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North Sea Transnational GridWork Package 5
Optimization of North Sea TransnationalGrid solutions
AuthorsTUDelft: Silvio RodriguesTUDelft: Pavol BauerECN: Edwin WiggelinkhuizenECN: Jan Pierik
October, 2013
This document was prepared using LATEX.
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
1 Introduction 1
1.1 Work Package 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Main conclusions of the Work Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Reports and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Transmission Systems Design 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Component Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Offshore Wind Farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.2 AC Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.3 DC Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.4 Voltage Source Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.5 Other Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 MTdc Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 MTdc network Power Flow Optimization 15
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Optimal power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 MTdc Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 MTdc network single objective optimization: load flow control to minimize losses . . . . . 18
3.4 MTdc network multi-objective optimization: load flow control for minimal losses and max-
imum social welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
i
4 Conclusions 33
4.0.1 MTdc Control: Single-Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.0.2 MTdc Control: Multi-Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.0.3 Transmission Systems Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.0.4 Multi-Objective Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . 34
Bibliography 39
Appendix A MTdc network technical and economic aspects 40
A.1 MTdc networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.1.1 Applications of a MTdc grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.1.2 Challenges towards a VSC-based MTdc network . . . . . . . . . . . . . . . . . . . 42
A.2 Market Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.2.1 Pool Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.2.2 Transmission Loss Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.3 MTdc Network Control Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.3.1 Information flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.3.2 Social Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Appendix B Multi-Objective Optimization Algorithms 50
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
B.2 Proposed Multi-Objective Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . 52
B.2.1 Population Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
B.2.2 Fitness evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.2.3 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.2.4 Population Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
B.2.5 Population Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B.2.6 Archive Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B.2.7 Ending criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
B.2.8 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.2.9 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.2.10 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B.2.11 Regeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
B.2.12 Chromosome composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ii
B.2.13 Population composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Appendix C Benchmark Results 64
iii
List of Tables
3.1 System Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 MTdc Network Cables length and Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Description of the analyzed case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 MTdc Network Rated Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Description of the analyzed case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Power flow results and loss allocation coefficients for the analyzed case studies. . . . . . . 25
3.7 OWFs market bids, capacity factors and power productions. . . . . . . . . . . . . . . . . . 28
3.8 Onshore ac networks market bids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.9 Description of the analyzed case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.10 Direct voltages, Power flow and loss allocation coefficients for the analyzed case studies. . 30
B.1 Possible rank values and their description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
B.2 Description of the population sorting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B.3 Description of the field Origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
C.1 Contribution of each genetic operator for the non dominated solutions in the final Archive. 68
iv
List of Figures
1.1 Multi-criteria decision making process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Optimization platform for the design of transmission systems. . . . . . . . . . . . . . . . . 5
2.2 Weibull distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Offshore wind farm normalized power curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Losses variation of three different AC Cables for a 100 km transmission system. . . . . . . 7
2.5 Losses variation of three different DC Cables for a 100 km transmission system. . . . . . . 8
2.6 Losses variation of the voltage-source converter. . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Configuration of the HVAC and HVDC transmission systems. . . . . . . . . . . . . . . . . 9
2.8 Variation of the transmission system losses percentage for both technologies with the trans-
mission distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9 Transmission system losses percentage breakdown through the components of the DC
system when a distance of 100 km and minimization of losses are considered. . . . . . . . 11
2.10 Transmission system losses percentage breakdown through the components of the AC sys-
tem when a distance of 100 km and minimization of losses are considered. . . . . . . . . . 11
2.11 Variation of the total investment cost for both technologies with the transmission distance. 12
2.12 Transmission system losses percentage breakdown through the components of the DC
system when a distance of 100 km and minimization of total investment costs are considered. 12
2.13 Optimal Pareto Fronts for both transmission technologies when a distance of 50 km is
considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.14 Optimal Pareto Fronts for both transmission technologies when a distance of 100 km is
considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.15 Optimal Pareto Fronts for both transmission technologies when a distance of 150 km is
considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Fitness evaluation flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Constraints incorporated in the genetic algorithm. . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The 19 node-meshed connected offshore MTdc network used in the simulations. . . . . . . 18
3.4 Wind farms power production during the simulation. . . . . . . . . . . . . . . . . . . . . . 19
v
3.5 Transmission power losses during the simulations for the three tested cases. . . . . . . . . 21
3.6 Direct voltage at the OWFs and onshore nodes during the simulations for the three tested
cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.7 Active power at the onshore nodes during the simulations for the three tested cases. . . . 22
3.8 Meshed MTdc network used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . 24
3.9 Network power losses and social welfare. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.10 Pareto Front and Market Clearing Price (disregarding losses) for the analyzed case studies. 26
3.11 Pareto Fronts between Social Welfare and Power Losses. . . . . . . . . . . . . . . . . . . . 29
3.12 Available Pareto front between social welfare and power losses and respective buyers and
sellers welfare for the considered case studies. . . . . . . . . . . . . . . . . . . . . . . . . . 29
A.1 Diagram of a pool market applied to a future offshore MTdc network. . . . . . . . . . . . 44
A.2 Flow chart of the distributed voltage control method. . . . . . . . . . . . . . . . . . . . . . 47
A.3 Information flow between the OWFs, the GA and the onshore VSC stations. . . . . . . . 47
A.4 Social welfare of two areas, 1 and 2, for three distinctive cases: (a) no trade; (b) limited
trade; (c) unlimited trade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B.1 Pareto front of a Multi-Objective Minimization Problem. . . . . . . . . . . . . . . . . . . 51
B.2 Flow chart of MOOA implemented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
B.3 Population initialization process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.4 Pareto front of a Multi-Objective Minimization Problem. . . . . . . . . . . . . . . . . . . 55
B.5 Archiving process of the non dominated solutions. . . . . . . . . . . . . . . . . . . . . . . 57
B.6 Crowding competition among the non dominated solutions present in the archive. . . . . . 58
B.7 Selection of solutions from the current population and archive to population the mating
pool. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.8 Possible effects of the crossover operator on the genes of the new solution. . . . . . . . . . 60
B.9 Desired effect of the mutation operator on the genes of the new solution. . . . . . . . . . . 61
B.10 Possible effects of the renegeration operator on the genes of the new solution. . . . . . . . 62
B.11 Composition of the chromosomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
B.12 Composition of the population after the genetic operators. . . . . . . . . . . . . . . . . . . 63
C.1 Contribution of each genetic operator to the population with new solutions. . . . . . . . . 64
C.2 Non dominated solutions of the final Archive obtained by the developed MOOA. . . . . . 68
vi
Chapter 1
Introduction
1.1 Work Package 5
The design of complex engineering systems, such as the North Sea Transnational Grid, requires application
of knowledge from several disciplines (multidisciplinary) of engineering and economy (electrical aspects
of multi-terminal converter, control aspects, grid aspects, costs). The interdisciplinary nature of complex
systems presents challenges associated with modeling, simulation, computation time and integration of
models from different disciplines. The increased complexity of systems as well as the increased number of
design parameters in the optimization results in the necessity to use mathematical system models and the
application of optimization techniques. An adequate system model is built and optimization algorithms
are employed to determine the optimal parameters for the system based on model results.
A well-known optimization method used gradient search algorithms. A major barrier to the use of gradient
based search methods is that complex multidisciplinary design spaces tend to have many apparent local
optima. Therefore a new method, based on multi-objective optimization and multi-criteria decision
making, called Progressive Design Methodology (PDM), will be used.
The Progressive Design Methodology is a three-step design process. In the first step a simple model of the
components of a system is developed and the problem is reformulated as a Multi-Objective Optimization
Problem (MOOP). In the second step the results obtained in the MOOP process are analyzed and a small
set of feasible solutions is selected. In the final step, a detailed model of the variants of the system, as
selected from the previous set, is developed and the optimization variables of the system are fine-tuned.
The objective of optimization will be to determine a configuration of the North Sea Transnational Grid
with low levelized production cost, high power quality, high reliability and high low voltage ride through
capability. The set of optimization objectives can be extended and the optimization variables will be the
result of inventory. Fig. 1.1 shows a diagram of the three-step Progressive Design Methodology.
1
System requirementsanalysis
Definition of boundaries
Development of a system’s model
Multi-objective optimization
Set of Pareto Fronts
New constrains added
Multi-criteria decision making
Reduced set of solutions
Detailed model
Reduced set of solutions
Multi-objective optimization
Final decision
Synthesis Intermediate analysis Final design
Figure 1.1: Multi-criteria decision making process.
1.2 Main conclusions of the Work Package
• A multi-objective optimization algorithm was developed and applied in different case studies;
• Single- and multi-objective optimal power-flows inside a large MTdc network were achieved using
the proposed control strategy;
• The performed dynamic numerical simulations show that the power flow control is accurate and
fast, i.e. a new operating point can be achieved within 200 ms;
• Optimal pareto fronts were achieved for the design of a wind farm transmission system for both
AC and DC options.
2
1.3 Reports and Publications
1. S. Rodrigues, “Multi-Objective Optimization Problems and Algorithms - A Survey,” Report, EPP
- TU Delft, December 2011.
2. S. Rodrigues, “Market in Electric Power Systems and Transmission Losses Allocation,” Report,
EPP - TU Delft, February 2012.
3. S. Rodrigues, R. Teixeira Pinto, P. Bauer, and J. Pierik, “Optimization of social welfare and
transmission losses in offshore mtdc networks through multi-objective genetic algorithm,” in Power
Electronics and Motion Control Conference (IPEMC), 2012 7th International, 2012, pp. 1287-1294.
4. S. Rodrigues, P. Bauer, and J. Pierik, “Comparison of offshore power transmission technologies:
A multi-objective optimization approach,” in 15th International Power Electronics and Motion
Control Conference (EPE/PEMC), 2012.
5. S. Rodrigues, R. Pinto, P. Bauer, E. Wiggelinkhuizen, and J. Pierik, “Optimal power flow of vsc-
based multi-terminal dc networks using genetic algorithm optimization,” in Energy Conversion
Congress and Exposition (ECCE), 2012.
6. R. Teixeira Pinto, P. Bauer, S. Rodrigues, E. Wiggelinkhuizen, J. Pierik, and B. Ferreira, “A
novel distributed direct-voltage control strategy for grid integration of offshore wind energy systems
through mtdc network,” IEEE Transactions on Industrial Electronics, vol. 60, no. 6, pp. 2429-2441,
2013.
7. R. Teixeira Pinto, S. Rodrigues, E. J. Wiggelinkhuizen, R. Scherrer, P. Bauer and J. Pierik: Opera-
tion and Power Flow Control of Multi-Terminal DC Networks for Grid Integration of OffshoreWind
Farms using Genetic Algorithms; ”Energies Journal,” submitted.
8. R. Teixeira Pinto, S. Rodrigues and P. Bauer: Optimal Control Tuning of Grid-Connected Voltage-
Source Converters using a Multi-Objective Genetic Algorithm; abstract submitted to PCIM 2013
Conference.
9. S. Rodrigues, R. T. Pinto , P. Bauer, and J. Pierik, “Optimal power flow control of vsc-based
multi-terminal dc network for offshore wind integration in the north sea,” IEEE Transactions on
Emerging and Selected Topics in Power Electronics, 2013.
3
Chapter 2
Transmission Systems Design
2.1 Introduction
Most of the existing offshore wind farms make use of AC transmission technology [1]. The Horns Rev
wind farm, built in 2002, in Denmark, has an installed capacity of 160 MW and was the first wind farm
to make use of an offshore transformer substation. The connection to shore is made via a three-core AC
cable with a rated voltage of 150 kV. On the other hand, the first Dutch offshore wind farm, Egmond aan
Zee (108 MW), built in 2006; and the first UK offshore wind farm, North Hoyle (60 MW), are examples
of systems which did not require an offshore transformer substation. The latter is connected to the UK
onshore grid through 33-kV AC cables [2].
Offshore wind farms are increasingly being built further from the coast and in deeper waters. In 2011 the
average water depth of offshore wind farms was circa 23 m and the average distance to shore was around
24 km [1]. For projects under construction, the average depth is 25 m, whereas the distance to shore is
33 km. According to the study in [1] this trend will continue in the short future.
In this way, it is expected that investment costs of future large offshore wind farms will increase with
the longer distances to shore. Therefore, it is important to study and compare the possible transmission
technologies in order to reduce investment costs and energy losses as much as possible.
Several studies which compared the costs of AC and DC transmissions systems for different wind farm
installed capacities and distances to shore are possible to be found in the related literature [3, 4, 5, 6].
However, they only optimize one criterion at a time, e.g. either energy losses or investment costs or LPC.
Examples of MOOAs application to the optimization design of electrical power transmission systems and
design of offshore structures may be found in [7] and [8], respectively.
The main objective of the this chapter is to implement a multi-objective approach to compare HVAC
and HVDC transmission alternatives. The components of the transmission system will be selected from
a database which contains real component data and costs.
4
2.2 Optimization Method
In order to apply the MOOA to the optimization of the design of offshore transmission systems it is
necessary to identify the variables of the problem. In this way, each solution X of the population is a
vector composed by several variables as shown as follows:
X =[α c1 · · · cn
](2.1)
The first field of each solution, α, determines whether x represents a HVAC or HVDC transmission
system. If α = 1, an AC transmission system is considered. On the other hand, if α = 2, the solution X
corresponds to a DC system. Every chromosome X holds one variable per component of the respective
transmission system. Such variables indicate which components of the database are used. Finally, a
variable representing the amount of elements for each component is present in the chromosomes. In Fig.
2.1 it is show the flow chart of the optimization method and its integration in the MOOA.
6. Terminate?
1. Population Initialisation 2. Finess Evaluation
3. RankPopulation
4. SortPopulation
5. ArchiveUpdate
7. Selection 8. Crossover 9. Mutation
11. Output Archive
Yes
No
10. Regeneration
Database with component information
Component models
Constraints
Total Investment Costs and Energy Losses
Figure 2.1: Optimization platform for the design of transmission systems.
2.2.1 Constraints
In order to obtain feasible solutions, constraints were implemented in the MOOA. For the AC configura-
tion the following constraints were used:
⎧⎪⎨⎪⎩
0.7V LT1 ≤ V cable
AC ≤ 1.3V LT1
0.7V HT1 ≤ V cable
HVAC ≤ 1.3V HT1
0.7V HT2 ≤ V cable
HVAC ≤ 1.3V HT2
(2.2)
For the DC technology the constraints used were as follows:
5
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
0.7V LT1 ≤ V cable
AC ≤ 1.3V LT1
0.7V HT1 ≤ V REC
AC ≤ 1.3V HT1
0.7V HT2 ≤ V INV
AC ≤ 1.3V HT2
mconvertersi ≤ 1.1
0.7V RECDC ≤ V cable
HVDC ≤ 1.3V RECDC
0.7V INVDC ≤ V cable
HVDC ≤ 1.3V INVDC
(2.3)
Margins are given to the transformers’ voltages. In this way, it is considered that the transformers have
a tap changer which allows a variation of 30% in the voltage. Moreover, it is made sure that the power
ratings of all the components are respected.
2.3 Component Models
In order to optimize the design of offshore transmission systems it is necessary to model every component.
2.3.1 Offshore Wind Farm
In order to obtain realistic results for the energy produced at the offshore wind farm, a Weibull distribution
for the offshore wind and a normalized wind farm power curve are used (see Fig. 2.2).
The Weibull distribution used was based on measurement data collected during 4 years in a platform,
Fino 1, located in the North Sea [9].
The wind farm power curve used was obtained by smoothing the normalized turbine power curve and
it is represented in Fig. 2.3. The power curve includes factors such as the turbine design and future
developments in this field up to the year 2030 [10].
