4 Work package 1: Merit-Order
The aim of the first step is to realize the merit order principle with a model. The available power plants are sorted in ascending order according to marginal costs and switched on until the load profile is covered in the time step. This ensures that the load profile is cov- ered at minimal cost. The result is an hourly power plant timetable for each plant. The merit-order curve and the power plant schedules are then to be graphically prepared as bar diagrams.
4.1 Input data
An exemplary power plant park is included in the model. There are different types of power plants in this model, which can be taken from Table 1. The hourly load profile for a day with 𝑇 = 24 is assumed according to Table 2:
Max. power plant power 𝑷𝒋 [MW]
Marginal costs 𝒄𝟏,𝒋 [€/MWh]
Max. power plant power 𝑷𝒋 [MW]
Marginal costs 𝒄𝟏,𝒋 [€/MWh]
Table 1: Parameters for power
16,19 6 80 17,26 7 85 16,60 8 55 16,50 9 55 19,70 10 55 plants
22,26 27,74 25,94 27,27 27,79
1 2 3 4 5 6 7 8 9 10 11 12
700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500
13 14 15 16 17 18 19 20 21 22 23 24
1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800 Table 2: Load profile over 24 hours
4.2 Description of the objective function
Several factors influence the objective function. These differ on the one hand in exoge- nously given parameters and on the other hand in variables of the optimization model. Exogenously given parameters are the time span (𝑡), the number of plants in the power plant park (𝑗) and the variable costs (𝑐1,𝑗). These are assigned to the decision variables of the objective function as coefficients. The objective function of the model is:
𝑚𝑖𝑛{∑∑(𝑐1,𝑗 ∙𝑝𝑗,𝑡)} 𝑗∈𝐽 𝑡∈𝑇
Power of power plant 𝑗 in time step t ∈R Marginal costs of power plant 𝑗 ∈R
Modelling of plant characteristics
Each unit of a thermal power plant has a technical maximum output limit. This perfor- mance limit is given by the physical resilience of the materials. The power variable 𝑝𝑗,𝑡 used in the model can be represented as follows:
0 ≤ 𝑝 ≤ 𝑃 , ∀𝑡 ∈ 𝑇, 𝑗,𝑡 𝑗
Power of power plant 𝑗 in time step 𝑡 Max. power of power plant 𝑗
Further tasks
a.) Create the merit order based on the given data on paper.
b.) Implement the merit order using the given data in Matlab.
c.) Formulate the merit order implemented in Matlab as a linear optimization model
(LP) and solve the optimization problem with the CPLEX solver.
Control results
455011513000 0000 455 35 130130 0 0 0000 45513513013000 0000 45523513013000 0000 45528513013000 0000 45538513013000 0000 45543513013000 0000 455455130130300 0000 455 455 130 130 130 0 0000 455 455 130 130 162 68 0000 455 455 130 130 162 80 03800 455 455 130 130 162 80 055330 455 455 130 130 162 68 0000 455 455 130 130 130 0 0000 455455130130300 0000 45533513013000 0000 45528513013000 0000 45538513013000 0000 455455130130300 0000 455 455 130 130 162 68 0000 455 455 130 130 130 0 0000 45538513013000 0000 45518513013000 0000 455 85 130130 0 0 0000
Costs per time step
Total costs of power plant
5 Work package 2a: Consideration of lower power limits, fixed operating costs and other operating conditions of thermal power plants
5.1 Extension of the input data
An exemplary power plant park is included in the model. There are different types of power plants in the model in terms of capacity and costs (see Table 3).
