Master in Energy Engineering

Master in Energy Engineering

Optimal power flow for hybridAC/DC systems in Europe

Gabriel Esteban Rojas Dueñas

Spring 2018

Universitat Politècnica de CatalunyaEscola Tècnica Superior d’Enginyeria Industrial de Barcelona

(ETSEIB)Barcelona, Spain

DirectorsProf. Dr. Mónica Aragües PeñalbaProf. Dr. Oriol Gomis Bellmunt

Contents

Abstract 5

1 Preface 61.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Introduction 72.1 Project objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Scope of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Previous requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 State of the art 93.1 HVDC transmission systems . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Core technologies . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Sequential method . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Unified method . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 AC optimal power flow . . . . . . . . . . . . . . . . . . . . . . 193.2.4 DC optimal power flow . . . . . . . . . . . . . . . . . . . . . . 203.2.5 AC/DC optimal power flow . . . . . . . . . . . . . . . . . . . 21

4 AC-DC OPF Methodology 244.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Objective function . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Software implementation . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Examples and verification . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Study Case 345.1 European Transmission Network . . . . . . . . . . . . . . . . . . . . . 345.2 Database characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 HVDC transmission system characteristics . . . . . . . . . . . . . . . 395.4 Data management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Simulation and results 466.1 Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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6.2 Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3 Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.4 Scenario 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.5 Scenario 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Conclusions 61

Bibliography 62

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Abstract

The study presents the procedure and results of an AC-DC optimal power flow(OPF) of the European Transmission Network. It provides a detailed explanationof how the OPF is applied to a problem in order to minimize the power losses ofa power system. The document presents an overview of the hybrid power systemsthat consist of a Direct Current (DC) and an Alternate Current (AC) grid that arecoupled with a bidirectional converter that allows the power exchange between thetwo technologies.To fulfill the project purposes is fundamental to understand the basic principlesof the HVDC technology, which is emerging as a solution to connect renewablegeneration to the grid and to transmit big quantities of power over long distances.The main characteristics are used to build the optimization problem, which finalobjective is to perform the load flow of the system while minimizing the losses inthe transmission network.Consequently, a model is implemented in the computational software GAMS. Thismodel is validated using as reference research papers that solved the AC-DC OPFfor certain power systems. This project is based on a data-set of the EuropeanTransmission Network that includes the main characteristics of the grid such as lineparameters, power plants characteristics and load demand for the winter of 2009.Since the database is not enough as an input of the problem, the data is processedusing MATLAB and some technical assumptions were made in order to obtain moreaccurate results.Different scenarios are proposed to analyze the impact of the HVDC lines in theEuropean Transmission Network. The results obtained show that setting the mini-mization of the power losses as the objective function implies that the power flowingthrough the HVDC lines increase and the losses are almost 4.5 times lower thanin the normal scenario. Furthermore, when the cross-border AC lines are replacedwith DC technology, the simulations show that the percentage of losses in the systemdrops below the value of the country with minimum transmission losses in Europe.On the other hand, simulating a system with numerous HVDC lines requires morecomputational capabilities.

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1 Preface

This section presents the motivation and the requirements to perform adequatelythe project. It shows the starting point and the focus of the research.

1.1 Motivation

The main motivation of this project is to evaluate the interconnection of high volt-age direct current (HVDC) transmission system with an electrical system mainlycomposed by the classic links where the power is transmitted using high voltagealternating current (HVAC). The HVDC technology allows bulk power transmissionover long distances with fewer power losses, which means that is ideal for offshorewind farms since some of them are far from land and require a large interconnectionto transmit the power that is generated without significant losses.Furthermore, another motivation is to interact with the European Transmission Sys-tem in order to analyse its power flows and optimal power flows taking into accountthe actual grid topology. The installation of multiple HVDC links to connect pro-duction to consumption centers that are located in different countries and progresstowards the construction of multi-terminal systems[1].

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2 Introduction

The following document shows the performance analysis of the European Trans-mission System. It is based on the AC/DC optimal power flow that is used tostudy interconnected networks with high voltage direct current (HVDC) transmis-sion lines, such as large off shore wind power plants in the North Sea. This leadsto a viability investigation where is determined whether or not the HVDC lines areconvenient to the transmission network.First of all, the project presents the state of the art, which includes researches aboutthe methodology of the Optimal Power Flow (OPF) and how to solve the problemidentifying objective function and constraints. Also, different computational solu-tions are studied in order to choose the best for the final purpose of the project.Furthermore, a brief study of the European Transmission System is presented torecognize the magnitude of the network and familiarize with the variables that aregoing to be used to perform the OPF.Secondly, the methodology to solve the OPF with AC/DC is described and someminor simulations and examples are presented. In addition to this, the ENTSO-Enetwork is fully depicted, emphasizing in the interconnections between the countriesand the direct current transmission lines that are present in the system nowadays.Considering this items, the implementation of the OPF with the European Transmis-sion System data is described and the algorithm to solve the optimization problem isshown. Taking into account the size of the network, just the most significant resultsare presented and analyzed, leading to a viability study that highlight the pros andcons of HVDC lines in the transmission grid.

2.1 Project objectives

The general objective of the study is the following:• Develop a tool to analyze the optimal operation of hybrid AC-DC power sys-

tems and apply it to the European Transmission Network.The specific objectives are the following:

• Model HVDC and HVAC power flows• Compare databases of the European Transmission Network to determine the

best one in order to obtain reliable results

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Chapter 2 Introduction

• Determine the viability of implementing HVDC transmission lines in a systemsuch as the European Transmission Network

• Analyze the impact of HVDC transmission lines in an electrical network

2.2 Scope of the project

The final purpose of this project is to deliver an approach to the actual stateof the European Transmission Network and how this grid can be improved byadding HVDC transmission links or replacing the current HVAC interconnectionsfor HVDC. One of the main results of the thesis is a generic code implemented inMatlab and GAMS that allows the user to perform an optimization problem thatsolves the optimal power flow of an electrical grid despite the dimensions accordingto the specified objective function..However, there are some boundaries that affected the accuracy of the results andthe final deliverable, and are the following:

• Reliability of the databases: There is no an unified database of the grid dataso the results depend on which one is selected. Some of them have betterapproximations than others and are more recent

• Simulation: A comparison of the performance between Matlab and GAMS isimpossible to make because the OPF in the first one requires a bigger capacityand more time than the computer can provide.

• Objective function: Is defined to minimize the power losses, but the othervariables that may affect an optimal approach are neglected

• Converters: Since there’s no available information about the converters used inthe substations where the HVDC lines are connected to the AC transmissionsystem, the accuracy of the losses is not the real one.

2.3 Previous requirements

To develop effectively the project, the main requirements where the following:• Data of the European Transmission System• GAMS Software: To solve the optimal power flow• Matlab: Data processing and analysis.• Parameters of active HVDC transmission lines in Europe and converter char-

acteristics• Optimal power flow knowledge

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3 State of the art

This chapter shows an overview of the needed concepts and methodologies to achievethe project objectives. It introduces HVDC transmission systems and it presentsthe state of the art of optimal power flow.

