Operation of electric Power Systems Chapter 0: Course ... · •FC = 72 * 2 500 000 = 180 M€ •VC = 30 * 2 500 * 8760 = 657 M€ Operation of Electric Power Systems ICAI –Master

Operation of Electric Power SystemsICAI – Master en Ingeniería Industrial (MII)

Chapter 1: Simplified centralized operation and investment planning exerciseCourse 2019-2020

1

ICAI – Master Ingeniería industrial (MII)

Operation of Electric Power SystemsChapter 1: Simplified centralized operation and investment planning exercise

Michel Rivier

Javier García

Efraím Centeno

Andrés Díaz Casado



2

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

• Optimal mix with a simplified representation• Demand by means of the load-duration curve

• Generation: annualized investment and variable costs

The basic centralized context

Optimal long-term planning

Determine:- Optimal mix- Total cost

Technology N C F

Fixed cost (annualised investment cost)

210 €/kW 114 €/kW 72 €/kW

Variable cost (operation cost)

6 €/MWh 18 €/MWh 30 €/MWh



3

Content

• Stylized representation of gen. and dem.

• Thermal mix• Centralized short-term operation

• Centralized long-term investment planning

• Hydrothermal mix• Centralized short-term operation


• Mathematic formulation of the stylizedproblems



4

Comparing costs

• Comparing total costs of each technology forcovering each slice• Slice 1: 2500 MW, 8760 h.

• N: 656,4 M€• FC = 210 * 2 500 000 = 525 M€• VC = 6 * 2 500 * 8760 = 131,4 M€

• C: 679,2 M€• FC = 114 * 2 500 000 = 285 M€• VC = 18 * 2 500 * 8760 = 394,2 M€

• F: 837 M€• FC = 72 * 2 500 000 = 180 M€• VC = 30 * 2 500 * 8760 = 657 M€



5

Comparing costs

• Therefore:• Slice 1: 2500 MW, 8760 h.

• N: 656,4 M€ 29,97 €/MWh

• Slice 2: 1500 MW, 6000 h• C: 333 M€ 37 €/MWh

• Slice 3: 1000 MW, 4500 h• C: 195 M€ 43,33 €/MWh

• Slice 4: 2000 MW, 1500 h• F: 234 M€ 78 €/MWh

• Slice 5: 2000 MW, 500 h• F: 174 M€ 174 €/MWh

• Total: 39 400 GWh at 1 592,4 M€: 40,4 €/MWh



6

Results

N

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

F

C



7

Analytical comparison

• T1 substitutes T2 if:FC1 + VC 1 * h FC2 + VC 2 * h

h (FC1 - FC2) / (VC 2 - VC 1) = h*

Then, T1 substitutes T2 if the running hours are greater

than h*.

• N vs. C:

• h* = (210 000 – 114 000) / (18 - 6) = 8000 h

• C vs. F:• h* = (114 000 – 72 000) / (30 - 18) = 3500 h



8

F

C

N

Hours

600 1000 2000 3000 4000 5000 6000 7000 8000 9000

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

4

fuel

carbon

nuclear

To

tal

cost

s€

/MW

/yea

r

72

114

210

x103

Total annualized unitary (per MW) costs curves as a function of the running hours (screening curves)



9

N

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

F

C

F

C

N

60 0 1000 2000 3000 4000 5000 6000 7000 8000 90001

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

4

fuel

carbon

nuclear

72

114

210

x103



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Production Structure

• N: produces 21 900 GWh:• produces at 2500 MW for 8760h.

• C: produces 13 500 GWh:• produces at 2500 MW for 4500 h• and at 1500 MW for another 1500 h.

• F: produces 4 000 GWh:• produces at 4000 MW for 500 h,• and at 2000 MW for another 1000 h.



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N

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

F

C

Energy produced

Installed capacity



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• How to include in the study the NSE (non-servedenergy) cost?• The maximum price the demand is willing to pay for the

electricity

• Consider a non-served energy cost of 80 €/MWh

• How would the solution change if…• …we change the rate of return of the project

• …we introduce a price for CO2 (provided technologyspecific emission rates)





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Content




• Hydrothermal mix• Centralized short-term operation





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The basic centralized contextBasic hydrothermal coordination

• Economic hydro thermal dispatch based on variable costs

•Data•Consider the same demand and thermal units•Consider the following stylized representation of hydro units:• First case: Pmax 4 GW, Energy that can produce 4 GWh• Second case: Pmax2 GW, Energy that can produce 5 GWh

N C F

CV (€/MWh) 6 18 30

Pmax (MW) 3000 4000 2000



15

The basic centralized contextBasic hydrothermal coordination

• Economic hydro thermal dispatch based on variable costs• When to use the hydro resources

Pmax

PmaxEnergy



16

The basic centralized contextOptimal thermal mix (with pre-existing hydro)

• Let us reconsider the previous problem of determininig the optimal mix

• But now we have in the system (before installingthe thermal generation) a hydro plant (theinvestment has alrealdy been done)• Maximum power output: 500 MW

• The size of the reservoir is 1500 GWh

• Which is the optimal mix now?• What is it more economical when considering the

investment costs? • Dispatching the hydro in the peak or as a base load?