11 21 3 0 4 0 5 0 6 0 7 0 8 0 9 0
8 150 455 16,19 1000 4500 8 8
8 150 455 17,26 970 5000 8 8 -5 20 130 16,60 700 550 5 5 -5 20 130 16,50 680 560 5 5 -6 25 162 19,7 450 900 6 6 -3 20 80 22,26 370 170 3 3 -3 25 85 27,74 480 260 3 3 -1 10 55 25,94 660 30 1 1 -1 10 55 27,27 665 30 1 1 -1 105527,796703011
Table 3: Power plant park with system-specific parameters
5.2 Modification of the objective function
In the objective function from AP1, only the marginal costs of the various power plants are taken into account. Now it is also relevant in the objective function whether the power plants are in operation (vj,t = 1) or are switched off (vj,t = 0). To the already considered term of the marginal costs of a power plant the plant-specific fixed operating costs
𝑐 are added and multiplied with the indicator variable 𝑣 . The fixed operating costs are
composed 𝑐0,𝑗 of
– Staff cost,
– Costs for maintenance (e.g. lubricants)
– Lighting, pumps, etc.
Status befo- re optimiza- tion period
Operating intervals be- fore optimisa- tion period [h]
Min. PP- power 𝑷𝒋 [MW]
Max. PP- power 𝑷𝒋 [MW]
Marginal costs 𝒄𝟏,𝒋 [€/MWh]
Fixed op- erating costs
Start-up cost 𝒄𝐬,𝒋 [€]
Downtime DT [h]
Uptime UT [h]
The costs 𝑐0,𝑗 are power-independent and are taken into account in the objective func- tion, as soon as the corresponding power plant is in operation.
The objective function of the model is now given by:
𝑝𝑗,𝑡 power of power plant 𝑗 in time step 𝑡 c1,j marginal costs of the power plant 𝑗
𝑣 indicator variable, if power plant 𝑗 is online in time step 𝑗,𝑡 𝑡
𝑐0,𝑗 fixed operating costs of power plant 𝑗
5.3 Modification of the power constraint
𝑚𝑖𝑛{∑∑(𝑐 ∙𝑣 +𝑐 ∙𝑝 )}
0,𝑗 𝑗,𝑡 1,𝑗
In order to ensure stable operation of the power plant, it must be operated at a minimum technical power level. Therefore, the lower power limit 𝑃 of a power plant block is intro-
duced here. The variable 𝑝𝑗,𝑡 corresponds to the generated power of a power plant 𝑗 in
the time span 𝑡 and can be either the value zero or a value between 𝑃 and 𝑃 . The con- 𝑗𝑗
straint (5-2) takes account of the fact that the power plant j provides power as soon as
the value of the switching variable 𝑣 is one. If a power plant is at standstill (𝑝
= 0), (5-2)
𝑣 = 0 applies. The power constraint is: 𝑗,𝑡
𝑣𝑗,𝑡 𝑝𝑗,𝑡 𝑃
𝑃∙𝑣 ≤𝑝 ≤𝑃∙𝑣 ∀𝑡∈𝑇,∀𝑗∈𝐽. 𝑗 𝑗,𝑡 𝑗,𝑡 𝑗 𝑗,𝑡
Indicator variable, if power plant 𝑗 in time step 𝑡 is online Power of power plant 𝑗 in time step 𝑡
Min power of power plant 𝑗
∈ {0,1} ∈R ∈R
Max power of power plant 𝑗
Modelling of temporal operating restrictions
In order to prevent increased attrition of the systems due to thermomechanical stresses, minimum operating and downtimes must be observed. The inequality (5-3) represents the mathematical implementation of the minimal downtime of a thermal power plant.
∈ {0,1} ∈R
Minimal downtime (bucMinDown)
∑𝑣 +𝑠𝑜𝑛 ∙𝐷𝑇 ≤𝑏𝑢𝑐
𝑗,(𝑡−𝑢) 𝑗,𝑡 𝑗 𝑀𝑖𝑛𝐷𝑜𝑤𝑛,𝑗
𝐷𝑇 Minimal downtime of power plant j
= 0) for the du-
𝑗 ∈N 𝑣𝑗,𝑡 Indicator variable, if power plant 𝑗 is online in time step 𝑡 ∈ {0,1} 𝑠𝑜𝑛𝑗,𝑡 Indicator variable, if power plant 𝑗 starts in time step 𝑡 ∈ {0,1}
The sum of 𝑣 is used to check whether a power plant was inactive (𝑣 𝑗,𝑡 𝑗,𝑡
ration of the minimum downtime and may be switched on at time step 𝑡 (𝑠𝑜𝑛 𝑗,𝑡
Three different cases can occur in this consideration:
1) The power plant was off for longer than the minimal downtime.
