒﺘﺔ ﺨﻼل ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﺤﺎﻝﻴﺔ ﻭﺍﻝﺘﻲ ﻫﻲ ﻓﻲ ﺍﻝﺤﻘﻴﻘﺔ ﻤﺤﺴﻭﺒﺔ ﻓﻲ ﻤﻨﺘﺼﻑ ﻫﺫﻩ‬
‫ﺍﻝﻔﺘﺭﺓ‪.‬‬

‫ﻭﺘﺠﺩﺭ ﺍﻹﺸﺎﺭﺓ ﺇﻝﻰ ﺃﻥ ﻜﻠﺘﺎ ﺍﻝﻔﺭﻀﻴﺘﻴﻥ ﻏﻴﺭ ﺼﺤﻴﺤﺔ ﻭﺫﻝﻙ ﻻﻥ ‪ δ‬ﺘﺘﻐﻴﺭ ﺒﺸﻜل ﻤﺴﺘﻤﺭ ﻭﺍﻥ ﻜ ﹰ‬
‫ﻼ‬
‫ﻤﻥ ‪ Pa‬ﻭ ‪ ω‬ﻫﻲ ﺃﻴﻀﺎ ﺩﺍﻝﺔ ﻝﻠﺯﺍﻭﻴﺔ ‪ . δ‬ﻭﻝﻜﻥ ﻴﻼﺤﻅ ﺍﻨﻪ ﻜﻠﻤﺎ ﻜﺎﻨﺕ ‪ ∆t‬ﺍﻗﺼﺭ ﺍﻗﺘﺭﺏ ﻤﻨﺤﻨﻰ ﺍﻝﺘﺄﺭﺠﺢ ﻤﻥ‬
‫ﺍﻝﻤﻨﺤﻨﻲ ﺍﻝﺤﻘﻴﻘﻲ‪ .‬ﻭﺫﻝﻙ ﻋﻠﻰ ﺍﻝﺭﻏﻡ ﻤﻥ ﻋﺩﻡ ﻤﻁﺎﺒﻘﺔ ﺃﻱ ﻤﻥ ﺍﻝﻔﺭﻀﻴﺘﻴﻥ ﺍﻝﻤﺫﻜﻭﺭﺘﻴﻥ ﻓﻲ ﺃﻋﻼﻩ ﻝﻠﺤﻘﻴﻘﺔ‪ ،‬ﻭﻝﻜﻥ‬
‫ﺍﻝﻤﻌﺎﻝﺠﺔ ﺍﻝﺤﺴﺎﺒﻴﺔ ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪-‬ﺨﻁﻭﺓ ﻫﻲ ﺍﻝﺘﻲ ﺘﺅﺩﻱ ﺇﻝﻰ ﺍﺴﺘﺨﺭﺍﺝ ﻗﻴﻡ ﻏﻴﺭ ﺒﻌﻴﺩﺓ ﻋﻥ ﺍﻝﻘﻴﻡ ﺍﻝﺤﻘﻴﻘﻴﺔ‬
‫]‪. [7,2‬‬

‫ﺘﻘﻴﻴﻡ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ‪:‬‬

‫ﻴﻤﻜﻥ ﺘﻘﻴﻴﻡ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻤﻥ ﺨﻼل ﻤﻨﺤﻨﻴﺎﺕ ﺍﻝﺘﺎﺭﺠﺢ ﻝﺩﻭﺍﺭ ﺍﻝﻤﻜﺎﺌﻥ ﺍﻝﺘﺯﺍﻤﻨﻴﻪ ﻓﻲ ﻤﻨﻅﻭﻤﺎﺕ ﺍﻝﻘﺩﺭﺓ‬
‫ﺍﻝﻜﻬﺭﺒﺎﺌﻴﻪ‪ ،‬ﻝﻠﺤﺼﻭل ﻋﻠﻰ ﻤﻨﺤﻨﻴﺎﺕ ﺃﻝﺘﺄﺭﺠﺢ ﻫﻨﺎﻙ ﻁﺭﻴﻘﺘﺎﻥ ﻴﺘﻡ ﺒﺭﻤﺠﺘﻬﺎ ﺒﺎﻝﺤﺎﺴﺒﺔ ﺍﻝﺭﻗﻤﻴﺔ ﻭﻫﻤﺎ‪:‬‬

‫‪.١‬ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ ﺍﻝﺒﺴﻴﻁﺔ‪:‬‬

 ‫ﻴﺘﻡ ﺃﻴﺠﺎﺩ ﻤﻨﺤﻨﻴﺎﺕ ﺃﻝﺘﺄﺭﺠﺢ ﻓﻲ ﻫﺫﻩ ﺍﻝﻁﺭﻴﻘﺔ ﻝﻤﺩﻯ ﻤﻥ ﺃﺯﻤﻨﺔ ﺇﺯﺍﻝﺔ ﺍﻝﻌﻁل ﺒﺘﻁﺒﻴﻕ ﻤﻌﺎﺩﻝﺔ ﺍﻝﻔﺭﻕ‪:‬‬

‫‪180 f‬‬
‫‪∆δ n = ∆δ n−1 +‬‬ ‫)‪.(∆t ) 2 Pa ( n−1‬‬ ‫)‪...(6‬‬
‫‪H‬‬

‫ﻋﺒﺭ ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ‪ t‬ﺍﻝﻤﺘﻀﻤﻨﺔ ﺍﻝﻔﺘﺭﺍﺕ ﺍﻝﻤﺘﻌﺎﻗﺒﺔ ﻭﺘﺼﺎﻍ ﻓﻲ ﺒﺭﺍﻤﺞ ﺍﻝﺤﺎﺴﺒﺔ ﻝﻜﺘﺎﺒﺔ‬ ‫‪δ‬‬ ‫ﺘﺤﺴﺏ ﺍﻝﺯﺍﻭﻴﺔ‬
‫ﺠﺩﻭل ﺒﺎﻝﻨﺘﺎﺌﺞ ﺍﻝﻤﺴﺘﺨﺭﺠﺔ‪ ،‬ﻭﻴﺴﺘﺨﺩﻡ ﻓﻴﻪ ﺒﺭﻨﺎﻤﺞ ﻓﺭﻋﻲ ﻝﺭﺴﻡ ﻤﻨﺤﻨﻴﺎﺕ ﺃﻝﺘﺄﺭﺠﺢ‪ .‬ﻭﺘﺅﺨﺫ ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ‬
‫ﻼ ‪ ∆t = 0.01‬ﻭﺘﻜﻭﻥ ﺃﺯﻤﻨﺔ ﺇﺯﺍﻝﺔ ﺍﻝﻌﻁل ﻓﻲ ﺍﻝﻤﺩﻯ ﺒﻴﻥ ‪ 0.05‬ﺇﻝﻰ ‪ 0.5‬ﺜﺎﻨﻴﺔ ﻭﻓﻲ ﻀﺅ ﺃﺴﺘﻤﺭﺍﺭ‬
‫ﺒﺎﻝﺜﺎﻨﻴﺔ ﻤﺜ ﹰ‬
‫ﺘﺯﺍﻴﺩ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ ﺍﻭ ﺘﻨﺎﻗﺼﻬﺎ ﻴﺘﻡ ﺍﻝﺤﻜﻡ ﻋﻠﻰ ﺍﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻤﻨﻅﻭﻤﺔ ]‪. [8,1‬‬

