ارائه يک روش خطي جديد جهت تخمين حالت سيستم هاي توزيع

‫ﺍﺭﺍﺋﻪ ﻳﮏ ﺭﻭﺵ ﺧﻄﻲ ﺟﺪﻳﺪ ﺟﻬﺖ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺳﻴﺴﺘﻢﻫﺎﻱ ﺗﻮﺯﻳﻊ‬ ‫ﺻﺎﺩﻕ ﺟﻤﺎﻟﻲ‬

‫ﺩﺍﻧﺸﮕﺎﻩ ﻋﻠﻢ ﻭ ﺻﻨﻌﺖ ﺍﻳﺮﺍﻥ‬

‫ﻋﻠﻴﺮﺿﺎ ﺟﻠﻴﻠﻴﺎﻥ‬

‫ﺩﺍﻧﺸﮕﺎﻩ ﻋﻠﻢ ﻭ ﺻﻨﻌﺖ ﺍﻳﺮﺍﻥ‬

‫ﭼﮑﻴﺪﻩ‪ :‬ﺑﺎﺗﻮﺟﻪ ﺑﻪ ﺗﻮﺳﻌﻪ ﺭﻭﺯ ﺍﻓﺰﻭﻥ ﺍﺗﻮﻣﺎﺳـﻴﻮﻥ ﺳﻴـﺴﺘﻢﻫـﺎﻱ‬ ‫ﺗﻮﺯﻳﻊ‪ ،‬ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﺑـﻪ ﻋﻨـﻮﺍﻥ ﻳﮑـﻲ ﺍﺯ‬

‫ﻗﺴﻤﺖﻫﺎﻱ ﺍﺻﻠﻲ ﺍﺗﻮﻣﺎﺳﻴﻮﻥ ﺳﻴﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﻣـﻮﺭﺩ ﺗﻮﺟـﻪ‬

‫ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳـﻦ ﻣﻘﺎﻟـﻪ ﺭﻭﺵ ﺟﺪﻳـﺪﻱ ﺑـﺮﺍﻱ ﺗﺨﻤـﻴﻦ‬ ‫ﺣﺎﻟﺖ ﺳﻴﺴﺘﻢﻫﺎﻱ ﺗﻮﺯﻳﻊ ﺍﺭﺍﺋـﻪ ﻣـﻲﺷـﻮﺩ‪ .‬ﺍﻳـﻦ ﺭﻭﺵ ﺑﺮﺍﺳـﺎﺱ‬ ‫ﺭﻭﺵ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﺑﻨﺎ ﻧﻬﺎﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺭﻭﺵ ﭘﻴﺸﻨﻬﺎﺩ ﺷـﺪﻩ‬

‫ﻣﺸﮑﻼﺕ ﺭﻭﺵ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﺩﺭ ﻣﻮﺭﺩ ﺷـﺒﮑﻪﻫـﺎﻱ ﺩﺍﺭﺍﻱ ﻣـﺶ‬

‫ﺑﺮ ﻃﺮﻑ ﺷﺪﻩ ﺍﺳﺖ ﻭ ﻫﻤﭽﻨﻴﻦ ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﻣﻘـﺎﺩﻳﺮ ﺍﻧـﺪﺍﺯﻩﮔﻴـﺮﻱ‬ ‫ﺷﺪﻩ ﻭﻟﺘﺎﮊ ﻧﻴﺰﺩﺭﻣﻮﺭﺩ ﺷﺒﮑﻪﻫﺎﻱ ﺩﺍﺭﺍﻱ ﻣﺶ ﺍﺳـﺘﻔﺎﺩﻩ ﮐـﺮﺩ‪ .‬ﻳـﮏ‬

‫ﺭﻭﺵ ﺟﺪﻳﺪ ﻧﻴﺰ ﺑﺮﺍﻱ ﻗﻄﺮﻱ ﮐﺮﺩﻥ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﺑـﻪ ﻣﻨﻈـﻮﺭ‬ ‫ﮐﺎﻫﺶ ﺣﺠﻢ ﻣﺤﺎﺳﺒﺎﺕ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﺭﻭﺵ ﺷﺎﺧﻪﻫـﺎ‬

‫ﻭ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺑﻪ ﮔﻮﻧﻪﺍﻱ ﻧﺎﻣﮕـﺬﺍﺭﻱ ﻣـﻲﺷـﻮﻧﺪ ﺗـﺎ‬

‫ﺣﺘﻲ ﺍﻻﻣﮑﺎﻥ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻗﻄﺮﻱﺗﺮ ﺷﻮﺩ‪.‬‬

‫ﻭﺍﮊﻩﻫﺎﻱ ﮐﻠﻴـﺪﻱ ‪ :‬ﺗﺨﻤـﻴﻦ ﺣﺎﻟـﺖ‪ ،‬ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ‪،‬‬ ‫ﺍﺗﻮﻣﺎﺳﻴﻮﻥ‪ ،‬ﭘﺨﺶ ﺑﺎﺭ‬

‫ﻋﺒﺎﺱ ﺩﻫﻘﺎﻧﻲ‬

‫ﺩﺍﻧﺸﮕﺎﻩ ﻋﻠﻢ ﻭ ﺻﻨﻌﺖ ﺍﻳﺮﺍﻥ‬

‫ﺑﺎ ﮐﻤﮏ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺳﻴﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ‪ ،‬ﺍﭘﺮﺍﺗـﻮﺭ ﻣـﻲﺗﻮﺍﻧـﺪ‬

‫ﻣﺤﺎﺳﺒﺎﺕ ﺗﻠﻔﺎﺕ ﻭ ﺑﻬﻴﻨﻪ ﺳﺎﺯﻱ ﻭﻟﺘـﺎﮊ ﻭ ﺗـﻮﺍﻥ ﺭﺍﮐﺘﻴـﻮ ﺭﺍ ﺍﻧﺠـﺎﻡ‬ ‫‪٣‬‬

‫ﺩﻫﺪ‪ .‬ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺭﺍ ﻫﻤﭽﻨﻴﻦ ﻣﻲﺗﻮﺍﻥ ﺩﺭ ﺗﺠﺪﻳﺪ ﭘﻴﮑﺮﺑﻨﺪﻱ ﻭ‬ ‫ﺗﺸﺨﻴﺺ ﺍﺿﺎﻓﻪ ﺑﺎﺭ ﺑﻪ ﮐﺎﺭ ﮔﺮﻓﺖ‪ .‬ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺗﺨﻤـﻴﻦ ﺣﺎﻟـﺖ‬

‫ﻣﻲ ﺗﻮﺍﻥ ﻗﺎﺑﻠﻴﺖ ﭘﺎﻳﺶ ﺳﻴﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﺭﺍ ﺍﻓـﺰﺍﻳﺶ ﺩﺍﺩ ﻭ ﺩﺭ‬

‫ﻧﺘﻴﺠﻪ ﮐﻴﻔﻴﺖ ﺗـﻮﺍﻥ ﻭ ﻗﺎﺑﻠﻴـﺖ ﺍﻃﻤﻴﻨـﺎﻥ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﺭﺍ‬

‫ﺍﻓﺰﺍﻳﺶ ﺩﺍﺩ‪ .‬ﺗﺨﻤﻴﻦ ﺣﺎﻟـﺖ ﺍﺯ ‪ ٣٠‬ﺳـﺎﻝ ﭘـﻴﺶ ﺩﺭ ﺳﻴـﺴﺘﻢﻫـﺎﻱ‬ ‫ﻗﺪﺭﺕ ﺑﻪ ﮐﺎﺭ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﻣﻬﻤﺘـﺮﻳﻦ ﺭﻭﺵ ﻣـﻮﺭﺩ ﺍﺳـﺘﻔﺎﺩﻩ‬

‫ﻗــﺮﺍﺭ ﮔﺮﻓﺘ ـﻪ ﺩﺭ ﺗﺨﻤــﻴﻦ ﺣﺎﻟــﺖ‪ ،‬ﺭﻭﺵ ﺣــﺪﺍﻗﻞ ﻣﺮﺑﻌــﺎﺕ ﻭﺯﻥ‬ ‫ﺩﺍﺭ‪ (WLS) ٤‬ﺍﺳﺖ‪ .‬ﻣﻘﺪﺍﺭ ﺑﺎﻻﻱ ﻧﺴﺒﺖ‪ r‬ﺑﻪ ‪ ،x‬ﺗﻌﺪﺍﺩ ﮐـﻢ ﺣﻠﻘـﻪ‬ ‫ﻫﺎﻱ ﺷﺒﮑﻪ‪ ،‬ﭘﺨﺶ ﺷﺪﻥ ﺑﺎﺭﻫﺎ ﺩﺭ ﻃﻮﻝ ﻓﻴﺪﺭ‪ ،‬ﮐﻢ ﺑﻮﺩﻥ ﺗﺠﻬﻴﺰﺍﺕ‬

‫ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ‪ ،‬ﺍﺯ ﺧﺼﻮﺻﻴﺎﺕ ﺳﻴﺴﺘﻢﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺴﺘﻨﺪ ﮐـﻪ ﺑﺎﻳـﺪ‬ ‫ﺩﺭ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﺩﺭ ﻧﻈـﺮ ﮔﺮﻓﺘـﻪ ﺷـﻮﻧﺪ‪.‬‬

‫ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﺘﻐﻴﻴﺮ ﺣﺎﻟـﺖ ﺩﺭ ﺣـﻞ ﻣـﺴﺄﻟﻪ‬ ‫ﺗﺨﻤــﻴﻦ ﺣﺎﻟــﺖ ﺍﺯ ﻣﺘــﺪﺍﻭﻟﺘﺮﻳﻦ ﺭﻭﺷﻬﺎﺳــﺖ ]‪ .[۴،۳،۲،۱‬ﺩﺭ ﻧﻈــﺮ‬

‫ﮔﺮﻓﺘﻦ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪﻫﺎ ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﺘﻐﻴﺮﻫﺎﻱ ﺣﺎﻟﺖ ﺩﺭ ﺳﻴﺴﺘﻢﻫﺎﻱ‬

‫ﺗﻮﺯﻳﻊ ﺩﺍﺭﺍﻱ ﻣﺰﺍﻳﺎﻱ ﻓﺮﺍﻭﺍﻧﻲ ﻣﻲﺑﺎﺷﺪ؛ ﺍﺯ ﺟﻤﻠﻪ ﻣﻲﺗـﻮﺍﻥ ﻣـﺴﺄﻟﻪ‬

‫‪ -۱‬ﻣﻘﺪﻣﻪ‬

‫ﺩﺭ ﺍﺗﻮﻣﺎﺳﻴﻮﻥ ﻭ ﻣﺪﻳﺮﻳﺖ ﻣﺪﺭﻥ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ‪(DMS)١‬‬

‫ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺭﺍ ﺑﺮﺍﻱ ﻫﺮ ﻓﺎﺯ ﺟﺪﺍﮔﺎﻧﻪ ﺑﮑﺎﺭ ﺑﺮﺩ ﮐﻪ ﺑﺎ ﺗﻮﺟـﻪ ﺑـﻪ‬

‫ﻧﺎﻣﺘﻌﺎﺩﻝ ﺑـﻮﺩﻥ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﺑـﺴﻴﺎﺭ ﻣﻄﻠـﻮﺏ ﺍﺳـﺖ‪ .‬ﻭ‬

‫ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﻧﻘﺶ ﻣﻬﻤـﻲ ﺭﺍ ﺑـﺎﺯﻱ ﻣـﻲﮐﻨـﺪ‪ .‬ﺗﺨﻤـﻴﻦ ﺣﺎﻟـﺖ‬

‫ﻫﻤﭽﻨﻴﻦ ﺑﻪ ﻋﻠﺖ ﺷﻌﺎﻋﻲ ﺑﻮﺩﻥ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﺑـﺎ ﺩﺍﺷـﺘﻦ‬

‫ﻣﺤﺪﻭﺩﻱ ﺍﺯ ﺗﺠﻬﻴﺰﺍﺕ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﻭ ﺍﻃﻼﻋﺎﺕ ﺑﺪﺳﺖ ﺁﻣﺪﻩ ﺍﺯ‬