2) The power plant is already in operation.
3) The power plant was not in operation, but the minimal downtime has not yet been
It is also necessary to consider the states before the optimization period, as these are
not included in the sum of the power plant states∑DTj vj,(t−u). For this purpose the u=1
bucMinDown,j vector is introduced, which reflects the three possible states before the op- timization period.
𝑏𝑢𝑐𝑀𝑖𝑛𝐷𝑜𝑤𝑛,𝑗(𝑡) = {
𝐷𝑇 Minimal downtime of power plant j
𝑤𝑒𝑛𝑛 |𝐵𝑣𝑂 | + 𝑡 ≤ 𝐷𝑇 𝑗 𝑗
This must be generated accordingly for each case. The correct functioning of the re- striction should take place based on the results with variation of the input data (BvO).
Indikator variables (𝒔𝒐𝒇𝒇𝒋,𝒕 𝒂𝒏𝒅 𝒔𝒐𝒏𝒋,𝒕)
The variables 𝑠𝑜𝑛 and 𝑠𝑜𝑓𝑓 are derived as follows from the indicator variable
∀𝑡 ∈ 𝑇,𝑗 ∈ 𝐽 𝑗 ∈N
Operating intervals that the power plant was in operation BvO (BvO>0) or was not in operation (BvO<0) prior to optimiza-
𝑣 (𝑡) − 𝑣 (𝑡 − 1) = 𝑠𝑜𝑛 (𝑡) − 𝑠𝑜𝑓𝑓 (𝑡), ∀𝑡 ∈ 𝑇, ∀𝑗 ∈ 𝐽 (5-5) 𝑗𝑗𝑗𝑗
𝑣𝑗,𝑡 Indicator variable, if power plant j is online in time step t 𝑠𝑜𝑛𝑗,𝑡 Indicator variable, if power plant j starts up in time step t
𝑠𝑜𝑓𝑓 Indicator variable, if power plant j shuts down in time step t 𝑗,𝑡
∈ {0,1} ∈ {0,1} ∈ {0,1}
Due to the consideration of the value 𝑣 (𝑡 − 1), which lies outside the optimization period 𝑗
in the first time step, the right- and lefthandside vectors in 𝑡 = 1 must be adjusted accord- ing to the operating status of the power plant.
It should also be noted that the newly introduced variables 𝑠𝑜𝑛 and 𝑠𝑜𝑓𝑓 must be tak- 𝑗,𝑡 𝑗,𝑡
en into account in the target function. Since only switching on has a financial effect in the form of start-up costs, the variable 𝑠𝑜𝑛 is assigned fixed start-up costs of 𝑐 . The vari-
able 𝑠𝑜𝑓𝑓 gets the objective function coefficient zero. 𝑗,𝑡
∈ {0,1} ∈ {0,1}
The following constraint ensures that both indicator variables do not assume the value one:
𝑠𝑜𝑛 +𝑠𝑜𝑓𝑓 ≤1, ∀𝑡∈𝑇,∀𝑗∈𝐽 𝑗,𝑡 𝑗,𝑡
𝑠𝑜𝑛𝑗,𝑡 Indicator variable, if power plant 𝑗 starts up in time step 𝑡 𝑠𝑜𝑓𝑓 Indicator variable, if power plant 𝑗 shuts down in time step 𝑡
The objective function of the model is therefore:
𝑚𝑖𝑛{∑∑(𝑐 ∙𝑣 +𝑐 ∙𝑝 +𝑐 ∙𝑠𝑜𝑛 )}
𝑣𝑗,𝑡 Indicator variable, if power plant j is online in time step t
𝑝𝑗,𝑡 Power of power plant j in time step t
𝑠𝑜𝑛𝑗,𝑡 Indicator variable, if power plant j starts up in time step t
𝑐0,𝑗 Fixed operating costs of power plant j
0,𝑗 𝑗,𝑡 1,𝑗 𝑗,𝑡 𝑠,𝑗
Programming Help
Marginal costs of power plant j ∈R Warm start costs of power plant j ∈R
Further tasks
Discuss the statements of the above-mentioned constraint and their effect. Formu- late these in writing and describe the implementation of the 𝑏𝑢𝑐𝑀𝑖𝑛𝐷𝑜𝑤𝑛 vector. Formulate the 𝑏𝑢𝑐𝑀𝑖𝑛𝐷𝑜𝑤𝑛 vector algebraically. Please note how long the power plant has been in operation/stillstand before.