0 5 10 15 20 25 30 350
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Wind Speed [m/s]
Rela
tive
Freq
uenc
y
Offshore Wind Weibull Distribution
Figure 2.2: Weibull distribution.
0 5 10 15 20 25 30 350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Wind Speed [m/s]
Win
d Fa
rm O
utpu
t Pow
er [p
u]
Wind Farm Normalized Power Curve
Figure 2.3: Offshore wind farm normalized powercurve.
When the multi-turbine power curve and the wind distribution are known, the transmission system loss
percentage can be calculated as in (2.4). For the sake of simplification the wind farm was considered to
be always available.
6
Elosses[%] =
∑υf(υ)× Plosses(υ)∑
υf(υ)× Pprod(υ)
× 100 (2.4)
Using (2.4), it is possible to fairly compare both HVAC and HVDC technologies in terms of the energy
lost in the transmission system itself.
2.3.2 AC Cables
The losses variation with the transmitted power for three AC cables with different rated voltages is shown
in Figure 2.4. The losses are present in pu, having the rated power of each cable as the power base.
The cable with lowest rated voltage (30-kV) is the one that presents higher power losses. The 150-kV
cable shows higher losses when compared to the cable of 138-kV one. This is due to the fact that the
150-kV cable present in the database has a higher resistance per unit of length when compared with the
138-kV cable. However, the 150-kV cable has a higher rated power and, in this way, it can transport
more power per cable ashore; thus, less cables are needed.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
Transmitted Power [pu]
Pow
er L
osse
s [%
]
AC Cables Losses (100 km)
ACcable - 30 kV (40 MVA)
ACcable - 138 kV (140 MVA)
ACcable - 150 kV (200 MVA)
Figure 2.4: Losses variation of three different AC Cables for a 100 km transmission system.
Due to the high transmission distances, in order to model the HVAC cables, shorter sections of the cable
were considered to be connected to each other. Thereafter, each section was modeled through a π-model.
Submarine and underground cables present larger distributed capacitances when compared to overhead
lines. The HVAC cable is modeled as several sections, each constituted of a lumped-π-model, in order to
minimize the error due to simplifications.
2.3.3 DC Cables
The DC cables were also modeled through a π-model. The power losses were scaled by a factor of two
since all the DC cables present in the database are bipolar. In Figure 2.5, the variation of the power
losses with the transmitted power is shown for three DC cables with different power ratings. Once again
the losses are present in pu, having the rated power of each cable as the power base. All the cables have
a rated voltage of 320 kV.
7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Transmitted Power [pu]
Pow
er L
osse
s [%
]
DC Cables Losses (100 km)
DCcable - 320 kV (500 MW)
DCcable - 320 kV (800 MW)
DCcable - 320 kV (1500 MW)
Figure 2.5: Losses variation of three different DC Cables for a 100 km transmission system.
2.3.4 Voltage Source Converters
The power losses of the voltage-source converter are due to the switching components; the conduction
losses, which vary in a quadratic proportion with the current; and the “no-load” losses. In Figure 2.6,
the contribution of each one of these factors and the total power losses of the converter are shown. The
power losses are shown in percentage of the converter rated power (570 MW).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Transmitted Power [pu]
Pow
er L
osse
s [%
]
Converter Losses
Plosses
Psw itchingPconduction
Pno load
Figure 2.6: Losses variation of the voltage-source converter.
2.3.5 Other Components
Offshore wind farms need a collection network. The medium voltage AC cables that constitute the
offshore wind farm collection network are modeled through a π-model. Since the distance considered
for the cables is much shorter than the ones for the HV cables, one π-section is applied to model the
collection system.
HVAC cables produce reactive power due to their dominant capacitance. The reactive charging current
redistributes over the cable length depending on the voltage amplitudes and angles at both ends of
the cable. In this way, inductors were installed at both ends of the cables to provide reactive power
compensation.
8
2.4 MTdc Design
In this case study, the objective functions used to compare the offshore transmission systems are the total
investment cost and the percentage of energy lost during transmission.
F
⎧⎪⎪⎨⎪⎪⎩
Investment cost =n∑
i=1
Costcompi
Energy losses [%] =n∑
i=1
Elosses[%]
where n is the number of components of each transmission system.
Case Study
The objective is to compare the AC and DC transmission systems in a multi-objective basis. The layout
and the components, of both HVAC and HVDC transmission systems, are depicted in Figure 2.7. The
AC cables that constitute the collection network are 5 km long. The rated power of the wind farm is 500
MW.
OffshoreWind farm AC cables Trafo 1 HVAC Cable
AC cables Trafo 1 HVDC Cable Inverter
==
RectifierOffshoreWind farm
...
...
...
...
...
...
Trafo 2
Trafo 2
Inductor 1 Inductor 2
Figure 2.7: Configuration of the HVAC and HVDC transmission systems.
Results
Initially a comparison between the two transmission systems was made to take into consideration one
criterion at the time. In the last part, a comparison based on multi-objective optimization was performed.
Single-Objective Optimization Comparison
Firstly, the transmission system losses percentage was optimized. Thereafter, the investment costs was
minimized once again for both AC and DC systems.
9
Energy Losses
In Figure 2.8, it is shown the transmission system losses percentages according to the distance between
the offshore wind farm to the shore. The break-even point, i.e. the distance for which the DC technology
presents lower energy losses when compared to the AC option, is circa 100 km. This result is in tune with
the findings of several other studies on the subject, which serves as a confirmation of the model accuracy
[2, 3].
The losses for the DC technology increase slowly with the distance. This is due to the fact that the
genetic algorithm chose the same configuration for all the considered distances. In this way, the losses
only depend on the increase of the DC cable’s length. The DC technology presents an energy loss of circa
5.44% at 100 km.
The AC alternative presents lower percentage losses for shorter distances. However, the increase rate is
much higher when compared to the DC one. This has to due with the fact that the reactive power in the
HVAC cables are highly dependent on cable rated voltage. When a 100 km distance is considered, the
AC transmission system presents an energy loss of 5.48%.
Differently from the DC solutions, the genetic algorithm for distances shorter than 120 km chose the AC
cable with a rated voltage of 138 kV. However, for longer distances the 150-kV AC cable was chosen
instead. Although as previously shown in Figure 2.4, the 138-kV cable presents lower losses, for distances
higher than 120 km, the cable does not respect the power rate constraint and therefore it becomes
unfeasible.
AC voltages higher than 150 kV have not been used so far in practical submarine installations [11].
Moreover, such voltage level is a standard choice for submarine cables in Europe [12] and it was used, for
instance, to connect the wind farm on the Thornton Bank and the Danish Horns Rev wind farm [1].
50 60 70 80 90 100 110 120 130 140 1503.5
4
4.5
5
5.5
6
6.5
7
7.5
8
Length [km]
Ener
gy L
osse
s [%
]
HVAC and HVDC Transmission System Losses (500 MVA)
HVACHVDC
Figure 2.8: Variation of the transmission system losses percentage for both technologies with the trans-mission distance.
An interesting study that can be done is to observe how the energy losses are distributed over the
components of each transmission system for the break-even distance (100 km). In Figure 2.9, the energy
losses breakdown for the components of the DC option is shown. The HVDC cable is the component with
the lowest share of losses. This has to do with the fact that the genetic algorithm chose the DC cable
with 1500 MW rated power. Even though the wind farm has only 500 MW of installed capacity, since the
investment costs are not being minimized in this first approach, the HVDC cable with the lowest losses
was chosen. On the other hand, the converters present the higher percentages. As previously shown in
Figure 2.6, the converter losses increase with the power transferred but they always present switching
10
and “no-load” losses. Furthermore, most of the time the offshore wind farm is not producing at its full
capacity and, thus, the converters will not be operating at their maximum efficiency.
AC Cables Trafo 1 Rectifier HVDC Cable Inverter Trafo 20
5
10
15
20
25
30
DC Transmission System Components
Ener
gy L
osse
s [%
]
Energy Losses Breakdown: Losses Minimization (100km)
Figure 2.9: Transmission system losses percentage breakdown through the components of the DC systemwhen a distance of 100 km and minimization of losses are considered.
In Figure 2.10 the energy losses breakdown for the components of the AC system is presented. The
138-kV cable was the one chosen when a distance of 100 km was considered and it is the component with
the highest share of the energy losses.
AC Cables Trafo 1 Inductor 1 HVAC Cable Inductor 2 Trafo 20
10
20
30
40
50
60Energy Losses Breakdown: AC Transmission (100km)
AC Transmission System Components
Ener
gy L
osse
s [%
]
Figure 2.10: Transmission system losses percentage breakdown through the components of the AC systemwhen a distance of 100 km and minimization of losses are considered.
Total investment costs
When the total investment costs are considered as the optimization function, the results are the ones
depicted in Figure 2.11. Once again the DC configuration chosen by the genetic algorithm was kept
constant for all the considered distances. However, it is a different configuration when compared with the
one obtained previously. In order to demonstrate it, the percentage of energy lost per component when a
distance of 100 km is considered is shown in Figure 2.12. It can be observed that the HVDC cable chosen
has higher losses percentage than the one chosen when only the losses were optimized. The HVDC cable
selected for this case has a rated power of 500 MVA, which is the power installed at the offshore wind
farm. Although the cost of the HVDC cable is lower, it presents higher losses.
For the AC configuration, the total investment cost for distances below 120 km raises with the increase of
the HVAC cables length since the configuration was kept constant. However, similarly for the case when
the losses were minimized, the HVAC cable chosen for distances longer than 110 km has a higher rated
voltage (150 kV). Such cable is much more expensive due to insulation requirements and; thus, the cost
of the transmission system is also higher.
11
50 60 70 80 90 100 110 120 130 140 15040
60
80
100
120
140
160
180
200
220
240
Length [km]
Tran
smis
sion
Sys
tem
Cos
t [M
€]
HVAC and HVDC Transmission System Costs (500 MVA)
HVACHVDC
Figure 2.11: Variation of the total investment cost for both technologies with the transmission distance.
AC Cables Trafo 1 Rectifier HVDC Cable Inverter Trafo 20
5
10
15
20
25
DC Transmission System Components
Ener
gy L
osse
s [%
]
Energy Losses Breakdown: Cost Minimization (100km)
Figure 2.12: Transmission system losses percentage breakdown through the components of the DC systemwhen a distance of 100 km and minimization of total investment costs are considered.
12
Multi-Objective Optimization Comparison
With the previous single-criterion optimization it is difficult to have a clear picture of all the possible
solutions. When the energy losses are being optimized, a set of transmission systems are obtained. On the
other hand, when the investments costs are minimized, a different set of solutions is reached. Therefore,
it is not possible, in a simple way, to have a clear picture of all the possible solutions for this particular
problem.
In Figures 2.13 to 2.15, the solutions obtained through the MOOA implemented are shown. All the non-
dominated solutions, for both AC and DC systems, are plotted as filled squares and dots, respectively.
When a distance of 50 km is used, as shown in Figure 2.13, the non-dominated AC system is the one
that presents the lowest losses and investment costs. Therefore, it is the best option for the transmission
system.
3.5 4 4.5 5 5.5 6 6.5 740
60
80
100
120
140
160
180
200
220
Energy Losses [%]
Tran
smis
sion
Sys
tem
Cos
t [M
€]
HVAC and HVDC Transmission Systems (50 km)
HVACHVDC
Figure 2.13: Optimal Pareto Fronts for both transmission technologies when a distance of 50 km isconsidered.
In Figure 2.14, the solutions for a 100 km distance are shown. One of the non-dominated DC solutions
is the one with the lowest energy losses. However, the non-dominated AC solution presents very similar
losses. Moreover, such an AC system is circa three times cheaper compared to the DC system with similar
losses. In this way, the AC system is still the best option to connect the offshore wind farm to shore.
5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 760
80
100
120
140
160
180
200
220
240
260
Energy Losses [%]
Tran
smis
sion
Sys
tem
Cos
t [M
€]
HVAC and HVDC Transmission Systems (100 km)
HVACHVDC
Figure 2.14: Optimal Pareto Fronts for both transmission technologies when a distance of 100 km isconsidered.
13
Finally, when a distance of 150 km is considered, the DC configurations which belong to the OPF present
lower energy losses and total investment costs when compared to the AC solutions (see Figure 2.15). In
this way, for this distance the DC technology is the most advantageous.
5.5 6 6.5 7 7.5 8 8.5 9140
160
180
200
220
240
260
280
300
Energy Losses [%]
Tran
smis
sion
Sys
tem
Cos
t [M
€]
HVAC and HVDC Transmission Systems (150 km)
HVACHVDC
Figure 2.15: Optimal Pareto Fronts for both transmission technologies when a distance of 150 km isconsidered.
14
Chapter 3
MTdc network Power Flow
Optimization
3.1 Introduction
There are several reasons contributing to promote the development of an European offshore grid in the
North Sea. The main drives are, amongst others, the need to integrate large amounts of renewable
energy, the will to boost transnational electricity trade, the European Union’s targets on renewable
energy generation and desire for security of supply [13][14].
Several European projects have performed studies on market, regulatory and policy challenges related
with the development of a North Sea transnational grid [15][16]. Nonetheless, hitherto, very few have
concentrated on its technical aspects, especially the challenges involved with controlling and operating
such networks [17][18].
The main goal of the present chapter is to show how to control and operate large multi-terminal dc
networks with the aid of a MOOA in order to obtain optimal and controllable power flows according to
different optimization targets.
3.2 Optimal power flow
The idea of obtaining optimal power flows with the aid of genetic algorithms is not new. Several studies
may be found in the literature [19, 20]. In [21] the OPF applied to IEEE test systems is achieved via
a genetic algorithm that makes use of both continuous and discrete control variables. The optimization
of social welfare was considered in [22] and [23]. Multi-Objective optimization to account the system
security in electricity markets may be found in [24]. The application of OPF for smart grid is more
recent, having as examples the works performed in [25] and [26].
The Independent System Operator (ISO) makes use of an OPF in order to dispatch the entities involved
in the pool market. During the optimization process, all the network constraints are taken in to account.
These constraints are usually described as a set of equality and inequality functions [27]. Using the
following security-constrained optimization problem, an OPF-based market model can be represented as
[28]:
OPF equation: f(PS , PD) (3.1)
15
Constraints: ⎧⎪⎪⎪⎨⎪⎪⎪⎩
V minDCi
≤ VDCi≤ V max
DCi
PS =∑
PWFs
0 ≤ PDi≤ Pmax
V SCi
−ImaxDCi
≤ IDCi ≤ ImaxDCi
(3.2)
where VDC represents the nodal DC voltages inside the MTdc network; PS and PD stand for supply and
demand power bids; and IDC is the current flowing in the DC cables.
Additionally to respecting the network’s voltage and current constraints, the OPF algorithm here imple-
mented will make sure that the ISO is presented with solutions for the MTdc network power flow which
are N-1 secure. In this way, the reliability and security of the grid are enhanced. As a matter of fact,
in case of an outage in any of the DC network terminals, if the OPF solution is guaranteed to be N-1
secure, the power balance inside the MTdc grid will be maintained [29].
3.2.1 MTdc Control Scheme
The variables encoded in the chromosomes, besides the other fields presented in Fig. B.11, are the direct
voltage references of all the onshore nodes without restrictions on their active power. All the encoded
variables are real valued and the composition of each individual chromosome is given in (3.3).
X =[V ∗DC1 · · · V ∗
DCn
](3.3)
where V ∗DCn is the dc system voltage reference of the n-th onshore VSC.