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• Baseload dispatch

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW Hydro: 500 MW and 1500 GWh



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• Baseload dispatch

• P = 1 500 000/8760 = 171.233 MW

• Savings (nuclear):• CF: 210 * 1000 * 171.233 = 35,96 M€• CV: 6 * 8760 * 171,233 = 9 M€• Savings = 44,96 M€

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

Hydro: 500 MW and 1500 GWh



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• Peak-load dispatch

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

Hydro: 500 MW and 1500 GWh



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•Peak-load dispatch

•Slice 5: P5 = 500 MW ; 500 h E5 = 250 GWh•Slice 4: P4 = 500 MW ; 1 000 h E4 = 500 GWh•Slice 3: E3 = 750 GWh; 3 000 h P3 = 250 MW

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

Hydro

4750 MW

6500 MW

8500 MW

• What are the savings?



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0 1000 2000 3000 4000 5000 6000 7000 8000 90001

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

fuel

Horas

Co

ste

to

tal (

pts

/kW

)

carbon

nuclear

Curvas de coste total

N

C

F•Peak-load dispatch

•Slice 5: P5 = 500 MW ; 500 h E5 = 250 GWh•Slice 4: P4 = 500 MW ; 1 000 h E4 = 500 GWh•Slice 3: E3 = 750 GWh; 3 000 h P3 = 250 MW

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

Hydro

4750 MW

6500 MW

8500 MW



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N

C

500 h 1500 h 4500 h 6000 h 8760 h

2500 MW

4000 MW

5000 MW

7000 MW

9000 MW

F

Agua

4750 MW

6500 MW

8500 MW

0 1000 2000 3000 4000 5000 6000 7000 8000 90001

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

fuel

Horas

Coste

to

tal (p

ts/k

W)

carbon

nuclear

Curvas de coste total

N

CF

Hydro



23


• Peak-load dispatch

• Investment cost savings• 500 MW of fuel (slices 5 and 4)

• 500 000 * 72 = 36 M€• 250 MW of C (slice 3)

• that are subtituted with hydro during 3000 h

• But that need to be supplyied with F during 1500 h: CF3 = 250 000 * (114-72) = 10,5 M€

• Variable cost savings• VC of F: (500*1500 - 250*1500 )* 30 = 11,25 M€• VC de C: 250 * 4500 * 18 = 20,25 M€

• Total savings: 78 M€ (> 44,96 M€)



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Content




• Hydrothermal coordination• Centralized short-term operation





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The basic centralized contextMathematical formulation

• The simplified setting:• Two periods with different demand values and duration

• Three technologies (N, C, F)• Variable costs (€/MWh)• Installed capacity (MW)• Investment costs (€/MW)

• Formulate• Minimum cost dispatch

• Capacity is already installed

• Minimum cost mix• No capacity is installed• There is a certain amount of capacity installed

• Reformulate including the non-served energy

t1 t2

d1

d2

𝑣𝑐𝑁, 𝑣𝑐𝐶 , 𝑣𝑐𝐹

𝑃𝑁, 𝑃𝐶, 𝑃𝐹

𝑖𝑐𝑁, 𝑖𝑐𝐶 , 𝑖𝑐𝐹



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• Short-term operation

• Variables:

Objective function:

𝑂𝐹 = 𝑣𝑐𝑁 ∙ (𝑝1𝑁 ∙ 𝑡1 + 𝑝2

𝑁 ∙ 𝑡2) +𝑣𝑐𝑐 ∙ (𝑝1

𝑐 ∙ 𝑡1 + 𝑝2𝑐∙ 𝑡2) + 𝑣𝑐𝐹 ∙ (𝑝1

𝐹 ∙ 𝑡1 + 𝑝2𝐹 ∙ 𝑡2)

𝑝1𝑁, 𝑝2

𝑁 , 𝑝1𝐶 , 𝑝2

𝐶 , 𝑝1𝐹 , 𝑝2

𝐹

Constraints:

𝑑1 = 𝑝1𝑁 + 𝑝1

𝐶 + 𝑝1𝐹

𝑑2 = 𝑝2𝑁 + 𝑝2

𝐶 + 𝑝2𝐹

0 ≤ 𝑝1𝑁 ≤ 𝑃𝑁, 0 ≤ 𝑝2

𝑁 ≤ 𝑃𝑁

0 ≤ 𝑝1𝐶 ≤ 𝑃𝐶 , 0 ≤ 𝑝2

𝐶 ≤ 𝑃𝐶

0 ≤ 𝑝1𝐹 ≤ 𝑃𝐹 , 0 ≤ 𝑝2

𝐹 ≤ 𝑃𝐹

t1 t2

d1

d2



27


• Now including non-served energy value

• Variables: Objective function:

𝑂𝐹 = 𝑣𝑐𝑁 ∙ (𝑝1𝑁 ∙ 𝑡1 + 𝑝2

𝑁 ∙ 𝑡2) +𝑣𝑐𝑐 ∙ (𝑝1

𝑐 ∙ 𝑡1 + 𝑝2𝑐∙ 𝑡2) + 𝑣𝑐𝐹 ∙ (𝑝1

𝐹 ∙ 𝑡1 + 𝑝2𝐹 ∙ 𝑡2) +

𝑛𝑠𝑒𝑣 ∙ (𝑝1𝑁𝑆𝐸 ∙ 𝑡1 + 𝑝2

𝑁𝑆𝐸∙ 𝑡2)

𝑝1𝑁, 𝑝2

𝑁 , 𝑝1𝐶 , 𝑝2

𝐶 , 𝑝1𝐹 , 𝑝2

𝐹, 𝑝1𝑁𝑆𝐸 , 𝑝2

𝑁𝑆𝐸

Constraints:

𝑑1 = 𝑝1𝑁 + 𝑝1

𝐶 + 𝑝1𝐹 + 𝑝1

𝑁𝑆𝐸

𝑑2 = 𝑝2𝑁 + 𝑝2

𝐶 + 𝑝2𝐹 + 𝑝2

𝑁𝑆𝐸

𝑛𝑠𝑒𝑣

0 ≤ 𝑝1𝑁 ≤ 𝑃𝑁, 0 ≤ 𝑝2

𝑁 ≤ 𝑃𝑁

0 ≤ 𝑝1𝐶 ≤ 𝑃𝐶 , 0 ≤ 𝑝2

𝐶 ≤ 𝑃𝐶

0 ≤ 𝑝1𝐹 ≤ 𝑃𝐹 , 0 ≤ 𝑝2

𝐹 ≤ 𝑃𝐹

0 ≤ 𝑝1𝑁𝑆𝐸 , 0 ≤ 𝑝2

𝑁𝑆𝐸

t1 t2

d1

d2



28


• Optimal mix

• Variables:

Objective function:

Constraints:

𝑂𝐹 = 𝑖𝑐𝑁 ∙ 𝑝𝑁+ 𝑖𝑐𝑐 ∙ 𝑝𝑐+ 𝑖𝑐𝐹 ∙ 𝑝𝐹 +𝑣𝑐𝑁∙ (𝑝1

𝑁 ∙ 𝑡1 + 𝑝2𝑁 ∙ 𝑡2) + 𝑣𝑐𝑐 ∙ (𝑝1

𝑐 ∙ 𝑡1 + 𝑝2𝑐∙ 𝑡2) +

𝑣𝑐𝐹 ∙ (𝑝1𝐹 ∙ 𝑡1 + 𝑝2

𝐹 ∙ 𝑡2) + 𝑛𝑠𝑒𝑣 ∙ (𝑝1𝑁𝑆𝐸 ∙ 𝑡1 + 𝑝2

𝑁𝑆𝐸∙ 𝑡2)

