Level 4 Economic studies

A technically correct installation is only useful if it is the cheapest of the solutions that meet the need, or if it is profitable. This level gives the tools to demonstrate it.
Tutorial Version 2.0.0 November 2026 EN / FR Click for the French version
Level 4 — Economic studies
NRJXpert 2.0.0 · November 2026

📘Introduction

Objectives

  • Discount cash flows, account for inflation, calculate an annuity.
  • Calculate the discounted cost of a project (investment, operation, replacements, residual value) and the cost of the kWh (LCOE).
  • Assess profitability: NPV, IRR, simple and discounted payback; determine the minimum tariff and subsidy.
  • Carry out a sensitivity analysis and compare solutions (PV, diesel, grid).

Prerequisites: level 3. Duration: 10 to 12 h.

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🔹4.1 Time value of money

A franc available today is worth more than a franc in ten years: it can be invested, and the future is uncertain. The discount rate a expresses this preference (cost of capital of the investor, 6 to 12 % in West Africa).

Present value of an amount Fn paid in year n: Fn/(1 + a)n
Future cost with inflation i: Cn = C₀·(1 + i)n    real rate: (1 + a)/(1 + i) − 1
Constant annuity repaying I in N years: A = I·CRF,   CRF = a(1 + a)N/((1 + a)N − 1)

Example 4.1

(a) Present value of 1 MF spent in 10 years at 8 %? (b) Annuity of a 3 000 kF investment over 20 years at 8 %? (c) Real rate for a = 8 % and i = 3 %?

Show the solution

(a) 1 000/1.08¹⁰ = 463 kF. (b) CRF = 0.10185; A = 305.6 kF/yr. (c) 1.08/1.03 − 1 = 4.85 %.

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💰4.2 Life-cycle cost and discounted cost

The costs of a system over its operating period N include:

  • the investment I₀: equipment, installation, wiring, civil works;
  • annual operation and maintenance (O&M), increased by inflation;
  • the replacements of equipment whose lifetime is shorter than N (batteries 5 years, electronics 10 years);
  • the fuel for a genset;
  • deducted, the residual value of recent equipment at the end of the project (straight-line depreciation).
Discounted cost (NPV of costs): Cdisc = I₀ + Σn=1..N (O&Mn + Rn + fueln)/(1 + a)n − Vres/(1 + a)N

NRJXpert also shows the « life-cycle cost » of the original program (depreciation annuities over the depreciation period + undiscounted operating costs): useful to compare variants of the same project, but only the discounted cost is comparable with published studies.

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💰4.3 Levelised cost of energy (LCOE)

LCOE = Cdisc / Σn=1..N En/(1 + a)n

Energy is discounted like money: a kWh produced in 15 years is « worth » less than a kWh produced next year. The LCOE is the constant selling price that makes the project just profitable (NPV = 0). It allows technologies with different lifetimes and cost profiles to be compared. For a stand-alone system, E is the useful energy (consumed by the appliances), not the energy produced.

Example 4.2 — 10 kWp grid-connected plant

Investment 5 000 kF (500 kF/kWp); O&M 150 kF/yr (year-0 prices, inflation 3 %); inverters replaced at 10 years (600 kF); production 15 810 kWh in the first year (example 3.5), decreasing by 0.5 %/yr; 20 years; discount rate 8 %.

Show the solution

Present value of O&M: Σ 150 × 1.03n/1.08n = 1 892.6 kF; of the replacement: 600 × 1.03¹⁰/1.08¹⁰ = 373.5 kF. Cdisc = 5 000 + 1 892.6 + 373.5 = 7 266 kF.

Discounted energy: Σ 15 810 × 0.995n−1/1.08n = 149 901 kWh. LCOE = 7 266/149 901 = 0.0485 kF/kWh = 48.5 F/kWh.

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📙4.4 NPV, IRR and payback

Cash flow of year n: Fn = revenuen − costsn (F₀ = −I₀)    NPV = Σn=0..N Fn/(1 + a)n
IRR: rate that cancels the NPV    payback: year when the cumulated cash flows (discounted or not) become positive
  • NPV > 0: the project earns more than an investment at rate a; between two mutually exclusive projects, the larger NPV is preferred.
  • IRR > a: equivalent to NPV > 0 for a single project; convenient to compare with the cost of credit.
  • Payback: risk (liquidity) indicator; it ignores what happens afterwards.

Example 4.3 — Profitability of the 10 kWp plant

The energy is sold (or saved) at 60 F/kWh, constant tariff.

Show the solution

Cash flow of year 1: 15 810 × 0.060 − 150 × 1.03 = 794 kF; year 10: negative (−101 kF) because of the inverter replacement. NPV = +1 728 kF; IRR = 12.8 %; payback 6.5 years (discounted: 11.0 years). The project is profitable at the 8 % rate.

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🧩4.5 Minimum tariff and subsidy

The NPV is a linear function of the tariff and the subsidy; they are therefore obtained exactly (NRJXpert does not scan):

Minimum tariff (NPV = 0) = LCOE (if revenue follows the energy)    Minimum subsidy = max(0, −NPVwithout subsidy/I₀)

Example 4.4

For the 10 kWp plant: minimum tariff? If the tariff is only 40 F/kWh, what share of the investment must be subsidised?

Show the solution

Minimum tariff = LCOE = 48.5 F/kWh. At 40 F/kWh, NPV = −1 270 kF: minimum subsidy = 1 270/5 000 = 25.4 %. Check: NPV(50 F/kWh) = +229 kF, NPV(60) = +1 728 kF, NPV(70) = +3 227 kF: the NPV increases by 150 kF per F/kWh (= discounted energy × 1 F).