It is important to refer that in the distributed voltage control method, even when receiving a preestab-
lished amount of power, an onshore VSC station will always be operating as a direct voltage regulating
node. However, its voltage reference will be determined by the MOOA after solving the optimal MTdc
network load flow. The fitness evaluation (step 2 of Fig. B.2) flowchart of the MOOA applied to the
control of MTdc networks is depicted in Fig. 3.1.
The step 2.3 from Fig. 3.1 is where the population of solutions are tested to verify their performance
with respect to the optimization targets set by the system operator.
Constraints
In order to guarantee that the solutions provided are feasible, several constraints were added to the
MOOA. These constraints guarantee that there are no dc cables overloaded, that the dc voltages of all
the nodes inside the MTdc grid respect the predefined boundaries and that the load flow solutions are N-1
secure, i.e. the power control will still be achieved even if an outage occurs in any of the VSC terminals
[30]. In Fig. 3.2 the implemented constraints are depicted.
16
6. Terminate?
1. Population Initialisation 2. Finess Evaluation
3. RankPopulation
4. SortPopulation
5. ArchiveUpdate
7. Selection 8. Crossover 9. Mutation
11. Output Archive
Yes
No
10. Regeneration
Population checked?2.1 Individual of the Population
YesNo
2.2 Determination of the Power Flow
2.3 Fitness evaluation 2.4 Verification of constraints
2.5 Augmented fitness function
2. Fitness evaluation
Step 3
Figure 3.1: Fitness evaluation flowchart.
Genetic Algorithm
VSC stations powerratings
DC voltageboundaries
N-1redundancy
Load flowOWFs powerproduction
DC cablescapability
Figure 3.2: Constraints incorporated in the genetic algorithm.
17
UK1
UK2
BE1
NL1
NL2
DE1DE2
DK1
DK2
HUB1
HUB2
HUB3NL
UK
BE
DK
DE
HUB4
HUB5
Figure 3.3: The 19 node-meshed connected offshore MTdc network used in the simulations.
3.3 MTdc network single objective optimization: load flow con-
trol to minimize losses
The performance of the algorithm and the DVC strategy in operating a MTdc network is tested via
numerical simulations. The MTdc network used is depicted in Fig. 3.3. It is composed of 19 nodes, inter-
connecting 9 OWFs to 5 different onshore ac networks representing each a northern European country;
namely, the UK, Belgium, the Netherlands, Germany and Denmark. Out of the total number of nodes, 5
are hub nodes, which means there are no converters installed in these locations and their main function
is to allow for cable joining.
As previously mentioned, there are 9 OWFs installed in the MTdc network and their aggregated installed
capacity is of 3 GW, or 15 pu (1 pu corresponds to 200 MVA) in the system base (see Table 3.4). The dc
system voltage is of ±250 kV and all the onshore VSCs are assumed to be connected at ac substations
with short-circuit ratio three times higher than the converter installed capacity.
In Table 3.2 the characteristics of the transmission lines of the MTdc network are given. It shows the
cable length in kilometers and also the installed capacity of the departing VSC node. For instance, Table
3.2 shows that the UK1 installed capacity is 3 pu, whereas 2 pu for UK2 and 5 pu for the UK.
The dc cables rated power was chosen to match the rated power of the nodes that they are connected
to. Hence, the dc cables have either the installed capacity of their respective OWF or onshore node. An
exception is made for the dc cables connecting the hubs, which have an installed capacity of 1 pu.
Table 3.1: System Parameters.
Parameter Unit Value
System base Rated Power Sb MVA 200DC System Voltage base Rated Voltage Vb kV ±250Short-circuit Ratio SCR - - 3
MTdc NetworkRated DC System Voltage Vdc kV ±250
Cable impedance ZdcΩ/km 0.02+j0.06nF 220
In performing the case studies real wind data was used to represent the power production of the different
18
Table 3.2: MTdc Network Cables length and Capacity.
Lines Length [km] Size [pu] Lines Length [km] Size [pu]
UK1 - HUB1 100 3 HUB3 - HUB4 250 1UK2 - HUB1 40 2 DE1 - HUB4 40 2UK - HUB1 120 5 DE2 - HUB4 70 2HUB1 - HUB2 300 1 DE - HUB4 150 4BE1 - HUB2 50 1 HUB4 - HUB5 120 1BE - HUB2 100 1 DK1 - HUB5 40 1HUB2 - HUB3 120 1 DK2 - HUB5 50 1NL1 - HUB3 100 2 DK - HUB5 150 2NL2 - HUB3 40 1 HUB1 - HUB5 380 1NL - HUB3 70 3
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
time [s]
Act
ive
Pow
er [p
u]
Wind Farms Power ProductionPUK1PUK2PBE1PNL1PNL2PDE1PDE2PDK1PDK2
Figure 3.4: Wind farms power production during the simulation.
19
OWFs. Wind data measured from a wind turbine was smoothed in order to simulate the power production
of the complete wind farm [31]. In Fig. 3.4 the OWFs power production employed during the simulation
is given. The wind power production curves are plotted in the system power base as given in Table 3.4.
The three different case studies analyzed are described in details in Table 3.3.
Table 3.3: Description of the analyzed case studies
Case Description
Case i The MTdc network will be controlled accordingto a flat voltage profile, i.e. all the onshore VSCvoltage references are set to 1.0 pu.
Case ii The MTdc network voltage references are set inorder to minimize the power losses during the en-tire simulation.
Case iii In the first part (5 ≤ t ≤ 10 s), priority in receiv-ing power is given to the UK and Germany. TheUK is willing to receive 1.5 pu and Germany 2pu. In the second part (10 ≤ t ≤ 20 s), the VSCvoltage references are set such that Belgium willreceive 1 pu; while the Netherlands and Germanywill receive 2 pu each. In the last part (t ≥ 20 s),the Netherlands and Denmark will receive 1 pu ofpower each.
In all case studies a three-phase fault was applied in the ac grid of the British converter station. The
fault occurs at t = 25 s and it is cleared one second later. The fault lasts for a complete second to show
that, when a converter station is definitely lost, the system is able to reach a new feasible operating point
after the transients decay. After 1 second, the fault is cleared to demonstrate that the system is again
able return to its previous operating point.
Results
The case studies described in Table 3.3 were analyzed in a simulation model. For the dynamic simulations
the ac grids are modeled as ideal voltage sources behind short circuit impedances whereas the MTdc
network is modeled as lumped impedances in a state-space matrix representation [32, 33]. The VSC
model employed is an averaged model which includes the dynamics of the HVdc terminals controls but
neglects the switching phenomena. The OWFs are modeled as current sources with the output power
given by Fig. 3.4 [34].
The transmission losses comparison between the three analyzed cases is displayed in Fig. 3.5. The MTdc
network voltages are shown in Fig. 3.6, while the actual active power at the onshore nodes is given in
Fig. 3.7.
Case i - No optimization
As described in Table 3.3, during Case i, the MTdc network is controlled with a flat voltage profile. In
that case, the MTdc network losses stay at around 2.75% of the OWFs produced power (see Fig. 3.5).
The first column in Fig. 3.6 displays the MTdc network onshore node voltages and the upper plot in
Fig. 3.7 shows the onshore nodes dc power for Case i. As seen from the results, all onshore VSCs are
able to keep the direct voltage constant at 1 pu by slowly varying the input dc power according to the
variations at the OWFs side. However, in this scenario the load flow distribution is not controlled and it
is not assured whether the MTdc network is N-1 secure.
20
0 5 10 15 20 25 302
2.53
3.54
4.55
5.56
6.57
7.58
time [s]
P loss
es [%
]
Transmission Power losses
Case iCase iiCase iii
Figure 3.5: Transmission power losses during the simulations for the three tested cases.
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case i - United Kingdom
VUK1
VUK2
VUK
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case ii - United Kingdom
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case iii - United Kingdom
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case i - Belgium
VBE1
VBE
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case ii - Belgium
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case iii - Belgium
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case i - Netherlands
VNL1
VNL2
VNL
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case ii - Netherlands
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case ii - Netherlands
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case i - Germany
VDE1
VDE2
VDE
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case ii - Germany
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case iii - Germany
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case i - Denmark
VDK1
VDK2
VDK
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case ii - Denmark
0 5 10 15 20 25 300.8
0.840.880.920.96
11.041.081.121.16
time [s]
V DC [p
u]
Case iii - Denmark
Figure 3.6: Direct voltage at the OWFs and onshore nodes during the simulations for the three testedcases.
21
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30−5.5−5
−4.5−4
−3.5−3
−2.5−2
−1.5−1
−0.50
0.51
1.5
time [s]
Act
ive
Pow
er [p
u]
Case iPUK PBE PNL PDE PDK
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30−5.5−5
−4.5−4
−3.5−3
−2.5−2
−1.5−1
−0.50
0.51
1.5
time [s]
Act
ive
Pow
er [p
u]
Case ii
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30−5.5−5
−4.5−4
−3.5−3
−2.5−2
−1.5−1
−0.50
0.51
1.5
time [s]
Act
ive
Pow
er [p
u]
Case iii
Figure 3.7: Active power at the onshore nodes during the simulations for the three tested cases.
Case ii - Losses optimization
From the transmission losses optimization results, as displayed in Fig. 3.5, it is straightforward to realize
that the MTdc network losses are only marginally lower than the case with no optimization. The dc
system voltages and active power for the flat profile are very close to the optimal load flow found in Case
ii (see Fig. 3.7). However, to minimize the transmission losses, the genetic algorithm reaches a load-flow
scenario with higher dc system voltage reference values (circa 1.02 pu). In this way, the transmission
power losses are slightly decreased. Additionally, with higher operating direct voltages it is expected
that the converter losses would decrease. Finally, the system security is also enhanced, since the genetic
algorithm assures that the network flow is N-1 secure, which is not certain with a flat profile.
Case iii - Controlled power flow
In this last case study, the power flow inside the MTdc network is controlled according to the specifications
given in last row of Table 3.3. It is possible to observe, from the lower plot in Fig. 3.7, that the DVC
strategy implemented via the genetic algorithm has correctly set the onshore node voltages to establish
the desired power flow. The load-flow results obtained are as expected, even though the optimal dc
power flow implemented by the genetic algorithm works with the wind power production averaged over
5 seconds. As previously mentioned, the active power variability is highly reduced since several OWFs
are interconnected over a wide area.
In order to control the power flow inside the MTdc network, the dc system voltage at the onshore nodes
has to be varied. The last column of Fig. 3.6 displays the variations in the direct voltages; which, for the
analyzed case, always lie inside the operational margin of ± 10% the rated value. Additionally, as shown
by the simulation results, if so desired, changes from one load-flow operating point to another can be
22
accomplished in less than 200 ms. This is due to the fast direct voltage control capabilities of the onshore
VSC terminals. Finally, in the last case study, the optimization goal is to minimize the losses while still
respecting the load-flow constraints set by a possible TSO. As a result of these extra constraints, the
MTdc network transmission losses in this case will always be higher than optimal, as shown in Fig. 3.5.
The results shown in the right column of Fig. 3.6 demonstrate that the onshore VSC stations are able
to withstand the onshore ac three-phase fault at the British station while keeping the power balance
inside the MTdc grid. The steady-state direct voltages inside the MTdc network are kept under the
pre-established 1.10 pu limit. The British onshore and offshore converter stations are, as expected, the
ones that suffer the highest variations in the direct voltage, while the remaining onshore nodes present
lower variations due to their higher distance to the fault. Between 25 s ≤ t ≤26 s, the power flow inside
the network is not optimized to minimize the power losses. However, it is guaranteed that the converter
stations rated power is respected.
During the fault the British converter station is not receiving any power, whereas the Belgian onshore
node does not alter its power transmission during the fault since it was previously at its maximum
transfer capacity of 1 pu. The Dutch onshore VSC station was the node that increased the most its
power reception, from circa 1.5 pu to 2.5 pu because the Netherlands is the closest onshore node with
available margin to increase its power reception. The remaining two countries, Denmark and Germany,
also increased their power consumption to maintain the direct voltage stable inside the MTdc network.
After fault clearance, at t = 26 s, the onshore stations are able to once again operate under the TSO’s
desired power flow.
3.4 MTdc network multi-objective optimization: load flow con-
trol for minimal losses and maximum social welfare
The main objective of this case study is to obtain a Pareto front showing a trade-off between the mini-
mization of the transmission losses and the maximization of the social welfare. The latter is explained in
A.3.2 and be calculated as in (A.6), while the transmission losses can be calculated by (A.3).
MTdc network 1
The MTdc network displayed in Fig. 3.8 is the one used during the simulations. It is a parallel-connected
meshed network, composed by 11 nodes and 11 DC cables. There are a total of 5 offshore wind farms
connected to 3 different onshore networks, which represent countries. In the case study these countries
are considered to be the United Kingdom, the Netherlands and Norway, respectively.
All the offshore wind farms are connected to hubs instead of being directly connected to its rightful owner
country. Furthermore, each hub is connected to its respective country, but also to adjacent hubs. This
type of DC network connection enhances the electricity trade amongst the interconnected onshore grids
[35].
The MTdc network parameters are given in Table 3.4. The system power base is 500 MVA. During
steady-state operation all DC voltages must lie within ±5% of the nominal value. The DC line distances
are given in Fig. 3.8. The wind farms cables (lines 2,3,5,6 and 9) are rated at 500 MW, whereas all
the other DC cables have two times the capacity of the wind farms, i.e. 1000 MW. The extra capacity
in the offshore hub cables is designed in order to increase the network flexibility and to enhance power
trade between the countries. The power being produced at the offshore wind farms during the simulation
scenarios is also given in Fig. 3.8.
23
ACnetwork 1
ACnetwork 3
ACnetwork 2
N2
WF 3
WF 4
WF 5WF 1
WF 2
N1
N6 N9
N10
N3
N7
N4
N11
N5
N8line 1 line 8 line 10
line 2
line 3line 4 line 11
line 5line 6
line 7
line 9
125 km
20 km
20 km
400 km
200 km200 km
15 km
100 km
15 km
125 km
15 km
VSC 1 VSC 3
VSC 2
1 0.5WFP pu
2 0.4WFP pu
3 0.5WFP pu
4 0.7WFP pu
5 0.9WFP pu
Figure 3.8: Meshed MTdc network used in the simulations.
In this study, the market clearing prices for the 3 interconnected countries are considered to be 60 e/MWh
in Norway, 72.8 e/MWh in the UK and 77.5 e/MWh in the Netherlands [35]. The offshore DC network
is operated in such a way that there are no congestions due to electricity trade between the countries
and, therefore, all the power being produced at the wind farms can be injected in the MTdc network.
Table 3.4: MTdc Network Rated Parameters.
Parameter Unit Value
Offshore VSC Rated Power Sowf MVA 500
Onshore VSC Rated Power Svsc MVA 1000
MTdc NetworkRated DC Voltage Vdc kV ± 200Cable impedance Zdc Ω/km 0.02
DC lines 2,3,5,6 & 9 Pdc MW 500DC lines 1,4,7,8,10 & 11 Pdc MW 1000
Table 3.5: Description of the analyzed case studies
Case Description
Case i No restrictions applied.
Case ii At least some of the offshore power being pro-duced at the Norway wind parks has to beconsumed locally.
Case iii Idem as Case ii. In addition the Netherlandsis willing to buy power for an extra 0.5 pu ontop of its offshore wind farm production of 1.2pu.
Results
A trade-off – or Pareto Front – for the operation of the MTdc network under study is obtained by solving
the OPF problem with the MOOA previously presented. The resulting network power losses and social
welfare are shown in Fig. 3.11 (a).
