1. 1 LRIC Updated Mathematical & Computer Modelling DCMF – 16/1/08

2. 2 LRIC in analytic form LRIC = i b (A/C) (D/C) i/r-1 /r £/kVA p.a. = i b (A/C) exp(-i t) exp(r t)/r £/kVA p.a. where b is the annuity factor i is discount rate r is growth rate D is demand

3. 3 Annuity Factors Incorrect: b based on lifetime of asset Corrected: b based on time of payment, T, from initial capacity D 0 to C b = exp(-i T)/T

4. 4 LRIC corrected LRIC2 = i (A/C) exp(-2i t) exp(r t)/rT = i (A/C) (D/C) 2i/r-1 /Log(C/D 0 ) if D 0 = C/2 = 1.44 i (A/C) (D/C) 2i/r-1

5. 5 Empirical Methods In empirical methods the cost function is selected and scaled to match the accumulated total revenue to the cost by the time of reinforcement. FCP uses the kernel exp(-i t)

6. 6 Mathematically Consistent Methods 1 For MCM the kernel is assumed to be initially unknown and take the form: p(t) where t is the time to reinforcement

7. 7 MCM 2 The cost, W, recovered by the time of reinforcement is: W (T) =  0 T p(t) D(t) exp(i t) dt This is set to equal the change in NPV: PV(0) - PV(T) = A(1 - exp(-i T))

8. 8 MCM 3 For mathematical consistency it is required that this equality is valid for all T. Hence: p(t) D(t) exp(i t) = i A exp(-i t) and p(t) = i A exp(-2i t)/D(t) = i (A/C) (D/C) 2i/r –1 The identical functional form to LRIC2

9. 9 MCM 4 A small proportion of the asset cost is not recovered in the above formula: PV 0 = A exp(- i T) = A(D 0 /C) i/r Scaling to recover this cost gives: MCM = i (A/C) (D/C) 2i/r -1 /(1-(D 0 /C) i/r ) for D 0 =C/2 and i = 6.9% Scaling factor =1.01 for r = 1%, =1.44 for r = 4%

10. 10 FCP 2 MCM provides a basis for a revised FCP, by taking the initial demand D 0 to be the demand N = 10 years prior to reinforcement FCP2 = i (A/C) (D/C) 2i/r -1 /(1- exp(-i N) )

11. 11 Conclusions  LRIC corrected, MCM, and FCP2 all have the same functional form with similar scalar multipliers.  This form is based on a more rigorous mathematical basis than hereto.  The variation with demand varies more strongly than earlier methods, delivering a more powerful message as full capacity is reached.