𝑑1 = 𝑝1𝑁 + 𝑝1

𝐶 + 𝑝1𝐹

𝑑2 = 𝑝2𝑁 + 𝑝2

𝐶 + 𝑝2𝐹

𝑝1𝑁, 𝑝2

𝑁 , 𝑝1𝐶 , 𝑝2

𝐶 , 𝑝1𝐹 , 𝑝2

𝐹, 𝑝1𝑁𝑆𝐸 , 𝑝2

𝑁𝑆𝐸, 𝑝𝑁, 𝑝𝐶, 𝑝𝐹

0 ≤ 𝑝1𝑁 ≤ 𝑝𝑁, 0 ≤ 𝑝2

𝑁 ≤ 𝑝𝑁

0 ≤ 𝑝1𝐶 ≤ 𝑝𝐶 , 0 ≤ 𝑝2

𝐶 ≤ 𝑝𝐶

0 ≤ 𝑝1𝐹 ≤ 𝑝𝐹 , 0 ≤ 𝑝2

𝐹 ≤ 𝑝𝐹

t1 t2

d1

d2



29


• Investment under uncertainty: two potentialscenarios (with associated probabilities)• Consider the non-served energy cost

ta1

da1

ta2

da2

tb1

db1

tb2

db2

Probability pa

Probability pb



30


• Variables:

Objective function:

Constraints:

𝑂𝐹 = 𝑖𝑐𝑁 ∙ 𝑝𝑁+ 𝑖𝑐𝑐 ∙ 𝑝𝑐+ 𝑖𝑐𝐹 ∙ 𝑝𝐹 +𝑝𝑎 ∙ [𝑣𝑐𝑁 ∙ (𝑝𝑎1

𝑁 ∙ 𝑡𝑎1 + 𝑝𝑎2𝑁 ∙ 𝑡𝑎2) + 𝑣𝑐𝑐 ∙ (𝑝𝑎1

𝑐 ∙ 𝑡𝑎1 + 𝑝𝑎2𝑐 ∙ 𝑡𝑎2)

+ 𝑣𝑐𝐹 ∙ (𝑝𝑎1𝐹 ∙ 𝑡𝑎1 + 𝑝𝑎2

𝐹 ∙ 𝑡𝑎2)]+𝑝𝑏 ∙ [𝑣𝑐𝑁 ∙ (𝑝𝑏1

𝑁 ∙ 𝑡𝑏1 + 𝑝𝑏2𝑁 ∙ 𝑡𝑏2) +𝑣𝑐𝑐 ∙ (𝑝𝑏1

𝑐 ∙ 𝑡𝑏1 + 𝑝𝑏2𝑐 ∙ 𝑡𝑏2)

+ 𝑣𝑐𝐹 ∙ (𝑝𝑏1𝐹 ∙ 𝑡𝑏1 + 𝑝𝑏2

𝐹 ∙ 𝑡𝑏2)]

𝑑a1 = 𝑝a1𝑁 + 𝑝a1

𝐶 + 𝑝a1𝐹


𝐶 + 𝑝a2𝐹

𝑝𝑎1𝑁 , 𝑝𝑎2

𝑁 , 𝑝𝑎1𝐶 , 𝑝𝑎2

𝐶 , 𝑝𝑎1𝐹 , 𝑝𝑎2

𝐹

0 ≤ 𝑝a1𝑁 ≤ 𝑝𝑁, 0 ≤ 𝑝a2

𝑁 ≤ 𝑝𝑁

0 ≤ 𝑝a1𝐶 ≤ 𝑝𝐶 , 0 ≤ 𝑝a2

𝐶 ≤ 𝑝𝐶

0 ≤ 𝑝a1𝐹 ≤ 𝑝𝐹 , 0 ≤ 𝑝a2

𝐹 ≤ 𝑝𝐹

ta1

da1

ta2

da2

pa

db2

tb1

db1

tb2

pb

𝑝𝑏1𝑁 , 𝑝𝑏2

𝑁 , 𝑝𝑏1𝐶 , 𝑝𝑏2

𝐶 , 𝑝𝑏1𝐹 , 𝑝𝑏2

𝐹 𝑝𝑁, 𝑝𝐶, 𝑝𝐹

𝑑b1 = 𝑝b1𝑁 + 𝑝b1

𝐶 + 𝑝b1𝐹


𝐶 + 𝑝b2𝐹

0 ≤ 𝑝b1𝑁 ≤ 𝑝𝑁, 0 ≤ 𝑝b2

𝑁 ≤ 𝑝𝑁

0 ≤ 𝑝b1𝐶 ≤ 𝑝𝐶 , 0 ≤ 𝑝b2

𝐶 ≤ 𝑝𝐶

0 ≤ 𝑝b1𝐹 ≤ 𝑝𝐹 , 0 ≤ 𝑝b2

𝐹 ≤ 𝑝𝐹



31

• Variables:

• Objective function:

• Constraints:

𝑂𝐹 = 𝑖𝑐𝑁 ∙ 𝑝𝑁+ 𝑖𝑐𝐶 ∙ 𝑝𝐶+ 𝑖𝑐𝐹 ∙ 𝑝𝐹 +𝑝𝑎 ∙ [𝑣𝑐𝑁 ∙ (𝑝𝑎1

𝑁 ∙ 𝑡𝑎1 + 𝑝𝑎2𝑁 ∙ 𝑡𝑎2) + 𝑣𝑐𝐶 ∙ (𝑝𝑎1

𝐶 ∙ 𝑡𝑎1 + 𝑝𝑎2𝐶 ∙ 𝑡𝑎2)

+ 𝑣𝑐𝐹 ∙ (𝑝𝑎1𝐹 ∙ 𝑡𝑎1 + 𝑝𝑎2

𝐹 ∙ 𝑡𝑎2) + 𝑛𝑠𝑒𝑣 ∙ (𝑝a1𝑁𝑆𝐸 ∙ 𝑡1 + 𝑝a2

𝑁𝑆𝐸∙ 𝑡2)]+𝑝𝑏 ∙ [𝑣𝑐𝑁 ∙ (𝑝𝑏1

𝑁 ∙ 𝑡𝑏1 + 𝑝𝑏2𝑁 ∙ 𝑡𝑏2) +𝑣𝑐𝐶 ∙ (𝑝𝑏1

𝐶 ∙ 𝑡𝑏1 + 𝑝𝑏2𝐶 ∙ 𝑡𝑏2)

+ 𝑣𝑐𝐹 ∙ (𝑝𝑏1𝐹 ∙ 𝑡𝑏1 + 𝑝𝑏2

𝐹 ∙ 𝑡𝑏2) + 𝑛𝑠𝑒𝑣 ∙ (𝑝b1𝑁𝑆𝐸 ∙ 𝑡1 + 𝑝b2

𝑁𝑆𝐸∙ 𝑡2)]]


𝐶 + 𝑝a1𝐹 + 𝑝a1

𝑁𝑆𝐸


𝐶 + 𝑝a2𝐹 + 𝑝a2

𝑁𝑆𝐸

𝑝𝑎1𝑁 , 𝑝𝑎2

𝑁 , 𝑝𝑎1𝐶 , 𝑝𝑎2

𝐶 , 𝑝𝑎1𝐹 , 𝑝𝑎2

𝐹 , 𝑝a1𝑁𝑆𝐸 , 𝑝a2

𝑁𝑆𝐸

0 ≤ 𝑝a1𝑁 ≤ 𝑝𝑁, 0 ≤ 𝑝a2

𝑁 ≤ 𝑝𝑁

0 ≤ 𝑝a1𝐶 ≤ 𝑝𝐶 , 0 ≤ 𝑝a2

𝐶 ≤ 𝑝𝐶

0 ≤ 𝑝a1𝐹 ≤ 𝑝𝐹 , 0 ≤ 𝑝a2

𝐹 ≤ 𝑝𝐹

ta1

da1

ta2

da2

pa

db2

tb1

db1

tb2

pb

𝑝𝑏1𝑁 , 𝑝𝑏2

𝑁 , 𝑝𝑏1𝐶 , 𝑝𝑏2

𝐶 , 𝑝𝑏1𝐹 , 𝑝𝑏2

𝐹 , 𝑝b1𝑁𝑆𝐸 , 𝑝b2

𝑁𝑆𝐸 𝑝𝑁, 𝑝𝐶, 𝑝𝐹


𝐶 + 𝑝b1𝐹 +𝑝b1

𝑁𝑆𝐸


𝐶 + 𝑝b2𝐹 +𝑝b2

𝑁𝑆𝐸

0 ≤ 𝑝b1𝑁 ≤ 𝑝𝑁, 0 ≤ 𝑝b2

𝑁 ≤ 𝑝𝑁

0 ≤ 𝑝b1𝐶 ≤ 𝑝𝐶 , 0 ≤ 𝑝b2

𝐶 ≤ 𝑝𝐶

0 ≤ 𝑝b1𝐹 ≤ 𝑝𝐹 , 0 ≤ 𝑝b2

𝐹 ≤ 𝑝𝐹


0 ≤ 𝑝a1𝑁𝑆𝐸 , 0 ≤ 𝑝𝑎2

𝑁𝑆𝐸 0 ≤ 𝑝b1𝑁𝑆𝐸 , 0 ≤ 𝑝b2

𝑁𝑆𝐸

Documents

Operation of electric Power Systems Chapter 0: Course ... · •FC = 72 * 2 500 000 = 180 M€ •VC = 30 * 2 500 * 8760 = 657 M€ Operation of Electric Power Systems ICAI –Master