‫ﺍﺯﻧﻈﺮﮐﺎﺭﺑﺮﺩﻫﺎﻱ ﻋﻤﻠﻲ‪ ،‬ﺗﺨﻤﻴﻦ ﺣﺎﻟـﺖ ﺑﺎﻳـﺪ ﺑﺘﻮﺍﻧـﺪ ﺍﺯ ﻣﻘـﺎﺩﻳﺮ‬

‫ﻭﺿﻌﻴﺖ ﺳﻴﺴﺘﻢ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺯﻣﺎﻥ ﺣﻘﻴﻘﻲ‪٢‬ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺗﻌﺪﺍﺩ‬

‫ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﻫﺎ‪ ،‬ﺗﻤﺎﻣﻲ ﭘﺎﺭﺍﻣﺘﺮﻫﺎﻱ ﺳﻴﺴﺘﻢ ﻗﺎﺑﻞ ﺗﺤﺼﻴﻞ ﺍﺳـﺖ‪.‬‬

‫ﭘﻴﺶﺑﻴﻨﻲ ﺑﺎﺭ ﺗﺨﻤﻴﻦ ﻣﻲﺯﻧﺪ‪.‬‬

‫ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ‪ (MV) ٥‬ﻭﻟﺘﺎﮊ‪ ،‬ﺗﻮﺍﻥ ﻭ ﺟﺮﻳﺎﻥ ﺍﺳﺘﻔﺎﺩﻩ ﻧﻤﺎﻳﺪ‪ .‬ﺩﺭ‬

‫‪١-Distribution Management System‬‬ ‫‪٢- Real Time‬‬

‫‪٣-Reconfiguration‬‬ ‫‪۴-Wighted Least Square‬‬ ‫‪۵-Measurement Value‬‬

‫ﺭﻭﺵ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﻋﻼﻭﻩ ﺑﺮ ﻣﻘﺎﺩﻳﺮ ﺍﻧـﺪﺍﺯﻩﮔﻴـﺮﻱ ﺷـﺪﺓ ﺟﺮﻳـﺎﻥ ﻭ‬ ‫ﺗﻮﺍﻥ‪ ،‬ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﺓ ﻭﻟﺘﺎﮊ ﺭﺍ ﻧﻴﺰ ﻣـﻲﺗـﻮﺍﻥ ﺩﺭ ﺗﺨﻤـﻴﻦ‬ ‫ﺣﺎﻟﺖ ﺑﮑﺎﺭﮔﺮﻓﺖ‪ .‬ﺩﺭﻣﺮﺟﻊ ]‪ [۳‬ﺭﻭﺷـﻲ ﺑـﺮﺍﻱ ﺍﺳـﺘﻔﺎﺩﻩ ﺍﺯ ‪MV‬‬

‫ﻭﻟﺘﺎﮊ ﺩﺭ ﻣﻮﺭﺩ ﺷﺒﮑﻪﻫﺎﻱ ﺷﻌﺎﻋﻲ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟـﻪ‬

‫ﺑـــﺮﺍﻱ ﺑﺪﺳـــﺖ‬

‫ﺁﻭﺭﺩﻥ ‪∆B‬‬

‫ﻻﺯﻡ ﺍﺳـــﺖ ﺩﺭ ﻫـــﺮ ﺑـــﺎﺭ ﺗﮑـــﺮﺍﺭ‬

‫‪ G = H WH‬ﻣﻌﮑﻮﺱ ﺷﻮﺩ‪ G .‬ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻧﺎﻡ ﺩﺍﺭﺩ ﮐﻪ ﺍﮐﺜـﺮ‬ ‫ﺣﺠﻢ ﻣﺤﺎﺳﺒﺎﺕ‪ ،‬ﻣﺮﺑﻮﻁ ﺑﻪ ﻣﻌﮑﻮﺱ ﮐﺮﺩﻥ ﺍﻳﻦ ﻣﺎﺗﺮﻳﺲ ﻣﻲﺑﺎﺷﺪ‬ ‫‪t‬‬

‫ﻭﺗﻤﺎﻣﻲ ﺭﻭﺷﻬﺎ ﺳﻌﻲ ﺩﺭ ﮐﻢ ﮐﺮﺩﻥ ﺯﻣﺎﻥ ﻣﺤﺎﺳﺒﻪ ﻣﻌﮑـﻮﺱ ﺍﻳـﻦ‬

‫ﺭﻭﺷﻲ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﮐﻪ ﺑﺘﻮﺍﻥ ‪ MV‬ﻭﻟﺘـﺎﮊ ﺭﺍ ﺩﺭ ﻣـﻮﺭﺩ ﺷـﺒﮑﻪﻫـﺎﻱ‬

‫ﻣﺎﺗﺮﻳﺲ ﺭﺍ ﺩﺍﺭﻧﺪ‪.‬‬

‫ﻫﺮ ﻣﺮﺣﻠﻪ ﺗﮑﺮﺍﺭ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ‪ ،‬ﮐﻪ ﺩﺭ ﺍﺩﺍﻣـﻪ ﺗﻌﺮﻳـﻒ ﻣـﻲﺷـﻮﺩ‪،‬‬

‫ﻣﻲﺗﻮﺍﻥ ﻭﻟﺘﺎﮊ ﻭ ﺗﻮﺍﻥ ﮔﺮﻩﻫﺎ ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﺩ‪.‬‬

‫ﺩﺍﺭﺍﻱ ﻣﺶ ﻫﻢ ﺑﻪ ﮐﺎﺭﺑﺮﺩ‪ .‬ﺩﺭ ﺭﻭﺵ ﺣﺪﺍﻗﻞ ﻣﺮﺑﻌﺎﺕ ﻭﺯﻥ ﺩﺍﺭ ﺩﺭ‬

‫ﻣﻌﮑﻮﺱ ﻣﻲﺷﻮﺩ ﮐﻪ ﺍﻏﻠﺐ ﺯﻣﺎﻥ ﻣﺤﺎﺳﺒﺎﺗﻲ ﺭﺍ ﺑﻪ ﺧﻮﺩ ﺍﺧﺘﺼﺎﺹ‬

‫ﻣﻲﺩﻫﺪ‪ .‬ﻫﺮ ﭼﻪ ﻗﺪﺭ ﻣﺎﺗﺮﻳﺲ ﺑﻬـﺮﻩ ﻗﻄـﺮﻱﺗـﺮ ﺑﺎﺷـﺪ ﺯﻣـﺎﻥ ﻻﺯﻡ‬

‫ﺑﺮﺍﻱ ﻣﻌﮑﻮﺱ ﺷﺪﻥ ﻣﺎﺗﺮﻳﺲ ﻧﻴﺰ ﮐﻤﺘـﺮ ﻣـﻲﺷـﻮﺩ‪ .‬ﺗﺮﺗﻴـﺐ ﻗـﺮﺍﺭ‬ ‫ﮔﺮﻓﺘﻦ ﻋﻨﺎﺻﺮ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﺑﻪ ﻧﺎﻣﮕﺬﺍﺭﻱ ﺷﺎﺧﻪﻫﺎ ﻭ ﻣﻘـﺎﺩﻳﺮ‬ ‫ﺍﻧـﺪﺍﺯﻩﮔﻴـﺮﻱ ﺷــﺪﻩ ﺑـﺴﺘﮕﻲ ﺩﺍﺭﺩ‪ .‬ﺩﺭ ﺍﻳـﻦ ﻣﻘﺎﻟــﻪ ﺭﻭﺷـﻲ ﺑــﺮﺍﻱ‬

‫ﻧﺎﻣﮕﺬﺍﺭﻱ ﺷﺎﺧﻪ ﻫﺎ ﻭ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴـﺮﻱ ﺷـﺪﻩ ﺍﺭﺍﺋـﻪ ﺷـﺪﻩ ﺗـﺎ‬ ‫ﻣﺎﺗﺮﻳﺲ ﺗﺎ ﺟﺎﻱ ﻣﻤﮑﻦ ﺑﻪ ﻓﺮﻡ ﻗﻄﺮﻱ ﺗﺒﺪﻳﻞ ﺷﻮﺩ‪.‬‬

‫‪ -۲‬ﺭﻭﺵ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺟﺮﻳﺎﻥ‬

‫)‪(1‬‬

‫ﺷﺎﺧﻪ]‪[۱،۲‬‬

‫‪mea‬‬

‫ﮐﻪ ﺩﺭ ﺭﺍﺑﻄﻪ ﻓﻮﻕ‪:‬‬

‫‪ : Vr‬ﻭﻟﺘﺎﮊ ﮔﺮﻩ ﺑﺎﻻﺗﺮ‬

‫‪ : Vs‬ﻭﻟﺘﺎﮊ ﮔﺮﻩ ﭘﺎﻳﻴﻦﺗﺮ‬

‫‪ Bl‬ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﻭ ‪ Zl‬ﺍﻣﭙﺪﺍﻧﺲ ﺷﺎﺧﻪ‪ l‬ﻣﻲ ﺑﺎﺷﻨﺪ‬

‫‪-۳‬ﻭﻳﮋﮔﻲﻫﺎﻱ ﻣﻄﻠﻮﺏ ﻳﮏ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ‬ ‫ﺗﺎﻣﻴﻦ ﮔﺮﺩﻧﺪ‪:‬‬

‫‪ -١‬ﺗﻮﺍﻧﺎﻳﻲ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺗﻤﺎﻣﻲ ﻣﻘﺎﺩﻳﺮ ﺍﻧـﺪﺍﺯﻩﮔﻴـﺮﻱ ﺷـﺪﻩ ﺍﻋـﻢ ﺍﺯ‬ ‫ﺗﻮﺍﻥ‪ ،‬ﺟﺮﻳﺎﻥ ﻭ ﻭﻟﺘﺎﮊ ﺭﺍ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪ .‬ﺩﺭ ﻋﻴﻦ ﺣﺎﻝ ﻣﺎﺗﺮﻳﺲ ﺑﻬـﺮﻩ‬

‫ﺧﻄﻲ ﻭ ﺛﺎﺑﺖ ﺑﺎﺷﺪ‪.‬‬

‫‪ -٢‬ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻃﻮﺭﻱ ﺗﺸﮑﻴﻞ ﺷﻮﺩ ﮐﻪ ﺍﺣﺘﻴﺎﺝ ﺑﻪ ﺣﺪﺍﻗﻞ ﺯﻣﺎﻥ‬

‫‪ :B‬ﻣﺘﻐﻴﻴﺮﻫﺎﻱ ﺣﺎﻟﺖ ﻳﻌﻨﻲ ﺟﺮﻳﺎﻥﻫﺎﻱ ﺷﺎﺧﻪ‬

‫ﺑﺮﺍﻱ ﻣﻌﮑﻮﺱ ﺷﺪﻥ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪.‬‬

‫‪ : Zimea‬ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﺷﺪﻩ )‪(MV‬‬

‫‪ -٣‬ﻗﺎﺑﻞ ﺗﺠﺰﻳﻪ ﺑـﻪ ﻗـﺴﻤﺖﻫـﺎﻱ ﺣﻘﻴﻘـﻲ ﻭ ﻣﻮﻫـﻮﻣﻲ ﺑﺎﺷـﺪ ﻭ‬

‫‪ : h‬ﻣﺎﺗﺮﻳﺲ ﺗﺎﺑﻊ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ‬

‫ﻫﻤﭽﻨﻴﻦ ﻣﺴﺄﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺭﺍ ﺑﺘﻮﺍﻥ ﺟﺪﺍﮔﺎﻧـﻪ ﺑـﺮﺍﻱ ﻫـﺮ ﻓـﺎﺯ‬

‫‪ : w‬ﻣﺎﺗﺮﻳﺲ ﻭﺯﻥ ﺩﻫﻲ‬

‫ﺣﻞ ﻧﻤﻮﺩ )ﮐﺎﻫﺶ ﺍﺑﻌﺎﺩ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ(‪.‬‬

‫ﻣﻲ ﺑﺎﺷﻨﺪ‪.‬‬

‫ﻣﺎﺗﺮﻳﺲ ﺗﺎﺑﻊ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ )‪ (h‬ﻣﻘـﺎﺩﻳﺮ ﺍﻧـﺪﺍﺯﻩﮔﻴـﺮﻱ ﺷـﺪﻩ ﺭﺍ ﺑـﻪ‬