3.1 HVDC transmission systems

High voltage direct current (HVDC) technology has increased the coverage in the lastyears becoming an advantageous form of transmiting the electrical energy for longdistances, asynchronous interconnections, underground and submarine cables amongothers. Currently there are more than 200 HVDC systems around the world[2],and most of them are located in Europe, Asia and North America. Despite ofthe numerous projects, there’s not a complete DC grid (three or more converterswith two or more transmission lines) yet. The growth of this technology is dueto the new necessities that the transmission networks are facing nowadays such aslonger distances between substations, offshore wind power plants, voltage controland reliability. This section deepen into the main characteristics of the HVDCtransmission technology, showing strengths,weaknesses, type of technology, viability,applications and reliability.

3.1.1 Core technologies

In the modern HVDC transmission systems there are two types of technologiesthat are used and it depends on the type of converter used for the interconnectionbetween the AC and DC systems[3]. The first one and more traditional, is the line-commutated current source converter and the second one is the self-commutatedvoltage source converter which represents a newer and more used technology.

Current source converters (CSC): Is a mature technology that is used basedon the principle of point-to-point connection. The working principle is to maintainthe DC current constant using a large inductor, creating an imaginary current sourceon the DC side. The direction of the current is always the same, but the power flowis determined by the polarity of the DC voltage at the terminals. The electricalenergy conversion is done by a three-phase full-wave bridge (Graetz bridge) using

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Chapter 3 State of the art

thyristor valves that perform the commutation. Depending on the complexity andthe necessities of the system, six-pulse or twelve-pulse bridges can be used, thesecond one has lower harmonics because it just have the 12n+1 harmonics[3].

The main drawback of this type of converter is that it needs a strong synchronousvoltage source to accomplish the commutation between phases using the thyristorvalves. Furthermore, this type of converter requires an injection of reactive powerin order to work in a lagging current state; AC filters are used to overcome thelack of reactive power. Usually, shunt banks or series capacitors are installed in thesubstations to provide the surplus or consume the deficit of the reactive power. Thereactive power compensation and exchange depends on the dc load current flowingthrough the converter[3].

The following figure shows the scheme of an HVDC current source converter.

Figure 3.1: HVDC Current source converter (CSC) [4]

It can be seen the topology of the three-phase converter, identifying the six thyristorvalves (modeled as diodes). To have a six-pulse commutation, each diode statefunction is different and all combined allows the rectification of the signal into aDC output voltage. The AC filters are located between the AC load/generator andground, so there’s a reactive power compensation.

The largest project that uses this technology is the Itaipu system in Brazil with apower level of 6,4GW. A bigger project is under development in Shanghai, is a 2071km line at a 800kV DC rating that will be able to transmit 6,4GW between an hydropower plant and the urban area of Shanghai.

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Self-commutated voltage source converter (VSC): Is a system that can per-form independent control of the active and reactive power at both ends of the trans-mission line, which is ideal to connect HVDC transmission lines with weak ACpower systems. Contrary to the CSC, the variable that determines the direction ofthe power flow through the system is the DC current while the polarity of the DCvoltage is being determined by the transmission system. The working principle isbased in control through sinusoidal pulse width modulation (PWM) of the insulated-gate bipolar transistor (IGBT) valves that allows higher voltages and power ratingscompared to the CSCs[3, 5].

To deliver a higher blocking voltage capability and an increased voltage rating ofthe HVDC system, the IGBT are connected in series, so there are faster and moreeffective commutations between the different states. Thus, the harmonics dependon the frequency of commutation of the IGBT and the filters are tuned accordingto this behavior. To provide a four-quadrant operation of the VSC, an anti-paralleldiode is needed.

The following picture shows the circuit of a VSC converter

Figure 3.2: HVDC Voltage Source Converter (VSC)[4]

The figure shows six IGBT used for the commutation of states in order to effectivelyconvert the AC signal to DC. Depending on the desired level of accuracy of the signaland the harmonic penetration, different configurations of IGBT could be used.

The introduction of voltage source converters introduced great advantages comparedto the HVDC systems that use CSCs, which the most important ones are[3]:

• Reliability: Disturbances in the ac network don’t affect the commutationscheme

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• Controlability: The active and reactive power consumed or generated by theconverter can be controlled by just changing the voltage

• The HVDC is able to connect to a weak ac network without a major impactto the power system performance. The short-circuit level is low

• Dynamic response: It is faster, so the harmonics are lower. This is becausethe switching frequency generated by the PWM. Implies that the size of theac filter is smaller

• There’s no need of transformers in the commutation process because it iscontrolled by the IGBT valves

Different topologies: There exist different multilevel topologies of VSCs and themain difference between them is the number of PWM voltage waveform used for thecommutation process. The figure 3.3 shows how the topology can change dependingon the number of submodules (formed by two IGBT). The main advantage of usingmultilevel converters is that the harmonic performance improves and the noise ofthe signal decreases, leading to a more reliable signal at the output of the VSC.However, a disadvantage of using multilevel topologies is that the voltage may beunbalanced across the dc bus capacitors[6].

Figure 3.3: Multilevel topology of a VSC

To overcome the problems due to the multilevel converters, different solutions wereimplemented and the most useful one was the introduction of a static synchronous

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compensator (STATCOM) to the converter. This device is used to compensate theconnection between two ac systems when there is no synchronous generation in themain grid. It offers a robust performance and a fast dynamic response that leads toa low frequency switching.According to the data provided by Flourentzou et. al., most of the VSC HVDCprojects nowadays are for the interconnection of offshore wind power plants in theNorth Sea. The distances covered by the actual transmission lines are greater than100 km and some of them are placed underground or as submarine cables.

3.1.2 Applications

The main reason to introduce the HVDC technology into the electrical grids isbecause in some cases it is cheaper than the HVAC systems. However, there arenumerous technical applications to consider the DC transmission lines as a betteralternative. This section shows the present and future applications of the HVDCsystems that are beneficial for the power networks.

Long-Distance Bulk Power Transmission

To deliver the power generated of remote generators to the final consumers (for ex-ample a rural power plant providing electrical energy to a big city) it is necessarylong distance transmission lines. An alternate current circuit could be connectedto interconnect the remote supplier but the losses would be considerable and theproject expensive. Thus, a solution arise to overcome this problem and is the im-plementation of the HVDC technology to connect the consumption center with theproduction plant. Usually, it is a more economical alternative for the scenario con-sidered above[2].For extra high voltage, HVDC lines gives the system a controllability that is use-ful for the parallel transmission because it blocks the loop flow and frees up thetransmission capacity in order to provide an outlet for local generation.Since the decisive factor to choose a HVDC system over a HVAC is the cost, there’sa concept known as “break-even distance” that helps the investor to make a decision.This is where the savings in line costs offset the higher converter station costs[7].Usually, for lines of a length of 500 km, the savings in the line construction areup to 30% and if the minimization of the losses during it lifetime is considered, thesavings are higher. Furthermore, the long-distance ac lines require small substations(switching substations) and reactive power compensation (shunt reactors), whichincrease the cost of the project for an HVAC system.The following figures show how the losses and the total cost of a transmission line(HVAC and HVDC) change according to the total distance of the project. It canbe seen that for short distances, the ac lines are better but at certain distance

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(break-even distance), the HVDC technology is more convenient because the powertransmission losses are less and it is cheaper.