24
35 40 45 50 55 60 65 70 75 801
2
3
4
5
6
7
Social Welfare [€/h]
Pow
er L
osse
s [%
]
Power losses and Social Welfare
All solutions found
35 40 45 50 55 60 65 70 75 801
2
3
4
5
6
7
Social Welfare [€/h]
Pow
er L
osse
s [%
]
Power losses and Social Welfare: N-1 secure and Pareto Front
N-1 Secure SolutionsPareto-Front Solutions
Figure 3.9: Network power losses and social welfare.
However, not all the obtained solutions are guaranteed to be N-1 secure. Therefore, all solutions are tested
against a DC load-flow algorithm. The algorithm systematically applies a fault to one network node at
a time. Afterwards, it excludes solutions that do not respect the DC voltage and current constraints for
all the possible N-1 scenarios. The new set of solutions is displayed in Fig. 3.11 (b), where the optimal
Pareto front is also highlighted.
Depending on the restrictions selected by the ISO, some MTdc operating points on the Pareto front
may become unavailable. Fig. 3.10 shows, for the selected case studies, the available operating points
on the Pareto front and the market clearing price. The hollow points on the Pareto fronts (Fig. 3.10),
represent unavailable operating points due to the ISO and Pool Market restrictions. In addition, Fig.
3.10 also displays, for each analyzed case study, the MCP obtained for the two most extreme points on
the available Pareto front. These points are equivalent to the lower losses (point A) and the higher social
welfare (point B) scenarios.
Table 3.6 contains the power flow results for the lower losses (point A) and higher social welfare (point
B) scenarios for all the examined case studies. It displays the nodal DC voltages (in per unit), the loss
allocation coefficients (in percentage of the losses), the transmission losses (in percentage of the offshore
generation) and the resultant social welfare.
Table 3.6: Power flow results and loss allocation coefficients for the analyzed case studies.
Node
Case i Case ii Case iii
A B A B A B
VDC Lk VDC Lk VDC Lk VDC Lk VDC Lk VDC Lk
WF1 1.0272 5.05 0.9983 -2.29 1.0272 5.05 1.0141 1.71 1.0021 2.46 1.0141 1.71WF2 1.0269 3.83 0.9980 -1.89 1.0269 3.83 1.0138 1.25 1.0018 1.81 1.0138 1.25WF3 1.0268 4.62 1.0064 -0.18 1.0268 4.62 1.0095 -1.17 0.9992 0.16 1.0095 -1.17WF4 1.0271 7.02 1.0068 -0.12 1.0271 7.02 1.0098 -1.32 0.9996 0.63 1.0098 -1.32WF5 1.0271 9.03 1.0352 12.70 1.0271 9.03 1.0256 15.86 1.0105 16.491 1.0256 15.86VSC1 1.0128 19.39 0.9666 42.04 1.0128 19.39 0.9986 15.22 0.9863 19.61 0.9986 15.22VSC2 1.0118 28.78 0.9811 26.97 1.0118 28.78 0.9842 68.72 0.9771 61.39 0.9842 68.72VSC3 1.0118 22.27 1.0494 22.76 1.0118 22.27 1.0237 -0.26 1.0045 -2.55 1.0237 -0.26
PL [%] 1.46 6.42 1.46 2.63 2.08 2.63SW [e/h] 59.07 77.52 59.07 71.69 68.14 71.69
25
58 60 62 64 66 68 70 72 74 76 781
2
3
4
5
6
7
Social Welfare [€/h]
Pow
er L
osse
s [%
]
Case i - No restrictions
58 60 62 64 66 68 70 72 74 76 781
2
3
4
5
6
7
Social Welfare [€/h]
Pow
er L
osse
s [%
]
Case ii - One restriction
58 60 62 64 66 68 70 72 74 76 781
2
3
4
5
6
7
Social Welfare [€/h]
Pow
er L
osse
s [%
]
Case iii - Two restrictions
Unavailable operating pointsAvailable operating points
Unavailable operating pointsAvailable operating points
Unavailable operating pointsAvailable operating points
Figure 3.10: Pareto Front and Market Clearing Price (disregarding losses) for the analyzed case studies.
Case Study i
In the first case scenario, as stated in Table 3.5, there are no extra restrictions applied by the ISO to the
load flow. In this way, the entire Pareto front (Fig. 3.10 (b)) is reachable.
For the point A, where the lowest power losses are found, the resultant load flow shows that the WFs
are sending almost all of their generation to the respective countries. In this way, the power flow inside
the grid is minimized and, thus, the power losses are optimal. Since all countries are receiving power
produced offshore, the MCP, as shown in Fig. 3.10 (d), is equal to 60 e/MWh, which corresponds to the
bid lowest bid, made by Norway.
The loss components, Lk, are all positive for point A, meaning that all the market participants are
responsible to pay part of the power lost in the transmission system. The Netherlands has the highest
loss component (28.78%) since it is the node which receives the highest amount of power from the MTdc
grid. This is because the Dutch farms have the highest combined production. Among the offshore nodes,
the highest loss allocation is given to the Norwegian WF (9.03%) since it has the highest single production
and, therefore, it is the one that injects the most power in the network.
In point B, the social welfare is being maximized. In order to do so, the countries with the highest
electricity bids have to receive the most power from the grid. As shown in Fig. 3.10 (d), both the
Netherlands and the UK are receiving power at their maximum capacity, i.e. 2 pu. Since the total power
production at all the WFs is 3 pu, Norway is transmitting the extra power for allowing the maximization
of the social welfare. The MCP for this operating point is the British bid, i.e. 72.8 e/MWh (Fig. 3.10
(d)).
In this scenario, the loss components of the Dutch and British WFs are negative. Negative loss compo-
nents represent monetary incentives to market participants that are positioned in strategic places in the
network [36], helping, in this way, to achieve the desired load flow. On the other hand, the loss component
of the Norwegian WF is positive. This has to do with the fact that this WF is transmitting power to the
UK and the Netherlands. Consequently, since the WF is located far from both countries, such injection
of power into the MTdc grid has a significant impact in the total transmission losses. Nevertheless, all the
26
onshore nodes are still allocated the highest loss components. The UK is the node receiving the highest
amount of power that is not being produced at its WFs (1.1 pu), consequently it is the node with the
highest share of power losses (42.04%).
Case Study ii
In Case ii, an additional constraint is added; at least some of the offshore power being produced at
the Norwegian wind park has to be consumed locally. Due to such constraint, approximately half of
the previous Pareto front becomes unavailable operating points. The load flows with the highest social
welfare values are not tangible in this case since they require Norway to transmit power to the MTdc
network.
The operating point A for Case ii is the same one as found in Case i. This is due to the fact that the
unavailable part of the Pareto front is located on the region where the social welfare is maximized.
On the other hand, point B is now shifted along the Pareto front. In this operating point, Norway is
neither receiving or injecting power in the MTdc network (see Fig. 3.10 (e)). Since the Dutch bid is the
highest one, the maximization of social welfare is now made through the transmission of 2 pu of power
to the Netherlands (maximum capacity of the node) and 1 pu to the UK. The MCP is once again defined
by the bid made by the UK.
Regarding the loss components for point B, it is interesting to notice that both the Dutch WFs have
negative values. The Netherlands is the node that needs to receive the most power in order to maximize
the social welfare. Consequently, since both WFs are the closest generating nodes from the Dutch grid,
they are the ones that contribute most efficiently the load flow. On the other hand, the Dutch onshore
node is allocated the highest share of transmission losses.
Once again the Norwegian WF is allocated a positive Lk. Since barely no power is being transmitted
to Norway, its offhshore production is being delivered to the Netherlands. In this way, the power has to
be transported from the WF to the Dutch onshore node. Due to this additional flow, the power losses
increase and, therefore, the loss component of the Norwegian WF have to account for the additional
power losses.
Case Study iii
In the last case study, on top of the restriction imposed in Case ii, the Netherlands requests, besides the
power being produced at its wind farms, an extra 0.5 pu of power extra from the MTdc network.
The operation point A is now shifted to the right along the subset of the Pareto front of the Case ii. In
this new point A, the Netherlands is now receiving 1.7 pu of power from the MTdc network (see Fig.
3.10 (f)). Since point A from Case i is no longer feasible due to the constraints imposed by the ISO, the
closest operation point is achieved when 1.7 pu of power is being delivered at the Dutch onshore node.
For such power flow scenario, the highest loss component is once again attributed to the Netherlands.
The Dutch WFs are producing a total 1.2 pu of power, thus, the additional 0.5 pu has to be produced
and transported from somewhere else in the MTdc network. Through the power flow results it is possible
to observe that this additional power is coming from the Norwegian WF. Hence, the latter is assigned
a positive and higher Lk, when compared to the other WFs, which are sending their generation to their
respective countries.
The extra constraint added in Case iii rules out operation points from the Pareto front were the lowest
power losses can be found. Therefore, the point that maximizes the social welfare (point B) is kept
constant and equal to the one from Case ii.
27
MTdc network 2
The analyzed MTdc network is shown in Fig. 3.3. Table 3.7 shows the OWFs bids, power production
and capacity factors [37]. In the time frame considered, the OWFs power production is assumed to be
its installed capacity times its capacity factor. The OWFs bid values were chosen in order to be within
the range given in [38]. The bids of the interconnected countries are given in Table 3.8 [39].
Table 3.7: OWFs market bids, capacity factors and power productions.
OWF UK1 UK2 BE1 NL1 NL2 DE1 DE2 DK1 DK2
Bid [/eMWh] 80 80 75 70 70 85 85 90 90Capacity factors [%] 0.35 0.4 0.5 0.4 0.45 0.35 0.4 0.5 0.4Power production [pu] 1.05 0.8 0.5 0.8 0.45 0.7 0.8 0.5 0.4
Table 3.8: Onshore ac networks market bids.
AC Network UK BL NL DE DK
Bid [/eMWh] 98.95 98.99 91.38 114.17 97.74
Three different case studies were established to analyze the influence of extra market constraints on the
optimization results. Table 3.9 contains a detailed description of each case study.
Table 3.9: Description of the analyzed case studies
Case Description
Case i No restrictions applied.
Case iiThe dc current in the cable betweenthe Netherlands and HUB3 is atmost 1.2 pu
Case iiiIdem as Case ii. In addition, the dccurrent in the German cable (fromHUB4 to shore) has 1.5 pu as maxi-mum
Results
The Pareto fronts for the MTdc network operation with the trade-off between transmission losses and
social welfare are shown in Fig. 3.11. The fronts composed by hollow circles represent the intermediate
Pareto fronts found during the optimization process. The filled circles represent the best found Pareto
fronts.
When the N-1 network security constraint is active, the best found Pareto front is shifted towards lower
social welfare and higher transmission losses values. To make the dc network N-1 secure, as the maximum
dc voltage value is set to 1.15 pu, the MOGA has to lower the onshore dc voltage references – this explains
the higher losses – because when there is a fault in an onshore node, the dc voltages inside the grid will
rise. On the other hand, the social welfare values are lowered since, to maintain network security while
optimizing social welfare, less power goes to onshore nodes with higher prices. For instance, when
maximum social welfare is considered, the power going to the highest bid, Germany, is reduced from 2.19
pu to 1.55 pu if the MTdc network is N-1 secure. The power balancing among onshore nodes, when N-1
security is considered, happens so that in case of a fault, the offshore power can be shifted to converter
28
stations neighboring the defective one. This step in the optimization process guarantees that, in case of
a fault, the converter stations will not be overloaded.
Differently from the maximum direct voltage limitation, the 1-pu current limit in the hubs dc cables
demonstrated to be an inactive constraint. For the non-secure MTdc scenario, the maximum hub current
was 0.50 pu from HUB3 to HUB4, at the maximum social welfare operating point. For the N-1 secure
case, the same situation was encountered but now in the dc cable connecting HUB2 to HUB3. This
feature from the MOOA can help the ISO to optimize the hubs installed capacity, saving costs.
1050 1100 1150 1200 1250 1300 13506
7
8
9
10
11
12
Social Welfare [M€/year]
Pow
er L
osse
s [%
]
Network not secureNetwork N-1 secure
Figure 3.11: Pareto Fronts between Social Welfare and Power Losses.
1050 1075 1100 1125 1150 1175 1200 1225 12508
8.5
9
9.5
10
10.5
11
11.5
Social Welfare [M€/year]
Pow
er L
osse
s [%
]
Case iii
Available operating pointsUnavailable operating points
1050 1075 1100 1125 1150 1175 1200 1225 12508
8.5
9
9.5
10
10.5
11
11.5
Social Welfare [M€/year]
Pow
er L
osse
s [%
]
Case ii
Available operating pointsUnavailable operating points
1050 1075 1100 1125 1150 1175 1200 1225 12508
8.5
9
9.5
10
10.5
11
11.5
Social Welfare [M€/year]
Pow
er L
osse
s [%
]
Case i
Available operating points
UK1+UK2
BE1
NL1+NL2
DE1+DE2
=DK1+DK2
BE
UK
DK
DE
1
1
NL
UK1+UK2
BE1
NL1+NL2
DE1+DE2
BE
UK
DK
DE
NL
UK1+UK2
BE1
NL1+NL2
DE1+DE2
BE
UK
DK
DE
NL
2A 3A3B
2B
1A
1B
=DK1+DK2 =DK1+DK2
2
2
3
3
1Aq
1Bq
2Aq
2Bq
3 3A Bq q
Figure 3.12: Available Pareto front between social welfare and power losses and respective buyers andsellers welfare for the considered case studies.
The graphic on Fig. 3.12 (a) shows the original optimization N-1 secure Pareto front without any market
constrains. The extra market constrains result in some MTdc operating points becoming unavailable
(represented by a thin line). Additionally, the second row of Fig. 3.12 displays, for each case study,
the sellers and buyers welfares for the two extremities on the available Pareto fronts. These two points
correspond to the lowest losses (point A) and the highest social welfare (point B) scenarios. The total
power received by the onshore nodes for the three different case studies is also shown.
29
Table 3.10: Direct voltages, Power flow and loss allocation coefficients for the analyzed case studies.
Node
Case i Case ii Case iii
A1 B1 A2 B2 A3 B3
Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk Vdc Pdc Lk
UK1 1.05 1.05 14.2 1.03 1.05 9.1 1.05 1.05 14.0 1.03 1.05 9.1 1.05 1.05 14.0 1.05 1.05 13.8UK2 1.02 0.80 6.5 1.01 0.80 3.7 1.02 0.80 6.4 1.01 0.80 3.7 1.02 0.80 6.4 1.02 0.80 6.3BE1 0.98 0.50 -0.8 1.00 0.50 2.1 0.98 0.50 -0.1 1.00 0.50 2.1 0.98 0.50 -0.1 0.98 0.50 0.1NL1 1.00 0.80 3.4 1.05 0.80 8.5 1.01 0.80 4.7 1.05 0.80 8.5 1.01 0.80 4.7 1.01 0.80 5.2NL2 0.98 0.45 -0.5 1.02 0.45 3.3 0.99 0.45 0.4 1.02 0.45 3.3 0.99 0.45 0.4 0.99 0.45 0.7DE1 1.01 0.70 4.2 0.98 0.70 0.2 1.00 0.70 2.9 0.98 0.70 0.2 1.00 0.70 2.9 1.00 0.70 2.4DE2 1.02 0.80 6.6 0.99 0.80 1.7 1.01 0.80 5.1 0.99 0.80 1.7 1.01 0.80 5.1 1.01 0.80 4.5DK1 1.01 0.50 2.2 0.96 0.50 -0.9 1.00 0.50 1.8 0.96 0.50 -0.9 1.00 0.50 1.8 1.00 0.50 1.6DK2 1.01 0.40 1.8 0.96 0.40 -0.7 1.00 0.40 1.5 0.96 0.40 -0.7 1.00 0.40 1.5 1.00 0.40 1.3UK 0.94 -1.25 10.7 0.91 -1.44 14.6 0.94 -1.27 11.0 0.91 -1.44 14.6 0.94 -1.27 11.0 0.94 -1.28 10.8BE 0.92 -0.92 12.4 0.95 -0.95 3.7 0.93 -0.93 10.8 0.95 -0.95 3.7 0.93 -0.93 10.8 0.93 -0.93 10.1NL 0.93 -1.32 15.9 1.01 -0.20 -1.1 0.94 -1.13 9.3 1.01 -0.20 -1.1 0.94 -1.13 9.3 0.95 -1.05 7.0DE 0.93 -1.01 11.2 0.85 -1.55 34.2 0.90 -1.24 21.4 0.85 -1.55 34.2 0.90 -1.24 21.4 0.89 -1.33 26.0DK 0.93 -1.00 12.1 0.87 -1.19 21.7 0.93 -0.93 10.8 0.87 -1.19 21.7 0.93 -0.93 10.8 0.93 -0.90 10.1
PL [%] 8.10 11.12 8.29 11.12 8.29 8.47SW [M/eyear] 1080 1223 1128 1223 1128 1146
Table 3.10 displays the nodal direct voltages and power (in per unit), the loss allocation coefficients (in
percentage of the total losses), the transmission losses (in percentage of the offshore generation), and the
resultant social welfare (in M/eyear) for points A and B of all the examined case studies.