‫‪ .٢‬ﻁﺭﻕ ﺍﻝﺘﻜﺎﻤل ﺍﻝﻌﺩﺩﻱ‪:‬‬

‫ﺘﺴﺘﺨﺩﻡ ﺍﻝﻤﻌﺎﺩﻝﺘﺎﻥ ﺍﻝﺘﻔﺎﻀﻠﻴﺘﺎﻥ ﻤﻥ ﺍﻝﺩﺭﺠﺔ ﺍﻷﻭﻝﻰ )‪ (٤‬ﻭ)‪ (٥‬ﻜﻤﺎ ﺠﺎﺀ ﻓﻲ ﺍﻝﻔﻘﺭﺓ )‪ (٣‬ﻭﻴﺘﺒﻊ ﻓﻲ ﺤل‬
‫ﻫﺎﺘﻴﻥ ﺍﻝﻤﻌﺎﺩﻝﺘﻴﻥ ﺍﻝﻁﺭﻴﻘﺔ ﺍﻻﻋﺘﻴﺎﺩﻴﺔ ﻹﻴﺠﺎﺩ ﺴﺭﻋﺔ ﺍﻝﻤﺎﻜﻨﺔ ﻭﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ‪ ،‬ﻴﻜﺘﺏ ﺒﺭﻨﺎﻤﺞ ﻓﺭﻋﻲ ﺩﻭﺭﻱ‬
‫ﻝﻠﺤﺼﻭل ﻋﻠﻰ ﻤﻨﺤﻨﻴﺎﺕ ﺃﻝﺘﺄﺭﺠﺢ ﺒﺘﻁﺒﻴﻕ ﺇﺤﺩﻯ ﺍﻝﻁﺭﻕ ﺁﻻﺘﻴﺔ‪:‬‬

‫ﺃ‪ .‬ﻁﺭﻴﻘﺔ ﺃﻭﻴﻠﺭ ‪(Euler) .‬‬

‫ﺏ‪ .‬ﻁﺭﻴﻘﺔ ﺃﻭﻴﻠﺭ ﺍﻝﻤﺤﻭﺭﺓ ‪(Modified Euler) .‬‬
‫ﺕ‪ .‬ﻁﺭﻴﻘﺔ ﺭﺍﻨﺞ –ﻜﻭﺘﺎ ‪(Runge-Kutta) .‬‬

‫ﻭﺘﺘﻡ ﺍﻝﻤﻔﺎﻀﻠﺔ ﺒﻴﻥ ﺍﻝﻁﺭﻕ ﺃﻋﻼﻩ ﻋﻠﻰ ﻀﺅ ﺍﻝﻀﺒﻁ ﻓﻲ ﺍﻝﻨﺘﺎﺌﺞ ﺍﻝﻤﺴﺘﺨﺭﺠﺔ ﺒﻜل ﻤﻥ ﻫﺫﻩ ﺍﻝﻁﺭﻕ‬
‫ﺍﺨﺫﻴﻥ ﺒﻨﻅﺭ ﺍﻻﻋﺘﺒﺎﺭ ﻤﺎ ﻴﻠﻲ‪:‬‬
‫ﺃ‪ .‬ﺍﻝﺘﻌﻘﻴﺩ ﻓﻲ ﺍﻝﺒﺭﻤﺠﺔ ﻭﻴﻘﺼﺩ ﺒﻬﺎ ﺤﺠﻡ ﺍﻝﺒﺭﺍﻤﺞ ﻭﺍﻝﻭﻗﺕ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﻠﺤل ﻤﻥ ﻗﺒل ﺍﻝﺤﺎﺴﺒﺔ‪.‬‬
‫ﺏ‪ .‬ﺍﻝﻀﺒﻁ ﻭﻴﻘﺼﺩ ﺒﻪ ﻗﺼﺭ ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﻤﺴﺘﺨﺩﻤﺔ ﻭﺍﻝﻭﻗﺕ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﺤل ﺍﻝﺒﺭﺍﻤﺞ ﻤﻥ ﻗﺒل ﺍﻝﺤﺎﺴﺒﺔ‬
‫]‪. [9,3‬‬

‫‪ .٣‬ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ‪:‬‬

‫ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺘﺘﻌﺎﻤل ﻤﻊ ﺍﻝﻤﻨﻅﻭﻤﺎﺕ ﺍﻝﺨﻁﻴﻪ ﻭﻏﻴﺭ ﺍﻝﺨﻁﻴﻪ‪ .‬ﻭﻫﻨﺎﻙ ﺍﺴﻠﻭﺒﺎﻥ ﻝﻤﻌﺎﻝﺠﺔ ﻤﺸﻜﻠﺔ‬
‫ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﺴﻠﻭﺏ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻷﻭل ﻴﻁﺒﻕ ﻓﻘﻁ ﻓﻲ ﺤﺎﻝﺔ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻤﺴﺘﻘﺭﺓ ﻭﺍﺴﻠﻭﺏ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﺜﺎﻨﻲ‬
‫ﻴﻁﺒﻕ ﻭﻴﺩﻋﻰ ﺒﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻭﻴﺴﺘﺨﺩﻡ ﻓﻲ ﺤﺴﺎﺒﺎﺕ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻝﻠﻤﻨﻅﻭﻤﺎﺕ ﻏﻴﺭ‬
‫ﺍﻝﺨﻁﻴﺔ‪ .‬ﻭﻨﺸﻴﺭ ﻫﻨﺎ ﺍﻝﻰ ﺍﻻﻋﺘﺒﺎﺭﺍﺕ ﺍﻝﺘﺎﻝﻴﺔ ﻋﻨﺩ ﺘﻁﺒﻴﻕ ﻁﺭﻴﻘﻪ ﻝﻴﺎﺒﻭﻨﻭﻑ ﻝﺘﻘﻴﻴﻡ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻭﻜﻤﺎ ﻴﻠﻲ‪:‬‬

‫‪.١‬ﺍﻝﻤﻭﻝﺩ ﺍﻝﺘﺯﺍﻤﻨﻲ ﻴﻤﺜل ﺒﻔﻭﻝﺘﻴﺔ ﺜﺎﺒﺘﻪ ﺨﻠﻑ ﺍﻝﻤﻔﺎﻋﻠﻪ ﺍﻝﻌﺎﺒﺭﺓ‪.‬‬

‫‪.٢‬ﺍﻝﻘﺩﺭﺓ ﺍﻝﺩﺍﺨﻠﺔ ﻝﻠﻤﻭﻝﺩ ﺜﺎﺒﺘﺔ ﺨﻼل ﺍﻝﻔﺘﺭﺓ ﺍﻝﻌﺎﺒﺭﺓ‪.‬‬
‫‪.٣‬ﻋﺯﻡ ﺍﻝﺘﺨﻤﻴﺩ ﻴﻌﺘﺒﺭ ﺼﻔﺭﹰﺍ‪.‬‬
‫‪.٤‬ﺘﻤﺜﻴل ﺍﻝﺤﻤل ﺒﻤﺴﺎﻤﺤﺔ ﺘﻭﺍﺯﻱ ﺜﺎﺒﺘﺔ‪.‬‬