Figure 3.4: Transmission distance vs. Power losses for HVAC and HVDC [ABB]

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Figure 3.5: Transmission distance vs. cost for HVAC and HVDC [ABB]

Another benefit of implementing HVDC in long transmission lines is the reductionof the right-of-way. This is mainly because the towers used for DC lines are narrowerthan in AC systems, so it requires less space and is cheaper. Also, this reductionleads to a less visual impact. An interesting example is shown in the followingpicture, the top line shows HVDC lines and the bottom shows the HVAC for a systemthat transmit 3000 MW with a rated voltage of 500kV, is evident the reduction inspace[2].

Figure 3.6: Right-of-way of HVAC and HVDC transmission system

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Underground and submarine cables

For the HVDC technology, there’s no physical limitations about the distance or thepower in underground and submarine cables. Thus, the cost savings are considerablewhen comparing to ac systems because in this conditions, HVAC technology havecertain restrictions. The dc line losses are almost half than the ac cables becauseit uses less conductors and it avoids physical properties as the skin-effect, reactivecurrent, cable sheath and armor. This is really useful for offshore wind power plantssince many of them use submarine cables to deliver the power generated[2].

Asynchronous ties

In order to have a more reliable operation for interconnected asynchronous networks,HVDC systems are used. It provides a buffer for the two networks and avoids thepropagation of cascading outages from a system to another. These interconnectionsare usually at the border of the transmission systems, where networks are weak ifthey are compared with the power that they need to transfer. HVDC lines providesreliability and it is cheaper than an interconnection using HVAC technology, also itallows fast recoveries from faults and/or outages[2].

Offshore transmission

HVDC transmission lines represents a huge benefit for large offshore wind powerplants, the main advantages are the following:

• Self-commutation• VSCs can operate at variable frequency• Dynamic voltage control• Black start capability• Long-distance submarine cables• Reactive support to the wind generation complex

Power delivery to large urban areas

Usually, large cities have high energy demand and no space to place generators.Then, the power supply depends on the capability of importing the power gener-ated in other areas. However, the transmission of energy to large cities is difficultbecause of land-use and right-of-way limitations. The solution to this problem isthe installation of underground HVDC transmission lines that are capable to trans-mit large amounts of power between the generation and the consumption center. Isan effective way of dealing with this problem because it does not compromise thereliability, provides voltage support and is more economical[2].

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3.2 Power flow

3.2 Power flow

Since the introduction of HVDC technology into the power networks over the world,an effort to find the best method to solve the load flow of this hybrid system havebeen carried by different researchers. There are a variety of approaches to performthe power flow and it depends on the type of network and the application[8]. Thissection presents the literature review of the first methods used and explains theoptimal power flow (OPF), which is the selected technique for the project.

3.2.1 Sequential method

The AC/DC power flow sequential method consists on the separate solution of theac and dc equations each iteration. It is easy to implement, but in some cases itmay be convergence problems that won’s solve the algorithm. This methodology wasintroduced in the late 70s and early 80s when the HVDC technology was starting tobe considered for some applications. It offers simplicity and is based in the followingassumptions:

• Continuous converter transformer tap• Scheduled voltage control with a certain minimum control angle• Fixed voltage margins at those terminals with a scheduled current or power

control• Converter model is based on the relationship between ripple-free average quan-

tities and the fundamental frequency ac quantities• Multi-terminal dc system• DC and AC equation treated separately

The first step of the methodology is to introduce the data set and the initial condi-tions, then the dc equations are solved to determine the power factor angles. Thisis useful because it leads to obtain the active and reactive power of the dc system atthe ac buses (converters). Afterwards, the ac load flow is solved (it doesn’t matterthe method used to do it, Newton-Raphson recommended). After obtaining thevoltages, the taps are determined and the product a x V to identify if the tap limitsare being exceeded. If the limits are violated, the DC voltage needs to be rescheduledand the process has to be repeated until the tap constraint is fulfilled[8].It is important to state that this is not the only sequential methodology, is justthe one defined by the authors above and the more widely used. There are otherapproaches where the ac load flow is solved first and the finishing criteria is different,but the essence of solving separately the two systems is basically the same.The main advantages that this method offers is the simple and fast computationalimplementation, and errors do not accumulate during each iteration. As mentioned

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before, one of the drawbacks the algorithm may not work for certain systems becauseof the convergence, also the algorithm relays on the initial values introduced by theuser.

3.2.2 Unified method

The unified method is a extended variable method that solves the load flow obtaininga dc-variables vector. This means that a DC load flow result in a set of variables thatwill affect the active power flows of the transmission lines in the power system[9]. Itis more used than the sequential method because the behavior of the devices has amore accurate representation and the final results are more reliable. The followingassumptions are made:

• Linear, bilateral and balanced network• Lumped parameters• All bus voltage magnitudes are 1 p.u.• Transmission line resistance is negligible

It is important to linearize the system because having a non-linear problem result inslow simulations which is not desirable for certain studies, such as: planning, con-tingency analysis and ranking, risk assessment, transmission and congestion man-agement, and other optimization problems.To represent the effect of HVDC transmission system and Flexible Alternate CurrentTransmission Systems (FACTS) devices in the ac load flow, they must be consideredas branches between two nodes in the ac grid. and the HVDC transmission linemodel is shown in the next figure. It is represented by a generator at one terminaland a load at the other one, the power losses are taken into account and then thereis a difference between the energy being generated and consumed by this virtualelements.

Figure 3.7: HVDC line representation in the unified method[9]

After finding all the values of loads and generators for the dc elements, the powerflow is performed. This AC load flow could be solved by classic iterative methods

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3.2 Power flow

such as Gause-Seidel or Newton-Raphson. However, these methods take a lot oftime to obtain the final result for the load flow. Thus, faster approaches are usedand these can be decoupled or fast decoupled power flow solutions.The main advantages of the unified method are:

• Simplification of the implementation and maintenance of the program becausethe dc and ac systems can be handled separately

• DC links between the ac nodes can be modeled easily• Real and reactive powers consumed by converters are modeled as voltage de-

pendent loadsOn the other hand, an important drawback is that it is complex to program andhard to combine with ac solutions techniques.