Case Study i
In the first case scenario there are no extra market restrictions on the load flow, thus, the entire Pareto
front on Fig. 3.12 (a) is reachable. The network losses difference for the two Pareto front extremities
(points A1 and B1) is approximately 3% of the power being generated. Considering the offshore bid as
90 e/MWh (the Danish OWF bid), and the same OWFs power production, the yearly cost difference in
the transmission losses between the two points is circa 157 Me. On the other hand, the yearly difference
in social welfare is 143 M.e In this case, if the MTdc grid is operated in point A1 that would lead to a
saving of circa 14 Me, which comes from the fact that, although, the social welfare is higher in point B1,
it does not compensate the extra power losses.
Point A1
The onshore direct voltages are approximately the same (see Table 3.10), leading to a reduction of the
power trade between the countries. Consequently, this is the point where the power losses are minimum.
Since all OWFs have entered the market, the MCP in this case, as shown in Fig. 3.12 (d), is equal to
90 e/MWh, i.e. the Danish OWFs bid.
The onshore nodes losses coefficients for point A1, are evenly distributed. The Netherlands has the
highest loss coefficient (15.9%), as it receives the highest amount of power – 1.32 pu – from the MTdc
grid. The second higher loss coefficient is given to Belgium, as the Belgian OWF (BE1) power production
is 0.5 pu and, hence, an extra 0.42 pu needs to be transmitted there. Amongst the offshore nodes, the
highest loss allocation is given to UK1 (14.2%), the highest power injecting node. Finally, two offshore
nodes, BE1 and NL2, have a negative loss coefficient which, according to the Z-bus method, means their
power production and location in the network is helping to achieve the desired load flow.
Point B1
The countries with the highest bids receive most of the power coming from the MTdc network, thus,
the social welfare is at is maximum. Table 3.10 shows that Germany receives 1.55 pu of power at B1
in comparison with only 1 pu at A1. This extra power for Germany is, for its most part, being shifted
30
from the Netherlands, whose power reception dropped from 1.32 pu to 0.20 pu, since it is the lowest
bidder. Accordingly, the German onshore node is attributed the highest loss coefficient (34.2%); because,
to maximize the social welfare, it receives more power than other countries. Amongst the offshore nodes,
again the highest loss coefficient is given to UK1 (9.1%).
Case Study ii
In Case ii, due to the additional market constraint (see Table 3.9), part of the original Pareto front – the
operating points with low losses – becomes unavailable. The Pareto extremities points, A2 and B2, have
a difference of 2.83% in their power losses, which capitalized over a year represents circa 147 Me. On the
other hand, the difference in social welfare is 95 Me. Hence, the Pareto extremity point A2 represents,
from the market participants point of view, a more economical operating point. Point A2 in comparison
with A1, represents a gain in social welfare of 48 M/eyear, while the power losses represent a loss of
circa 10 M.e Thus, point A2 is more beneficial for the market participants than its counterpart A1.
Point A2
The transmission losses in this case are 8.29%, which is 0.19% higher than for point A1. The new
active constraint reduces the Dutch power to 1.13 pu, compared to previous 1.32 pu (see Table 3.9); and
the difference is once more transferred to Germany. The power at the other onshore nodes was kept
approximately constant.
Regarding the loss components, Germany is the node with the highest losses share with 21.4%. The
remaining countries are being apportioned circa the same losses coefficients. For the OWFs, once more,
UK1 has the highest share (14%).
Point B2
Since the Pareto front region, where the social welfare is being maximized, has not changed due to the
extra market constraint in Case ii, the point B remained constant, i.e. B1 = B2.
Case Study iii
The second additional market constraint (see Table 3.9) impairs the highest social welfare value in the
Pareto front by 77 M/eyear. The transmission losses are reduced by 2.65%, when comparing points B2
and B3, leading to savings of approximately 138 M/eyear. Therefore, for the market participants point
B3 is more economical than B2.
Point A3
The extra constraint only excludes Pareto front points where the highest social welfare values are found,
thus, point A2 is kept constant, i.e. A3 = A2.
Point B3
Germany now receives 1.33 pu of power, which is 0.22 pu lower than in point B2. The decrease is a result
of the extra active constraint limiting the maximum dc current to Germany at 1.5 pu. The dc power
which could not be transferred to Germany is injected in the Netherlands. Hence, as the Netherlands
31
is helping to achieved the optimal load flow, it is rewarded with the lowest losses share amongst the
onshore nodes. On the other hand, Germany and OWF UK1 are assigned, once more, with the highest
losses coefficients amongst the onshore and offshore nodes, for being the highest buyer and producer,
respectively.
32
Chapter 4
Conclusions
4.0.1 MTdc Control: Single-Objective
Multi-terminal dc networks can help pave the way for further penetration of remotely located renewable
energy sources, such as offshore wind and wave energy. Nevertheless, before MTdc networks can be
developed, various problems, such as the control of the power flow, need to be addressed and solved.
The power-flow control inside a large MTdc network can be achieved using the distributed voltage control
(DVC) strategy. In the present work, a genetic algorithm has successfully been applied to solve an
optimal dc load-flow problem and give the DVC strategy the necessary voltage references to share the
responsibility of controlling the MTdc system voltage between all the onshore nodes. Dynamic numerical
simulations performed show that the power flow control is accurate and fast, i.e. a new operating point
can be achieved within 200 ms. Moreover, the genetic algorithm makes sure that the MTdc network is
N-1 secure and, in the event of an ac fault in any network node, the network operation is not disrupted
as shown in the simulation results.
4.0.2 MTdc Control: Multi-Objective
Several studies have suggested the construction of a transnational multi-terminal dc network for the
interconnection of offshore wind farms, specially in the North Sea. However, optimal power flow control
in MTdc networks is still a challenge. In this work, an independent system operator, which controls the
MTdc network via a market pool structure, has been proposed. The introduced ISO is responsible for
receiving the pool market participants bids and, after solving an optimal dc power flow, sending the dc
voltage references for the onshore converters so that the optimal flow can be realized. The optimization of
the MTdc network power flow was performed by a multi-objective genetic algorithm set to optimize the
network transmission losses and social welfare. Additionally, a loss allocation procedure was implemented
to distribute the transmission power losses costs in a fair way amongst the generator and consumer entities.
Through this procedure, market participants with higher power supply and demand were allocated with
higher power losses coefficients.
The results from the obtained Pareto fronts indicate the maximum social welfare is achieved when more
power is transmitted to countries with higher bids, however, this leads to an increase in transmission
losses. The addition of extra constraints to the power flow prohibits the operation in some of the Pareto
front regions. Consequently, it may not be possible for the ISO to operate the MTdc network where the
best possible cases are found. In this way, the ISO has to decide which is the best operating point for
the offshore grid.
33
The proposed approach to control MTdc network offers a high flexibility to the ISO in obtaining optimal
trade-offs. Although, in this work only the transmission losses and social welfare were analyzed, more
optimization goals may be considered simultaneously.
4.0.3 Transmission Systems Design
The MOOA implemented was able to find feasible solutions for the proposed problem. Optimal Pareto
Fronts were achieved for both AC and DC transmission options when different transmission distances
were considered. The decision maker can then be presented with these OPFs for the transmission system;
and, based on the trade-off between the two goals, it will be possible for him to decide which configuration
better suits the purposes of the project.
It is important to refer that data from real components was used. Nevertheless, as with any database,
the one used is limited in information; and, therefore, the results obtained with the algorithm have
the database as a major drawback. Moreover, the MOOA itself does not guarantee the optimal set of
solutions [40].
The next application of the multi-objective optimization method would be to include component reliabil-
ity, power quality, control and grid code aspects to transmission systems for offshore grids, which cannot
be easily taken into account with other optimization methods.
4.0.4 Multi-Objective Optimization Algorithms
MOOAs are an important instrument since, among other characteristics, they do not require any infor-
mation about the targets to be optimized besides their own evaluation. Therefore, the goal functions
are not restrict with respect to differentiation, continuity or any other particular behavior in the search
space of the problem. MOOAs are able to work simultaneously with real and integer representation of
variables [41]. A very important feature for the success of such algorithms is the fact that they treat the
multi-objective problem with non-commensurable objectives, i.e. no a priori information is needed [42].
34
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39
Appendix A
MTdc network technical and
economic aspects
A.1 MTdc networks
For bulk transmission purposes (e.g. ≥ 500 MW), as well as for long submarine transmission distances
(e.g. ≥ 60 - 80 km), using HVdc systems for the transmission of the generated electricity is both
economically and technically more attractive than using HVac systems [43][44]. When it comes to HVdc
transmission two options are available: line-commutated current-source HVdc (CSC-HVdc), also referred
to as classic HVdc, or forced-commutated voltage-source HVdc transmission (VSC-HVdc). The latter
is a newer technology and, despite the fact it has higher converter losses, it has significant advantages
when it comes to bringing the generated offshore wind energy ashore [45][46]. Due to its smaller footprint
and improved controllability, the use of VSC-HVdc systems is being considered as suitable for a broad
range of multi-terminal dc applications, such as supergrids, electronic power distribution systems and
transnational networks for integration of offshore wind farms [32].
The advent of VSC-HVdc for transmission purposes has brought back the interest in building MTdc
networks; which has been proved difficult solely using CSC-HVdc technology [2]. The limitations to
2-quadrant operation, the need for reactive power supply at each terminal, harmonic content filtering
equipment size and its limited control capabilities are examples of drawbacks when it comes to the
construction of a MTdc network using classic HVdc technology [46]. Hence, most likely, high-voltage
MTdc networks will be developed using voltage-source converters, given their control flexibility and
hardware modularity [47, 48].
A.1.1 Applications of a MTdc grid
Multi-terminal HVdc transmission systems are characterized by having more than two converter stations
interconnected on the network dc side. There are several possibilities for the grid topology: shore-to-
shore, radial, meshed, cluster with multi-way interconnector, a combination of interconnectors, amongst
others [1]. The configuration choice may differ depending on the application of the MTdc network. Some
or the most prominent applications are briefly discussed next.
40
Offshore Wind & Ocean Energy
In Europe, the offshore potential consists both of wind and ocean energy. Ocean energy includes wave,
tidal current, tidal range, osmotic and ocean thermal energies. West European coasts have high wave
energy potential [49]. Since wave energy has greater predictability and less variability when compared to
wind energy; and because wave energy generally peaks 6 to 8 hours later than wind energy, a combination
of both resources would result in smoother variations of the generated power and better predictability
[49].
In the near future, offshore wind farms (OWFs) are expected to have higher power ratings and to be
situated further offshore [1]. This implies that the costs of the electrical infrastructure will constitute a
higher share in the total investment costs. New offshore projects may be interconnected with the already
existing MTdc network, which increases the existing connection utilization. Moreover, since it is not
necessary to build a new transmission systems to connect to shore, the initial capital expenditure of new
projects can be reduced [49].
Oil & Gas Platforms
In 2005, the Norwegian energy company Statoil became the first to have an offshore platform (Troll A)
powered via a VSC-HVdc transmission system [50]. In most offshore installations, generators and large
compressors are driven by onboard gas turbines or diesel engines, whose efficiencies are low, usually well
below 50% [50]. This results in large amounts of CO2 emissions and a high fuel consumption.
A MTdc network interconnecting OWFs, oil and gas platforms and onshore grids would result in reduced
operational costs, increased reliability, reduced CO2 emissions, lower maintenance requirements, longer
lifetime and higher availability when compared to gas turbines and diesel engines. Moreover, if the
transmission equipment can be located on decommissioned offshore platforms, their postponed removal
cost can be an important factor as well [50].
Energy Market
Due to the deregulation of the European energy sectors, there exists an increasing demand for cross-
border exchange capacity. The need for the higher interconnection capacity is led by market coupling
and for balancing fluctuations in the output from renewable energy. A MTdc network connecting offshore
wind power plants with inter-connectors between countries could increase the utilization of the electrical
infrastructure and, hence, reduce costs. Additionally, it could help to decrease the variability in wind
power production (which decays when aggregated over large areas), as well as enable the trade of power
imbalances, such as unscheduled surpluses or deficits of wind energy [51].
Asynchronous Systems & AC Grid Reinforcement
A dc network allows for the interconnection of asynchronous systems, differently from ac networks; where
a surplus of generating capacity in one grid it cannot be transmitted to others if they have different
frequencies (e.g. Brazil and Paraguay) or are not synchronous (e.g. the UK and mainland Europe). Every
network must have its own capacity of peak power generation, usually in the form of older, inefficient
power plants. Moreover, asynchronous HVdc links can act as a firewall, effectively protecting against
propagation of cascading outages in one network to another. Furthermore, with a dc transmission system
it is also possible to damp local area power oscillations, manage congestion and reinforce existing ac grids
[33].
41
Reliability and Redundancy
Most offshore wind power plants built so far use a single connection to shore [15]. The reliability of the
connection critically relies on some of its main components, most importantly the cable circuitry. When
malfunctioning, these components might take the complete installation out of service.
In the future, TSOs may increase the requirements on the connection availability. By having several con-
nection points to the onshore grids, properly designed MTdc networks are expected to improve reliability.
Even in the event of a MTdc network section fault, there can be an alternative path along which the
power, or part of it, can still be transmitted.
Flexibility and Controllability
Nowadays, large wind farms are required to actively participate in the power system control – both
frequency and voltage – just as any conventional power plant [52][53]. When an OWF is connected via
a dc transmission system, the responsibility of complying to the grid code requirements is passed onto
the onshore HVdc station. A VSC can generate or consume reactive power from the connected network
helping with voltage regulation. Additionally, a predefined power flow along a certain transmission cable
can be quickly achieved.
Therefore, when connecting offshore wind farms, a MTdc network may guarantee the necessary flexibility
and controllability to meet the main grid code requirements such as frequency operating range, active
and reactive power control, support during voltage dips and fault-ride through capabilities.
A.1.2 Challenges towards a VSC-based MTdc network
By 2030, circa 150 GW of wind power is predicted to be installed in the North Sea and offshore wind
farm constructions have already started. Out of a total of 76 offshore wind farms installations around
the world, 31 (circa 41%) are in the North Sea. From the latter, six OWFs – all planned to be delivered
in 2013/2014 – will use an HVdc transmission system [54].