‫ﻣﺘﻐﻴﻴﺮﻫﺎﻱ ﺣﺎﻟﺖ ﻣﺮﺗﺒﻂ ﻣﻲ ﮐﻨﺪ‪ .‬ﺑـﺮﺍﻱ ﺣـﻞ ﻣـﺴﺄﻟﻪ ﺑـﺎﻻ ﺭﻭﺵ‬ ‫ﺗﮑﺮﺍﺭ ﻧﻴﻮﺗﻮﻥ ﺑﮑﺎﺭ ﮔﺮﻓﺘﻪ ﻣﻲﺷﻮﺩ ﻭ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬ ‫ﮐﻪ ﺩﺭ ﺭﺍﺑﻄﻪ ﻓﻮﻕ ﺩﺍﺭﻳﻢ‪:‬‬

‫ﮐﻪ ﺩﺭ ﺭﺍﺑﻄﻪ ﻓﻮﻕ‪:‬‬

‫ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺍﺭﺍﺋﻪ ﻣﻲﮔﺮﺩﺩ‪ .‬ﺑﺮﺍﻱ ﺍﻳﻦ ﻣﻨﻈﻮﺭ ﺍﻫﺪﺍﻑ ﺯﻳـﺮ ﺑﺎﻳـﺪ‬

‫‪J ( x) = W ( Z i − h( B))^ 2‬‬

‫)‪(2‬‬

‫)‪(5‬‬

‫‪Vr = Vs + Bl Zl‬‬

‫ﺩﺭﺍﻳﻦ ﻗﺴﻤﺖ ﺑﻪ ﺻﻮﺭﺕ ﻓﻬﺮﺳﺖ ﻭﺍﺭ ﻭﻳﮋﮔﻲﻫﺎﻱ ﻣﻄﻠﻮﺏ ﻳـﮏ‬

‫ﺗﺎﺑﻊ ﻫﺪﻑ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﺑـﺮ ﺍﺳـﺎﺱ ﺭﻭﺵ ‪WLS‬‬

‫ﺑﺮﺍﺑﺮ ﺯﻳﺮ ﺍﺳﺖ‪:‬‬

‫ﺑﻌﺪ ﺍﺯ ﺑﺪﺳﺖ ﺁﻣﺪﻥ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪﻫﺎ ﺑﺎ ﺍﺳـﺘﻔﺎﺩﻩ ﺍﺯ ﺭﻭﺵ ﭘﻴـﺸﺮﻭ‬

‫‪H H∆B = H tW∆I‬‬ ‫‪t‬‬

‫‪ -٤‬ﺑﺘﻮﺍﻥ ﺗﻮﺳﻂ ﺁﻥ ﻭﺿﻌﻴﺖ ﺷﺒﮑﻪﻫﺎﻱ ﺩﺍﺭﺍﻱ ﻣـﺶ ﺿـﻌﻴﻒ ﺭﺍ‬ ‫ﺗﺨﻤﻴﻦ ﺯﺩ ﺩﺭ ﻋﻴﻦ ﺣﺎﻝ ﺳﻪ ﻭﻳﮋﮔﻲ ﺑﺎﻻ ﺭﺍ ﻧﻴﺰ ﺗﺄﻣﻴﻦ ﮐﺮﺩ‪.‬‬

‫‪ -۳-۱‬ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ‬

‫‪ -۳-۱-١‬ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺟﺮﻳﺎﻥ]‪[۲،۷‬‬

‫‪ :H‬ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﺗﺎﺑﻊ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ‬

‫ﺩﺭ ﺍﻳﻦ ﺣﺎﻟﺖ ﺍﮔﺮ‪ MV‬ﺑﻪ ﺻﻮﺭﺕ ﻓﺎﺯﻭﺭ ﺟﺮﻳﺎﻥ ﺑﺎﺷﺪ‪ ،‬ﺑـﺎ ﺗﻮﺟـﻪ‬

‫‪ : ∆B‬ﻣﻘﺎﺩﻳﺮ ﺑﺎﻗﻴﻤﺎﻧﺪﻩ ﺍﻱ‬

‫ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﺍﺯ ﺭﺍﺑﻄﻪ ﺯﻳﺮ ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ‪.‬‬

‫‪ :W‬ﻣﺎﺗﺮﻳﺲ ﻭﺯﻥ ﺩﻫﻲ‬

‫ﺑﻪ ﺗﻌﺮﻳﻒ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻋﻨﺎﺻﺮ ﻣﺮﺑـﻮﻁ ﺑـﻪ ‪ MV‬ﺟﺮﻳـﺎﻥ ﺩﺭ‬

‫‪ : ∆I‬ﻣﺎﺗﺮﻳﺲ ﺧﻄﺎﻱ ﻋﺪﻡ ﺗﻄﺎﺑﻖ‬

‫ﻣﻘﺎﺩﻳﺮ ﺑﺎﻗﻴﻤﺎﻧﺪﻩ ﺍﻱ ) ‪ ( ∆B‬ﺩﺭ ﻫﺮ ﺗﮑـﺮﺍﺭ‪ ،‬ﻣﺤﺎﺳـﺒﻪ ﻭ ﺑـﻪ ﺟﺮﻳـﺎﻥ‬ ‫ﺷﺎﺧﻪﻫﺎ ﺍﻓﺰﻭﺩﻩ ﻣﻲﺷﻮﻧﺪ‪:‬‬ ‫)‪(3‬‬

‫‪= B + ∆B‬‬ ‫‪k‬‬

‫‪k +1‬‬

‫‪B‬‬

‫ﻣﺎﺗﺮﻳﺲ ﺧﻄﺎﻱ ﻋﺪﻡ ﺗﻄﺎﺑﻖ ) ‪ ( ∆I‬ﺗﻔـﺎﻭﺕ ﻣﻘـﺎﺩﻳﺮ ﺟﺮﻳـﺎﻥﻫـﺎﻱ‬ ‫ﻣﻌﺎﺩﻝ ﺷﺎﺧﻪﻫﺎ )ﻳﺎ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ( ﺑﺎ ﻣﺤﺎﺳﺒﻪ ﺷﺪﻩ ﻣﻲﺑﺎﺷﺪ‪:‬‬ ‫)‪(4‬‬

‫‪∆I = I eq − I cal‬‬

‫)‪(6‬‬

‫‪∂I r ‬‬ ‫‪‬‬ ‫‪∂Bi  1 ‬‬ ‫=‬ ‫‪∂I i   1‬‬ ‫‪∂Bi ‬‬

‫‪ ∂I r‬‬ ‫‪ r‬‬ ‫‪∂B‬‬ ‫‪= i‬‬ ‫‪ ∂I‬‬ ‫‪ ∂B‬‬ ‫‪ r‬‬

‫‪H SUB‬‬

‫ﮐﻪ ﺩﺭ ﺍﻳﻦ ﺭﺍﺑﻄﻪ ‪ l‬ﺷﻤﺎﺭﻩ ﺷﺎﺧﻪ ﻭ ‪ I‬ﻣﻘﺪﺍﺭ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺑﻪ‬

‫ﺻﻮﺭﺕ ﺯﻳﺮ ﻣﻲﺑﺎﺷﺪ‪:‬‬

‫‪I = MV = I lr + jI li‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻗﻄﺮﻱ ﺑﻮﺩﻥ ﻣﺎﺗﺮﻳﺲ ﻣـﻲﺗـﻮﺍﻥ ﻣﻘـﺎﺩﻳﺮ ﻣﻮﻫـﻮﻣﻲ ﻭ‬

‫‪ MV -١‬ﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﺗﻮﺍﻥ ﺟﺎﺭﻱ ﺧﻄﻮﻁ ﻣﻲﺑﺎﺷـﺪ‪ :‬ﺩﺭ ﺍﻳـﻦ‬

‫ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ ،‬ﺍﻣﺎ ﺍﮔﺮ‪ MV‬ﺑﻪ ﺻﻮﺭﺕ ﺩﺍﻣﻨﻪ ﺟﺮﻳﺎﻥ ﺑﺎﺷـﺪ ﺧـﻮﺍﻫﻴﻢ‬

‫ﺗﺒﺪﻳﻞ ﮐﺮﺩ‪:‬‬

‫ﺣﻘﻴﻘﻲ ﺭﺍ ﺍﺯ ﻫﻢ ﺟﺪﺍ ﻧﻤﻮﺩ‪ .‬ﺩﺭ ﺿﻤﻦ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻧﻴـﺰ ﻗﻄـﺮﻱ‬ ‫ﺩﺍﺷﺖ‪:‬‬ ‫)‪(7‬‬

‫‪2‬‬ ‫‪2‬‬ ‫‪h( I ) = I r + I i‬‬

‫)‪(8‬‬

‫‪ ∂h‬‬ ‫‪ ∂I r = cos φ‬‬ ‫‪‬‬ ‫‪ ∂h = sin φ‬‬ ‫‪ ∂I i‬‬

‫ﻋﻨﺎﺻﺮﻣﺮﺑﻮﻁ ﺑﻪ ﺍﻳﻦ‪ MV‬ﺩﺍﻣﻨﻪ ﺟﺮﻳﺎﻥ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﺍﺳﺖ‪:‬‬

‫ﮐﻪ ﺩﺭ ﺁﻥ‬

‫‪Ii‬‬ ‫‪Ir‬‬

‫‪φ = tan −1‬‬

‫ﻣﻲ ﺑﺎﺷﺪ‪ .‬ﻭﺟﻮﺩ ﺍﻳﻦ ﻋﻨﺎﺻﺮ ﺩﺭ ﻣـﺎﺗﺮﻳﺲ‬

‫ﮊﺍﮐﻮﺑﻴﻦ ﺳﺒﺐ ﭘﻴﺪﺍﻳﺶ ﺟﻤﻼﺕ ﺯﻳﺮ ﺩﺭ ﻣـﺎﺗﺮﻳﺲ ﺑﻬـﺮﻩ ﺧﻮﺍﻫـﺪ‬ ‫ﺷﺪ‪:‬‬

‫ﺻﻮﺭﺕ ﻣﻲﺗﻮﺍﻥ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺭﻭﺍﺑﻂ ﺯﻳﺮ ﺁﻥ ﺭﺍ ﺑـﻪ ﺟﺮﻳـﺎﻥ ﻣﻌـﺎﺩﻝ‬ ‫)‪(14‬‬ ‫‪k‬‬

‫*‪S‬‬ ‫‪1 ‬‬ ‫‪flow‬‬ ‫‪r + jI r ⇒ H‬‬ ‫‪= I eq‬‬ ‫=‬ ‫‪eq‬‬ ‫‪sub  1‬‬ ‫*‬ ‫‪k‬‬ ‫‪V‬‬

‫‪ V‬ﻭﻟﺘﺎﮊ ﮔﺮﻩﺍﻱ ﺍﺳﺖ ﮐﻪ ﺟﺮﻳﺎﻥ ﻣﻌﺎﺩﻝ ﺍﺯ ﺁﻥ ﺧﺎﺭﺝ ﻣﻲﺷﻮﺩ‪.‬‬

‫‪ MV-۲‬ﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﺗﻮﺍﻥ ﺑﺎﺭ ﻣﺘﺼﻞ ﺑﻪ ﺑﺎﺳﻬﺎﻣﻲﺑﺎﺷﺪ‪:‬‬

‫ﺍﻳﻦ ﻧﻮﻉ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱﻫﺎ ﻣﻌﻤﻮﻻﹰ ﻧﺘﺎﻳﺞ ﺑﺪﺳﺖ ﺁﻣﺪﻩ ﺍﺯ ﭘﻴﺶﺑﻴﻨﻲ ﺑﺎﺭ‬