3.2.3 AC optimal power flow

This subsection presents the OPF used in AC networks, in order to have a generalidea about how it is done and to have a better understanding of how the hybrid AC-DC OPF works. It is a non-linear and non-convex problem which includes binaryand continuous variables. Is based on the minimization of an objective function,which usually refers to the cost of generating electrical energy, but it also worksfor system operation enhancements[10, 11]. This optimization problem is subject tophysical, operational and technical constraints which are:

• AC power flow equations (Kirchhoff and Ohm laws)• Generator active power limits• Generator reactive power limits• Voltage magnitude limits• Active power flow limits• Apparent power flow limits• Line current (I) limits• Voltage angle• Equipment constraints

The mathematical formulation of these constraints and the objective function areshowed in the following section, where the problem is addressed and explained. Thesolution obtained using this AC OPF is the power to be generated for each generatorat a certain time. This solution is not dynamic, so it just offers a snapshot of theactual state of the system, it may change depending on the time and the day.The main advantages of the AC OPF are:

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• It co-optimizes the real and the reactive power• Internalizes the losses, which are not analyzed in other approaches• Respect the voltage limits• Operation and control actions

However, there’s a major disadvantage which is the computational requirementsthat an ACOPF demands. Since the optimization problem takes into account aconsiderable number of constraints (depends on the size of the power system), itrequires more time to solve and in certain cases it is unfeasible. According to theFERC, the problem was formulated 50 years ago and there’s not a fast and robustsolution technique for this problem.

3.2.4 DC optimal power flow

Similarly to the AC optimal power flow, the DC OPF is based on the solutionof an optimization problem. Nevertheless, it is a convex and linear problem thatminimizes the cost of producing the electrical energy (objective function)[11]. Itneglects several physical and technical constraints that are present in the ACOPF,such as the voltage magnitude, power flow limits and the reactive power generatedby each generator. The main constraints of the optimization problem are:

• Generator active power limits• Difference between power generated and power demanded• Power limits of the transmission lines

At the end of the process, the optimized variables are the power generated by eachpower plant and the voltage angle at the nodes of the system. Some DC-OPFproblems are formulated considering the power transfer distribution factors (PTDF)of the lines, that relate the flows through the transmission system to the injectedpower by the generators. This is used in Europe for the zonal pricing in Europeand the bug advantage is that eliminates the angle variable from the optimizationproblem[11].The main advantages are:

• Market-clearing prices at a certain time• Convex and linear problem• Feasible for large networks, there’s not computational restrictions• The approximations are more suitable for transmission systems

In despite of this, the drawbacks of this methodology are:• The reactive power in the system is not known• Voltage magnitude is always fixed to 1 p.u.• Power losses are not determined

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3.2 Power flow

Economic dispatch

It is the simplest optimization problem because it does not consider the power flowsand the network constraints. The objective function is exactly the same as theDCOPF, but there’s not constraints related to the grid, it just takes into accountthe power limits of each power plant and the demand at the nodes of the powersystem. This type of problem can be solved with the help of the merit order curve,that is shown in the next figure.

Figure 3.8: Economic dispatch merit curve[11]

The figure shows where the generation meets the demand and the clearing priceof the market. However, this problem is more complex that it seems because isa mixed integer linear programming (MILP) optimization that requires completeinformation about all the generators and the loads in the system.

This is the basic principle about how the market operator deals with the dispatchof electrical energy in the day-ahead market.

3.2.5 AC/DC optimal power flow

This is an optimization problem that considers both AC and DC elements in anhybrid power system. In a certain way, it combines the ACOPF with the DCOPFto obtain an accurate solution of the load flow. The problem can be classified as a

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non-linear constrained optimization problem because it comprises the basics of theACOPF which is non-convex and non-linear[12].

As in the ACOPF, the objective function can change depending on the application:

• Minimize power losses

• Minimize generation costs

• Maximize the reactive power margin

• Minimize the voltage deviation related to a set value

• Minimize the deviation from another variable present in the system

Despite the objective function, the constraints in the problem are always the sameand are the following:

• DC Voltage limits

• Relation between DC and AC current (based on the conductance matrix)

• DC Current limits

• DC lines power flow limits

• DC Ohm law

• AC Voltage limits

• AC lines power flow limits

• Power entering the system through converters

• AC reactive and active power limits

• Active and reactive power exchange in the AC-DC converters

The equations and mathematical formulation of the problem is presented in the nextsection. The set of constraints shows that HVDC and HVAC systems are treatedseparately but the key element to integrate both systems is the converter becauseit connects the HVDC system with the HVAC. Thus, it is important to perform aprecise modeling of this device in order to obtain realistic results, it can be a CSCor a VSC[12].

Basically, what this constraints do is to obtain an AC equivalent circuit consideringall the characteristics of the DC network elements. Then, concepts from the unifiedmethod, ACOPF and DCOPF were used to built this methodology that solves thepower flow of an hybrid network. The figures shown below present a small AC/DCgrid and at the right there’s the AC network after the conversion of the DC elementsis performed[8].

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3.2 Power flow

Figure 3.9: AC-DC OPF conversion scheme [8]

It can be seen that for this case the converter is modeled as an ac source, whichallows the ACOPF. This is just one of the many approaches that can be made tosolve this problem and the one that is going to be used on this paper.The working paper “Optimal power flow tool for mixed high-voltage alternating cur-rent and high-voltage direct current systems for grid integration of large wind powerplants”[12] explain the procedure to address and solve the AC-DC OPF optimizationproblem. Firstly, it presents the general overview of the problem and the mathemat-ical formulation. Then, it explains how it is implemented computationally in orderto obtain the solution for certain networks. Finally, there is a comparison betweenthe computational efficiency of GAMS and Matlab in the solution of the OPF. Thesystem analyzed has 6 DC and 8 AC nodes with two wind power plants, the resultsshow similar results for both software but the minimization done in GAMS is wayfaster ans efficient. Also, a sensitivity analysis was performed, were the wind speedof both generators was changed in order to examine the effect in the transmissionpower losses.

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4 AC-DC OPF Methodology

This chapter presents the methodology to solve the AC-DC OPF and its implemen-tation. It recalls the hybrid and non-hybrid optimal power flow methods analyzed inthe Chapter 3 in order to propose an accurate approach that allows reliable resultsand computational efficiency.

4.1 Algorithm

In order to integrate the method with the topology of the electrical network, andbased on the model proposed by Schilling et al.[11], the following implementationsteps are considered:

1. Data Collection: This process consist on identify the electrical network tobe analyzed and to collect all the required data. It includes converter charac-teristics, transmission lines parameters, generation ratings, power demanded,base voltage and power. On the reliability of this data depends whether ornot the results are accurate. When data is unknown, assumptions are madeto have a realistic approach.

2. Import and process data: The data collected in the previous step is im-ported, and the interactions between the AC and DC grids are realized byconnecting the converters to the nodes. In most of the cases, there’s the needto manage the data of the line electrical parameters because the admittanceand susceptance matrices are an input of the problem. Then, this matricesare built taking into account the transmission lines characteristics and theelectrical grid configuration.

3. Determine optimization vector: This step aim to identify the optimiza-tion vector x that contains the state and control variables of the HVAC andHVDC system. Where the entries are the vectors of nodal angles, nodal ACvoltages, active and reactive power generation of power plants connected tothe AC system, active power demand of loads connected to the AC grid, nodalDC voltages, active power generation of the generators connected to the DCgrid (or DC current flowing through each node), active power demand of loadsconnected to the DC grid, converter active and reactive power injections tothe AC grid.

