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Derivations
Derivation of Equation 3.11
Given any finite continuous, discontinuous, or discrete density function f(x) or continuous or discontinuous cumulative distribution F(x), the convolution of a very small width rectangular block function of width (x and heighth 1/(x located at x = a with the f(x) or F(x) function causes a shift of the f(x) or F(x) function as shown in Figure A.1. For finite (x, the shift is aproximate. In the limit as (x(0, the shift of f(x) to the right by a is an exact process.
1/(x
f(x) ( f(x(a)
(x
0 a 0
Figure A.1 Convolution of an Impulse Function With f(x)
This can be written as an impulse function ((x(a) replacing the rectangular section. Convolving the impulse function with f(x) produces
( ((
f(x)+ = ( f(y)((x(y(a) dy = f(x(a) . (A.1)
(((
The f(x)+ is the function after convolution. The original f(x) function before convolution is simply shifted to the right by a as a result of the convolution.
The three state generator model (Table 3.1) can be written as an impulse function. For any generator k, let the state probabilities be pu=1(DFORk(FORk for the up state, pd=DFORk for the derated state, and po=FORk for the outaged state. The convolution of these generator k states into the cumulative distribution F(x) is given in Equation A.2 as
( ((
F(x)+ = ( F(y)[pu((x(y) + pd((x(y(Dk) + po((x(y(Ck)]dy . (A.2)
(((
Evaluating Equation A.2 produces
F(x)+ = puF(x) + pdF(x(Dk) + poF(x(Ck) . (A.3)
Equation A.3 is equivalent to Equation 3.11 which completes the derivation.
Derivation of Equation 3.13
A simple notation is used to develop the piecewise quadratic function shown in Equation 3.13. The resulting equation from the simpler notation derivation is shown to be equivalent to Equation 3.13. Figure A.2 shows a quadratic curve fit of any three consecutive evenly spaced points on F(x) with values F(1 , F0 , and F+1 . The discrete points are located at points x=(r, x=0, and x=r, respectively, where(1(r<1. A, B, and C are dummy coefficients in the quadratic equation Fr=Ar2+Br+C. Fr provides a smooth function interpolation of the discrete points.
F(1
F0
F+1
local range r : (1 0 +1
Figure A.2 Piecewise Quadratic Spline Fr
Solving for coefficients A, B, and C:
1. F(1 = A (B +C
2. F0 = C
3. F+1 = A +B +C (A.4)
Then C = F0 (A.5)
Adding 1 and 3 F(1 + F+1 = 2A + 2F0 (A.6)
Solving for A A = (F(1 ( F0 + (F+1 (A.7)
Subtracting 1 from 3 F+1 ( F(1 = 2B (A.8)
Solving for B B = (F+1 ( (F(1 (A.9)
Inserting the A, B, C coefficients into the original equation
Fr = ((F(1 ( F0 + (F+1)r2 + ((F+1 ( (F(1)r + F0 (A.10)
Collecting common F terms gives
Fr = ((r2 ( (r)F(1 + (1 ( r2)F0 + ((r2 + (r)F+1 (A.11)
Equation A.11 is the same as Equation 3.13 which completes the derivation.
Derivation of Equation 3.14
The convolution process uses an interpolation of Fr between the left most and central points as shown in Figure A.3. The shift of Fr in the convolution process is done by calculating the interpolated function values for D MW and C MW (shown below) for the derated and full outaged capacity states respectively. This is most convenient if the direction of r is reversed from that shown in Figure A.2. So Figure A.3 shows positive interpolation r to the left of discrete point Fi(j. New point Fi+ value is calculated from the original function to the left by C MW. The C and D MW consists of a number of integer steps j plus a fractional part of a step r. The j and r are calculated from the equation C = h(jc+rc) where h is the grid increment MW step size. The derated state is shifted by D = h(jd+rd) MW.
F i(j(1 C or D Fi+
Fi(j
Fi(j+1
r
j
local range r : +1 0 (1
Figure A.3 Piecewise Quadratic Interpolation
Inserting negative r into Equation A.11 produces
Fr = ((r2 + (r)F(1 + (1 ( r2)F0 + ((r2 ( (r)F+1 (A.12)
Let the generator state probabilities be pu, pd, and po for the up, derated, and outaged states. For any new x = ih discrete point on the F(ih)+ function, the convolution Equation A.3 becomes
Fi+ = puFi + pdF(ih(D) + poF(ih(C) . (A.13)
Inserting the j and r terms for the C and D MW shifts gives
Fi+ = puFi + pdF[ih((jd+rd)h] + poF[ih((jc+rc)h] . (A.14)
Expanding the shifted terms in the above equation using Equation A.12 produces
Fi+ = puFi
+ pd[((rd2 + (rd)Fi(jd(1 + (1 ( rd2)Fi(jd + ((rd2 ( (rd)Fi(jd+1]
+ po[((rc2 + (rc)Fi(jc(1 + (1 ( rc2)Fi(jc + ((rc2 ( (rc)Fi(jc+1]
for integers jc ( 0 and jd ( 0 and reals 0 ( rc ( 1 and 0 ( rd ( 1 . (A.15)
Equation B.15 is equivalent to Equation 3.14, which completes the derivation.
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