The North Sea MTdc network can serve as a platform for offshore wind and electricity market integration
in Northern Europe. Hence, power flow control inside MTdc networks will be of vital importance [55].
However, before multi-terminal dc networks can be developed, four key aspects need to be better studied
and developed: system integration, dynamic and contingency behavior, stability and power flow control.
System Integration
System integration aspects of multi-terminal dc networks have been tackled by several studies [56, 57, 58].
The main conclusion is that MTdc networks will have to organically grow with time; from an inherently
simple initial phase to a desired later form, a hopefully much more complex meshed topology [59].
Dynamic & Contingency Behaviour
The dynamic behavior of VSCs — including multi-level topologies — and multi-terminal dc networks has
been extensively analyzed in the literature [60, 61, 62]. The behavior of MTdc networks during contin-
gencies, specially on the dc-side of the transmission system, will be vital for its successful development.
After a dc contingency the system will most likely have only a few milliseconds to clear faults before high
42
dc currents start posing a threat to the MTdc network [63]. This is the reason why several efforts are
currently focusing on research and development of HVdc switch breakers [64, 65].
Dynamic Behaviour
The stability of MTdc networks has also been thoroughly analyzed in the literature [61, 66]. It has been
shown that MTdc networks formed by VSCs are inherently unstable systems which need feedback control
to compensate the unstable system poles [67].
Power Flow Control
Inside a MTdc transmission system, controlling the network direct voltage is one of the most important
tasks given to VSC-HVDC stations. A well-controlled direct voltage on a HVDC link is a guarantee of
power balance between all the interconnected nodes [47].
The control of point-to-point HVDC transmission systems is typically arranged so that one terminal
controls the dc network voltage, whereas the other operates in current or power regulation mode. This
control philosophy – of having only one converter controlling the direct voltage – can be extended to
MTdc networks. However, as the MTdc network grows, it will be increasingly difficult to assure power
balance by having only one terminal responsible for voltage regulation. Hence, for large MTdc networks,
controlling the voltage at a single terminal is not desirable. For its successful development and operation,
MTdc networks will require a control strategy capable of sharing the dc system voltage control amongst
more than one network node.
Different control strategies have been proposed in the literature to operate MTdc networks. Four different
direct voltage control methods were analyzed and compared according to their capability of performing
different market dispatch schemes in [33, 68]. A brief introduction to these control methods is given next.
• Voltage droop method: it uses a droop mechanism to control the dc voltage. Although this method
was initially thought for controlling MTdc networks using HVdc classic technology, it can also be
applied for VSC-based MTdc networks [69];
• Ratio control: a power ratio between onshore stations may be established. The power transmission
ratio between the stations can be set and varied by the system operator from time to time [70];
• Priority control: one onshore terminal has priority in terms of receiving the energy generated
offshore over the others. With this control strategy, the remaining onshore terminals will not
receive any energy until the capability or the active power limit of the first terminal has been
reached [71];
• Voltage margin method: in this method, each converter station inside the network is given a
dc voltage reference marginally offset, generating the voltage margin. The method was initially
developed to control a three-terminal back-to-back HVDC system in Japan [72].
The aforementioned control methods have drawbacks regarding their expandability, dynamic response
and flexibility, i.e. the capacity of steering the power flow inside large MTdc networks [68]. A nre
method of controlling VSC-MTdc networks called distributed voltage control (DVC) is proposed [73]. In
the DVC method the dc system voltage references are obtained through means of a dc Optimal Power
Flow.
43
A.2 Market Schemes
A restructuring process in the power industry, which started at the end of the seventies, gave birth to the
world’s first competitive market for electricity generation in Chile, in 1982 [74]. The order to privatize
the United Kingdom electricity industry in 1988, concluded with the creation, in 1992, of the England
and Wales Power Pool [75]. In the same year, the approval of the Electricity Policy Act (EPAct) ordered
an open access to transmission networks in the USA. Ever since, a variety of electricity pool markets have
been created; among them, the Pennsylvania-New Jersey-Maryland (PJM) Pool (1997), the Californian
market, and the New York Power Pool (1998) [75].
Several other countries have also implemented electricity markets, or are in the process of doing so; e.g.
Australia, Brazil, Germany, Norway and Spain. The world-wide restructuring process of the electricity
industry begun more than twenty years ago and has since deeply accelerated, creating a political and
technical turmoil [76]. The operation of large investors or state-owned utilities is being transformed from
an rate-of-return basis to a competitive basis by the creation of electricity markets.
Future offshore transnational offshore networks will most probably employ a centralized model imple-
menting a pool market scheme to regulate the wind farms output power [35]. Moreover, the possibility of
setting up an international independent system operator (ISO) to regulate a future offshore transnational
DC network in the North Sea is currently under discussion [77].
A.2.1 Pool Market
A Pool Market is defined as a centralized marketplace which clears the market for buyers and sellers. In
the case of an offshore MTdc network, the sellers will be represented by the offshore wind farms whereas
the buyers will be constituted by the onshore networks, somehow interconnected via the MTdc network.
The market participants submit their bids to the pool, stating the amount of power that they are willing
to trade. The sellers in a pool market compete for the right to supply energy to the grid, not to specific
customers. If an electricity seller would bid too high, it might not be included for dispatch. On the other
hand, buyers compete for purchasing power, therefore, if their bids would be too low, they may be left
out of the market. This market scheme is shown in Figure A.1.
POOL MARKET
Bids MCP
Bids MCP
Onshore Stations
Offshore Wind Farms
Figure A.1: Diagram of a pool market applied to a future offshore MTdc network.
ISO
A competitive electricity market needs an independent operational grid control. Such task cannot be
guaranteed without establishing an ISO [76]. The ISO administers transmission tariffs, maintains the
system security, coordinates maintenance scheduling, and has a decisive role in coordinating the long-term
planning. It functions independently of any market participants, such as transmission owners, generators,
distribution companies and end-users. Moreover, it should also provide non-discriminatory open access
to all transmission system users. The ISO has the authority to commit and dispatch some or all the
44
system resources; and to shed loads if necessary to maintain system security, e.g. remove transmission
violations or balance supply and demand.
By running an auction for the electricity trade, the ISO ensures a competitive marketplace. The spot
price, also designated market clearing price (MCP), is obtained based on the lowest bid, which is the bid
of the last demand entity which was admitted to enter the market. In order to obtain the spot price, the
ISO makes use of an optimal power flow (OPF) dispatch model, taking into consideration the constraints
of the network. Market participants must provide extensive data, such as cost data for every generator
and daily demand for every consumer or load. Utilizing these extensive data, the ISO obtains the unit
commitment and the correspondent dispatch, which maximizes the social welfare and sets transmission
congestion prices. The social welfare can be mathematically described as:
Social welfare =(CT
DPD − CTS PS
)(A.1)
where CD and CS are, respectively, demand and supply bids vectors in e/MWh, whereas PD and PS
represent the demand and supply power bids vectors in MW.
A.2.2 Transmission Loss Allocation
In a practical implementation of a MTdc network, the question of whom should be the responsible for the
system losses payment naturally arises [78]. Since system losses can be typically estimated as five to ten
percent of the total generation, if capitalized over the course of one year, the value of system losses can be
in the range of tenths of millions of euros [36]. Consequently, a fair power losses allocation among loads
and generators has an important impact on their benefits and needs to be included in the optimization
algorithm.
Although transmission losses are nonlinear functions of the line flows(Plosses ∝ I2line
), several studies
have tried to determine which amount of a line power flow is the responsibility of a given generator or
demand. In [79] and [80] new methods for tracing the flow of electricity in meshed electrical networks
are proposed.
Due to the lack of a perfect algorithm for tracing the power flow, its acceptance will mainly depend on the
perceived fairness as seen by each one of the pool participants [78]. In this work, the chosen transmission
losses allocation procedure is a circuit-based strategy known as Z-bus loss allocation method [36]. A brief
description of the procedure is given next.
Z-bus Loss Allocation Method
The classic Z-bus allocation method used mostly on AC networks [36], is here modified for its application
in MTdc networks. The Z-bus method emphasizes the currents, instead of the active power flow, during
the losses allocation. It also takes into consideration the network topology. Therefore, generators or
loads which are located in distant regions from the network center of gravity tend to proportionally be
allocated higher losses.
The Z-bus allocation method makes use of a previously solved power flow to systematically distributes the
power transmission losses, Plosses, amongst all the nodes present in the MTdc network. This procedure
can be mathematically expressed as:
Plosses =n∑
k=1
Lk (A.2)
45
where Lk stands for the power losses fraction allocated to the k-th DC node, and n is the total number
of nodes inside the MTdc network.
In this way, the responsibility of paying for its losses share is assigned to each individual market partici-
pant. The extra cost due to the loss allocation must then be deducted from the revenue of the generating
entities and added to the load payments in order to maintain the pool revenue null.
If a certain DC node has both generation and demand, its allocated loss component, Lk, may be divided
among the different players in a pro rata manner [36]. On the contrary, if a specific DC node has neither
load nor generation, its loss allocation must be zero.
In order to calculate the allocation fractions, Lk, the Z-bus method makes use of the network impedance
matrix, Z. The latter can be further separated into two matrixes: the resistance matrix, R, and the
reactance matrix, X. However, for the power losses calculation only the resistance matrix is of concern
[36]. Hence, the power losses can be achieved through:
Plosses =
n∑k=1
Ik
⎛⎝ n∑
j=1
RkjIj
⎞⎠ (A.3)
where Ik is the DC current injection at the node k, Rkj is the DC cable resistance between the nodes k
and j, and Ij is the DC current injection at node j.
Finally, the loss allocation terms, Lk, can be expressed as:
Lk = Ik
⎛⎝ n∑
j=1
RkjIj
⎞⎠ (A.4)
As can be seen from (A.4), the loss component Lk, is equivalent to a weighted function where the current
in node k, Ik, serves as weight to the sum of the currents in all the network nodes. With this approach,
no special assumptions or approximations are necessary to derive the loss allocation terms.
A.3 MTdc Network Control Principles
The control of point-to-point HVDC transmission systems is typically arranged so that one terminal
controls the dc network voltage, whereas the other operates in current or power regulation mode. This
control philosophy – of having only one converter controlling the direct voltage – can be extended to
MTdc networks. Mathematically, disregarding transmission losses and considering the magnitude of the
transmitted powers, this control strategy for MTdc networks can be translated as:
PmaxV dc ≥
N∑i=1
P iIdc(t) (A.5)
where PmaxV dc is the maximum rating of the VSC controlling the direct voltage (N-th node) and P i
Idc(t) is
the power production at the i-th VSC terminal operating in current regulation mode.
As it can be seen from (A.5), as the MTdc network grows, it will be increasingly difficult to assure power
balance by having only one terminal responsible for voltage regulation. Hence, for large MTdc networks,
controlling the voltage at a single terminal is not desirable. In addition, if an outage would affect the
only direct voltage controlling station, direct voltage control would be lost until somehow transferred
to another node in the network. Such scenario is, however, also not desirable. The fast nature of dc
46
phenomena could trigger protection equipment, within only a few cycles of the ac network [48].
For its successful development and operation, MTdc networks will require a control strategy capable of
sharing the dc system voltage control amongst more than one network node. A more suitable control
strategy for large multi-terminal dc networks, would be then to assign each dc-voltage-controlling VSC
terminal with a specific voltage set-point.
In this way, any predefined load flow scenario can be achieved while no single converter is left alone
with the responsibility of balancing the power inside the transmission system, i.e. the control of the dc
system voltage is distributed between several nodes inside the MTdc network. This method of controlling
VSC-MTdc networks is called distributed voltage control (DVC) [25] and it is depicted in Fig. A.2.
Distributed DC Load Flow
Algortihm
OPF Algorithm
No
Generation at OWFs
Voltage setpoints
to GS-VSCs
Optimum?
MTDC Secure?
Check for N-1 Security
No
Yes
Yes
DVC
SCADAsignal
SCADAsignal
Figure A.2: Flow chart of the distributed voltage control method.
A.3.1 Information flow
The information flow inside the system is as follows: firstly, according to the wind speeds, a certain power
is produced at the OWFs. The information about the generation is given to the MOOA, which, knowing
the MTdc network topology, will obtain optimal power flows according to the several constraints. Finally,
according to the DVC method, the dc system voltage reference values are transmitted to the onshore
VSC stations to generate the optimal power flow inside the MTdc grid. In Fig. A.3 it is shown how the
entire system is interconnected.
MOOA
MTDCnetwork
OnshoreVSCs
ACgrids
OffshoreVSCsOWFsWind
*1DCV
*DCjVWFi
DCP
1WFDCP
... ...
Figure A.3: Information flow between the OWFs, the GA and the onshore VSC stations.
It is important to point out that the MOOA is provided with the OWFs averaged power production over
a certain period. In this way, it is expected that the actual and the desired load flows will somehow
differ. However, since several OWFs will be connected to the MTdc network, it is anticipated that the
variability in the total power production will be smoothed out, due to the effect of integrating the wind
energy production over a large area [81].
47
A.3.2 Social Welfare
Social welfare is widely used as an efficiency indicator to evaluate liberalized power markets and may be
mathematically described as:
SW =
qλ∫0
(B − S) dq (A.6)
where B and S are, respectively, the buyers and sellers aggregated bid curves as a function of the power
quantity, and qλ is the traded power quantity.
The aggregated curves for sellers and buyers for two areas, 1 and 2, may be described as [82]:
S1,2 = α1,2 + β1,2 · q ; B1,2 = γ1,2 − φ1,2 · q (A.7)
where β and φ are the curves slopes and α, β, γ and φ > 0.
Without trade (Fig. A.4 (a)), the traded quantities and MCPs (λ) of both areas are:
qa1,2 =γ1,2 − α1,2
β1,2 + φ1,2; λa
1,2 =α1,2φ1,2 + β1,2γ1,2
β1,2 + φ1,2(A.8)
(b)
(c)
(a)Price
PriceBuyers WelfareSellers Welfare
S1
B1
S2
B2
Price
1aq 1
bq
1a1b
2b
2a
1a
2a
1aq
2bq 2
aq
2aq
1aq 1
cq 2aq2
cq
q q
qqq
qqq
1a
2a
1 2c c
2 1c c
1 2bq
1 2cq
Figure A.4: Social welfare of two areas, 1 and 2, for three distinctive cases: (a) no trade; (b) limitedtrade; (c) unlimited trade.
In order to maximize social welfare, the ISO establishes power flows from areas with lower MCPs to
areas with higher MCPs (see Figs. A.4 (b) and (c)) . In such way, costly and highly-pollutant units are
48
not dispatched. Moreover, cheaper energy suppliers will see their welfare increased, enhancing market
competition.
The social welfare might not be fully maximized if the transfer capacity between the areas is limited
(Fig. A.4 (b)). Mathematically this can be expressed as:
SWnt︸ ︷︷ ︸no
trade
≤ SWltc︸ ︷︷ ︸limitedtransfercapacity
≤ SWutc︸ ︷︷ ︸unlimitedtransfercapacity
(A.9)
49
Appendix B
Multi-Objective Optimization
Algorithms
B.1 Introduction
In many real-world situations there may be several objectives which must be optimized simultaneously
in order to solve certain problems. The main difficulty when considering multi-objective optimization
problems is that, in such cases, there is no definition of optimum solution. Therefore, it is difficult to
compare solutions and decide which ones are better. The quality of a solution is subjective and depends
on the needs of the designer or decision maker [83].