‫ﻝﺘﻭﻀﻴﺢ ﺫﻝﻙ ﺴﻨﺫﻜﺭ ﺩﺍﻝﺘﻲ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﺤﺩﻫﺎ ﺘﺎﺨﺫ ﺒﻨﻅﺭ ﺍﻻﻋﺘﺒﺎﺭ ﻤﻘﺎﻭﻤﺔ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻭﺤﺴﺏ‬
‫ﺍﻝﻤﻌﺎﺩﻝﻪ)‪ (٧‬ﻭﺍﻻﺨﺭﻯ ﺘﻬﻤل ﻤﻘﺎﻭﻤﺔ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻭﺤﺴﺏ ﺍﻝﻤﻌﺎﺩﻝﺔ )‪ (٨‬ﻭﻜﻤﺎ ﻴﺎﺘﻲ ‪:‬‬
 ‫‪n‬‬
‫‪1‬‬ ‫‪n‬‬
‫~‬ ‫‪~ n −1‬‬ ‫‪n‬‬
‫∑ ‪V = ∑ M i ω~i2 + ∑ Pai (δ i s − δ i ) +‬‬ ‫∑‬
‫‪i =1 2‬‬ ‫‪i =1‬‬ ‫‪i =1‬‬ ‫‪j = i +1‬‬
‫~‬ ‫~‬
‫‪‬‬ ‫‪δ i +δ j‬‬
‫‪~ ~ ~ ‬‬
‫‪Cij (cos δ ij − cos δ ij ) − ∫ Dij cos δ ij d(δ i + δ j )‬‬
‫‪s‬‬
‫)‪...(7‬‬
‫‪‬‬ ‫~ ~‬ ‫‪‬‬
‫‪‬‬ ‫‪δ i s + δ js‬‬ ‫‪‬‬

‫‪n‬‬
‫‪1‬‬ ‫‪n‬‬
‫~‬ ‫~‬ ‫‪n −1‬‬ ‫‪n‬‬
‫~‬ ‫~‬
‫∑ ‪V = ∑ M i ω~i2 + ∑ Pai (δ i s − δ i ) +‬‬ ‫∑‬ ‫)‪C ij (cos δ ijs − cos δ ij ) ...(8‬‬
‫‪i =1 2‬‬ ‫‪i =1‬‬ ‫‪i =1‬‬ ‫‪j = i +1‬‬

‫ﻨﺤﺴﺏ ﺤﺩﻭﺩ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ‪ V1‬ﻤﻥ ﺍﻝﻤﻌﺎﺩﻝﺔ ﺭﻗﻡ )‪ (9‬ﻭﻜﻤﺎ ﻴﺎﺘﻲ ‪:‬‬

‫‪s‬‬
‫‪V1 = − Pmi (π − 2δ is ) + 2 Peimax cos δ i‬‬ ‫)‪...(9‬‬

‫ﻭﺘﺘﺤﻘﻕ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﺫﺍ ﻜﺎﻨﺕ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ )‪ (V‬ﺍﻗل ﺍﻭ ﺘﺴﺎﻭﻱ ﺤﺩﻭﺩ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ )‪(V1‬‬

‫ﻝﺘﺤﺩﻴﺩ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ﻓﻲ ﻜل ﺤﺎﻝﺔ ﻴﺘﻡ ﺍﺠﺭﺍﺀ ﺍﻝﺨﻁﻭﺍﺕ ﺍﻵﺘﻴﻪ‪:‬‬

‫‪ .١‬ﺘﻜﻭﻴﻥ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ‪. V‬‬

‫‪ .٢‬ﻨﺤﺩﺩ ﻨﻘﻁﺔ ﺘﻭﺍﺯﻥ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﻝﻤﺎ ﺒﻌﺩ ﺍﻝﻌﻁل ﻤﻥ ﺨﻼل ﺤﺴﺎﺏ ﺤﺩﻭﺩ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﻪ )‪.(Vl‬‬
‫‪ .٣‬ﻨﻜﺎﻤل ﻤﻌﺎﺩﻝﺔ ﺍﻝﻤﻨﻅﻭﻤﻪ ﺍﺜﻨﺎﺀ ﺍﻝﻌﻁل ﻭﺨﻼل ﻜل ﺘﻜﺎﻤل ﻨﺤﺴﺏ ﺩﺍل ﻝﻴﺎﺒﻭﻨﻭﻑ ‪.V‬‬
‫‪ .٤‬ﻨﻘﺎﺭﻥ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ‪ V‬ﻤﻊ ‪ Vl‬ﺤﻴﺙ ﺘﻜﻭﻥ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻤﺴﺘﻘﺭﺓ ﻋﻨﺩﻤﺎ ‪.V<V1‬‬
‫‪ .٥‬ﻋﻨﺩﻤﺎ ﺘﺘﺴﺎﻭﻯ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ‪ V‬ﻤﻊ ‪ Vl‬ﻴﺘﻡ ﺘﺤﺩﻴﺩ ﺯﻤﻥ ﺍﻻﺯﺍﻝﻪ ﺍﻝﺤﺭﺝ ‪. [10,5] CCT‬‬

‫‪ .2‬ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺴﺘﺨﺩﻤﺔ‪:‬‬

‫ﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻭﺤﺴﺏ ﺍﻝﻤﻌﺎﺩﻻﺕ )‪ (9,8,7‬ﻋﻠﻰ ﻤﻨﻅﻭﻤـﺔ ﺍﻝﻘـﺩﺭﺓ ﺍﻷﺨﺘﺒﺎﺭﻴـﺔ‬
‫ﺍﻝﻤﻭﻀﺤﺔ ﻓﻲ ﺸﻜل )‪ . (1‬ﻭﺒﻴﺎﻨﺎﺕ ﺍﻝﻤﻨﻅﻭﻤﺔ ﺍﻻﺨﺘﺒﺎﺭﻴﺔ ﻜﻤﺎ ﻤﺒﻴﻥ ﻓﻲ ﺍﻝﻤﻠﺤﻕ )‪ (1‬ﻭﺘـﻡ ﺍﺴـﺘﺨﺩﺍﻡ ﺒﺭﻨـﺎﻤﺞ‬
‫)‪ (MATLAB‬ﻋﻠﻰ ﺤﺎﺴﺒﻪ ﻨﻭﻉ )‪ .(P4‬ﻭﻴﺘﻡ ﺤﺴﺎﺏ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ‪ Tcc‬ﺒﺎﻝﺨﻁﻭﺍﺕ ﺍﻝﻤﺒﻴﻨﺔ ﻜﻤﺎ ﻴﺄﺘﻲ‪:‬‬

‫‪ .١‬ﺒﻨﺎﺀ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ‪ V‬ﻝﻠﻤﻨﻅﻭﻤﺔ ﺒﻌﺩ ﺍﺯﺍﻝﺔ ﺍﻝﻌﻁل‪.‬‬

‫‪ .٢‬ﺍﻴﺠﺎﺩ ﺍﻝﻘﻴﻤﺔ ﺍﻝﺤﺭﺠﺔ ﻝﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ‪ V‬ﻋﻨﺩ ﺤﺎﻝﺔ ﻋﻁل ﻤﻌﻴﻥ ﻨﺭﻤﺯ ﻝﻬﺎ ‪.V1‬‬
‫‪ .٣‬ﻨﻜﺎﻤل ﻤﻌﺎﺩﻻﺕ ﺍﻝﻌﻁل ﺤﺘﻰ ﺘﺼﺒﺢ ﻗﻴﻤﺔ ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ‪V‬ﺘﺴﺎﻭﻱ ﺍﻝﻘﻴﻤﺔ ﺍﻝﺤﺭﺠﺔ ﻝﺩﺍﻝـﺔ ﻝﻴـﺎﺒﻭﻨﻭﻑ‬
‫‪.V1‬‬
‫ﻋﻨﺩ ﻫﺫﻩ ﺍﻝﻠﺤﻅﺔ ﻨﺤﺼل ﻋﻠﻰ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ‪.Tcc‬‬
‫‪ .٤‬ﻨﺤﺴﺏ ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﻠﻔﺭﻕ )‪ (% difference‬ﺒﻴﻥ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ﺍﻝﻤﺤﺴﻭﺏ ﺒﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‬
‫ﺍﻝﻤﺒﺎﺸﺭﺓ ﻭﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ﺍﻝﻤﺤﺴﻭﺏ ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ – ﺨﻁﻭﺓ ﻏﻴﺭ ﺍﻝﻤﺒﺎﺸﺭﺓ ﺒﺘﻁﺒﻴﻕ ﺍﻝﻌﻼﻗـﺔ‬
‫ﺍﻝﻤﺒﻴﻨﺔ ﻓﻲ ﺍﻝﻤﻠﺤﻕ )‪.(2‬‬
‫‪ .٥‬ﻨﺤﺴﺏ ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﺘﻘﻠﻴل )‪ (% reduction‬ﺯﻤﻥ ﺍﻝﺤل ﺍﻝﻤﺴﺘﻐﺭﻕ ﺒﺎﻝﻁﺭﻴﻘﻴﺘﻴﻥ ﺒﺘﻁﺒﻴـﻕ ﺍﻝﻌﻼﻗـﺔ‬
‫ﺍﻝﻤﺒﻴﻨﺔ ﻓﻲ ﺍﻝﻤﻠﺤﻕ )‪.(2‬‬