Control results
1 455245000000
2 455295000000
3 455395000000
4 4554550040000
5 4554550090000
6 455455013060000
7 45545501301100000022632
8 45545501301600000023617
0 0 13565 0 0 14428 0 0 16154 0 0 19328 0 0 19413 0 0 22207
Costs per time step
45513013013000000 455130130162680000 4551301301628038000 455 130 130 162 80 33 5500
13 4554551301301626800 00
14 455455130130130000 00
15 4554550130160000 00
16 455440013025000 00
17 455390013025000 00
18 455455013060000 00
19 4554550130160000 00
20 4554551301301626800 00
21 455455130130130000 00
22 455455013060000 00
23 455445000000 00
24 455345000000 00
Total costs of power plant
𝑣𝑗,𝑡 Indicator variable, if power plant j is online in time step t ∈ {0,1}
𝑠𝑜𝑓𝑓 Indicator variable, if power plant j is switched off in time step t ∈ {0,1} 𝑗,𝑡
Time status of power plant j
𝑈𝑇 Minimal operating time of power plant j
The structure of this inequality also corresponds to the same principle as inequality (5-3). To determine if a power plant can be shut down in time step 𝑡 (𝑠𝑜𝑓𝑓 = 1) , it is checked
if this power plant was in active state (𝑣 = 1) for the duration of the minimal operating 𝑗,𝑡
time. This is done by adding up all operating states 𝑣 over the minimal operating peri- 𝑗,𝑡
od. Two cases have to be considered:
1) The power plant was in operation long enough and may therefore be shut down. 2) The power plan was in operation, but has not reached the minimal operating time.
Since with the help of this sum ∑UTj vj,(t−u) only states within the optimization period can u=1
be included, it is necessary to introduce a vector blcMinUp.
6 Work package 2b: Consideration of lower power limits, fixed operating costs and other operating restrictions of thermal power plants
6.1 Continuation of the modelling of temporal operating re- strictions
The inequality (6-1) shows the minimum operating time of a thermal power plant. Similar to inequality (5-3), this applies to the prevention of component wear due to thermome- chanical stresses. The mathematical realization is:
Minimal operating time (blcMinUp)
𝑈𝑇𝑗 𝑏𝑙𝑐 ≤∑𝑣
−𝑠𝑜𝑓𝑓 ∙𝑈𝑇, ∀𝑡∈𝑇,∀𝑗∈𝐽. 𝑗,𝑡 𝑗
𝑏𝑙𝑐 𝑀𝑖𝑛𝑈𝑝,𝑗
−(𝑈𝑇 −𝑡+1), 𝑖𝑓 𝑡≤𝑈𝑇 ∧𝐵𝑣𝑂 +𝑡>𝑈𝑇 𝑗𝑗𝑗𝑗
(𝑡)={−𝐵𝑣𝑂, 𝑖𝑓𝑡≤𝑈𝑇 ∧𝐵𝑣𝑂 >0∧𝐵𝑣𝑂 +𝑡≤𝑈𝑇
𝑗 𝑗 𝑗 𝑗 𝑗 ∀𝑡 ∈ 𝑇, 𝑗 ∈ 𝐽
0, 𝑒𝑙𝑠𝑒 𝑈𝑇 Minimal operating time of power plant j
operating intervals, which the power plant was in operation (BvO>0)
or was not in operation (BvO<0) before optimization
Based on the operating intervals before the optimization period, this contains information on how long the power plant j has to run in the optimization period at the beginning be- fore it can be shut down.