24

4.2 Mathematical formulation

4. Specify the constraints: The equality and inequality constraints are deter-mined. The Jacobian matrix could be obtained to identify the partial deriva-tives of the equality constraints indicating sensitivities. The modeling of thisequations could lead to divergence while solving the optimization problem.

5. Define the objective function: The objective function f(x) is defineddepending on the requirements of the problem and depends on the optimizationvector mentioned in the step 3. Normally, this output is the active and reactivepower generated by each power plant in the system.

6. Prepare the optimal power flow: In this final step, the initial point of theoptimization problem is selected according to common values such as voltageangles 0° and magnitudes 1 p.u.. In addition to that, the solver is chosen, foran AC-DC OPF problem the interior point method fits with the optimizationrequirements.


Considering the AC-DC OPF explained in 3.2.5 and based on the “Optimal powerflow tool for hybrid AC/DC systems”[13], the optimization problem equations arestated and explained to have a better understanding of the problem and to identifylinear and non-linear constraints. Also, it is important to have an accurate approachof the converter because is the link between ac and dc technologies, and where thenon-linearity of the problem arises.First of all, the optimization vector that contains the state and control variables isspecified:

x =

VDC

IDC

VAC

θPAC

QAC

P genAC

QgenAC

P convAC

QconvAC

(4.1)

The size of this vector depends exclusively on the topology of the system, whichneeds to be specified at the beginning of the problem. Since x is represented by setsof variables, the following notation is used to differentiate them:

• iε (1, n), where n is the number of DC nodes• jε (1,m), where m is the number of AC nodes

25

Chapter 4 AC-DC OPF Methodology

4.2.1 Constraints

Following the steps presented in the previous section, the constraints are specifiedconsidering all the variables present in x.

First of all, the active and reactive power constraints of each node in the AC gridare defined taking into account the susceptance and admittance matrices of theHVAC system. Considering that the power transmission lines are modeled with theπ equivalent circuit shown in the figure[13], the matrices mentioned are constructed.

Figure 4.1: Pi model used for the AC system [4]

Now, the fundamental AC equations are presented:

PAC,j =m∑

k=1VAC,kVAC,j (GAC,jk cos (θjk) +Bjk sin (θjk)) (4.2)

QAC,j =m∑

k=1VAC,kVAC,j (GAC,jk sin (θjk) −Bjk cos (θjk)) (4.3)

GAC,jk represents the real part of the branch jk of the admittance matrix, whileBjk is the imaginary part; and θjk = θj − θk states the difference between the phaseangles in the nodes j and k. It can be appreciated that both equations are non-linear,so they have to be treated accordingly and be aware about convergence problems.From the equations 4.2 and 4.3, the node equations of the AC grid are determined:

PAC,j = P genAC,j − P load

AC,j (4.4)

26


QAC,j = QgenAC,j −Qload

AC,j (4.5)

The constraints show the active and reactive power injection into certain AC node,determined by the difference of generation and load at that node.On the other hand, the equations that show the characteristics of the DC networkare the following:

IDC = GDCVDC (4.6)

PDC,i = VDC,iIDC,i (4.7)

These two equations represent the existent power equality constraint at each nodein the DC grid. Where GDC is the ij value of the DC admittance matrix.As it was described in the Chapter 3, there are inequality constraints that representthe physical, technical and electrical limitations of the hybrid AC-DC system, whichincludes generators, branches, loads and nodes. These constraints are:

VDC,min ≤ VDC,i ≤ VDC,max (4.8)

IDC,min ≤ IDC,i ≤ IDC,max (4.9)

Pminkl ≤ Gkl (VDC,k − VDC,l)VDC,k ≤ Pmax

kl (4.10)

VAC,min ≤ VAC,j ≤ VAC,max (4.11)

θmin ≤ θj ≤ θmax (4.12)

PAC,min ≤ PAC,j ≤ PAC,max (4.13)

27


QAC,min ≤ QAC,j ≤ QAC,max (4.14)

P genAC,min ≤ P gen

AC,j ≤ P genAC,max (4.15)

QgenAC,min ≤ Qgen

AC,j ≤ QgenAC,max (4.16)

Considering that the power generated and consumed by the DC nodes is not avariable in the optimization vector, the equation 4.10 uses the Ohm law to find thepower flowing through each line of the DC system and set the boundaries for it.

AC-DC interface: Converters

So far, the constraints presented, analyze the AC and DC networks separately. Thus,an integration of both grids is necessary in order to model the AC-DC hybrid powersystem. In the Section 3.1.1 the converter technologies were depicted and character-ized, for this project purposes, the chosen technology was voltage source converters(VSC) because most of the HVDC transmission lines in Europe use them[8].Firstly, the VSC is modeled with two nodes that represent the AC and the DCnetworks. This means that there is a transfer of power between the two grids, whichleads to a converter balance constraint that relate the AC and DC power flows. Thefollowing figure shows a one line diagram of the VSC technology and the equation4.17 refers to the constraint.

Figure 4.2: One line diagram of VSC

P convAC,i + P conv

DC,i + Plossvsc,i = 0 (4.17)

The only unknown variable from the equation is the power losses in the converter.According to [8], the losses are a function of the AC current flowing through converterand follow a second order polynomial equation, which is:

28


Plossvsc,i = a+ bIconv,i + cI2conv,i (4.18)

The a, b and c values are constants determined by the two operating modes of theconverter, that are inverter and rectifier. In the validation section, these values areshown and explained according to the HVDC system. The value of the convertercurrent is determined from:

Iconv,i = Sconv,i

VAC,i

=

√P conv2

AC +Qconv2AC

VAC,i

(4.19)

On the other hand, there’s also a exchange of reactive power, which is a modificationof the equation 4.5 and it is described by:

QAC,i = QconvAC,i +Qgen

AC,i −QloadAC,i (4.20)

Finally, the limitation constraints of the converter are:

P convAC,min ≤ P conv

AC,i ≤ P convAC,max (4.21)

QconvAC,min ≤ Qconv

AC,i ≤ QconvAC,max (4.22)

4.2.2 Objective function

For the AC-DC optimal power flow, there’s the possibility to choose between differentobjective functions depending on the desired outcome[13]. These objective functionswere presented in the Chapter 3 and now they are formulated mathematically.Minimum power losses:

minm∑

k=1

(P gen

AC,k − P loadAC,k

)(4.23)

Minimum generation costs:

minnG∑k=1

CkPk (4.24)

Where Ck refers to the cost of producing one unit of electrical energy (MWh).

29


Maximum reactive power margin:

maxnG∑k=1

QAC,k (4.25)

Minimum voltage deviation related to a set value:

minm∑

k=1(VAC,k − Vset)2 (4.26)

Minimum deviation from another state:

minm∑

k=1(xk − xset)2 (4.27)

These five equations allow the user to perform a technical (losses, reactive power,voltage deviation) or economical (generation costs) analysis of the power system. Itis important to notice that all of the objective functions are non-linear, even if isnot explicit in the equations 4.23 and 4.25.For this project purposes and because of the availability of the information, theobjective function to be used is the one that refers to minimize the power losses.