‫‪G2‬‬ ‫‪G3‬‬
‫‪L2‬‬ ‫‪L3‬‬
‫~‬ ‫~‬
‫‪2‬‬ ‫‪3‬‬

‫‪L5‬‬ ‫‪L6‬‬ ‫‪L7‬‬

‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪9‬‬ ‫‪10‬‬

‫‪11‬‬ ‫‪5‬‬
‫~‬ ‫‪G5‬‬

‫‪12‬‬ ‫‪13‬‬ ‫‪14‬‬

‫‪4‬‬ ‫‪1‬‬

‫~‬ ‫~‬
‫‪L4‬‬ ‫‪L1‬‬
‫‪G4‬‬ ‫‪G1‬‬

‫ﺍﻝﺸﻜل )‪ :(١‬ﻤﻨﻅﻭﻤﺔ ﺍﺨﺘﺒﺎﺭﻴﺔ ﺨﻤﺴﺔ ﻤﻜﺎﺌﻥ‬

‫‪ -٣‬ﻫﻴﻜﻴﻠﻴﺔ ﺍﻝﺒﺭﻨﺎﻤﺞ‬

‫ﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ‪ MATLAB‬ﻝﻤﺤﺎﻜﺎﺓ ﺍﺴﺘﻘﺭﺍﺭﻴﺔ ﻤﻨﻅﻭﻤﺔ ﻗﺩﺭﺓ ﺍﺨﺘﺒﺎﺭﻴﺔ ﺘﺤﺘﻭﻱ ﺨﻤﺱ ﻤﻜﺎﺌﻥ ﺒﺨﻁﻭﺍﺕ‬
‫ﻤﺒﻴﻨﺔ ﻜﻤﺎ ﻴﺄﺘﻲ‪:‬‬
 ‫‪ .١‬ﺃﺩﺨﺎل ﺍﻝﺒﻴﺎﻨﺎﺕ‬
‫‪ .٢‬ﺒﻨﺎﺀ ﻤﺼﻔﻭﻓﺔ ﺍﻝﻤﺴﺎﻤﺤﺎﺕ ]‪.[Y‬‬
‫‪ .٣‬ﺤﺴﺎﺒﺎﺕ ﻗﺒل ﺍﻝﻌﻁل )‪.(Prefault‬‬
‫‪ .٤‬ﺤﺴﺎﺒﺎﺕ ﺍﺜﻨﺎﺀ ﺍﻝﻌﻁل )‪.(At fault‬‬
‫‪ .٥‬ﺤﺴﺎﺒﺎﺕ ﺒﻌﺩ ﺍﺯﺍﻝﺔ ﺍﻝﻌﻁل )‪.(post fault‬‬
‫‪ .٦‬ﺒﺩﺀ ﺍﻝﺘﻘﺎﺭﺏ )‪ (Iteration‬ﻤﻥ ﺨﻼل ﻋﺩﺍﺩ ﺍﻝﺯﻤﻥ )‪. (t‬‬
‫‪ .٧‬ﺯﻴﺎﺩﺓ ﺍﻝﺯﻤﻥ )‪ (t‬ﺒﻤﻘﺩﺍﺭ ﻤﻨﺎﺴﺏ ) ‪ ( ∆t‬ﻭﺘﺤﺩﻴﺩ ﻤﺼﻔﻭﻓﺔ ﺍﻝﻤﺴﺎﻤﺤﺎﺕ ]‪. [Y‬‬
‫‪ .٨‬ﺍﺴﺘﺨﺩﺍﻡ ﺍﻝﺘﻴﺎﺭﺍﺕ ﻭﻤﺼﻔﻭﻓﺎﺕ ﺍﻝﻤﺴﺎﻤﺤﺎﺕ ﻝﺤﺴﺎﺏ ﺍﻝﻘﺩﺭﺓ ﺍﻝﻜﻬﺭﺒﺎﺌﻴﺔ ﺍﻝﻤﺠﻬﺯﺓ ﻤﻥ ﻜل ﻤﻭﻝﺩ‪.‬‬
‫ﻭﺍﺴﺘﺨﺩﺍﻡ ﺍﻻﺨﻴﺭﺓ ﻝﺤل‬ ‫ﻝﻜل ﻤﻭﻝﺩ ﻻﻴﺠﺎﺩ ﺍﻝﻘﺩﺭﺓ ﺍﻝﻤﻌﺠﻠﺔ)‬ ‫‪ .٩‬ﺍﺴﺘﺨﺩﺍﻡ ﺍﻝﻘﺩﺭﺓ ﺍﻝﺤﻘﻴﻘﻴﻪ )‬
‫ﻝﻜل ﻤﻭﻝﺩ ﻋﻨﺩ ﻜل ﺯﻴﺎﺩﺓ ﻓﻲ‬ ‫ﻭﺍﻝﺴﺭﻋﺔ‬ ‫ﻤﻌﺎﺩﻝﺔ ﺍﻝﺘﺄﺭﺠﺢ ﻝﻜل ﻤﻭﻝﺩ ﻻﻴﺠﺎﺩ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ‬
‫‪.‬‬ ‫ﺍﻝﺯﻤﻥ‬
‫‪ .١٠‬ﺨﺯﻥ ﺍﻝﻨﺘﺎﺌﺞ ﻭﺘﺩﻗﻴﻕ ﺍﻝﺤﺎﻻﺕ )‪ (check‬ﻻﻴﻘﺎﻑ ﺍﻝﺘﻘﺎﺭﺏ )‪ (Iteration‬ﺍﺫﺍ ﻝﻡ ﻴﺘﺤﻘﻕ ﺍﻝﺘﻘﺎﺭﺏ ﺘﻌﺎﺩ‬
‫ﺍﻝﺨﻁﻭﺓ ‪ ٧‬ﻭﺍﻻ ﻴﺘﻡ ﺍﻻﻨﺘﻘﺎل ﺍﻝﻰ ﺍﻝﺨﻁﻭﺓ ‪.١١‬‬
‫‪ .١١‬ﻁﺒﻊ ﺍﻝﻨﺘﺎﺌﺞ ﻭﺍﺼﺩﺍﺭ ﺤﻜﻡ ﻋﻠﻰ ﺍﻻﺴﻘﺭﺍﺭﻴﺔ )ﻤﺴﺘﻘﺭﺓ ﺍﻡ ﻻ ( ﻤﻥ ﺨﻼل ﻤﻘﺎﺭﻨﺔ ﺩﺍﻝﺘﻲ ﻝﻴﺎﺒﻭﻨﻭﻑ‬
‫‪.V1,V‬‬
‫‪ .١٢‬ﺘﻐﻴﺭ ﺯﻤﻥ ﺍﺯﺍﻝﺔ ﺍﻝﻌﻁل )ﺯﻴﺎﺩﺓ ﺍﻭﻨﻘﺼﺎﻥ( ﻀﻤﻥ ﻓﺘﺭﺓ ﺍﻝﺤل ﻭﺍﻋﺎﺩﺓ ﺍﻝﺨﻁﻭﺍﺕ)‪(١٠-٦‬ﺤﺘﻰ ﺘﺘﻐﻴﺭ‬
‫ﺤﺎﻻﺕ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻓﻲ ﺍﻝﺨﻁﻭﺓ ‪ ١١‬ﻭﺍﻻ ﻓﺎﻥ ﺯﻤﻥ ﺍﻝﺤل ﺘﺠﺎﻭﺯ ﻓﺘﺭﺓ ﺍﻝﺤل‪.‬‬
‫‪ .١٣‬ﺘﺤﺩﻴﺩ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ‪.Tcc‬‬
‫‪ .١٤‬ﺍﻝﻨﻬﺎﻴﺔ‪.‬‬