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📘4.6 Sensitivity and comparison of solutions

Discount rate4 %6 %8 %10 %12 %
LCOE of the plant (F/kWh)40.044.148.553.158.1

A PV plant is a capital-intensive energy: its cost depends mainly on the discount rate (access to credit), unlike the genset, whose cost depends mainly on the price of diesel.

Solution (orders of magnitude, 20 years, 8 %)Cost of the kWhDominant item
Grid-connected PV plant (example 4.2)≈ 50 F/kWhinvestment
5 kVA genset, 0.35 l/kWh, diesel 750 F/l≈ 385 F/kWhfuel (262 F/kWh)
Stand-alone PV with batteries (health post, example 3.3)≈ 580 F/kWhbatteries (77 %)

Stand-alone PV looks expensive, but it must be compared with the real alternative on the site (genset, grid extension over several kilometres) and with the value of the service (vaccine cold chain). A hybrid PV-battery-genset solution, optimised at level 5, is often the cheapest.

Lab 9 — « Economic viability » tab of the PV window

  1. PV window, tab 6: grid-connected system, 10 kWp, module and inverter models from the database or imported.
  2. Enter the costs of example 4.2 (installation 500 kF/kW, operation 15 kF/kW/yr), tariff 60 F/kWh, 20 years, inflation 3 %, discount rate 8 %.
  3. Compare NPV, IRR, payback and LCOE with example 4.3. The differences come from the hourly calculation (I-V model, temperature, clipping, system losses): identify them in the report.
  4. Minimum tariff and Minimum subsidy buttons; Sensitivity sub-tab: export the table NPV = f(tariff, power).
  5. Repeat the study as stand-alone: how do the LCOE and the share of the batteries change?
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🎯Exercises

Exercise 1 — Replacing a genset

A genset consumes 4 000 l/yr of diesel (750 F/l) and costs 400 kF/yr in maintenance. A PV-battery system of 18 MF would replace it entirely, with 300 kF/yr of maintenance and 6 MF of batteries to replace in years 7 and 14. Over 20 years at 8 % (without inflation), what is the NPV of the operation (savings − costs)?

Show the solution

Annual saving: 3 000 + 400 − 300 = 3 100 kF; Σ 1/1.08n (20 years) = 9.818 → 30 437 kF. Replacements: 6 000 × (1/1.08⁷ + 1/1.08¹⁴) = 6 000 × (0.5835 + 0.3405) = 5 544 kF. NPV = −18 000 + 30 437 − 5 544 = +6 893 kF. Residual value of the year-14 batteries (1/7 of life remaining) neglected: it would improve the result further.

Exercise 2 — Effect of the rate

With the calculator above (investment 10 000 kF, O&M 200 kF/yr, 15 000 kWh/yr, 20 years), from which discount rate does the LCOE exceed 100 F/kWh?

Show the solution

LCOE = (10 000·CRF + 200)/15 000 ≤ 0.1 kF/kWh ⇔ CRF ≤ 0.13. CRF(20 years) = 0.13 for a ≈ 11.5 %.

Exercise 3 — Interpreting a payback period

Two projects have the same NPV; the first pays back in 4 years, the second in 9 years. Which one should be chosen and why?

Show the solution

For equal NPV, the one that pays back faster exposes the investor for a shorter time (risk of failure, of tariff change, of exchange rate). The payback period separates projects that the NPV does not distinguish.

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📋Level 4 quiz

1. At 8 %, 1 000 kF paid in 10 years is worth today:

  1. 800 kF
  2. 463 kF
  3. 1 000 kF
  4. 2 159 kF
1 000/1.08¹⁰ = 463 kF; 2 159 kF is the future value of 1 000 kF invested for 10 years.

2. The LCOE is:

  1. the investment divided by the power
  2. the price of grid electricity
  3. the discounted cost divided by the discounted energy
  4. the energy cost of the first year
It is the constant tariff that cancels the NPV.

3. For a stand-alone system, the LCOE is calculated on:

  1. the useful energy consumed by the appliances
  2. the energy produced by the array
  3. the battery capacity
  4. the peak power
The surplus that is not consumed has no value.

4. A project has an IRR of 12 % and the discount rate is 8 %. Then:

  1. NPV < 0
  2. NPV = 0
  3. nothing can be said
  4. NPV > 0
NPV(a) > 0 for a < IRR (conventional project: an investment followed by positive cash flows).

5. The residual value at the end of the project:

  1. is added to the costs
  2. is deducted from the costs
  3. only affects the IRR
  4. is always zero
Equipment bought shortly before the end of the project still has value.

6. The cost of the kWh of a PV plant is mainly sensitive to:

  1. the price of diesel
  2. inflation
  3. the discount rate
  4. the power factor
Capital-intensive energy: the cost of capital dominates.

7. If the NPV without subsidy is −1 270 kF for an investment of 5 000 kF, the minimum subsidy is:

  1. 25.4 %
  2. 12.7 %
  3. 74.6 %
  4. 100 %
The subsidy reduces the year-0 investment, not discounted: 1 270/5 000.

8. The discounted payback period is:

  1. always shorter than the simple payback
  2. equal to the IRR
  3. independent of the rate
  4. longer than the simple payback
Future cash flows are reduced by discounting: more years are needed to offset the investment.
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← Level 3 — Renewable energyLevel 5 — DER and smart grids →

👤Author

LT
Prof. Lamine THIAW
Department of Electrical Engineering
École Supérieure Polytechnique
Cheikh Anta Diop University, Dakar
lamine.thiaw@ucad.edu.sn
lamine.thiaw@esp.sn

November 2026