4.3 Software implementation

Considering the mathematical formulation and the methodology proposed to solvethe optimization problem, the solution is proposed using two different software,which are Matlab and GAMS. Both are capable of solving non-linear optimizationproblems involving several variables. However, for an AC-DC OPF is less timeconsuming the second one, as it was proved in [13].The initial step consists of preparing all the input values for both software. This isdone with Matlab, where the admittance and conductance matrices are constructedtaking into account the grid characteristics. In addition to that, the necessaryvectors and matrices are created depending on the requirements of each software(for example, GAMS require a table that specifies the connections between differentnodes).

Matlab

A generic function was implemented in order to run the model, no matter the sizeof the system or the topology of the hybrid AC-DC grid. The optimization is basedon the function fmincon(), which solves a non-linear program based on the interiorpoint method. The inputs are the following:

30

4.3 Software implementation

• Objective function: Is represented as a Matlab function due to the non-linearity characteristic of the equation.

• Initial point: It is defined by the user and is a vector with the start point forall the variables listed in the objective function.

• Equality constraints matrix• Equality vector• Inequality constraints matrix• Inequality vector• Lower boundary: Minimum values for all variables• Upper boundary: Maximum values for all variables• Non-linear constraints: Is a function that obtains all the non-linear constraints

considering the variable vector x. The computation time depends on how thisfunction is implemented and how the constraints are stated.

The outputs are:• Optimal values of the optimization vector x• Value of the objective function at the optimal value• Exitflag: Describes the exit condition of the function fmincon()

The code is presented in the annexes section

GAMS

This software offers more versatility than Matlab because it allows the user to changethe grid topology without doing major changes in the code. It uses the function ’nlp’to solve the minimization problem and the input parameters are the following:

• AC nodes connected to DC nodes through VSC• Demanded active and reactive power in each node• Conductance and susceptance matrices• Base values• Vector x of optimization variables• Start point• Constraint equations• Objective function

The outputs (File summary) are:• Optimal value of vector x

31


• Lower, upper and marginal levels• Value of the objective function at the minimum

Another advantage of using GAMS is that it provides a more detailed analysis ofthe minimum value.

4.4 Examples and verification

Finally, the last step of this chapter is to prove that the algorithm is working properlyand the solutions obtained by using computational tools are accurate. In order toverify this, the topology used was the one specified in the paper [13], it consists ofa 5 node AC and 3 node DC network where the objective function is defined bythe minimization of the power losses in the electrical network. This working papersolve the optimization problem just using Matlab. However, for the validation, bothsoftware (Matlab and GAMS) are used because the results are expected to be thesame.The following table shows the results and a comparison with the values obtained inthe mentioned paper.

Table 4.1: Validation of the AC-DC OPF code

[13]Reference Paper Matlab GAMSVAC,1 1,1 1,0999 1,1VAC,2 1,1 1,1 1,1VAC,3 1,0809 1,0854 1,082VAC,4 1,0807 1,0845 1,0817VAC,5 1,0811 1,0843 1,0816VDC,2 1,003 1,0026 1,0028VDC,3 1 1 1VDC,5 0,999 0,999 0,998θ1 0 rad 0 rad 0 radθ2 0 rad 0 rad 0 radθ3 -0,043 rad -0,0434 rad -0,043 radθ4 -0,0458 rad -0,046 rad -0,0453 radθ5 -0,0468 -0,0465 rad -0,0466 rad

From the results shown in the table, it is evident that the algorithm was correctlyimplemented because the difference between the values is not considerable and itcould be the result of external errors (use of decimal numbers, admittance matrix,etc.) and not an error of the program.In despite of the results presented above, it is necessary to test the algorithm with abigger system. Then, the network introduced in the paper [12] that has 8 AC nodes

32

4.4 Examples and verification

and 6 DC nodes is considered for the analysis. It was implemented in Matlab andGAMS and the optimal values are presented in the following table:

Table 4.2: Validation of codes in Matlab and GAMS

Reference[12] Paper Matlab Reference[12] Paper GAMS Matlab GAMSVAC,1 0,998 0,998 0,999 0,998VAC,2 1 1 1 1VAC,3 0,997 0,997 0,997 0,997VAC,4 1 1 1 1VAC,5 1,099 1,1 1,1 1,1VAC,6 1,098 1,099 1,099 1,099VAC,7 1,099 1,1 1,098 1,099VAC,8 1,098 1,099 1,098 1,099VDC,1 1,099 1,099 1,099 1,098VDC,2 1,098 1,098 1,099 1,098VDC,3 1,097 1,099 1,098 1,098VDC,4 1,099 1,099 1,098 1,099VDC,6 1,098 1,1 1,099 1,099VDC,8 1,099 1,1 1,099 1,1θ1 -0,059 rad -0,059 rad -0,059 rad -0,058 radθ2 -0,019 rad -0,018 rad -0,019 rad -0,019 radθ3 -0,064 rad -0,064 rad -0,064 rad -0,064 radθ4 0 rad 0 rad 0 rad 0 radθ5 0,006 rad 0 rad 0 rad 0 radθ6 -0,006 rad -0.012 rad -0,007 rad -0,012 radθ7 0.006 rad 0 rad 0 rad 0 radθ8 -0.006 rad -0,012 rad -0,006 rad -0,011 rad

From the table, it can be seen that the implemented code is accurate and the resultsare almost the same than the ones obtained in the paper reviewed. However, thesimulation times differ because for this project purposes it takes longer in bothsoftware to solve the optimization problem. For example, in Matlab it takes 32seconds while in the paper it took 8 seconds.

33

5 Study Case

The selection of an accurate database is as important as the coding of the AC-DC optimal power flow, in order to obtain a realistic solution of the optimiza-tion problem. Taking into account that it doesn’t exists an official data-set of thesystem, an approximation has to be used. On the web-page https://wiki.openmod-initiative.org/wiki/Transmission_network_datasets, different data-sets for the trans-mission network can be found and each one with different characteristics and pur-poses. Up to 20 databases are available, but at the end just one was selectedconsidering the suitability of it to perform an optimal power flow.

5.1 European Transmission Network

Considering the general objective of the project, there’s the need to familiarizewith the European Transmission Network. This system is an AC-DC network thathas a total of 3523 nodes and 5145 transmission lines[14]. The system has fournon-synchronized high-voltage electricity grids that work at the following voltages:150kV, 220kV, 300kV and 380kV. The interconnection between this networks consistsof twelve HVDC transmission lines. The total installed capacity of the member statesof the European Union (EU-28) in 2015 was of 982 GW and the gross electricityproduction was of 3234 TWh. The European countries that generate more electricalenergy are Germany, France, United Kingdom, Spain and Italy, and the ones witha higher energy export are France, Germany and Sweden respectively. On the otherhand, Italy, Belgium and the United Kingdom represent the countries that importmore electricity[14].