‫و‪ YZAD‬ا=‪ WXNO‬ا‪ P@IQERS= TJIUKQV‬ا=‪;NFKO‬م =‪IJIKL‬ت ا‪E<FG‬ار‪ ?@ABC@ ?D‬ا=<;رة‪.‬‬

‫‪@A‬ا>=‬

‫‪PQ‬اءة ‪DLDMA‬ت ا‪ =EIJG‬وا‪BCDEFG‬‬

‫‪DWX‬ب ا‪ TMUG‬ا‪@RAS‬ا‪=EIJYG =MC‬‬

 ‫ﺍﻝﺸﻜل )‪ :(٢‬ﻤﺨﻁﻁ ﻜﺘﻠﻲ ﻤﺒﺴﻁ ﻴﻭﻀﺢ ﺘﻨﻅﻴﻡ ﺤﺴﺎﺒﺎﺕ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ‬

‫ﺠﺩﻭل )‪ :(١‬ﺯﻤﻥ ﺍﻹﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ‪ CCT‬ﺒﻁﺭﻴﻘﺘﻲ ﻝﻴﺎﺒﻭﻨﻭﻑ ﻭﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ‬

‫ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ‬ ‫ﺯﻤﻥ ﺍﻷﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ‪ CCT‬ﺜﺎﻨﻴﺔ‬ ‫ﺍﻝﻤﻨﻅﻭﻤﺔ ﺍﻷﺨﺘﺒﺎﺭﻴﺔ‬

‫ﻝﻠﻔﺭﻕ‬
‫ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‬ ‫ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ ‪ -‬ﺨﻁﻭﺓ‬
‫)‪(%difference‬‬
‫‪6.5‬‬ ‫‪0.58‬‬ ‫‪0.62‬‬ ‫ﻤﻨﻅﻭﻤﺔ ﺘﺤﺘﻭﻱ ﺨﻤﺴﺔ ﻤﻜﺎﺌﻥ‬
 ‫‪+‬‬
‫ﺠﺩﻭل )‪ :(٢‬ﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﻠﺤل ﺒﻁﺭﻴﻘﺘﻲ ﻝﻴﺒﻨﻭﻑ ﻭﺍﻝﺨﻁﻭﺓ‪-‬ﺨﻁﻭﺓ‬

‫ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ‬ ‫ﺜﺎﻨﻴﺔ ﺒﻭﺍﺴﻁﺔ‪ Ts‬ﺯﻤﻥ ﺍﻝﺤل‬ ‫ﺍﻝﻤﻨﻅﻭﻤﻪ ﺍﻻﺨﺘﺒﺎﺭﻴﺔ‬

‫ﻝﺘﻘﻠﻴل ﺯﻤﻥ ﺍﻝﺤل‬ ‫ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‬ ‫‪++‬‬
‫ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪-‬ﺨﻁﻭﺓ‬
‫)‪(% reduction‬‬
‫‪95.6‬‬ ‫‪0.7‬‬ ‫‪16‬‬ ‫ﻤﻨﻅﻭﻤﺔ ﺘﺤﻭﻱ ﺨﻤﺴﺔ ﻤﻜﺎﺌﻥ‬
‫‪+‬‬
‫ﺍﻝﺤﺎﺴﺒﺔ ﺍﻝﻤﺴﺘﺨﺩﻤﻪ ‪.P4‬‬
‫‪++‬‬
‫ﺯﻤﻥ ﺍﻝﺤل)‪ ،(T‬ﺜﺎﻨﻴﺔ ﻭﺍﺤﺩﺓ‪.‬‬

‫ﺠﺩﻭل )‪ :(٣‬ﻨﺴﺒﺔ ﺍﻝﺨﻁﺄ ﻓﻲ ﺤﺴﺎﺏ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ ‪ δ‬ﺤﺴﺏ ﻁﺭﻕ ﺍﻝﺤل ﺍﻝﻌﺩﺩﻴﺔ‬

‫ﻨﺴﺒﺔ ﺍﻝﺨﻁﺄ )‪ (% error‬ﻓﻲ ﺤﺴﺎﺏ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ ‪ δ‬ﺤﺴﺏ‬ ‫ﻓﺘﺭﺓ‬

‫ﺍﻝﺤل ﺒﻁﺭﻴﻘﺔ ﺭﺍﻨﺞ – ﻜﻭﺘﺎ‬ ‫ﺍﻝﺤل ﺒﻁﺭﻴﻘﺔ ﺍﻴﻭﻝﺭ ﺍﻝﻤﻌﺩﻝﺔ‬ ‫ﺍﻝﺤل ﺒﻁﺭﻴﻘﺔ ﺍﻴﻭﻝﺭ‬ ‫ﺍﻝﺤل ﺍﻝﺩﻗﻴﻕ‬ ‫ﺍﻝﺤل‬
‫‪Runge-Kutta‬‬ ‫‪modified Euler‬‬ ‫‪Euler method‬‬ ‫‪Exact sol.‬‬ ‫‪∆t‬‬
‫‪0.00012‬‬ ‫‪0.00051‬‬ ‫‪0.03600‬‬ ‫‪0.00000‬‬ ‫‪0.01‬‬