Verification should take place by varying the input data.
6.2 Further tasks
a.) Discuss the statements of the above-mentioned restrictions and their effects. Formulate these in writing and describe the implementation of the 𝑏𝑙𝑐𝑀𝑖𝑛𝑈𝑝 vector.
b.) Formulate the 𝑏𝑙𝑐𝑀𝑖𝑛𝑈𝑝 vector algebraically. Please note how long the power plant has been in operation/still stand before.
Control results
4552450000 0000 4552950000 0000 4553950000 0000 455365013000 0000 455415013000 0000 4554550130600 0000 45545501301100 0000 45545501301600 0000 455 455 130 130 130 0 0000 455 455 130 130 162 68 0000 455 455 130 130 162 80 03800 455 455 130 130 162 80 055330 455 455 130 130 162 68 0000 455 455 130 130 130 0 0000 45545513001600 0000 4554401300250 0000 4553901300250 0000 4554551300600 0000 45545513001600 0000 455 455 130 130 162 68 0000 455 455 130 130 110 20 0000 4554550130060 0000 455315013000 0000 455215013000 0000
Costs per time step
Total costs of power plant
7 Work package 3: Modelling of further temporal operating restrictions
In order to counteract life-shortening thermomechanical stresses of the power plant due to high frequent start-ups and shut-downs, its number of starting procedures and operat- ing intervals are limited in accordance with Table 4.
1 24 0 24 6 24 0 24 2 24 0 24 7 24 0 24 3 24 0 24 8 24 0 24 4 24 0 24 9 24 0 24 5 24 0 24 10 24 0 24
Table 4: System parameters for start procedures and operating intervals
The restriction of the maximum number of start procedures is implemented by restriction (7-1).
Max. amount of
Min. amount of
operating intervals
Max. amount of
operating intervals
Max. amount of
Start- ups
Min. amount of
Operating intervals
Max. amount of
operating intervals
0 ≤ ∑ 𝑠𝑜𝑛 (𝑡) ≤ A, ∀𝑡 ∈ 𝑇, ∀𝑗 ∈ 𝐽 𝑗
𝑠𝑜𝑛𝑗,𝑡 Indicator variable, if power plant j starts in time step t
Variable to limit the start processes
∈ {0,1} ∈N
The constraint (7-1) does not represent a serious impairment for a power plant park with large-scale thermal plants and an observation period of one day. This condition is given particular emphasis if a period of one month or more, or smaller plants, such as CHP plants, are considered.
The same applies to the minimum and maximum number of operating intervals that can be assigned to a power plant 𝑗 for the period. The inequality for this restriction is as fol- lows:
Indicator variable, if power plant j is online in time step t Minimum amount of operating intervals
∈ {0,1} ∈N ∈N
𝑟𝑓 ≤ ∑ 𝑣 (𝑡) ≤ 𝑟𝑓 , ∀𝑡 ∈ 𝑇, ∀𝑗 ∈ 𝐽
Maximum amount of operating intervalls
The variables 𝐴, 𝑟𝑓 , 𝑟𝑓 are represented by vectors with power plant-specific values.
7.1 Further tasks
In addition to the implementation, the effect of the new constraints has to be investigated. First, assume that a power plant can be active for a minimum of 0 hours and a maximum of 24 hours and that the maximum number of starts is 𝐴 = 24:
a.)Analyze how limiting the maximum number of starts of peak load power plants (𝐴 = 1 for power plant 6) affects the cost structure by modifying 𝐴 accordingly and reviewing the new results.
b.) How does limiting the operating intervals of baseload power plants affect the cost structure? To do this, limit the number of operating intervals. Check your results again (first 𝑟𝑓 = 6 for power plant 5, then 𝑟𝑓 = 4 for power plant 10; 𝐴 = 24
for power plant 6).