Although multi-objective genetic optimization algorithms do not guarantee the optimal trade-off between
the criteria taken into consideration, they provide good solutions within acceptable computation times
[84]. These algorithms mimic the behavior of natural evolution presented by Darwin in 1859 [85] and
treat solution candidates as individuals of a population that compete in a virtual environment. Each
solution is also designated as a chromosome, composed by genes.
The mathematical foundations for multi-objective optimization were laid down by Vilfredo Pareto more
than one hundred years ago [86]. Pareto optimality became an important notion in economics, game
theory, engineering, and social sciences. It defines the frontier of solutions that can be reached by
trading-off conflicting objectives in an optimal manner. From the Optimal Pareto Front (OPF) the
decision maker may choose a solution that best suits the problem considered [87].
MOOAs are an important instrument since, among other characteristics, they do not require any infor-
mation about the targets to be optimized besides their own evaluation. Therefore, the goal functions
are not restrict with respect to differentiation, continuity or any other particular behavior in the search
space of the problem. MOOAs are able to work simultaneously with real and integer representation of
variables [41]. A very important feature for the success of such algorithms is the fact that they treat the
multi-objective problem with non-commensurable objectives, i.e. no a priori information is needed [42].
Over the past two decades, several MOOAs have been suggest. The most famous and used algorithms
are, among others, the NSGA-II [88], Pareto-archived evolution strategy (PAES) [89] and the strength-
Pareto EA (SPEA) [90]. A comparison between the standard MOOAs is elaborated in [91]. Different
approaches from the genetic algorithms have been introduced in the literature, being the Particle Swarm
Optimization one of the most famous algorithms [92]. This algorithm, differently from the GAs does not
rely only in the information of the present population to create new solutions, since it keeps track of the
best result of each solution. Its multi-objective optimization form was presented in [93].
50
The purpose of a MOOA is to obtain the optimal Pareto-front (OPF) of a problem. This front gives
the optimal trade-off between the considered fitness functions. A MOOA can be generically described by
[91]:
MOOA
⎧⎪⎪⎪⎨⎪⎪⎪⎩
Optimize F
gi(x) ≤ 0 ; 1 ≤ i ≤ J
hi(x) = 0 ; J + 1 ≤ i ≤ J +K
LBi ≤ xi ≤ UBi ; 1 ≤ i ≤ m
where F consists of n objective functions fi, each representing one criterion to be optimized; gi(x) are
the inequalities of the problem; hi(x) are the equalities that the algorithm solutions need to respect and
LBi and UBi are the variables boundaries.
Optimal Pareto Front
Being X the solution space of the problem, an element x∗ ∈ X is Pareto optimal (and hence, part of
the optimal set X∗) if there is no other element in X that dominates x∗. Mathematically, this can be
expressed as:
x∗ ∈ X∗ ⇔ ∀x /∈ X∗ : x∗ � x (B.1)
In terms of Pareto optimization, X∗ is called the OPF.
An element x1 dominates x2 (x1 � x2) if, x1 is better than x2 in at least one objective function and not
worse with respect to all other objectives [94]:
x1 � x2 ⇔∀i : 1 ≤ i ≤ n ⇒ wifi(x1) ≤ wifi(x2) ∧∃j : 1 ≤ j ≤ n : wjfj(x1) < wjfj(x2)
(B.2)
where wi is 1 if fi is to be minimized and −1 if a maximization of fi is desired.
In Fig. B.1 it is possible to see an example of an OPF if the minimization of two functions is considered.
1 1( ,..., )nf x x
21
(,..
.,)
mfx
x
Optimal Pareto Front (X*)
SolutionSpace (X)
Figure B.1: Pareto front of a Multi-Objective Minimization Problem.
51
B.2 Proposed Multi-Objective Optimization Algorithm
The proposed multi-objective optimization algorithm resembles greatly the already referred standard
MOOAs. However, new ranking and sort strategies for the population are introduced. Moreover, a new
genetic operator, so-called regeneration, is also introduced. In Fig. B.2 it is shown a high-level flow chart
of the proposed MOOA while in (B.3) the pseudo-code is presented.
6. Terminate?
1. Population Initialisation
2. Finess Evaluation
3. RankPopulation
4. SortPopulation
5. ArchiveUpdate 7. Selection
8. Crossover
9. Mutation
10. Regeneration
11. Output Archive
Yes
No
Figure B.2: Flow chart of MOOA implemented.
52
Archive actual size = 0;
N generations = 1;
Population = initialization;
while (1)
Population( : , Aug F it1 · · ·Fitm) = fitness(Population);
Population( : , Aug F it1 · · ·Fitm) = fitness(Population);
Population( : , Rank) = rank(Population);
Population = sort(Population);
/ ∗Archive update ∗ /Archive = cat(Archive, Population( : , Rank == 1));
Archive( : , Rank) = rank(Archive);
Archive = Archive( : , Rank == 1);
Update Archive actual size;
if Archive actual size > Archive max size
Archive = crowding competition(Archive);
end
increment N generations;
if N generation > N generations max
break;
end
Mating Pool = selection(Archive, Population);
Population = crossover(Mating Pool);
Population = mutation(Mating Pool);
Population = regeneration(Mating Pool);
end
(B.3)
In the next subsections each step of the flow chart will be introduced and described.
B.2.1 Population Initialization
There are two ways to initialize the population, P(t=0), of a MOOA [83]. One of the options is to
randomly initialize all the variables of the chromosomes. However, it has to be assure that the random
values respect the boundaries of the respective genes. The initialization can be performed according to
the following formula:
genei = α× (UPi − LBi) + LBi (B.4)
where LPi and UPi represent the boundary limits of the i− th gene, and α is a random number between
0 and 1.
The second approach is to choose the initial variable values for the genes of the chromosomes. This
53
strategy may be mainly used if there is some a priori knowledge about the problem. In Fig. B.3 it
is shown the population initialization procedure implemented in the algorithm. Independently of the
variables initialization, the value 1 is attributed in the origin field of the chromosomes.
PopulationInitialization
User specified values
Bonded random values
Origin = 1
Figure B.3: Population initialization process.
B.2.2 Fitness evaluation
In this step the fitness values of the solutions are determined. In this way, the chromosomes are tested
in order to see how well they perform in the optimization goals that are being considered.
Most of the real world problems have constrains over the variables problems. This way, genetic algorithms
must have a way to handle constrains and, therefore, present feasible solutions at the end, i.e. solutions
that respect all the constrains and boundaries of the considered problem. In Appendix A a description
about constraint handling techniques is presented.
The penalty term is obtained according to the following formula:
pen(x) = φα × λ (B.5)
where φ =[1 · · · J +K
]and λ is the sorted vector with the constraint violations.
The terms λi(x) are calculated as:
λi(x) =
⎧⎪⎨⎪⎩
0 if x is feasible
|gi(x)| if x is infeasible ∧ 1 ≤ i ≤ J
|hi(x)| if x is infeasible ∧ J + 1 ≤ i ≤ J +K
B.2.3 Constraint Handling
There are several constrain handling strategies in the literature (for surveys regarding constraint handling
techniques in a multi-objective genetic algorithm see [95, 96]). In this report the two most common
strategies to handle constrains will be present and described.
Before presenting the strategies it is shown, once again, the constrains that a genetic algorithm can be
submitted to:
Constraints
⎧⎪⎨⎪⎩
gi(x) ≤ 0 ; 1 ≤ i ≤ J
hi(x) = 0 ; J + 1 ≤ i ≤ K
LBi ≤ xi ≤ UBi ; 1 ≤ i ≤ n
where gi(x) are the inequalities of the problem; hi(x) are the equalities that the solutions of the algorithm
need to respect and LBi and UBi are the variables boundaries.
54
In general, the solution space can be divided in two distinctive parts: the feasible and the infeasible areas.
This distinction in areas is depicted in Fig. B.4.
1 1( ,..., )nf x x
21
(,..
.,)
mfx
xOptimal Pareto
Front (X*)
Feasiblespace
SolutionSpace (X)
Figure B.4: Pareto front of a Multi-Objective Minimization Problem.
Preservation of feasible solutions
One of the simplest ways to assure that all the solutions are feasible at all times is to rejected offspring that
are infeasible. With this strategy, all offspring after their generation are tested in order to verify if they
respect all the constrains of the problem. Another approach is to try and change an infeasible solution
into a feasible one via a deterministic procedure - repairing strategy. Although these strategies have the
advantage of never generating infeasible solutions, in a highly constrained problem, infeasible solutions
may have an important role. The movement of solutions through infeasible areas tend to increase the
velocity of the optimization and to produce better solutions [97].
Penalty term
Penalty techniques are most probably the most common strategy to handle constrains in a genetic algo-
rithm [40]. With this approach, a constrained problem becomes unconstrained and the infeasible solutions
are penalized, having, this way, worse fitness values. The penalty term that is added to infeasible solu-
tions traduces their infeasibility. Therefore, if the penalty term is small that means that the solution is
infeasible but it is close to be the feasible area. On the other hand, if the penalty value is high, that
corresponds to a chromosome that is far away from the desired feasible region.
In general, there are two distinctive ways to obtain the fitness value of a chromosome with the penalty
term: addition or multiplication of the penalty with the fitness value. If the addiction approach is chosen,
the augmented value (total evaluation of the score of a chromosome) of the fitness of a chromosome is
obtained through:
Aug Fit = Fit + Penalty (B.6)
Since in a minimization problem a solution with a lower fitness value is a better solution, the penalty
term has to be derived as:
{pen(x) = 0; if x is feasible
pen(x) > 0; if x is infeasible(B.7)
There are several options to generate the penalty term. The strategy implement to obtain the penalty
value has to be carefully chosen since the guide has to be effectively guided towards a promising area
55
of the solution space. The penalty function can be classified in two groups: constant and dynamic
penalties. The approach that uses the constant penalty value is known to have a worse performance than
the strategies that implement a variable penalty scheme and, therefore, a higher attention is being given
to the last one by the researchers [98].
The dynamic penalty approaches can be separated in two groups: the variable penalty ratio and the
penalty value for the violation of the constraints [99]. The last one applies a penalty term that varies
with the “amount” of violation of the constraints by a chromosome. The variable penalty ratio can be
implemented according to the degree of the violation of the constraints or according to the generation
number of the genetic algorithm.
The dynamic penalty approaches can be separated in two groups. In the first group the penalty term is
determined by the degree of the violation of the constrains. In this way, a solution that violates severely
the constraints of the problem is given a higher penalty (if a minimization problem is considered). For
the second group of dynamic penalties, the generation number of the genetic algorithm is the variable
that controls the penalties attributed to the infeasible solutions of the population.
B.2.4 Population Ranking
After the fitness evaluation, the population is ranked. In Table B.1 the possible ranks are introduced and
distinguished, while in (B.8) the pseudo-code for the rank attributions is laid down.
Table B.1: Possible rank values and their description.
Rank Description
1 Non dominated and feasible solution2 Non dominated, feasible and repeated solution3 Non dominated and infeasible solution4 Dominated Solution
Pop(:, Rank) = 4;
for each solution of the Pop
if Pop(i, Rank) == 4
Index(1 : Popsize) = true;
Index2(1 : Popsize) = true;
for each optimization goal fi
Index(Index) = Pop(Index, fi) ≤ Pop(i, fi);
Index2(Index2) = Pop(Index2, fi) == Pop(i, fi);
end
if sum(Index) == sum(Index2)
if Pop(Index) are feasible
Pop(i, Index) = 2;
Pop(i, Rank) = 1;
else
Pop(i, Rank) = 3;
end
end
end
(B.8)
56
B.2.5 Population Sorting
The next step of the MOOA is the population sorting. The first step is to sort the population through the
rank value. Thereafter, each rank has its own sorting, being the maximum constraint violation the second
sort parameter. In this way, each rank gives priority to solutions that disrespect with lower severity the
problem constraints. The attempt of this principle is to reach feasible solutions. In Table B.2 it is shown
the aspect of the population after being sorted.
Table B.2: Description of the population sorting.
Rank Sorting
1 Lowest to Highest Max constraint2 Lowest to Highest Max constraint3 Lowest to Highest Max constraint4 Lowest to Highest Max constraint
B.2.6 Archive Update
The Archive represents what in a single-objective genetic algorithm is known as elitism, which is a
strategy to ensure that the best chromosomes found so far are not lost due to randomized operators such
as crossover and mutation. In the multi-objective context, the elitism is done through the preservation
of the non-dominated solutions found so far [91]. There are two approaches to keep stored the non-
dominated solutions: to keep all found so far or the pruning strategy where only part of the solutions
are kept [100, 101]. If the later is chose it is important to assure that it is achieved a balancing between
proximity and diversity of the OPF [102].
After sorting the population all the solutions that were assigned with Rank 1 are merged with the present
Archive. After an archive ranking is performed, and in this way, the solutions that are dominated will
be erased from the Archive. If the number of archived solutions is higher than the archive maximum
size, a crowding competition is performed in order to choose with chromosomes to keep. In Fig. B.5 the
archiving process is depicted.
Archive
Population Solutions with Rank=1
Crowding Competition
Yes
NoNumber of solutions > Elite size?
Archive
Merge Rank
Figure B.5: Archiving process of the non dominated solutions.
It is important to refer that a crowding competition will always be performed if the actual archive size is
higher than the number of solutions that are sent from the Archive to the mating pool (elite size). In this
way, the chromosomes sent to participate in the creation of new solutions are the ones with the biggest
cuboids. Such feature enhances the search of Pareto front zones that were the least explored.
Crowding competition
The crowding distance, originally from the NSGA-II algorithm [88], is used to estimate the solutions
density surrounding a particular solution i. To do so, the average distance of the two solutions that are
57
next to i is taken for all the objectives. The average distance, di , is an estimation of the perimeter of
the cuboid formed by the closest neighbors. In Fig. B.6 it is shown the cuboid for the solution i.
2f
1f1i 1
1i 1
i
Cuboid
Non dominated solution
Dominated solution
Figure B.6: Crowding competition among the non dominated solutions present in the archive.
In order to calculate the crowding distance of the archived solutions the following steps are used:
1. Obtain the number of solutions in the Archive: l = size(Archive).
2. For all the solutions assign d = 0.
3. For each fitness function, m = 1, 2, ...,M , find the sorted indices vector in worse order of fitness
value, Im = sort(fm), and assign dIm1
= dIml
= ∞.
4. For the other chromosomes, i = 2, ..., l − 1, the crowding distance is calculated as:
dImi
= dImi+
fImi+1
m − fImi−1
m
fmaxm − fmin
m
(B.9)
The elements Im1 and Iml represent the best and the worst fitness values for the fitness function m,
respectively. The last term of the equation shown above represents the difference in fitness function
values between the two neighbors of the solution Imi . The parameters fminm and fmax
m represent the
maximum and the minimum value of fitness for the m-th function.
B.2.7 Ending criteria
In a MOOA the presence of a termination criteria for the algorithm is needed. The most common
terminating conditions found in the literature are [103]:
• Maximum number of generations (iterations);
• Maximum time;
• The fitness value of at least one of the chromosomes is lower than a predefined minimum;
• Maximum number of fitness function evaluations;
• etc..
The conditions above can be used solo or in combinations. The choice for the terminating condition is
somehow problem related and also depends of the desire of the user. In the present work the maximum
number of iterations was used as ending criterion.
58
B.2.8 Selection
The selection operator resembles the process of natural selection and the survival of the fittest of Dar-
winian evolution theory. In this process, an intermediate population, called mating pool. The selected
chromosomes in the mating pool, also designated as parents, will take part in the subsequent genetic
operations [83].
In this step of the algorithm the mating pool is populated with solutions coming from the population
and from the archive. As it is possible to observe in Fig. B.7 the solutions coming from the Population
participate in a 4-th tournament selection [104]. The winner of each tournament goes to the mating pool.