‫ﺍﻝﻤﻨﺎﻗﺸﺔ‪:‬‬

‫ﻴﺒﻴﻥ ﺍﻝﺠﺩﻭل)‪ (١‬ﻤﺎ ﻴﺎﺘﻲ‪:‬‬

‫ﺘﻘﺎﺭﺏ ﺯﻤﻥ ﺍﻻﺯﺍﻝﻪ ﺍﻝﺤﺭﺝ ‪ CCT‬ﺒﻴﻥ ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻭﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ – ﺨﻁﻭﺓ ﻏﻴﺭ‬ ‫ﺃ‪-‬‬
‫ﺍﻝﻤﺒﺎﺸﺭﺓ‪.‬‬
‫ﺏ‪ -‬ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﻠﻔﺭﻕ )‪ (% difference‬ﺒﻴﻥ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ ‪ CCT‬ﺍﻝﻤﺤﺴﻭﺏ ﺒﺎﻝﻁﺭﻴﻘﺘﻴﻥ‬
‫ﺘﺴﺎﻭﻱ ‪ ،6.5‬ﻭﻋﻠﻰ ﺫﻝﻙ ﺘﻜﻭﻥ ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﺩﻗﻴﻘﺔ ﻓﻲ ﺤﺴﺎﺏ ﺯﻤﻥ ﺍﻻﺯﺍﻝﺔ ﺍﻝﺤﺭﺝ‪،‬‬
‫ﻭﺇﻥ ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﻠﻔﺭﻕ ﻴﻤﻜﻥ ﺍﻥ ﺘﺅﺩﻱ ﺍﻝﻰ ﺘﻨﺎﻗﺹ ﺍﻀﻁﺭﺍﺏ ﺍﻝﻤﻨﻅﻭﻤﺔ ﻋﻨﺩ ﺤﺩﻭﺙ ﺍﻝﻌﻁل‬
‫ﻭﺘﻜﻭﻥ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﺍﻗل ﻗﻠﻘﹰﺎ ﻭﺫﻝﻙ ﺒﺴﺒﺏ ﺯﻴﺎﺩﺓ ﻤﺠﺎل ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻤﺘﺭﺘﺏ ﻋﻠﻰ ﺍﻝﻨﺴﺒﺔ‬
‫ﺍﻝﺌﻭﻴﺔ ﻝﻠﻔﺭﻕ )‪ (% difference‬ﻤﻤﺎ ﻴﺅﺩﻱ ﺍﻝﻰ ﺍﻥ ﻴﻜﻭﻥ ﻤﺠﺎل ﺍﻝﺘﺨﻠﺹ ﻤﻥ ﺍﻝﻁﺎﻗﺔ ﺍﻝﺤﺭﻜﻴﺔ‬
‫ﺍﻝﺯﺍﺌﺩﺓ ﻓﻲ ﺍﻝﺩﻭﺍﺭ ﺍﻜﺒﺭ‪ ،‬ﻝﺫﻝﻙ ﺘﺴﺎﻫﻡ ﻫﺫﺓ ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﻠﻔﺭﻕ ﻓﻲ ﺘﻌﺯﻴﺯ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ‬
‫ﻝﻤﻨﻅﻭﻤﺔ ﺍﻝﻘﺩﺭﺓ ﻤﺘﻌﺩﺩﺓ ﺍﻝﻤﻜﺎﺌﻥ‪.‬‬

‫ﻭﻴﺒﻴﻥ ﺍﻝﺠﺩﻭل )‪ (٢‬ﻤﺎ ﻴﺎﺘﻲ‪:‬‬

‫ﺘﺤﻘﻴﻕ ﺍﻨﺨﻔﺎﺽ ﻭﺍﻀﺢ ﻝﺯﻤﻥ ﺍﻝﺤل ﻤﻥ ﺨﻼل ﺘﻁﺒﻴﻕ ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻝﻤﺒﺎﺸﺭﺓ ﻝﺘﻘﻴﻡ‬ ‫ﺃ‪-‬‬
‫ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻝﻤﻨﻅﻭﻤﺔ ﺍﻝﻘﺩﺭﺓ‪.‬‬
‫ﺏ‪ -‬ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﺘﻘﻠﻴل )‪ (% reduction‬ﺯﻤﻥ ﺍﻝﺤل ﺤﺴﺏ ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﻤﻘﺎﺭﻨﺔ ﺒﺯﻤﻥ ﺍﻝﺤل‬
‫ﺤﺴﺏ ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ – ﺨﻁﻭﺓ ﺘﺴﺎﻭﻱ ‪ ، 95.6‬ﻭﻋﻠﻰ ﺫﻝﻙ ﺘﻜﻭﻥ ﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ ﺍﻓﻀل ﻤﻥ‬
‫ﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ – ﺨﻁﻭﺓ ﻝﺘﻘﻴﻴﻡ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻝﻤﻨﻅﻭﻤﺔ ﺍﻝﻘﺩﺭﺓ ﻷﻫﻤﻴﺔ ﺘﻭﻓﻴﺭ ﺍﻝﻭﻗﺕ ﻓﻲ‬
‫ ﻝﻜﻲ ﺘﺘﻡ ﻋﻤﻠﻴﺔ ﺍﻝﻀﺒﻁ ﻝﺘﺤﻤل ﺍﻝﻌﻁل ﻭ ﺍﻝﻌﻭﺩﺓ ﺍﻝﻰ ﺍﻝﺘﺯﺍﻤﻥ ﻤﻥ ﺨﻼل ﻭﺤﺩﺓ‬،‫ﺘﺤﺩﻴﺩ ﺍﻷﺴﺘﻘﺭﺍﺭﻴﺔ‬
. (Controller) ‫ﺍﻝﻤﺴﻴﻁﺭ‬

‫ ﺜﺎﻨﻴﺔ ﺒﺘﻁﺒﻴﻕ‬0.01 ‫∆ ﺘﺴﺎﻭﻱ‬t ‫ ﻝﻔﺘﺭﺓ ﺤل‬δ ‫( ﻨﺴﺒﺔ ﺍﻝﺨﻁﺄ ﻓﻲ ﺤﺴﺎﺏ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ‬٣) ‫ﻭﻴﺒﻴﻥ ﺍﻝﺠﺩﻭل‬
‫ ﺍﻗل ﻨﺴﺒﺔ ﺨﻁﺄ ﻭﻫﻲ ﺍﻷﻗﺭﺏ‬Runge – Kutta ‫ﻁﺭﻕ ﺍﻝﺘﻜﺎﻤل ﺍﻝﻌﺩﺩﻱ ﺤﻴﺙ ﺃﻅﻬﺭﺕ ﻁﺭﻴﻘﺔ ﺭﺍﻨﺞ – ﻜﻭﺘﺎ‬
‫ ﺍﻓﻀل ﻁﺭﻴﻘﺔ ﺘﻜﺎﻤل ﻋﺩﺩﻱ‬Runge – Kutta ‫ ﻭﻋﻠﻰ ﺫﻝﻙ ﺘﻜﻭﻥ ﻁﺭﻴﻘﺔ ﺭﺍﻨﺞ – ﻜﻭﺘﺎ‬،‫ﺍﻝﻰ ﺍﻝﺤل ﺍﻝﺩﻗﻴﻕ‬
.(٣) ‫( ﺍﻝﺘﻲ ﻭﺭﺩﺕ ﻓﻲ ﺍﻝﻔﻘﺭﺓ‬٥) ‫( ﻭ‬٤) ‫ﻝﺤل ﺍﻝﻤﻌﺎﺩﻻﺕ‬

:‫ﺍﻝﻤﺼﺎﺩﺭ‬

1. C. A. Gross, Power System Analysis, Wiley, New York, 1979.

2. O. I. Elgerd, Electric Energy Systems Theory, McGraw-Hill, New York, 1982.
3. W. D. Stevenson, Jr., Elements of Power System Analysis, McGraw-Hill, New
York, 1982.
4. K. R. McClymont et al., “Experience with High- Speed Rectifier Excitation
Systems”, IEEE Trans PAS, vol. PAS-87 , pp. 1464-1470, 1986.
5. P. Kundur, Power System Stability and Control, McGraw-Hill, New York, 1994.
6. CIGRE Task force 38.02.16, “Impact of the Interaction Among Power System
Controls, Technical Report”, 2000.
7. D. J. Glover, Power System Analysis and Design, Joh Wiley & sons, New York,
2002.
8. A. Moore and C. Atkeson, “Prioritized Sweeping: Reinforcement learning with
less Data and less Real Time”, Machine Learning vol. 13, pp. 103-103, 1998.
9. C. Druet, D. Emst and L. Wehenkel, “Application of reinforcement Learning to
electrical Power system closed-loop emergency control”, in Proceedings of
PKDD’2000, pp. 86-95, 2000.
10. M. Glavic, D. Emst, and L. Wehekel, “A Reinforcement Learning Based
Discrete Supplementary Control for Power System Transient Stability
Enhancement", Vollos, Greece, 2003.