‫ﺍﺳﺖ ﻭ ﺩﺍﺭﺍﻱ ﺩﻗﺖ ﮐﻤﺘﺮﻱ ﻧﺴﺒﺖ ﺑﻪ ﺑﻘﻴـﻪ ‪MV‬ﻫـﺎ ﻣـﻲﺑﺎﺷـﺪ ﻭ‬

‫ﺍﮐﺜﺮ ﻭﺭﻭﺩﻳﻬﺎﻱ ﻣﺴﺄﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺭﺍ ﺗﺸﮑﻴﻞ ﻣﻲﺩﻫﺪ‪ .‬ﺑﺎ ﺗﻮﺟﻪ‬ ‫ﺑﻪ ﺷﮑﻞ ‪ ۱‬ﻭ ﺭﺍﺑﻄﻪ ‪ KCL‬ﻣﻲ ﺗﻮﺍﻥ ﻧﻮﺷﺖ‪:‬‬

‫*‪S‬‬ ‫` ‪r + jI i‬‬ ‫‪I‬‬ ‫=‬ ‫‪= I eq‬‬ ‫‪eq‬‬ ‫‪load V k‬‬ ‫‪I‬‬ ‫‪= B − B‬‬ ‫‪load ∑ in ∑ out‬‬

‫)‪(15‬‬ ‫‪cos φ sin φ ‬‬ ‫‪‬‬ ‫‪sin 2 φ ‬‬

‫)‪(9‬‬

‫‪ cos 2 φ‬‬ ‫‪Gsub = ‬‬ ‫‪ sin φ cos φ‬‬

‫= ‪eq‬‬ ‫‪I mea‬‬

‫)‪(16‬‬

‫ﺩﺭ ﺍﻳﻦ ﺣﺎﻟﺖ ﻗﺴﻤﺖﻫﺎﻱ ﻣﻮﻫﻮﻣﻲ ﻭ ﺣﻘﻴﻘﻲ ﻣـﺎﺗﺮﻳﺲ ﺑﻬـﺮﻩ ﺑـﺎ‬

‫ﻫﻢ ﺗﺮﮐﻴﺐ ﻣﻲﺷﻮﻧﺪ) ﺗﺰﻭﻳﺞ(‪ ،‬ﭼﻮﻥ ﺩﺭ ﻋﺒﺎﺭﺕ ﺑﺎﻻ ‪ φ‬ﮐﻪ ﻣﺮﺑـﻮﻁ‬ ‫ﺑﻪ ﻫﺮ ﺩﻭ ﻗﺴﻤﺖ ﻣﻮﻫﻮﻣﻲ ﻭ ﺣﻘﻴﻘﻲ ﻣﻲﺑﺎﺷﺪ ﻇﺎﻫﺮ ﺷـﺪﻩ ﺍﺳـﺖ‪.‬‬ ‫ﺑﺮﺍﻱ ﺣﻞ ﺍﻳﻦ ﻣﺸﮑﻞ ﻣﻲ ﺗﻮﺍﻥ ﺍﻗﺪﺍﻣﺎﺕ ﺯﻳﺮ ﺭﺍ ﺍﻧﺠﺎﻡ ﺩﺍﺩ‪:‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺍﻳﻨﮑﻪ ﻣﻴﺰﺍﻥ ﺗﻮﺍﻥ ﺗﺤـﻮﻳﻠﻲ ﭘـﺴﺖ ﺍﺻـﻠﻲ ﺑـﻪ ﺷـﺒﮑﻪ‬

‫ﻣﺸﺨﺺ ﻣﻲﺑﺎﺷﺪ‪ ،‬ﻣﻲﺗﻮﺍﻥ ﺭﻭﺍﺑﻂ ﺯﻳﺮ ﺭﺍ ﻧﻮﺷﺖ‪:‬‬ ‫)‪(10‬‬

‫‪=1<0 pu‬‬ ‫‪S = p + jQ = VI * v‬‬ ‫‪‬‬ ‫* ‪→ S = I‬‬ ‫) ‪⇒ angle( I ) = −angle( S‬‬

‫ﺍﺯ ﺍﻳﻦ ﻣﻘﺪﺍﺭ ﺯﺍﻭﻳﻪ ﻣﻲﺗﻮﺍﻥ ﺑﻪ ﻋﻨﻮﺍﻥ ﺯﺍﻭﻳﻪ ﺍﻭﻟﻴﻪ ‪ MV‬ﺩﺍﻣﻨﻪ‬

‫ﺟﺮﻳﺎﻥ ﺍﺳﺘﻔﺎﺩﻩ ﻧﻤﻮﺩ‪ ،‬ﺑﻨﺎﺑﺮﺍﻳﻦ‪:‬‬

‫‪i‬‬

‫)‪(11‬‬

‫‪I mea < angle( I ) = I + jI‬‬ ‫‪r‬‬

‫ﺑﺮﺍﻱ ﻓﺎﺯﻫﺎﻱ ‪ B‬ﻭ ‪ C‬ﻣﻲﺗﻮﺍﻥ ﺑﺎ ﺍﻓﺰﻭﺩﻥ ‪ +١٢٠‬ﻭ‪ -١٢٠‬ﺩﺭﺟﻪ ﺑـﻪ‬ ‫)‪ angle(I‬ﺭﺍﺑﻄﻪ ﻓﻮﻕ ﺭﺍ ﺑﮑﺎﺭ ﺑـﺮﺩ‪ .‬ﺍﻣـﺎ ﺑـﺮﺍﻱ ﺗﮑﺮﺍﺭﻫـﺎﻱ ﺑﻌـﺪﻱ‬

‫ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﺯﺍﻭﻳﻪ ﻣﺤﺎﺳﺒﻪ ﺷﺪﻩ ﺩﺭ ﺗﮑﺮﺍﺭ ﻗﺒﻞ ﺍﺳﺘﻔﺎﺩﻩ ﻧﻤﻮﺩ‪.‬‬ ‫)‪(12‬‬

‫‪Bik‬‬ ‫‪= I r + jI i‬‬ ‫‪Brk‬‬

‫‪k +1‬‬ ‫‪I mea‬‬ ‫‪= I mea < tan −1‬‬

‫ﺩﺭ ﺍﻳﻦ ﺣﺎﻟﺖ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬ ‫)‪(13‬‬

‫‪r‬‬ ‫‪ ∂I mea‬‬ ‫‪=1‬‬ ‫‪‬‬ ‫‪ ∂B r‬‬ ‫‪ i‬‬ ‫‪ ∂I mea = 1‬‬ ‫‪ ∂Bi‬‬

‫ﻋﻨﺎﺻﺮ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻣﺮﺑﻮﻁ ﺑﻪ ﺍﻳﻦ ‪ MV‬ﺩﺭ ﺻﻮﺭﺕ ﺗﺪﺍﺑﻴﺮ‬ ‫ﺑﺎﻻ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪:‬‬

‫‪ -۳-۱-۲‬ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺗﻮﺍﻥ‬

‫ﻣﻌﻤﻮﻻﹰ ‪ MV‬ﺗﻮﺍﻥ ﺑﻪ ﺩﻭ ﺻﻮﺭﺕ ﺍﺳﺖ‪:‬‬

‫‪1 ‬‬ ‫‪H =‬‬ ‫‪‬‬ ‫‪ 1‬‬

‫ﺷﮑﻞ ‪ :١‬ﺭﺍﺑﻄﻪ ‪KCL‬‬

‫ﺑﻪ ﻋﻨﻮﺍﻥ ﻣﺜﺎﻝ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻣﺮﺑﻮﻁ ﺑﻪ‪ MV‬ﺗﻮﺍﻥ ﺑﺎﺭ )ﺍﻧـﺪﺍﺯﻩ‬ ‫ﮔﻴﺮﻱﻫﺎﻱ ﮐﺎﺫﺏ( ﺩﺭﺷﮑﻞ ‪ ١‬ﺑﺮﺍﺑﺮ ﺯﻳﺮ ﺍﺳﺖ‪:‬‬

‫‪H‬‬ ‫]‪= [+1 + 1 −1 −1‬‬ ‫‪sub‬‬

‫‪ -۳-۱-۳‬ﻣﻘﺎﺩﻳﺮﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﺷﺪﻩ ﻭﻟﺘﺎﮊ]‪[۳‬‬

‫ﺍﮔﺮ‪ MV‬ﻭﻟﺘﺎﮊ ﺑﻪ ﺻﻮﺭﺕ ﺩﺍﻣﻨﻪ ﺑﺎﺷﺪ‪ ،‬ﻣﺜﻞ‪ MV‬ﺟﺮﻳﺎﻥ ﻣﻲﺗﻮﺍﻥ ﺍﺯ‬ ‫ﺯﺍﻭﻳﻪ ﻣﺤﺎﺳﺒﻪ ﺷﺪﻩ ﺩﺭ ﺗﮑﺮﺍﺭ ﻗﺒﻞ ﺍﺳﺘﻔﺎﺩﻩ ﮐﺮﺩ‪.‬‬

‫‪r + jV i‬‬ ‫)‪(17‬‬ ‫‪Veq = Vmea <θ vk −1 = Veq‬‬ ‫‪eq‬‬ ‫ﻣﻘﺎﺩﻳﺮ ﺍﻭﻟﻴﻪ ﺯﺍﻭﻳﺔ ‪ MV‬ﻭﻟﺘﺎﮊﻫﺎ ﺻﻔﺮ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﻣﻴـﺸﻮﺩ ‪ .‬ﺑـﺎ‬

‫ﺻﺮﻑ ﻧﻈﺮ ﺍﺯ ﻣﻘﺎﺩﻳﺮ ﺍﻣﭙﺪﺍﻧﺲ ﻣﺘﻘﺎﺑﻞ ﺑﺎﺳـﻬﺎ ﻣـﺎﺗﺮﻳﺲ ﺑﻬـﺮﻩ ﺑـﻪ‬ ‫ﺻﻮﺭﺕ ﺭﺍﺑﻄﻪ ﺯﻳﺮ ﺗﺸﮑﻴﻞ ﻣﻲﺷﻮﺩ‪:‬‬

‫‪‬‬ ‫‪G= T‬‬ ‫‪− X aa Wva Raa‬‬

‫‪r‬‬ ‫‪Ga‬‬ ‫‪T‬‬ ‫‪+ Raa Wva X aa‬‬

‫)‪(18‬‬

‫‪− RT W X + xT W R ‬‬ ‫‪aa va aa‬‬ ‫‪aa va aa ‬‬ ‫‪‬‬ ‫‪Gi‬‬ ‫‪a‬‬ ‫‪‬‬

‫ﻛﻪ ﺩﺭ ﺭﺍﺑﻄﻪ ﻓﻮﻕ‪:‬‬

‫‪r )T W H r + RT W‬‬ ‫‪Gar = ( H aa‬‬ ‫‪pa aa‬‬ ‫‪aa va R aa‬‬ ‫‪TW X‬‬ ‫‪+ X aa‬‬ ‫‪va aa‬‬ ‫‪i )T W H i + RT W R‬‬ ‫‪Gai = ( H aa‬‬ ‫‪qa aa‬‬ ‫‪aa va aa‬‬ ‫‪T‬‬ ‫‪+ X aa Wva X aa‬‬

‫‪ G ar , G ia‬ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﺣﻘﻴﻘﻲ ﻭ ﻣﻮﻫﻮﻣﻲ ﺗﺸﮑﻴﻞ ﺷﺪﻩ ﺗﻮﺳﻂ‬ ‫‪MV‬ﻫﺎﻱ ﻏﻴﺮ ﺍﺯ ﻭﻟﺘﺎﮊ ﻣﺮﺑﻮﻁ ﺑﻪ ﻓﺎﺯ ‪ a‬ﻫﺴﺘﻨﺪ‪.‬‬

‫‪ : R aa , X aa‬ﻣﺠﻤﻮﻋــﻪ ﻗــﺴﻤﺖﻫــﺎﻱ ﻣﻮﻫــﻮﻣﻲ ﻭ ﺣﻘﻴﻘــﻲ‬

‫ﺍﻣﭙﺪﺍﻧﺲﻫﺎﻱ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺑﻴﻦ ﺑﺎﺱ ﺍﺻﻠﻲ ﻭ ﺑﺎﺳـﻲ ﮐـﻪ ﻭﻟﺘـﺎﮊ ﺁﻥ‬ ‫ﺍﻧﺪﺍﺯﻩ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﻣﻲ ﺑﺎﺷـﻨﺪ‪.‬‬

