*This example proves that the direct solution recursive convolution method produces the same *results as a full expansion of a binary tree. However the recursive method problem size does *not grow exponentially with the number of generators convolved whereas the binary tree does. * Below is the math for F(x) = the probability capacity is available at x MW load. * generator i forced outage rate is FOR(i) and maximum capacity is Pmax(i) * initialize F(0)=1 and all other F(x>0)=0, F(0)=1 always INTEGER PMAX(10)/5,10,13,22,43,57,101,198,248,276/,MW(1024) REAL*8 F(0:1000)/1.D0,1000*0.D0/,PR(1024),F2(0:1000)/1001*0.D0/, &FOR(10)/.15D0,.10D0,.09D0,.07D0,.06D0,.05D0,.04D0,.04D0,.04D0, &.0454D0/ DO 1 I=1,10 ! generator I DO 1 IX=1000,1,-1 ! X MW on the F(x) curve J=IX-PMAX(I) ! shift F(x) to the right by PMAX MW (the up state) IF(J.LT.0) J=0 ! any F(X<=0)=1 by definition 1 F(IX)=(1.-FOR(I))*F(J)+FOR(I)*F(IX) ! add together the up and down state F's * Program vdc continues from here with building F2(x) function from a binary tree * which allows us to verify that the recursive convolution F(x) is the same curve. * MW(i) is the MW and PR(i) is probability of each i'th state on the binary tree. * Note that 2^10 = 1024 states or different generator up-down combinations. PR(1)=1.D0-FOR(1) ! first generator up state probability PR(2)=FOR(1) ! first generator down state probability MW(1)=PMAX(1) ! first generator up state MW MW(2)=0 ! first generator down state MW J=2 ! J is the size of the binary tree at this point DO 3 I=2,10 ! next generator I DO 2 K=1,J ! binary tree position K upper states and K+J lower states MW(J+K)=MW(K) ! enter the lower tree down states of generator I, i.e. 0 MW added PR(J+K)=PR(K)*FOR(I) ! and probabilities of the lower tree down states, i.e. the FOR(i) MW(K)=MW(K)+PMAX(I) ! the MW of up state of generator I is added to the upper tree 2 PR(K)=PR(K)*(1.D0-FOR(I)) ! the probability of the generator I up states are included 3 J=J*2 ! update the binary tree size DO 4 I=1,J IX=MW(I) 4 F2(IX)=F2(IX)+PR(I) * we know that the F2(0) when summed must be exactly 1. So lets start at the top and sum probabilities. * we have to shift F2 down one notch when summing the LOLPs, i.e. the reverse of summing from the bottom * note there is a small summation rounding error at the bottom now, the recursive method doesn't have this error DO 5 IX=1000-1,0,-1 5 F2(IX+1)=F2(IX) F2(0)=0.D0 ! by definition the LOLP has to be zero if there is no load to serve DO 6 IX=1,1000 6 F2(IX)=F2(IX)+F2(IX-1) OPEN(3,FILE='OP') * MW F(X) LOLP = 1 - F(X) LOLP2 = F2(X) * 0 1.0000000000000000 0.0000000000000000 0.0000000000000000 WRITE(3,*)' MW F(X) LOLP = 1 - F(X) LOLP2', &' = F2(X)' WRITE(3,'(I4,3F20.16)')(IX,F(IX),1.D0-F(IX),F2(IX),IX=0,1000) END recursive the binary tree MW F(X) LOLP = 1 - F(X) LOLP2 = F2(X) 0 1.0000000000000000 0.0000000000000000 0.0000000000000000 1 0.9999999999991762 0.0000000000008238 0.0000000000008237 2 0.9999999999991762 0.0000000000008238 0.0000000000008237 3 0.9999999999991762 0.0000000000008238 0.0000000000008237 4 0.9999999999991762 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