(١)‫ﺍﻝﻤﻠﺤﻕ‬

cdIe@ fOg ?D‫ر‬IRFg‫ ا‬: ?@ABC@ ‫ت‬IQIUJ

(‫;ول )أ‬i
k<C=‫ط ا‬AXg mJ‫ا‬An
 ‫ﻤﻭﻗﻊ ﻤﺎﺨﺫ ﺍﻝﻤﺤﻭل)ﻨﺴﺒﺔ ﻤﺌﻭﻴﺔ(‬ ‫ﺍﻝﻤﻤﺎﻨﻌﺔ )ﻨﺴﺒﺔ ﺍﻝﻭﺤﺩﺓ(‬ ‫ﺍﻝﺨﻁ‬
‫ﺍﻝﻤﺤﺎﺜﺔ‬ ‫ﺍﻝﻤﻘﺎﻭﻤﺔ‬ ‫ﺍﺴﺘﻼﻡ‬ ‫ﺃﺭﺴﺎل‬
‫‪٠,٩٥‬‬ ‫‪0.132‬‬ ‫‪١٢‬‬ ‫‪١‬‬
‫‪٠,٧٨٥‬‬ ‫‪0.11‬‬ ‫‪١‬‬ ‫‪٢‬‬
‫‪١,٢٩‬‬ ‫‪٠,١٧٨‬‬ ‫‪٣‬‬ ‫‪٢‬‬
‫‪٠,٣٨‬‬ ‫‪٠,٠٥٤٥‬‬ ‫‪١٠‬‬ ‫‪٣‬‬
‫‪٠,٩١‬‬ ‫‪٠,١٢٦‬‬ ‫‪٣‬‬ ‫‪٤‬‬
‫‪١٢,٥‬‬ ‫‪٠,٧٢‬‬ ‫‪٠,٠‬‬ ‫‪٥‬‬ ‫‪٤‬‬
‫‪١,٠٥٨‬‬ ‫‪٠,٣٤٩‬‬ ‫‪٦‬‬ ‫‪٥‬‬
‫‪٥,٠‬‬ ‫‪٠,٧٥‬‬ ‫‪٠,٠‬‬ ‫‪٨‬‬ ‫‪٦‬‬
‫‪١,٠٥٨‬‬ ‫‪٠,٠‬‬ ‫‪٦‬‬ ‫‪٧‬‬
‫‪١,٠٨‬‬ ‫‪٠,٠‬‬ ‫‪٧‬‬ ‫‪٩‬‬
‫‪٦,٠‬‬ ‫‪٠,٧٢‬‬ ‫‪٠,٠‬‬ ‫‪١١‬‬ ‫‪١٠‬‬
‫‪١,٠٨‬‬ ‫‪٠,٠٩٥‬‬ ‫‪١٣‬‬ ‫‪١٢‬‬
‫‪٠,٦٨٥‬‬ ‫‪٠,٠٧٣‬‬ ‫‪٢‬‬ ‫‪١٤‬‬
‫‪٠,٤٥٥‬‬ ‫‪٠,٠‬‬ ‫‪٤‬‬ ‫‪١٤‬‬
‫‪٠,٠١٧٤‬‬ ‫‪٠,٠‬‬ ‫‪١‬‬ ‫‪١‬‬
‫‪٠,٠٤٨‬‬ ‫‪٠,٠‬‬ ‫‪٢‬‬ ‫‪٢‬‬
‫‪٠,٠٢٥٧‬‬ ‫‪٠,٠‬‬ ‫‪٣‬‬ ‫‪٣‬‬
‫‪٠,٢٧٢‬‬ ‫‪٠,٠‬‬ ‫‪٤‬‬ ‫‪٤‬‬
‫‪٠,٠٠١٤‬‬ ‫‪٠,٠‬‬ ‫‪٥‬‬ ‫‪٥‬‬
‫‪٠,٠٠٢٧‬‬ ‫‪٠,٠‬‬ ‫‪٦‬‬ ‫‪٦‬‬
‫‪٠,٠٠١٤‬‬ ‫‪٠,٠‬‬ ‫‪٧‬‬ ‫‪٧‬‬
‫‪٠,٠٠٣٨‬‬ ‫‪٠,٠‬‬ ‫‪١٠‬‬ ‫‪١٠‬‬
‫‪٠,٠٠٩٥‬‬ ‫‪٠,٠‬‬ ‫‪١٢‬‬ ‫‪١٢‬‬

‫ﺠﺩﻭل )ﺏ(‬
‫ﺜﻭﺍﺒﺕ ﺍﻝﻤﻜﺎﺌﻥ ﺍﻝﺘﺯﺍﻤﻨﻴﺔ‬
‫ﻤﺤﺎﺜﺔ ﺍﻝﻤﺤﻭﺭ‬ ‫ﺜﺎﺒﺕ ﺍﻝﻤﺎﻜﻨﺔ ‪ H‬ﻜﻴﻠﻭ ﻭﺍﻁ ﺜﺎﻨﻴﺔ‪/‬‬ ‫ﻤﻘﻨﻥ ﺍﻝﻤﻭﻝﺩ ﻤﻴﻜﺎ ﻭﺍﻁ‬ ‫ﺍﻝﻤﻭﻝﺩ ﻋﻨﺩ ﺍﻝﻌﻤﻭﻤﻲ ﺭﻗﻡ‬
‫ﺍﻝﻤﺒﺎﺸﺭ ﻨﺴﺒﺔ ﺍﻝﻭﺤﺩﺓ‬ ‫ﻜﻴﻠﻭﻓﻭﻝﺕ ﺃﻤﺒﻴﺭ‬ ‫)‪(MW‬‬
‫)‪(P.U.‬‬ ‫)‪(KW sec/KVA‬‬
‫‪٢‬‬ ‫‪٣‬‬ ‫‪١٣٥‬‬ ‫‪٨‬‬
‫‪١,٥‬‬ ‫‪٣‬‬ ‫‪١٨٠‬‬ ‫‪٩‬‬
 ‫‪٠,٣‬‬ ‫‪٣‬‬ ‫‪٩٠٠‬‬ ‫‪١١‬‬
‫‪٢,٤‬‬ ‫‪٣‬‬ ‫‪١١٢,٥‬‬ ‫‪١٣‬‬
‫‪٠,٤٨‬‬ ‫‪٣‬‬ ‫‪٧٥٠‬‬ ‫‪١٤‬‬

‫‪;i‬ول )ج(‬
‫‪Av‬ز‪ tD‬ا=‪ kOL‬وا=‪ kRs ?@ABCOS= ;U=AF‬ا=‪kXw‬‬

‫ﻗﺩﺭﺓ ﺍﻝﺤﻤل ﺍﻝﻤﺘﻔﺎﻋﻠﺔ‬ ‫ﻗﺩﺭﺓ ﺍﻝﺤﻤل ﺍﻝﺤﻘﻴﻘﻴﺔ‬ ‫ﻗﺩﺭﺓ ﺍﻝﺘﻭﻝﻴﺩ ﺍﻝﻤﺘﻔﺎﻋﻠﺔ‬ ‫ﻗﺩﺭﺓ ﺍﻝﺘﻭﻝﻴﺩ ﺍﻝﺤﻘﻴﻘﻴﺔ‬ ‫ﺭﻗﻡ ﺍﻝﻌﻤﻭﻤﻲ‬
‫)ﻤﻴﻜﺎ ﻓﺎﺭ ‪(Mvar‬‬ ‫)ﻤﻴﻜﺎ ﻭﺍﻁ ‪(MW‬‬ ‫)ﻤﻴﻜﺎ ﻓﺎﺭ ‪(Mvar‬‬ ‫)ﻤﻴﻜﺎ ﻭﺍﻁ ‪(MW‬‬
‫‪١٥,٣‬‬ ‫‪٣٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪١‬‬
‫‪٣٧,٨‬‬ ‫‪٦٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٢‬‬
‫‪٤٥‬‬ ‫‪٩٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٣‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٤‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٥‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٦‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠٠‬‬ ‫‪٧‬‬
‫‪٨٢,٥‬‬ ‫‪١٨٠‬‬ ‫‪١٧٩,٠‬‬ ‫‪٧٥‬‬ ‫‪٨‬‬
‫‪١٥‬‬ ‫‪٦٠‬‬ ‫‪٦,٣‬‬ ‫‪١٠٥‬‬ ‫‪٩‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪١٠‬‬
‫‪٤٥٠‬‬ ‫‪٧٥٠‬‬ ‫‪٥٣٤‬‬ ‫‪٦٠٠‬‬ ‫‪١١‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪١٢‬‬
‫‪٦٣‬‬ ‫‪١٢٠‬‬ ‫‪٨١‬‬ ‫‪٥٢,٥‬‬ ‫‪١٣‬‬
‫‪٠,٠‬‬ ‫‪٠,٠‬‬ ‫‪٥٥٠‬‬ ‫‪٤٧٨‬‬ ‫‪١٤‬‬