‫‪Wva ,Wqa ,Wpa‬‬

‫ﺯﻳـﺮ ﻣـﺎﺗﺮﻳﺲ‬

‫ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻣﻲﺍﻧﺠﺎﻣﺪ‪ .‬ﺩﺭ ﺁﻥ ﺻﻮﺭﺕ ﺯﻣﺎﻥ ﻣﺤﺎﺳﺒﺎﺕ ﺑﻪ ﻃـﺮﺯ‬ ‫ﭼﺸﻤﮕﻴﺮﻱ ﮐﺎﻫﺶ ﻣﻲﻳﺎﺑﺪ‬

‫‪ -۳-۳-١‬ﻋﺪﻡ ﻭﺟﻮﺩ ‪MV‬ﻫﺎﻱ ﻭﻟﺘﺎﮊ‬

‫ﺩﺭ ﺍﻳــﻦ ﺻــﻮﺭﺕ ﻣــﺎﺗﺮﻳﺲ ﮊﺍﮐــﻮﺑﻴﻦ ﺑــﻪ ﺻــﻮﺭﺕ ﺯﻳــﺮ ﺍﺳــﺖ‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪i‬‬ ‫‪H bb ‬‬ ‫‪‬‬

‫‪r‬‬ ‫‪ H aa‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪HI ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬

‫‪i‬‬ ‫‪H aa‬‬

‫ﻫﺎﻱ ﻭﺯﻥﺩﻫﻲ ﻣﺮﺑﻮﻁ ﺑـﻪ ‪MV‬ﻫـﺎﻱ ﺗـﻮﺍﻥ ﻭ ﻭﻟﺘـﺎﮊ ﻣـﻲﺑﺎﺷـﻨﺪ‪.‬‬

‫)‪(19‬‬

‫‪ -۳-۲‬ﻗﻄﺮﻱ ﮐﺮﺩﻥ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ‬

‫ﻫﻤﻴﻨﻄﻮﺭ ﮐﻪ ﻣﺸﺎﻫﺪﻩ ﻣﻴﺸﻮﺩ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﮐﺎﻣﻼﹰ ﻗﻄﺮﻱ ﺍﺳﺖ‬

‫ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﺓ ﻓﺎﺯ‪ b‬ﻭ‪ c‬ﻧﻴﺰ ﺑﻄﻮﺭ ﻣﺸﺎﺑﻪ ﺗﺸﮑﻴﻞ ﻣﻲﺷﻮﻧﺪ‪.‬‬

‫ﺭﺍﺑﻄﻪ ﺷﻤﺎﺭﺓ ‪MV‬ﻫﺎ ﺑﺎ ﺷـﻤﺎﺭﻩ ﺷـﺎﺧﻪﻫـﺎ ﺗﺮﺗﻴـﺐ ﻗـﺮﺍﺭ ﮔـﺮﻓﺘﻦ‬

‫ﻋﻨﺎﺻﺮ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﺭﺍ ﻣﺸﺨﺺ ﻣﻲﮐﻨﺪ‪ .‬ﺍﮔﺮ ﺷﻤﺎﺭﻩ ‪MV‬ﻫـﺎ‬

‫‪r‬‬ ‫‪H cc‬‬

‫‪i‬‬ ‫‪H bb‬‬

‫‪r‬‬ ‫‪H bb‬‬

‫ﻭ ﻣﻲﺗﻮﺍﻥ ﺁﻧﺮﺍ ﺑﻄﻮﺭ ﮐﺎﻣﻞ ﺁﻧﺮﺍ ﺑﻪ ﺷﺶ ﺯﻳﺮﻣﺎﺗﺮﻳﺲ ﺗﺠﺰﻳﻪ ﻧﻤﻮﺩ‪.‬‬

‫‪ -۳-۳-٢‬ﻭﺟﻮﺩ ‪MV‬ﻫﺎﻱ ﻭﻟﺘﺎﮊ‬

‫ﺑﺎ ﺷﻤﺎﺭﺓ ﻣﺘﻐﻴﻴﺮﻫﺎﻱ ﺣﺎﻟﺖ ﻣﺮﺑﻮﻃﻪ ﻧﺰﺩﻳﮏ ﺑﺎﺷﺪ‪ ،‬ﻣﺎﺗﺮﻳﺲ ﺑﻬـﺮﻩ‬

‫ﺑﺎ ﻓـﺮﺽ ﺍﻳﻨﮑـﻪ ﺩﺭ ﺳﻴـﺴﺘﻢﻫـﺎﻱ ﺗﻮﺯﻳـﻊ ﻧـﺴﺒﺖ ‪ r x‬ﺑﺎﻻﺳـﺖ‬ ‫ﻣﻲ ﺗﻮﺍﻥ ﺍﺯ ﻋﻨﺎﺻﺮ ﻣﻮﺟﻮﺩ ﺩﺭ ﻗﻄﺮ ﻓﺮﻋﻲ ﺭﺍﺑﻄﻪ )‪(۱۸‬ﺻﺮﻑ ﻧﻈـﺮ‬

‫‪ -١‬ﺷــﺒﮑﻪ ﺭﺍ ﺑــﻪ ﺻــﻮﺭﺕ ﻻﻳــﻪ ﺑﻨــﺪﻱ ﺩﺭ ﻧﻈــﺮ ﮔﺮﻓﺘــﻪ ﺳــﭙﺲ‬

‫ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﺭﺍ ﺑﻪ ﻗﺴﻤﺖﻫﺎﻱ ﺣﻘﻴﻘﻲ ﻭ ﻣﻮﻫﻮﻣﻲ ﺗﺠﺰﻳﻪ ﻧﻤـﻮﺩ‬

‫ﺑﻪ ﻓﺮﻡ ﻗﻄﺮﻱ ﻧﺰﺩﻳﮏ ﻣﻲﺷﻮﺩ‪.‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺑﺤﺚ ﺑﺎﻻ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺯﻳﺮ ﭘﻴﺸﻨﻬﺎﺩ ﻣﻲﺷﻮﺩ‪:‬‬

‫ﺷﺎﺧﻪ ﻫﺎﻱ ﻣﻮﺟﻮﺩ ﺩﺭ ﻫﺮ ﻻﻳﻪ ﺑـﻪ ﺗﺮﺗﻴـﺐ ﺷـﻤﺎﺭﻩ ﮔـﺬﺍﺭﻱ ﻣـﻲ‬

‫ﻧﻤﺎﻳﻴﻢ‪ .‬ﺳﭙﺲ ﺑﻪ ﻻﻳﻪ ﺑﻌﺪﻱ ﺭﻓﺘﻪ ﻭ ﺷﺎﺧﻪﻫﺎﻱ ﻻﻳﻪﻫﺎﻱ ﺑﻌـﺪﻱ ﺭﺍ‬ ‫ﺑﻪ ﻫﻤﻴﻦ ﺗﺮﺗﻴﺐ ﻧﺎﻣﮕﺬﺍﺭﻱ ﻣﻲﮐﻨﻴﻢ ﺗﺎ ﺑﻪ ﻻﻳﻪ ﺁﺧﺮ ﺑﺮﺳﻴﻢ‪.‬‬

‫‪-۲‬ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺭﺍ ﺑﻪ ﺩﻭ ﺩﺳﺘﻪ ﺗﻘﺴﻴﻢ ﻣﻲﮐﻨﻴﻢ‬

‫ﮐﺮﺩ‪ .‬ﺩﺭ ﺁﻥ ﺻﻮﺭﺕ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻗﻄـﺮﻱ ﻣـﻲﺷـﻮﺩ ﻭ ﻣـﻲﺗـﻮﺍﻥ‬

‫ﻭ ﻣﺴﺄﻟﻪ ﺭﺍﻣﻲﺗﻮﺍﻥ ﻃﺒﻖ ﺭﻭﺍﺑﻂ ﺯﻳﺮ ﺣﻞ ﻧﻤﻮﺩ‪:‬‬

‫‪r )T W ∆I r − RT W ∆V r‬‬ ‫‪Gar ∆Bar = ( H aa‬‬ ‫‪aa va a‬‬ ‫‪pa a‬‬ ‫‪i‬‬ ‫‪T‬‬ ‫‪− XaaWva ∆Va‬‬ ‫‪i‬‬ ‫‪i‬‬ ‫‪i )T W ∆I i + XT W ∆V r‬‬ ‫‪Ga ∆Ba = ( H aa‬‬ ‫‪qa a‬‬ ‫‪aa va a‬‬ ‫‪T W ∆V i‬‬ ‫‪− Raa‬‬ ‫‪va a‬‬

‫)‪(20‬‬

‫ﺍﻟﻒ(ﻣﻘﺎﺩﻳﺮ ﺟﺮﻳﺎﻥ ﻭ ﺗﻮﺍﻥ ﺟﺎﺭﻱ ﺷﺎﺧﻪﻫﺎ‪ ،‬ﺏ(ﺍﻃﻼﻋﺎﺕ ﺑﺎﺭ‬

‫‪ -٣‬ﺍﮔﺮ ‪ MV‬ﺟﺮﻳﺎﻥ ﻳﺎ ﺗﻮﺍﻥ ﺟﺎﺭﻱ ﺷﺎﺧﻪﻫﺎ ﺑﻮﺩ‪ ،‬ﺷـﻤﺎﺭﺓ ﻣﻘـﺪﺍﺭ‬

‫ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺭﺍ ﺑﺮﺍﺑﺮ ﺑﺎ ﺷﻤﺎﺭﻩ ﺷﺎﺧﻪ ﻣﺮﺑﻮﻃﻪ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﺩ‪.‬‬

‫‪ -٤‬ﺍﮔﺮ ﻣﻘﺪﺍﺭ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺍﻃﻼﻋﺎﺕ ﺑﺎﺭ ﺑـﻮﺩ‪ ،‬ﺑﺮﺭﺳـﻲ ﻣـﻲ‬ ‫ﺷﻮﺩ ﮐﻪ ﺑﺎﺭ ﺑﻪ ﭼﻪ ﺑﺎﺳﻲ ﻣﺘﺼﻞ ﺍﺳﺖ ﻭﭼﻪ ﺷﺎﺧﻪﻫـﺎﻳﻲ ﺑـﻪ ﺍﻳـﻦ‬

‫ﺑﺎﺱ ﻣﺘﺼﻞ ﻫﺴﺘﻨﺪ‪ .‬ﺍﮔﺮ ﺗﻨﻬﺎ ﻳﮏ ﺷﺎﺧﻪ ﻣﺘﺼﻞ ﺑﻮﺩ ﺷـﻤﺎﺭﻩ ‪MV‬‬

‫‪ -۳-۴‬ﻭﺟﻮﺩ ﻣﺶﻫﺎﻱ ﺿﻌﻴﻒ]‪[۲,۳,۹‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺷﮑﻞ ‪ ۲‬ﻭ ﺭﺍﺑﻄﻪ ‪ KVL‬ﻣﻲﺗﻮﺍﻥ ﻧﻮﺷﺖ‪:‬‬ ‫‪y1‬‬ ‫‪y2‬‬

‫ﺭﺍ ﺑﺮﺍﺑﺮ ﺑﺎ ﺷﻤﺎﺭﻩ ﺍﻳﻦ ﺷﺎﺧﻪ ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﻭﮔﺮﻧﻪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ‬

‫‪y3‬‬

‫ﻣﺤﺪﻭﺩﻳﺖﻫﺎﻱ ﺯﻳﺮ ﺑﺮﺍﺑﺮ ﺑﺎ ﺷﻤﺎﺭﻩ ﻳﮑﻲ ﺍﺯ ﺷﺎﺧﻪﻫﺎﻱ ﻣﺘﺼﻞ ﺑـﻪ‬ ‫ﺑﺎﺱ ﻣﺮﺑﻮﻃﻪ ﻳﺎ ﻧﺰﺩﻳﮏ ﺑﻪ ﺁﻧﻬﺎ ﺩﺭ ﻧﻈﺮﻣﻲﮔﻴﺮﻳﻢ‪.‬‬

‫‪ -١‬ﺍﻳﻦ ﺷﻤﺎﺭﻩ ﻗﺒﻼﹰ ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﻧﮕﺮﻓﺘﻪ ﺑﺎﺷﺪ‪.‬‬