Mating PoolArchive
Population Tournament Selection
Elite size
Figure B.7: Selection of solutions from the current population and archive to population the mating pool.
If the Archive in not empty, the archived solutions with the biggest cuboids are sent to the mating pool.
The ratio between solutions coming from the population and the archive is user specified. If the number
of chromosomes in the archive is not enough to cover its share part in the mating pool, extra solutions are
picked from the Population. However, the number of solutions coming from the Archive to the Mating
Pool is never higher than the user specified value (elite size).
B.2.9 Crossover
In the nature, although it may be much more complicated, crossover basically occurs as follows: chromo-
somes of both parents are randomly divided from the same gene positions into a number of segments and
the corresponding segments are exchanged and copied to the chromosome of the newly created offspring.
Therefore, the offspring inherit traits from both parents.
The idea behind crossover is that the new chromosomes may be better than both the parents if they
take the best characteristics from each of the parents. Crossover occurs during evolution according to a
user-definable crossover probability. Some popular crossover methods, when a binary representation for
the chromosome genes is used, are single-point, two point and uniform crossover [104].
In the single-point crossover a crossover point is randomly chosen and then the two parents, also ran-
domly chosen from the population, are interchanged at this point in order to produce two new offspring.
Variations of this technique such as two-point (can also be generalized to n-point) were already presented
[83].
In real-valued approaches, one of the major challenges is how to use the real-valued genetic information of
both parents to create offspring. However, the techniques to perform crossover for binary representations
can also be used for this case, most of the time that does not occur. This is due to the fact that there is
a large diversity of blending operators, e.g. the BLX-α technique which is able to create offspring out of
the gap formed by the parents [40] and the quadratic crossover where the information of three solutions
is used to produce one offspring [105].
The crossover technique used is the heuristic crossover [106]. This approach tries to create better off-
59
spring via extrapolation. After the random selection of two parents from the mating pool, their genetic
information is blended according to the following formula:
offpringi = α× (x2i − x1i ) + x1i (B.10)
where α is a random value between 0 and 2 and x2i having a better rank than x1
i . If both parents have
the same rank they are randomly assigned.
With this approach it is expected that the offspring will have better fitness values than their progenitors
and in this way tend to the true Pareto front of the problem. Although most of the offspring produced
might represent worse solutions since only two parents are considered at a time, after a large number
of tryouts the predictions will hopefully lead the population towards the Pareto front. In Fig. B.8 it is
depicted the possible result of the heuristic crossover. Differently from other crossover strategies, with
this method only one child is created from the matting of two parents.
Gene valueParent 2
Possible value for the gene value of the child
Gene valueParent 1
Figure B.8: Possible effects of the crossover operator on the genes of the new solution.
B.2.10 Mutation
The mutation in a genetic algorithm is also a process that was inspired in nature. During the transfer of
genetic contents from parents to offspring, something may go wrong, and therefore, random changes may
occur in the new chromosome. As a result, the quality of the individual might degrade considerably. On
the other hand, it can occur that the quality of the new chromosome is higher. Moreover, this process
helps to keep diversity in population, and therefore, it helps the algorithm to abstain from getting stuck
onto a local minimum of the problem search space. The concept of mutation is employed by introducing
a mutation probability parameter. Before inserting an individual to the next generation, random changes
on its chromosome is performed according to this probability. This parameter is most of the times set
low in order to avoid the algorithm of becoming a primitive random search.
In binary chromosome representation, the random change that may occur in the new chromosome is the
inversion of the value of a random gene position from 0 to 1 (and vice-versa). On the other hand, if a
real-valued representation is used, one of the chromosome’s genes may be altered and the new value is
a randomly generated real number between the upper and lower bounds of that specific gene. Another
option is the non-uniform mutation technique which enhances diversity in the first generations of the
population. However, with the increase of the number of generations, the mutation process starts to
converge in a way that there is a high probability of creating a mutated chromosome close to the initial
value focusing in this way the search [40]. Other widely known strategy is to perform mutation in a
real-valued chromosome by adding a normally distributed random value [107]. The desired effect of the
mutation process can be visualized in Fig. B.9 [108].
In the algorithm a point mutation was employed with three different possibilities of mutation with the
same probability of occurrence. The mutated gene might receive the value of its upper or lower bound
and a random value between its boundaries. Such mutations are described as:
60
Global minimum
Local minimum
Fitness value
Search space
Figure B.9: Desired effect of the mutation operator on the genes of the new solution.
genei
⎧⎪⎨⎪⎩
α(UBi − LBi) + LBi
LBi
UBi
where α is a random value between 0 and 1.
The first point random mutation process is not only completely independent of the gene’s value of the
parent but is also able to create a value from the entire interval set by the boundaries of the particular
variable being mutated.
B.2.11 Regeneration
The genetic operator regeneration can be seen as a severe mutation since all the genes of the new
chromosome are altered when compared to the solution from where it was originated. However, it can
also be seen as a crossover with only one parent that alters, regenerates, itself in order to, hopefully,
create a new better solution.
There are two possibilities for the regeneration operator. In the first approach, each gene of a randomly
chosen solution, x1 , from the mating pool is added of itself times a random value that lies in the interval
[−0.05, 0.05]. In this way, the genes with higher values will suffer a higher increase when compared to
the ones with lower gene values. Such approach is benefic in the cases where the distance between the
genes is the key to reach better areas of the search space.
offpring = x1 + α× x1 (B.11)
In the second approach, once again one chromosome is randomly chosen from the mating pool. However,
differently from the first strategy, now a random variable of the gene is chosen and multiplied by α (with
α ∈[−0.05, 0.05
]). After, each gene is summed with this value. In this way, all the genes are
modified with the same amount and, therefore, the difference between the genes is kept constant.
offpring = x1 + α× x1(β) (B.12)
This strategy is benefic when the difference between the genes is already optimal but it is still necessary
to change the absolute values.
It is important to refer one feature of this genetic operator. If the population converges to only one point
the effect of the crossover operator will disappear. In such cases the algorithms rely in the mutation
operator to introduce diversity in the population. The regeneration operator does not share this feature
61
[%]
GeneValue
5-5
ExpansionContraction
0
Figure B.10: Possible effects of the renegeration operator on the genes of the new solution.
with the crossover operator since it is able to change one chromosome into another one without information
of any other solution.
A study on the effectiveness of the regeneration operator was conducted and may be found in Appendix
C.
B.2.12 Chromosome composition
The first n entries of each chromosome are the variables of the problem. They are real-valued and
lower and upper bounded. For each optimization goal there are two entries in the solutions. One is the
augmented fitness, which is composed by the fitness value summed with the penalty term, while the other
entry corresponds to the fitness value itself. The last three entries correspond to the absolute value of
the maximum constraint violation, the rank of the solution and its origin. In Fig. B.11 the encoding of
each solution is shown.
X [var1 var Aug_Fit 1 Aug_Fit Fit 1 Fit Rank Max_constraint Origin]n m mvar Aug_Fit 1 Aug_Fit Fit 1 Fit RaFit 1 Fit Au Fit 1 Au Fit nk Maxk Aug Fit 1 Aug Fit Fit 1 Fit Aug Fit 1 Aug Fit Fit 1 Fit Aug Fit 1 Aug Fit
LB variables UBn nFitness + penalty
termFitness values Highest
constraint violation1 Rank 4
1 Origin 4
Figure B.11: Composition of the chromosomes.
In Table B.3 it is explained the possibilities for the last gene of each chromosome.
Table B.3: Description of the field Origin.
Value Origin
1 Initial Population2 Crossover3 Mutation4 Regeneration
B.2.13 Population composition
After the creation of solutions via the genetic operators, the population is once again tested for the
fitness value of its chromosomes. The amount of solutions coming from each genetic operator, crossover,
mutation and regeneration are user specified. The population after the genetic operators is illustrated in
the next picture.
62
Crossover
Mutation
Regeneration
Population
Figure B.12: Composition of the population after the genetic operators.
In Appendix B, the presented MOOA is tested via several benchmark functions.
63
Appendix C
Benchmark Results
In this appendix the performance of the developed MOOA is tested through well-known benchmark
functions for multi-objective optimization algorithms.
Before introducing the test function it is important to present how the MOOA was set. For all the
benchmarks the population size was set to 100, while the Archive size was defined to 250. The contribution
of each genetic operator was also kept constant throughout the benchmark session and defined as shown
in Fig. C.1. The termination criteria used was the number of fitness evaluations allowed by each test
function. A 4− th tournament selection was implement to select solutions from the Population to enter
the Mating Pool. The size of the Mating Pool was set to 20 and the maximum contribution of the Archive
to the Mating Pool was determined to be 10.
Crossover
Mutation
Regeneration
Population
80%
10%
10%
Figure C.1: Contribution of each genetic operator to the population with new solutions.
Test function 1
The test function 1 is a problem proposed by Kita [109]:
Minimize F = (−f1(x, y), −f2(x, y)), where
f1(x, y) = −x2 + y, f2(x, y) =1
2x+ y + 1
subject to:
• 1
6x+ y − 13
2≤ 0,
1
2x+ y − 15
2≤ 0, 5x+ y − 30 ≤ 0
64
• 0 ≤ x, y ≤ 7
• Maximum number of fitness evaluations: 5000
Test function 2
The test function 2 can be found in [93]:
Minimize F = (f1(x, y), f2(x, y)), where
f1(→x) =
n−1∑i=1
(−10 exp
(−0.2
√x2i + x2
i+1
)), f2(
→x) =
n∑i=1
(|xi|0.8 + 5 sin (xi)
3)
subject to:
• −5 ≤ x1, x2, x3 ≤ 5
• Maximum number of fitness evaluations: 12000
Test function 3
The test function 3 was proposed by K. Deb [110]:
Minimize F = (f1(x1, x2), f2(x1, x2)), where
f1(→x) = x1, f2(
→x) = g(
→x) · h(→x)
where
g(→x) = 11 + x2
2 − 10 cos(2πx2)
h(→x) =
⎧⎪⎨⎪⎩ 1−
√f1(
→x)
g(→x)
, if f1(→x) ≤ g(
→x)
0, otherwise
subject to:
• 0 ≤ x1 ≤ 1, −30 ≤ x2 ≤ 30
• Maximum number of fitness evaluations: 4000
Test function 4
The test function 4 was also proposed by K. Deb [110]:
Minimize F = (f1(x1, x2), f2(x1, x2)), where
f1(→x) = x1, f2(
→x) =
g(x2)
f1
where
g(x2) = 2.0− exp
{−(x2 − 0.2
0.004
)2}
− 0.8 exp
{−(x2 − 0.6
0.4
)2}
subject to:
65
• 0.1 ≤ x1 ≤ 1, 0.1 ≤ x2 ≤ 1
• Maximum number of fitness evaluations: 10000
Test function Kita
This test function was proposed by Kita [109]:
Minimize F = (−f1(x1, x2), −f2(x1, x2)), where
f1(x1, x2) = x2 − x21, f2(x1, x2) = 0.5x1 + x2 + 1
subject to:
• x1, x2 ≥ 0
• −(6.5− x1
6− x2
)≤ 0
• −(7.5− x1
2− x2
)≤ 0
• − (30− 5x1 − x2) ≤ 0
• Maximum number of fitness evaluations: 5000
Test function OSY
This test function may be found in [111]:
Minimize F =(f1(
→x), f2(
→x)), where
f1(→x) = −25
[2∑
i=1
(xi − 2)2+ (x3 − 1)
2+ (x4 − 4)
2+ (x5 − 1)
2
], f2(
→x) =
6∑i=1
x2i
subject to:
• 0 ≤ x1, x2, x6 ≤ 10, 1 ≤ x3, x5 ≤ 5, 0 ≤ x4 ≤ 6
• − (x1 + x2 − 2) ≤ 0
• − (6− x1 − x2) ≤ 0
• − (2 + x1 − x2) ≤ 0
• − (2− x1 + 3x2) ≤ 0
• −(4− (x3 − 3)
2 − x4
)≤ 0
• −((x5 − 3)
2+ x6 − 4
)≤ 0
• Maximum number of fitness evaluations: 25000
66
Test function SRN
This test function may be found in [111]:
Minimize F =(f1(
→x), f2(
→x)), where
f1(→x) = 2 + (x1 − 2)
2+ (x2 − 1)
2, f2(
→x) = 9x1 − (x2 − 1)
2
subject to:
• −20 ≤ x1, x2 ≤ 20
• − (225− x21 − x2
2
) ≤ 0
• − (10− x1 + 3x2) ≤ 0
• Maximum number of fitness evaluations: 5000
Test function TNK
This test function was first proposed in [112]:
Minimize F =(f1(
→x), f2(
→x)), where
f1(x) = x1, f2(x) = x2
subject to:
• 0 ≤ x1, x2 ≤ π
• −(x21 + x2
2 − 1− 0.1 cos
(16 arctan
(x1
x2
)))≤ 0
• −(0.5−
2∑i=1
(xi − 0.5)
2)
≤ 0
• Maximum number of fitness evaluations: 20000
Test function DTZL9
The test function DTLZ9 can be found in [113]:
Minimize F =(f1(
→x), f2(
→x), f3(
→x),), where
f1(→x) =
1
10
10∑i=1
xi, f2(→x) =
1
10
20∑i=10
xi, f3(→x) =
1
10
30∑i=20
xi
subject to:
• 0 ≤ →x ≤ 1
• −(f23 (
→x) + f2
1 (→x)− 1
)≤ 0
• −(f23 (
→x) + f2
2 (→x)− 1
)≤ 0
• Maximum number of fitness evaluations: 50000
67
The Pareto fronts found by the MOOA, for the test cases during a normal run described above, are shown
in Fig. C.2. It is possible to observe that the algorithm was able to found the true Pareto front of each
problem and, moreover, the entire front was achieved.
-20 -19 -18 -17 -16 -15 -14 -13-12
-10
-8
-6
-4
-2
0
2Test function 2
f1
f2
-6 -4 -2 0 2 4 6 87.4
7.6
7.8
8
8.2
8.4
8.6
8.8Test function 1
f1
f2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4Test function 3
f1
f2
-4 -2 0 2 4 6 87.4
7.6
7.8
8
8.2
8.4
8.6
8.8Test function Kita
f1
f2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8Test function 4
f1
f2
-300 -250 -200 -150 -100 -50 00
10
20
30
40
50
60
70
80Test function OSY
f1
f2
0 50 100 150 200 250-250
-200
-150
-100
-50
0
50Test function SRN
f1
f2
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4Test function TNK
f1
f2
0
0.5
1 0 0.2 0.4 0.6 0.8 1
0.4
0.5
0.6
0.7
0.8
0.9
1
f2
Test function DTLZ9
f1
f3
Figure C.2: Non dominated solutions of the final Archive obtained by the developed MOOA.
Due to the Origin filed present in each chromosome it is possible to track which genetic operator generated
them. In Table C.1 the percentages of solutions generated per each genetic operator is shown for the
different test function. The crossover is the genetic operator that most contributes to the non dominated
solutions present in the final Archive in all functions. The regeneration operator, as shown in the table, for
some functions generated up to 33% of the non dominated solutions. However, for two of the benchmark
function it did not contribute with any solution. In this way, it is possible to conclude that the contribution
of the regeneration operator is problem dependent.
Table C.1: Contribution of each genetic operator for the non dominated solutions in the final Archive.
Test Function Crossover [%] Mutation [%] Regeneration [%]
F1 95 0 5F2 64 3 33F3 73 8 19F4 88 12 0Kita 100 0 0OSY 79 15 6TNK 84.42 0.65 14.94DTLZ9 64 19 17
68