‫ﺍﻝﻤﻠﺤﻕ )‪(٢‬‬

‫) ‪ CCT‬ﺍﻝﻤﺤﺴﻭﺏ ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ(– ) ‪ CCT‬ﺍﻝﻤﺤﺴﻭﺏ ﺒﻁﺭﻴﻘﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ(‬

‫ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﻠﻔﺭﻕ ﺒﻴﻥ ‪= (%difference) CCT‬‬
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
 ‫ﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﻠﺤل ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ‬

‫)ﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﻠﺤل ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ(–) ﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﻠﺤل ﺒﻁﺭﻴﻘﺔ‬
‫ﻝﻴﺎﺒﻭﻨﻭﻑ(‬
‫ﺍﻝﻨﺴﺒﺔ ﺍﻝﻤﺌﻭﻴﺔ ﻝﺘﻘﻠﻴل ﺯﻤﻥ ﺍﻝﺤل)‪= (% reduction‬‬
‫ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ‬
‫ﺍﻝﺯﻤﻥ ﺍﻝﻤﺴﺘﻐﺭﻕ ﻝﻠﺤل ﺒﻁﺭﻴﻘﺔ ﺍﻝﺨﻁﻭﺓ‪ -‬ﺨﻁﻭﺓ‬

‫ﻗﺎﺌﻤﺔ ﺍﻝﺭﻤﻭﺯ‪:‬‬

‫‪ = ∆δ n‬ﺍﻝﺘﻐﻴﺭ ﻓﻲ ﺍﻝﺯﺍﻭﻴﺔ ﺨﻼل ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ‪. ∆t‬‬

‫‪ = t‬ﺍﻝﺯﻤﻥ )ﺒﺎﻝﺜﺎﻨﻴﺔ(‪.‬‬
‫‪ = H‬ﺜﺎﺒﺕ ﺍﻝﻤﺎﻜﻨﺔ‪.‬‬
‫‪ = ω sm‬ﺍﻝﺘﺭﺩﺩ ﺍﻝﺯﺍﻭﻱ ﻝﻠﻤﺎﻜﻨﺔ ﺒﺎﻝﺩﺭﺠﺎﺕ ﻨﺼﻑ ﺍﻝﻘﻁﺭﻴﺔ ‪.‬‬
‫‪ = Pm‬ﺍﻝﻘﺩﺭﺓ ﺍﻝﺩﺍﺨﻠﺔ ﺇﻝﻰ ﻋﻤﻭﺩ ﺍﻹﺩﺍﺭﺓ‪.‬‬
‫‪ = Pe‬ﺍﻝﻘﺩﺭﺓ ﺍﻝﻜﻬﺭﺒﺎﺌﻴﺔ ﺍﻝﻌﺎﺒﺭﺓ ﻝﻠﻔﺠﻭﺓ ﺍﻝﻬﻭﺍﺌﻴﺔ ﻓﻲ ﺍﻝﻤﺎﻜﻨﺔ‪.‬‬
‫‪ = Pa‬ﺍﻝﻘﺩﺭﺓ ﺍﻝﺘﻌﺠﻴﻠﻴﺔ ﺍﻝﻨﺎﺠﻤﺔ ﻋﻥ ﻓﻘﺩﺍﻥ ﺍﻝﺘﻭﺍﺯﻥ ﺒﻴﻥ ‪ Pm‬ﻭ ‪. Pe‬‬
‫‪ = ω s‬ﺍﻝﺴﺭﻋﺔ ﺍﻝﺘﺯﺍﻤﻨﻴﺔ ﻝﻠﻤﺎﻜﻨﺔ ﺒﺎﻝﺩﺭﺠﺎﺕ ﻨﺼﻑ ﺍﻝﻘﻁﺭﻴﻪ‪.‬‬
‫‪ = f‬ﺍﻝﺘﺭﺩﺩ ﺫﺒﺫﺒﺔ‪ /‬ﺜﺎﻨﻴﺔ‪.‬‬
‫‪ = x‬ﻤﻔﺎﻋﻠﺔ ﺍﻻﻨﺘﻘﺎل‪.‬‬
‫‪ = ∆t‬ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﻤﺘﻌﺎﻗﺒﺔ‪.‬‬
‫‪ = ∆δ n−1‬ﺍﻝﺘﻐﻴﺭ ﻓﻲ ﺍﻝﺯﺍﻭﻴﺔ ﺨﻼل ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﺴﺎﺒﻘﺔ‪.‬‬
‫)‪ = Pa ( n−1‬ﺍﻝﻘﺩﺭﺓ ﺍﻝﺘﻌﺠﻴﻠﻴﺔ ﺨﻼل ﺍﻝﻔﺘﺭﺓ ﺍﻝﺯﻤﻨﻴﺔ ﺍﻝﺴﺎﺒﻘﺔ‪.‬‬
‫‪ = ω‬ﺴﺭﻋﺔ ﺍﻝﺩﻭﺍﺭ )ﻨﺼﻑ ﻗﻁﺭﻴﺔ – ﺜﺎﻨﻴﺔ(‪.‬‬
‫‪ = π‬ﺍﻝﻨﺴﺒﺔ ﺍﻝﺜﺎﺒﺘﺔ‪.‬‬
‫‪ = V‬ﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‪.‬‬
‫‪ = V1‬ﺤﺩﻭﺩ ﺍﻻﺴﺘﻘﺭﺍﺭﻴﺔ ﺃﻭ ﺍﻝﻘﻴﻤﺔ ﺍﻝﺤﺭﺠﺔ ﻝﺩﺍﻝﺔ ﻝﻴﺎﺒﻭﻨﻭﻑ‪.‬‬
‫‪ = Pmax‬ﺍﻝﻁﺎﻗﺔ ﺍﻝﻘﺼﻭﻯ ﺍﻝﻤﻨﻘﻭﻝﺔ‪.‬‬
‫~‬
‫‪ = δ i‬ﺍﻝﻔﺭﻕ ﺒﻴﻥ ﺯﺍﻭﻴﺔ ﺍﻝﺩﻭﺍﺭ ﻝﻠﻤﺎﻜﻨﺔ ‪ i‬ﻭﺍﻝﺯﺍﻭﻴﺔ ﺍﻻﺒﺘﺩﺍﺌﻴﺔ ﻝﻠﺩﻭﺍﺭ ) ‪. (δ i − δ o‬‬
‫‪ = ω~i‬ﺍﻝﻔﺭﻕ ﺒﻴﻥ ﺴﺭﻋﺔ ﺍﻝﺩﻭﺍﺭ ﻝﻠﻤﺎﻜﻨﺔ ‪ i‬ﻭﺍﻝﺴﺭﻋﺔ ﺍﻝﺘﺯﺍﻤﻨﻴﻪ ) ‪. (ωi − ωo‬‬