‫‪ -٢‬ﺩﺭ ﻧﻬﺎﻳﺖ ﻧﺒﺎﻳﺪ ﺷﻤﺎﺭﻩﺍﻱ ﺑﺰﺭﮔﺘﺮ ﺍﺯ ﺗﻌﺪﺍﺩ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ‬ ‫ﺷﺪﻩ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﻮﺩ‪.‬‬

‫ﻧﮑﺘﻪ ﻣﻮﺭﺩ ﺗﻮﺟﻪ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺻﻮﺭﺕ ﻭﺟـﻮﺩ ﻣـﺶ ﻳـﺎ ‪MV‬‬

‫‪y4‬‬

‫‪y5‬‬

‫ﺷﮑﻞ ‪۲‬‬

‫‪yiφ Biφ = 0‬‬

‫)‪(21‬‬

‫‪ nl‬ﺗﻌﺪﺍﺩ ﺷﺎﺧﻪﻫﺎﻱ ﻣﻮﺟﻮﺩ ﺩﺭ ﺣﻠﻘﻪ ﻣﻲﺑﺎﺷﺪ‬ ‫‪nl‬‬

‫‪Birφ − xiφ Biiφ + j ∑ riφ Biiφ + xiφ Birφ = 0‬‬ ‫‪i =1‬‬

‫ﻫﻢ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎﻻ ﻣﻲﺗﻮﺍﻥ ﺷﺮﺍﻳﻂ ﺭﺍ ﺑﻬﺒﻮﺩ ﺑﺨﺸﻴﺪ‪.‬‬

‫ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺭﺍ ﺩﺭ ﻣﻮﺭﺩ ﻫﺮ ﻓﺎﺯ ﺟﺪﺍﮔﺎﻧﻪ ﺑﻪﮐﺎﺭ ﺑـﺮﺩ‪ .‬ﻫﻤﭽﻨـﻴﻦ‬ ‫ﺗﺠﺰﻳﻪ ﻣﺴﺎﻟﻪ ﺑﻪ ﺩﻭ ﻗﺴﻤﺖ ﺣﻘﻴﻘﻲ ﻭ ﻣﻮﻫﻮﻣﻲ ﺑـﻪ ﮐـﺎﻫﺶ ﺍﺑﻌـﺎﺩ‬

‫‪i =1‬‬

‫‪nl‬‬

‫‪∑r‬‬

‫‪iφ‬‬

‫‪B − xiφ B = 0‬‬

‫‪iφ‬‬

‫‪B + xiφ B = 0‬‬

‫‪iφ‬‬

‫‪i‬‬ ‫‪iφ‬‬

‫‪r‬‬ ‫‪iφ‬‬

‫‪r‬‬ ‫‪iφ‬‬

‫‪i‬‬ ‫‪iφ‬‬

‫‪i =1‬‬ ‫‪nl‬‬

‫)‪(22‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻧﺎ ﻣﺘﻌﺎﺩﻟﻲ ﺳﻴﺴﺘﻢﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻣﻨﺎﺳﺐ ﺍﺳﺖ ﮐﻪ ﻣﺴﺎﻟﻪ‬

‫∑‬

‫ﺭﺍﺑﻄﻪ)‪(۲۱‬ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﻧﻮﺷﺖ‪:‬‬

‫ﻭﻟﺘﺎﮊ ﺩﺭ ﺷﺒﮑﻪ ﺣﺎﻟﺖ ﻗﻄﺮﻱ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﺍﺯ ﺑﻴﻦ ﻣﻲﺭﻭﺩ ﺍﻣﺎ ﺑـﺎﺯ‬

‫‪ -۳-۳‬ﺗﺠﺰﻳﻪ‬

‫م‪n‬‬

‫‪∑r‬‬ ‫‪i =1‬‬ ‫‪nl‬‬

‫‪∑r‬‬ ‫‪i =1‬‬

‫ﺩﺭ ﺣﻘﻴﻘﺖ ﺭﻭﺍﺑﻂ)‪ (۲۲‬ﻗﻴﺪﻫﺎﻱ ﻣﺴﺎﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﻣﻲﺑﺎﺷﻨﺪ‪.‬‬

‫ﺑﺮﺍﻱ ﺣﻞ ﻣﺴﺎﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺑﺎ ﻭﺟﻮﺩ ﻗﻴﺪ ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﺿﺮﺍﻳﺐ‬

‫ﻻﮔﺮﺍﻧﮋ ﺍﺳﺘﻔﺎﺩﻩ ﻧﻤﻮﺩ‪ .‬ﺑﻌﺪ ﺍﺯ ﺣﻞ ﻣﺴﺎﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺑﺎ ﺗﻮﺟﻪ‬

‫ﻗﻴﻮﺩ‪،‬ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺿﺮﺍﻳﺐ ﻻﮔﺮﺍﻧﮋ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﺑﻪ ﻓﺮﻡ ﺯﻳﺮ‬

‫ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ]‪:[۹،۶‬‬

‫‪Ct ‬‬ ‫‪‬‬ ‫‪0‬‬

‫)‪(23‬‬

‫‪Gm‬‬ ‫‪‬‬ ‫‪C‬‬

‫‪r‬‬ ‫‪r‬‬ ‫‪raaT  ∆Ba   MIS ‬‬ ‫‪ r  =  r ‬‬ ‫‪0  λ  C a ‬‬

‫‪G aar‬‬ ‫‪‬‬ ‫‪ raa‬‬

‫‪r   ∆B‬‬ ‫‪ i‬‬ ‫‪ λ‬‬

‫‪G‬‬ ‫‪‬‬ ‫‪ raa‬‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬

‫)‪(27‬‬

‫‪i‬‬

‫‪  MIS‬‬ ‫‪= i‬‬ ‫‪ C a‬‬

‫‪i‬‬ ‫‪a‬‬

‫‪T‬‬ ‫‪aa‬‬

‫‪i‬‬ ‫‪aa‬‬

‫ﮐﻪ ‪ Gm‬ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻗﺴﻤﺖ ﺷﻌﺎﻋﻲ ﺷﺒﮑﻪ ﻭ‪ Cx‬ﻣﺎﺗﺮﻳﺲ‬

‫ﻧﮑﺘﻪ ﻣﻮﺭﺩ ﺗﻮﺟﻪ ﺍﻳـﻨﺴﺘﮑﻪ ﺩﺭ ﺗﺸـﮑﻴﻞ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ‪،‬ﺷﺎﺧﻪ‬

‫ﺍﻣﭙﺪﺍﻧﺲﻫﺎﻱ ﻣﺘﻘﺎﺑﻞ ﺍﺯ ﺭﺍﺑﻄﻪ ﺯﻳﺮ ﺣﺎﺻﻞ ﻣﻲﺷﻮﺩ‪:‬‬

‫ﺩﺭ ﺣﻘﻴﻘﺖ ﻭﺟﻮﺩ ﻣﺶ‪ ،‬ﺳﺎﺧﺘﺎﺭ ﻣﺎﺗـﺮﻳﺲ ﮊﺍﮐـﻮﺑﻴﻦ ﺭﺍ ﺑﻪ‬

‫ﮊﺍﮐﻮﺑﻴﻦ ﻗﻴﻮﺩ ﻣﻲﺑﺎﺷﻨﺪ‪ .‬ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻗﻴﻮﺩ ﺑﺎ ﺻﺮﻑ ﻧﻈﺮ ﺍﺯ‬

‫)‪(24‬‬

‫‪0 ‬‬ ‫‪0 ‬‬ ‫‪0 ‬‬ ‫‪‬‬ ‫‪0 ‬‬ ‫‪− x bb ‬‬ ‫‪‬‬ ‫‪rbb ‬‬

‫‪0‬‬

‫‪0‬‬

‫‪0‬‬

‫‪− x aa‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪− x bb‬‬ ‫‪rbb‬‬

‫‪0‬‬ ‫‪rbb‬‬ ‫‪x bb‬‬

‫‪raa‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪raa‬‬ ‫‪x bb‬‬

‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪0‬‬

‫‪ raa‬‬ ‫‪x‬‬ ‫‪ aa‬‬ ‫‪ 0‬‬ ‫‪HC = ‬‬ ‫‪ 0‬‬ ‫‪ 0‬‬ ‫‪‬‬ ‫‪ 0‬‬

‫‪ rcc,rbb,raa‬ﺑﻠﻮﮎﻫﺎﻱ ﻗﺴﻤﺖﻫﺎﻱ ﺣﻘﻴﻘﻲ ﺍﻣﭙﺪﺍﻧﺲﻫﺎﻱ ﺷﺎﺧﻪ‬

‫ﻫﺎﻱ ﺗﺸﮑﻴﻞ ﺩﻫﻨﺪﻩ ﻣﺶ ﻭ ‪ xcc,xbb,xaa‬ﺑﻠﻮﮎﻫﺎﻱ ﻗﺴﻤﺖﻫﺎﻱ‬

‫ﻣﻮﻫﻮﻣﻲ ﺍﻣﭙﺪﺍﻧﺲﻫﺎﻱ ﺷﺎﺧﻪﻫﺎﻱ ﺗﺸﮑﻴﻞ ﺩﻫﻨﺪﻩ ﻣﺶ ﻣﻲﺑﺎﺷﻨﺪ‪.‬‬ ‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻗﻄﺮﻱ ﺑﻮﺩﻥ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ‪ ،‬ﻣﺴﺎﻟﻪ ﻗﺎﺑﻞ ﺗﺠﺰﻳﻪ ﺑﻪ‬

‫ﻫﺮ ﻓﺎﺯ ﻣﻲﺑﺎﺷﺪ‪ .‬ﭘﺲ ﻣﺴﺎﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺑﺮﺍﻱ ﻓﺎﺯ ‪ a‬ﺭﺍ ﻣﻲﺗﻮﺍﻥ‬

‫ﺑﻪ ﻓﺮﻡ ﻣﺎﺗﺮﻳﺴﻲ ﺯﻳﺮ ﻧﻮﺷﺖ‪:‬‬

‫)‪(25‬‬

‫‪r‬‬ ‫‪T‬‬ ‫‪ ∆Ba   MIS r ‬‬ ‫‪xaa‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪i‬‬ ‫‪raaT  ∆Ba   MIS i ‬‬ ‫=‪= r ‬‬ ‫‪0  λ   C ar ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪0  λi   C ai ‬‬ ‫‪‬‬ ‫‪‬‬

‫‪raaT‬‬

‫‪hr‬‬

‫‪T‬‬ ‫‪− xaa‬‬

‫‪i‬‬ ‫‪G aa‬‬ ‫‪− xaa‬‬

‫‪0‬‬

‫‪raa‬‬

‫‪0‬‬

‫‪G aar‬‬ ‫‪ i‬‬ ‫‪h‬‬ ‫‪ raa‬‬ ‫‪‬‬ ‫‪ xaa‬‬

‫ﮐﻪ ‪ λi ,λr‬ﺿﺮﺍﻳﺐ ﻻﮔﺮﺍﻧـﮋ ﻭﺳـﺎﻳﺮ ﭘﺎﺭﺍﻣﺘﺮﻫـﺎ ﺑـﻪ ﺻـﻮﺭﺕ ﺯﻳـﺮ‬

‫ﻣﻲﺑﺎﺷﻨﺪ‪:‬‬

‫ﻫﻢ ﻧﻤﻲﺯﻧﺪ‪.‬‬

‫‪-۴‬ﺍﺟﺮﺍﻱ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺮﺭﻭﻱ ﻳﮏ ﺷﺒﮑﻪ ﻧﻤﻮﻧﻪ‬

‫ﺷﺒﮑﻪ ﻧﻤﻮﻧﻪ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﻳﮏ ﺷﺒﮑﻪ ‪ ۲۵‬ﺷﻴﻨﻪ ‪IEEE‬‬