The following map shows the electrical grid in Europe:

34

5.1 European Transmission Network

Figure 5.1: European Transmission Network [ENTSO-E]

It can be seen where are the transmission lines all over the continent. The conven-tions are the following:

• Red: 380 kV - 400 kV line• Green: 220 kV line• Yellow: 300 kV - 330 kV line• Blue: 750 kV line• Purple: DC line

The voltages of the transmission system at the Eastern countries (excluding Russia)are different from the one of the West. Also, it is evident the presence of HVDCtechnology to connect transmission systems that are separated by water like inthe North Sea or the Mediterranean Sea. This can be checked by looking UnitedKingdom in the map, since it is an island, the interconnections between the systemand the rest of Europe are with the HVDC technology[14].

ENTSO-E

In 1999, The European Network of Transmission System Operators (ENTSO-E) wascreated, it includes 36 countries across Europe, 43 electricity transmission system

35

Chapter 5 Study Case

operators (TSOs) and over 500 million final consumers. It was established to lib-eralize the energy market and promote mechanisms that facilitate the interactionbetween different countries in the continent. However, the most important objec-tive of the members of this organization is the integration of renewable energies inEuropean Transmission Network and develop flexibility for users[15].Furthermore, this organization provide data about the electrical system in Europe.So the power statistics are available in their website and are free to access (netcapacity, monthly load, power flows, inventory of generation and transmission, etc.).

5.2 Database characteristics

The chosen database was the one created by Janusz Bialek and is available online[16].It consists of an approximate model of the European interconnected system and themain purpose was to analyze the effects of the cross-border trades in the continent[17].Only publicly available information was considered and the results of the load flowwere compared with the actual power transfer in the system to corroborate thevalidness of the obtained values. The Database was first published in 2005[17] andthen was updated and validated in 2009[16] adding some missing regions. It wascreated in order to give researchers the tools and information to perform a correctanalysis of the European Electricity Market.The database presents the main electrical parameters of all the system for the winterof 2009, so it is possible that the AC-DC OPF are different compared to others thatwork with different characteristics.The data was classified in three different categories, which are: transmission networkdata; power plant locations, fuel types and capacities; and the load center locationsand capacities. The description of each is shown below.

Transmission network

Since the electrical parameters of the transmission lines are not available to thepublic, they were estimated from the lengths and voltage of the lines. Consideringthat two voltage levels were used for the construction of the data-set (220 kV and380 kV), the typical reactance level per kilometer was used for this purpose, andthe values are 0,31 ohm/km (220 kV) and 0,28 ohm/km (380 kV). The authorsemphasize in the following assumptions for the transmission data:

• Only the transmission lines with a voltage level of 220 kV or higher, are takeninto account because these are the lines that transmit most of the electricalenergy in the network.

• The shunt admittance, capacitance and the resistance were ignored becausethe authors considered a dc power flow model.

36

5.2 Database characteristics

• All the circuit breakers are closed during normal operation, even if in thereal network some of them are kept open depending on the necessities of thenetwork.

The figure below show the transmission lines in the continental Europe.

Figure 5.2: Map of the interconnections in the continental Europe [16]

As stated before, the database just provide the reactance for each transmission lineof the system (it neglects the resistance, capacitance and maximum power capacity)because the authors were interested in solving a DC power flow. However, to solvethe AC-DC OPF, the conductance and susceptance matrices are needed. Consider-ing the value of the reactance per distance presented in the data-set, the resistancewas found for an overhead three phase transmission line which characteristics corre-sponds to the one used by the authors. The chosen conductor is the ACSR Joree[4][4]that has a resistance of 22, 7mΩ/km and the same reactance mentioned before. Forthis project purposes, the capacitance was ignored and the maximum capacity ofeach line is assumed as infinite due to the lack of information and the Bialek data-set just presents the maximum power of the interconnections between the differentcountries in the continent.

37


Generation

The generators data is private because it can affect the marginal cost of competi-tors. Then, the authors refer to historical information and data of the present realnetwork. The generators are classified as hydraulic, fossil thermal and nuclear. Con-sidering that it was not possible to obtain the dispatch procedure of every countryin the system, it was assumed that the actual power outputs are proportional to thepower plants installed capacity. This gives a rough estimation of the power injectedto the grid by the generator. Obviously, the difference between peak and base loadplants is specified. Every power plant is related to the bus (substation) where isconnected and the main characteristics such as country, installed capacity, etc.

Besides the technical characteristics, the cost curve information was also obtainedin order to perform an optimal power flow which objective function is based on thecost of generating electrical energy by each power plant in the system. A typicalcost curve is used depending on the type of generator.

For this project purposes, the Borkum Riffgrund offshore wind power plant wasintroduced to the system, it has a generation capacity of 800MW and is located inthe North Sea connected to Germany with a HVDC underground transmission line.It is not a part of the data-set because the wind-farm was constructed after 2009(year of the last update of the database).

Demand

Considering that there’s not enough information about the loads in the Europeaninterconnected system, the authors estimated the demand levels by relating themto the population and the industry. However, the total demand of each country isavailable online, then the dis-aggregated demand was distributed among the loadbuses located in the country considering the items mentioned before. After obtainingthe demands with that approach, a comparison with Italy real demand was madeand there was a correlation of 91% between them, so it’s a good approximation ofthe loads. The demand buses are also associated to the different nodes of the systemwith the respective information and characteristics.

The following figure shows how the energy exchange between the different countriesoccurs during the winter season.

38

5.3 HVDC transmission system characteristics

Figure 5.3: Electrical energy exchange in Europe[16]

The resulting model has the following characteristics:• 1515 buses• 2323 transmission lines• 570 power stations• 1092 loads• Voltage regulators (AVR) are not considered• Maximum capacities of only the cross-border transmission lines


The Bialek model presented in the previous section does not include the HVDCtransmission links of the European interconnected network. Thus, there is thenecessity of identifying and characterizing all the existing interconnections in thecontinent that use the HVDC technology.

39


There are 18 existing dc transmission lines nowadays and none of them are locatedin the continental Europe. This means that the main purpose of the HVDC inter-connections is to connect two different substations separated by the sea (North orMediterranean). However, there are planned links to connect different countries inEurope that are separated by a long distance (for example, Belgium and the southof France). In the following table and map these lines are presented with theircharacteristics.

Figure 5.4: HVDC Interconnections in Europe [ENTSO-E]

The red lines represent the existing projects, the blue discontinuous are the optionsunder consideration and the green line is a project under construction.