7.2 Control results
[𝒓𝒇𝒎𝒊𝒏(𝑷𝑷𝟏𝟎) = 𝟎; 𝒓𝒇𝒎𝒂𝒙(𝑷𝑷𝟓) = 𝟐𝟒; 𝑨(𝑷𝑷) = 𝟏]
1 455245000
2 455295000
3 455395000
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 38 0 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 55 0 0 0 0 0 0 0 0
33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0
Costs per time step
4 455 365 0
5 455 415 0
6 455 455 0
7 455 455 0
8 455 455 0
9 455 455 130
10 455 455 130
11 455 455 130
12 455 455 130
13 455 455 130
14 455 455 130
15 455 455 0
16 455 440 0
17 455 390 0
18 455 455 0
19 455 455 0
20 455 455 130
21 455 455 130
22 455 385 130
23 455 315 130
24 455 215 130
130 600 130 110 0 130 160 0 130 130 0 130 162 68 130 162 80 130 162 80 130 162 68 130 130 0 130 160 0 130 25 0 130 25 0 130 60 0 130 160 0 130 162 0 130 130 0 130 0 0
Total costs of power plant
Code Help
[𝒓𝒇𝒎𝒊𝒏(𝑷𝑷𝟏𝟎) = 𝟒; 𝒓𝒇𝒎𝒂𝒙(𝑷𝑷𝟓) = 𝟐𝟒; 𝑨(𝑷𝑷𝟔) = 𝟐𝟒]
Costs per time step
10 455 455 130 130 162
11 455 455 130 130 162
12 455 455 130 130 162
13 455 455 130 130 162
14 455 455 130 130 130
15 455 455 0 130 160
16 4554400130250 0
17 4553900130250 0
18 4554550130600 0
1 455245000
2 455295000
3 455395000
4 4554550040
5 4554550090
6 45545500162
7 455 455 130 0 110
8 455 455 130 0 160
9 455 455 130 130 130
00013565 00014428 00016154 00019328 00019413 002822309 00023215 00023650 00026444 00028568 003830421
80 0 55 0 33 32369
68 0 0 0 0 0
00028398 00025884 00023617 00020698 00019835 00021647 00023617 0 55 13 30320 00025884 00021665 00017631 00015905
19 455 455 0 130 160
20 455 455 130 130 162
21 455 455 130 130 130
22 45538513013000 0
23 455315130 0 0 0 0
24 455215130 0 0 0 0
0 0 0 0 0 0
Total costs of power plant
[𝒓𝒇𝒎𝒊𝒏(𝑷𝑷𝟏𝟎) = 𝟎; 𝒓𝒇𝒎𝒂𝒙(𝑷𝑷𝟓) = 𝟔; 𝑨(𝑷𝑷𝟔) = 𝟐𝟒]
Costs per time step
1 45524500
2 45529500
3 45539500
4 455 365 0 130
5 455 415 0 130
6 455 385 130 130
7 455 435 130 130
8 455 455 130 130
9 455 455 130 130
10 455 455 130 130
11 455 455 130 130
12 455 455 130 130
13 455 455 130 130
14 455 455 130 130
15 455 455 130 130
16 455 335 130 130
17 455 285 130 130
18 455 385 130 130
19 455 455 130 130
20 455 455 130 130
21 455 455 130 130
22 455 385 130 130
23 4554450000
24 4553450000
0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 0300 0 000 0 000 0 3800 0 55330 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 55 55 40 0 50 0 0 0 000 0 000 0 000
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
130 0 162 68 162 80 162 80 162 68 110 20
030 00 00 00 030 080 080 00
Total costs of power plant
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