‫‪ ۴/۱۶‬ﮐﻴﻠﻮ ﻭﻟﺖ ﻣﻲﺑﺎﺷﺪ ﮐﻪ ﺩﺍﺭﺍﻱ ‪ ۲‬ﻣﺶ ﻣﻲﺑﺎﺷﺪ‪ .‬ﺩﺭ ﻗﺴﻤﺖ‬

‫ﺍﻭﻝ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﻭﻟﺘﺎﮊ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﻧﻤﻲﺷﻮﺩ ﻭﻟﻲ ﺩﺭ‬ ‫ﻗﺴﻤﺖ ﺑﻌﺪﻱ ﺍﺯ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﻭﻟﺘﺎﮊ ﻧﻴﺰ ﺑﻪ ﻋﻨﻮﺍﻥ‬ ‫ﻭﺭﻭﺩﻱ ﻣﺴﺎﻟﻪ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﺷﻮﺩ‪ .‬ﺩﺭ ﺟﺪﻭﻝ ‪ ۱‬ﺗﻌﺪﺍﺩ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩ‬

‫ﮔﻴﺮﻱ ﺷﺪﻩ ﺁﻣﺪﻩ ﺍﺳﺖ‪ .‬ﻣﻮﻗﻌﻴﺖ ﺍﻳﻦ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻳﻬﺎ ﻧﻴﺰ ﺩﺭ ﺷﮑﻞ‬ ‫‪ ۳‬ﻣﺸﺨﺺ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﺍﻳﻦ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻳﻬﺎ ﻧﺘﺎﻳﺞ ﺑﺨﺶ ﺑﺎﺭ ﺷﺒﮑﻪ‬ ‫ﻣﻮﺭﺩ ﻧﻈﺮ ﻫﺴﺘﻨﺪ ﮐﻪ ﺑﺎ ﺍﻓﺰﻭﺩﻥ ‪۱۰‬ﺩﺭﺻﺪ ﺧﻄﺎ ﺑﻪ ﻣﻘﺎﺩﻳﺮ ﺟﺮﻳﺎﻥ‬ ‫ﻭ ﻭﻟﺘﺎﮊ ﻭ ‪ ۳۰‬ﺩﺭﺻﺪ ﺧﻄﺎ ﺑﻪ ﻣﻘﺎﺩﻳﺮ ﺗﻮﺍﻥ ﺗﺰﺭﻳﻘﻲ ﺑﺪﺳﺖ ﺁﻣﺪﻩ‬

‫ﺍﺳﺖ‪ .‬ﻣﺘﻮﺳﻂ ﻧﺴﺒﺖ ‪ r‬ﺍﻣﭙﺪﺍﻧﺲ ﺷﺎﺧﻪ ﻫﺎ ﺑﺮﺍﺑﺮ ‪ ۱/۲‬ﻣﻲ‬ ‫‪x‬‬ ‫ﺑﺎﺷﺪ‪ .‬ﻧﺘﺎﻳﺞ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺩﺭ ﺷﮑﻞ ‪ ۴‬ﻭ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ‬ ‫‪MV‬ﻫﺎﻱ ﻭﻟﺘﺎﮊ ﻭ‪MV‬ﻫﺎﻱ ﺟﺮﻳﺎﻥ ﻭ ﻣﻌﺎﺩﻝ ﺟﺮﻳﺎﻥ ﺩﺭ ﺷﮑﻞ‬

‫‪ ۵‬ﻭ‪ ۶‬ﺁﻣﺪﻩ ﺍﺳﺖ‪ .‬ﻫﻤﺎﻧﻄﻮﺭ ﮐﻪ ﻣﺸﺎﻫﺪﻩ ﻣﻲﺷﻮﺩ ﺑﺎ ﺍﺣﺘﺴﺎﺏ‬

‫‪nl‬‬

‫‪C ar = −∑ ri ∆Bir − xi ∆Bii‬‬ ‫‪i =1‬‬ ‫‪nl‬‬

‫‪C a = −∑ ri ∆Bii + xi ∆Bir‬‬ ‫‪i‬‬

‫‪i =1‬‬

‫‪h = − RaaT Wva X aa + X aaT Wva Raa‬‬ ‫‪r‬‬

‫‪h i = − X aaT Wva Raa + RaaT Wva X aa‬‬ ‫‪MIS r = ( H aar ) T Wpa ∆I ar − RaaT Wva ∆Var − xaaT Wva ∆Vai‬‬ ‫‪MIS i = (H aai ) Wqa ∆I ai + xaaT Wxa ∆Var − RaaT Wva ∆Vai‬‬ ‫‪T‬‬

‫ﮐﻪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﭘﺎﺭﺍﻣﺘﺮﻫﺎﻱ ﻓﻮﻕ‪ ،‬ﺭﺍﺑﻄﻪ)‪ (۲۵‬ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺑﻪ ﻓﺮﻡ‬ ‫ﺯﻳﺮ ﻧﻮﺷﺖ‪:‬‬

‫)‪(26‬‬

‫ﻫﺎﻳﻲ ﮐﻪ ﺑﺎﻋﺚ ﺍﻳﺠﺎﺩ ﻣﺶ ﺷﺪﻩﺍﻧﺪ ﺩﺭ ﻧﻈﺮ ﮔـﺮﻓﺘﻪ ﻧﻤﻲ ﺷﻮﻧﺪ‪.‬‬

‫‪T  ∆B r ‬‬ ‫‪ MIS r ‬‬ ‫‪‬‬ ‫‪xaa‬‬ ‫‪a‬‬ ‫‪ r   r ‬‬ ‫‪0  λ  C a ‬‬ ‫=‬ ‫‪raaT  ∆Bai   MIS i ‬‬ ‫‪‬‬ ‫‪0  λi  C br ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬

‫‪hr‬‬ ‫‪− xaa‬‬

‫‪raaT‬‬ ‫‪0‬‬

‫‪G aai‬‬

‫‪T‬‬ ‫‪− xaa‬‬

‫‪raa‬‬

‫‪0‬‬

‫‪G aar‬‬ ‫‪‬‬ ‫‪ raa‬‬ ‫‪ hi‬‬ ‫‪‬‬ ‫‪ xaa‬‬

‫ﻣﻘﺎﺩﻳﺮ ‪ xaa , h r , h i‬ﻧﺴﺒﺖ ﺑﻪ ﻣﻘﺎﺩﻳﺮ ﻣﻮﺟﻮﺩ ﺩﺭ ﻗﻄﺮ ﺍﺻﻠﻲ ﺑﺴﻴﺎﺭ‬

‫ﮐﻮﭼﮑﺘﺮ ﻫﺴﺘﻨﺪ‪ .‬ﺑﺎ ﺻﺮﻓﻨﻈﺮ ﺍﺯ ﺁﻧﻬﺎ ﻣﺴﺎﻟﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ‪ ،‬ﺑﻪ ﺩﻭ‬ ‫ﻗﺴﻤﺖ ﻭ ﺣﻘﻴﻘﻲ ﻭ ﻣﻮﻫﻮﻣﻲ ﺗﺠﺰﻳﻪ ﻣﻲﺷﻮﺩ‪.‬‬

‫ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﻭﻟﺘﺎﮊ ﺩﻗﺖ ﻣﺴﺎﻟﻪ ﺍﻓﺰﺍﻳﺶ ﭘﻴﺪﺍ ﻣﻲﮐﻨﺪ ﺍﻣﺎ‬

‫ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻧﻴﺰ ﺍﺯ ﺣﺎﻟﺖ ﻗﻄﺮﻱ ﺧﺎﺭﺝ ﻣﻲﺷﻮﺩ‪ .‬ﺩﺭ ﻗﺴﻤﺖ‬ ‫ﺍﻭﻝ ﻣﺴﺎﻟﻪ ﺑﺎ ‪ ۲‬ﺗﮑﺮﺍﺭ ﻭ ﺩﺭ ﻗﺴﻤﺖ ﺩﻭﻡ ﻣﺴﺎﻟﻪ ﺑﺎ ‪ ۴‬ﺗﮑﺮﺍﺭ ﺑﻪ‬

‫ﻫﻤﮕﺮﺍﻳﻲ ﻣﻲﺭﺳﺪ‪ .‬ﺩﺭ ﻗﺴﻤﺖ ﺍﻭﻝ‪ ،‬ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻓﻘﻂ ﺍﺯ‬ ‫ﻋﻨﺎﺻﺮ ‪۱‬ﻭ‪ -۱‬ﺗﺸﮑﻴﻞ ﺷﺪﻩ ﻭﻟﻲ ﺩﺭ ﻗﺴﻤﺖ ﺩﻭﻡ‪ ،‬ﻣﻘﺎﺩﻳﺮ ﺍﻣﭙﺪﺍﻧﺲ‬

‫ﺷﺎﺧﻪ ﻫﺎ ﻧﻴﺰ ﺩﺭ ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﻇﺎﻫﺮ ﺷﺪﻩ ﺍﻧﺪ ﺍﻣﺎ ﺩﺭ ﻫﺮ ﺩﻭ‬

‫ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ﺩﺍﺭﺍﻱ ﻣﻘﺎﺩﻳﺮ ﺛﺎﺑﺖ ﻣﻲﺑﺎﺷﺪ‪ .‬ﺑﻪ ﻋﻠﺖ ﺗﻘﺮﻳﺒﻬﺎﻱ‬

‫ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﺪﻩ ﺩﺭ ﻗﺴﻤﺖ ‪ ۲‬ﺗﻌﺪﺍﺩ ﺗﮑﺮﺍﺭﻫﺎ ﻧﻴﺰ ﺍﻓﺰﺍﻳﺶ ﻳﺎﻓﺘﻪ‬

‫ﺍﺳﺖ‪.‬‬

‫ﺟﺪﻭﻝ‪:۱‬ﺗﻌﺪﺍﺩ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻳﻬﺎ‬

‫ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﮐﺎﺫﺏ‬ ‫‪۲۳‬‬

‫ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﺟﺮﻳﺎﻥ‬ ‫‪۳‬‬

‫ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﻭﻟﺘﺎﮊ‬ ‫‪۵‬‬

‫‪١‬‬ ‫‪٣‬‬ ‫‪٧‬‬ ‫‪١٤‬‬ ‫‪٢٠‬‬

‫‪٦‬‬

‫‪٥‬‬ ‫‪١٠‬‬

‫‪١١‬‬

‫‪١٢‬‬

‫‪١٣‬‬ ‫‪٢۶‬‬

‫‪٢‬‬

‫‪٢۵‬‬

‫‪١٨‬‬

‫‪١٩‬‬

‫‪٩‬‬

‫‪١٧‬‬

‫‪٤‬‬ ‫‪٨‬‬ ‫‪١٦‬‬

‫‪١۵‬‬

‫‪٢١‬‬

‫‪٢٢‬‬ ‫‪٢۴‬‬

‫ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮ ﺟﺮﻳﺎﻥ‬

‫ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮ ﺗﻮﺍﻥ)ﺍﻃﻼﻋﺎﺕ ﮐﺎﺫﺏ(‬

‫‪٢٣‬‬

‫ﺷﮑﻞ‪ :٣‬ﺷﺒﮑﻪ ﻧﻤﻮﻧﻪ‬

‫ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮ ﻭﻟﺘﺎﮊ‬

‫ﻓﺎﮐﺘﻮﺭﻳﺰﻩ ﺷﻮﺩ‪ .‬ﻫﻤﭽﻨﻴﻦ ﺍﻟﮕﻮﺭﻳﺘﻤﻲ ﺑﺮﺍﻱ ﻧﺎﻣﮕﺬﺍﺭﻱ ﺷﺎﺧﻪﻫﺎ ﻭ‬

‫ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺍﺭﺍﺋﻪ ﺷﺪ ﺗﺎ ﺑﺎ ﮐﻤﮏ ﺁﻥ ﺑﺘﻮﺍﻥ ﺑﻪ ﻳﮏ‬

‫ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻗﻄﺮﻱ ﺩﺳﺖ ﻳﺎﻓﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﺣﺠﻢ‬ ‫ﻣﺤﺎﺳﺒﺎﺕ ﻧﻴﺰ ﺑﻪ ﺧﺎﻃﺮ ﻗﻄﺮﻱ ﺷﺪﻥ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﮐﺎﻫﺶ‬

‫ﻣﻲﻳﺎﺑﺪ‪ .‬ﻫﻤﭽﻨﻴﻦ ﺭﻭﺵ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺷﺒﮑﻪﻫﺎﻱ ﺩﺍﺭﺍﻱ ﻣﺶ ﺿﻌﻴﻒ‬ ‫ﻧﻴﺰ ﺭﺍ ﺗﺤﺖ ﭘﻮﺷﺶ ﻗﺮﺍﺭ ﻣﻲﺩﻫﺪ‪ .‬ﺑﺎ ﮐﻤﮏ ﺿﺮﺍﻳﺐ ﻻﮔﺮﺍﻧﮋ‬

‫ﻣﻲﺗﻮﺍﻥ ﻗﻴﻮﺩ ﺍﻳﺠﺎﺩ ﺷﺪﻩ ﺗﻮﺳﻂ ‪ KVL‬ﺭﺍ ﻟﺤﺎﻅ ﻧﻤﻮﺩ‪ .‬ﺍﺯ ﺩﻳﮕﺮ‬

‫ﻣﺰﺍﻳﺎﻱ ﺭﻭﺵ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﻣﻲﺗﻮﺍﻥ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ‬

‫ﺷﺪﻩ ﻭﻟﺘﺎﮊ ﺩﺭ ﻣﻮﺭﺩ ﺷﺒﮑﻪ ﺩﺍﺭﺍﻱ ﻣﺶ ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﺩﺍﺩ‪ .‬ﺩﺭ‬

‫ﺭﻭﺵ ﭘﻴﺸﻨﻬﺎﺩ ﺷﺪﻩ ﻓﺎﺯﻫﺎﻱ ﺳﻪﮔﺎﻧﻪ ﺟﺪﺍﮔﺎﻧﻪ ﺗﺨﻤﻴﻦ ﺯﺩﻩ‬

‫ﻣﻲﺷﻮﻧﺪ ﮐﻪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻧﺎﻣﺘﻌﺎﺩﻝ ﺑﻮﺩﻥ ﺳﻴﺴﺘﻢﻫﺎﻱ ﺗﻮﺯﻳﻊ ﺑﺴﻴﺎﺭ‬

‫ﻣﻄﻠﻮﺏ ﻣﻲﺑﺎﺷﺪ‪ .‬ﺣﺘﻲ ﻗﺴﻤﺘﻬﺎﻱ ﺣﻘﻴﻘﻲ ﻭ ﻣﻮﻫﻮﻣﻲ ﻣﺴﺎﻟﻪ ﻧﻴﺰ‬

‫ﺷﮑﻞ ‪ :۴‬ﻧﺘﺎﻳﺞ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺷﺒﮑﻪ ﻧﻤﻮﻧﻪ ‪ ۲۵‬ﺷﻴﻨﻪ‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 Z 23‬‬ ‫‪0 0‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 Z19‬‬

‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 Z17‬‬ ‫‪0 Z17‬‬ ‫‪0 0‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 Z12‬‬

‫‪0 0‬‬ ‫‪0 0‬‬ ‫‪0 Z10‬‬ ‫‪0 Z10‬‬ ‫‪0 0‬‬

‫‪0‬‬ ‫‪Z8‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪Z7‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬

‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪Z6‬‬

‫‪0 0‬‬ ‫‪0 Z5‬‬ ‫‪0 Z5‬‬ ‫‪0 Z5‬‬ ‫‪0 0‬‬

‫‪Z3‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪Z3‬‬

‫‪0‬‬ ‫‪Z2‬‬ ‫‪Z2‬‬ ‫‪Z2‬‬ ‫‪0‬‬

‫‪ Z1‬‬ ‫‪Z‬‬ ‫‪ 1‬‬ ‫‪H V =  Z1‬‬ ‫‪‬‬ ‫‪ Z1‬‬ ‫‪ Z1‬‬ ‫‪‬‬

‫ﺷﮑﻞ‪ :۵‬ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ ‪MV‬ﻫﺎﻱ ﻭﻟﺘﺎﮊ‬

‫‪-۵‬ﻧﺘﻴﺠﻪ ﮔﻴﺮﻱ‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺍﻳﻦﮐﻪ ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ ﺑﺎﻳﺪ ﺑﻪ ﺻﻮﺭﺕ ﺯﻣﺎﻥ ﺣﻘﻴﻘﻲ‬

‫ﻋﻤﻞ ﻧﻤﺎﻳﺪ‪ .‬ﺑﺎﻳﺪ ﻣﻨﺎﺳﺒﺘﺮﻳﻦ ﻭ ﺩﻗﻴﻘﺘﺮﻳﻦ ﺟﻮﺍﺏ ﺭﺍﺩﺭ ﮐﻤﺘﺮﻳﻦ‬

‫ﺯﻣﺎﻥ ﻣﻤﮑﻦ ﺍﺭﺍﺋﻪ ﺩﻫﺪ‪ .‬ﺗﻤﺎﻣﻲ ﺍﻟﮕﻮﺭﻳﺘﻢ ﻫﺎ ﺑﻪ ﻧﻮﻋﻲ ﺩﺭ ﺑﻬﺒﻮﺩ‬

‫ﻫﺪﻑ ﺑﺎﻻ ﺍﺭﺍﺋﻪ ﻣﻲﺷﻮﻧﺪ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﻳﮏ ﺭﻭﺵ ﻓﻮﻕ ﺍﻟﻌﺎﺩﻩ‬ ‫ﺧﻄﻲ ﺑﺮ ﺍﺳﺎﺱ ﺟﺮﻳﺎﻥ ﺷﺎﺧﻪ ﻫﺎ ﺍﺭﺍﺋﻪ ﺷﺪ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺍﻳﻦ‬ ‫ﺧﺼﻮﺻﻴﺖ ﺩﻳﮕﺮ ﺍﺣﺘﻴﺎﺝ ﻧﻴﺴﺖ ﮐﻪ ﺩﺭ ﻫﺮ ﺗﮑﺮﺍﺭ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ‬

‫ﺗﺠﺰﻳﻪ ﻣﻲﺷﻮﻧﺪ ﮐﻪ ﻧﺘﻴﺠﻪ ﺁﻥ ﮐﺎﻫﺶ ﺍﺑﻌﺎﺩ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﻭ ﺑﻪ ﺗﺒﻊ‬

‫ﺁﻥ ﮐﺎﻫﺶ ﺣﺠﻢ ﻣﺤﺎﺳﺒﺎﺕ ﻣﻲﺷﻮﺩ‪ .‬ﺩﺭ ﺭﻭﺵ ﭘﻴﺸﻨﻬﺎﺩ ﺷﺪﻩ‬

‫ﻣﻲﺗﻮﺍﻥ ﺍﺯ ﺗﻤﺎﻣﻲ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩﮔﻴﺮﻱ ﺷﺪﻩ ﺩﺭ ﺟﻬﺖ ﺑﻬﺒﻮﺩ ﭘﺎﺳﺦ‬

‫ﻣﺴﺎﻟﻪ ﺍﺳﺘﻔﺎﺩﻩ ﻧﻤﻮﺩ‪ .‬ﺍﻣﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﺷﺪﻩ‬ ‫ﻭﻟﺘﺎﮊ ﺑﺎﻋﺚ ﺍﻓﺰﺍﻳﺶ ﺯﻣﺎﻥ ﻣﺤﺎﺳﺒﺎﺕ ﻣﻲﺷﻮﺩ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻧﺘﺎﻳﺞ‬

‫ﻋﻤﻠﻲ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻣﻘﺎﺩﻳﺮ ﺍﻧﺪﺍﺯﻩ ﮔﻴﺮﻱ ﺷﺪﻩ ﻭﻟﺘﺎﮊ ﺩﺭ ﺗﻤﺎﻣﻲ ﺑﺎﺳﻬﺎ‬

‫ﻫﻢ ﺑﻪ ﺩﻻﻳﻞ ﺍﻗﺘﺼﺎﺩﻱ ﻭ ﻫﻢ ﺑﻪ ﺩﻻﻳﻞ ﺑﺎﻻ ﺑﻪ ﺻﺮﻓﻪ ﻧﻴﺴﺖ‪.‬‬ ‫ﻭﺟﻮﺩ ﻣﺶ ﺩﺭ ﺷﺒﮑﻪﻫﺎ ﺑﺎﻋﺚ ﻣﻲﺷﻮﺩ ﮐﻪ ﻣﺎﺗﺮﻳﺲ ﺑﻬﺮﻩ ﺑﺎ ﺗﻮﺟﻪ‬

‫ﺑﻪ ﺿﺮﺍﻳﺐ ﻻﮔﺮﺍﻧﮋ ﺍﺯ ﻓﺮﻡ ﻗﻄﺮﻱ ﺧﺎﺭﺝ ﺷﻮﺩ‪ .‬ﺗﺨﻤﻴﻦ ﺣﺎﻟﺖ‬ ‫ﺷﺒﮑﻪ ﺩﺍﺭﺍﻱ ﻣﺶ ﺑﺪﻭﻥ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺿﺮﺍﻳﺐ ﻻﮔﺮﺍﻧﮋ ﺍﺯ ﺍﻫﺪﺍﻑ‬

‫ﺗﺤﻘﻴﻘﺎﺕ ﺁﻳﻨﺪﻩ ﻣﻲﺑﺎﺷﺪ‪.‬‬

1 −1 −1

1

−1 −1

1 1

−1 −1 −1

1

−1 −1

1

−1 −1

1

−1 −1

1 1 1 1 1 HI =

−1

1

−1

1 1 1

−1

1

−1

1 1

1 −1

1

1 1 1 −1

1 1

−1 −1

‫ﻫﺎﻱ ﺟﺮﻳﺎﻥ ﻭ ﻣﻌﺎﺩﻝ ﺟﺮﻳﺎﻥ‬MV ‫ﻣﺎﺗﺮﻳﺲ ﮊﺍﮐﻮﺑﻴﻦ‬: ۶‫ﺷﮑﻞ‬

‫ﻣﺮﺍﺟﻊ‬ [1] M. E. Baran and A. W. Kelly,"A Branch-CurrentBased State Estimation for Distribution Systems", IEEE Trans. on Power Systems, vol. 10, no.1, pp. 483-491, Feb. 1995 [2] Whei-Min Lin,Jen-Hao,and Shi-Jaw Chen,"A Highly Efficient Algorithm in Treating-CurrentBased Distribution StateEstimation", IEEE Trans on Power Delivery, vol. 16, no. 3, July 2001 [3] J-H. Teng,"Using Voltage Measurements to Improve the Results of Branch-Current-Based State Estimators for Distribution Systems", IEE proc Trns. Distrib, vol. 149, No. 6, November 2002 [4] Youman Deng, Ying He, and Boming Zhang"A Branch-Estimation-Based State Estimation Method for Radial Distribution Systems", IEEE Transactions on Power Delivery, vol. No.4, October 2002 [5] C. L. Lawson and R. J. Hanson, Solving Least Square Problems: Prentice-Hall, 1974. [6] G.H.Golub,"Numerical Methods for Solving Linear Least Square Problems", Numersche Mathematik, vol, 7, 1965 [7] Lin, W. M, and Teng, J. H "Distribution FastDecoupled State Estimation by Measurement Pairing ", IEEE proc, Gener. Transm. Distrib,1996, 143, pp. 43-48 [8] F. F. Wu, W.H.E.Liu,and S.-M.Lun,"Observability Analysis and Bad Data Processing for State Estimation With Equality Constraints", IEEE Trans. on Power Systems, vol. 3, no. 2, pp. 541548, May 1988 [9] W. M. Lin and J. H. Teng, "Stat Estimation for Distribution Ststems With Zero-Injection Constraint", IEEE Trans. on Power Systems, vol. 11, no. 1, pp. 518-524, Feb. 1996. [10] M.E.Baran,"State Estimation for Real-Time Monitoring of Distribution Systems", IEEE Trans on. Power Systems, vol. 9, no.3, pp.1601-1609, Aug [11] L. Holten, A. Gjelsvik, S. Aam, F.F. Wu,and W.-H. E. Liu, "Comparison of Different Methods for State Estimation", IEEE Trans. on PAS, vol. 3, no. 4, pp. 1798-1806, Nov. 1988