40


Table 5.1: HVDC Interconnections in Europe

Name From To Length (km) Voltage (kV) Power (MW) Year Converter Supplier

Moyle Scotland N. Ireland 63,5 250 250 2001 Thyr Nexans

Fenno-Skan Sweden Finland 233 400 550 2011 Thyr ABB

Vyborg Russia Finland 0 85 4 x 355 2001 Thyr -

Estlink Estonia Finland 105 150 350 2006 Thyr ABB

NorNed Norway Netherlands 580 450 700 2008 Thyr ABB

Skagerrak Norway Denmark 240 350 1632 2009 IGBT Alcatel

Konti-Skan Denmark Sweden 149 300 300 2006 Thyr ABB

Gotland Sweden Sweden 99 150 130 1987 Thyr ABB

NordE.On 1 Germany North Sea 200 300 800 2015 IGBT Siemens

StoreBaelt Denmark Denmark 58 400 600 2010 Thyr Siemens

SwePol Sweden Poland 254 450 600 2000 Thyr ABB

Baltic Cable Germany Sweden 262 450 600 1994 Thyr ABB

Kontek Denmark Germany 170 400 600 1995 Thyr ABB

BritNed England Netherlands 260 450 1000 2011 Thyr Siemens

Cross Channel France England 73 400 2000 1986 Merc -

SACOI Italy Italy island 385 200 300 1992 Thyr Alstom

Sapei-Sardinia Italy Italy 435 500 1000 2010 Thyr ABB

Italy-Greece Italy Greece 313 400 500 2001 Thyr -

According to the data in the table and taking into account that most of this trans-mission lines have submarine power cables, the line parameters were calculated.

Line parameters

Since there is not available information about the parameters of the transmissionlines used for HVDC projects, a estimation is required. Considering the typicalcharacteristics of an HVDC transmission line, the reactance and resistance are0, 2112Ω/km, 0, 0176Ω/km respectively[4]. Then, the line parameters for each trans-mission line in the European interconnected network is calculated and is shown inthe following table:

41


Table 5.3: HVDC transmission lines parameters

HVDC Line Resistance (Ω) Reactance (Ω)Moyle 1,1176 13,411

Fenno-Skan 4,1008 49,21Vyborg 0 0Estlink 1,848 22,176NorNed 10,208 122,496Skagerrak 4,224 50,688Konti-Skan 2,622 31,469Gotland 1,742 20,909

NordE.On 1 3,52 42,24StoreBaelt 1,02 12,25SwePol 4,47 53,645

Baltic Cable 4,611 55,334Kontek 2,992 35,904BritNed 4,576 54,912

Cross Channel 1,285 15,418SACOI 6,776 81,312

Sapei-Sardinia 7,656 91,872Italy-Greece 5,509 66,106

However, since the database does not include some Eastern-Europe and Scandina-vian (except Denmark) countries, from the previous 17 existent HVDC lines, only 10are suitable for the data-set. Thus, the lines included in the optimization problemare:

• Skagerrak

• Konti-Skan

• NordE.On 1

• StoreBaelt

• Baltic cable

• Kontek

• Cross Channel

• SACOI

• Sapei-Sardinia

• Italy - Greece

42

5.4 Data management

Converter model

Considering the converter model specified in the sections 3.1.1 and 4.2.1, the pa-rameters to be introduced in the optimization problem are defined. The selectedtopology is the same for every transmission line because there’s not available infor-mation about the characteristics of each converter. Thus, the values of the variablesa, b and c in the equation 4.18, are based on the research paper [12], and are pre-sented in the following table.

Table 5.5: Converter loss parameters[12]

VSC a b cRectifier 11, 033 ∗ 10−3 3, 464 ∗ 10−3 5, 4 ∗ 10−3

Inverter 11, 033 ∗ 10−3 3, 464 ∗ 10−3 7, 67 ∗ 10−3

5.4 Data management

The data-set obtained has to be modified in order to solve the minimization problemin GAMS. So, there is an initial step where the data is manipulated to satisfy theinput requirements of the software. The Bialek database consists of an Excel file of6 sheets, which are:

• Line records: This sheet contains the information about the transmission linesin the European network. It specifies the interconnected buses, the status ofthe line, the electrical parameters (resistance, reactance and capacitance), andin some cases it shows the maximum capacity.

• Generation records: It presents the main characteristics of the power plantsin the system. The following items are included: Name, zone, fuel type, busnumber, ID, status, generation, default voltage, AGC characteristics, mini-mum and maximum active/reactive power, and participation factors.

• Load records: Presents the demand parameters for each load. It includesthe geographical location, the number of bus, zone number, and the active,reactive and apparent power.

• Area records: Shows how the author classify the areas in the system (bycountries), having 32 in total. It presents the load and generation of eachparticipant country and the net exchange of electrical energy.

• Power flows: The sheet presents the results of the power flow performed withthe PowerWorld tool.

• Bus records: Shows the voltages, net power and angle of each bus of the systemafter solving the power flow.

43


Having this information, the Excel file is imported to Matlab for the data processingin order to obtain the inputs shown in the section 4.3 Is important to state that thebase values of the system are Sbase = 1000MVA and Vbase = 380kV and all the datais converted to the per unit system.Then, with the data of the transmission lines, the conductance and susceptancematrices are calculated for both typologies (AC and DC). The input is the firstsheet of the Excel document and the algorithm implemented in Matlab build theimpedance matrix, then it is inverted to obtain the admittance matrix, which isdecomposed in real and imaginary part. Finally, the two matrices are converted toa GAMS readable format which extension is .gdx.Besides these matrices, the generating units characteristics are organized in a tablethat relates each node to the maximum and minimum values of the generated power.Since the input values only present the nodes that have a generator and consideringthat GAMS asks for the information of every node, even if it doesn’t generate anypower, the table was built using conditionals and the first two rows are shown.

Also, the power demanded by each load is processed to obtain it in a single tablethat relates it to a specific node. It occurs the same as with the generator values,but the demand is a ’parameter’ in the program, so the active and reactive powerare obtained in separate tables and the following images show how this data isintroduced to GAMS.

Then, the table that introduce the power boundaries of each node to the problem iscreated. Taking into account that most of the lines don’t specify the maximum powerthat can flow through them, the minimum and maximum values for these branchesare -10 p.u. and 10 p.u., this value was given to avoid convergence problems whensolving. On the other hand, the branches where the boundaries are specified areassociated to their respective nodes and the minimum and maximum values aredetermined. The result of the first two rows is shown.

44

5.4 Data management

With the data obtained in the Table 5.2, the DC conductance matrix is calculatedand saved as a .gdx file. Finally, the matrix that relates each AC node to corre-spondent DC nodes (location of the VSCs) is created and the AC buses with noconnection to the HVDC network are filled with zeros.Considering that the calculated vectors and matrices are an input of the GAMSfile, these set of values were converted to a .gdx file that can be properly read andprocessed in order to solve the optimization problem in GAMS.The dimension of the problem is given by the calculated matrices, and is:

• AC conductance and susceptance matrices: 1515 x 1515• Generation limits: 1515 x 5• Power flow boundaries: 1515 x 5• Active power demand: 1515 x 2• Reactive power demand: 1515 x 2• AC-DC relation: 1515 x 2• DC conductance matrix: 20 x 20

Besides these input values, some constraints were written using Matlab. Consideringthat GAMS does not allow the user to import the equations that define the opti-mization problem and having a large system, the following equations were writtenin Matlab and pasted to GAMS:

• Active and reactive power balance (Equations 4.4 and 4.5)• Converter current (Equation 4.19)• Generated power is equal to zero for some nodes• Active and reactive power in the converter (Equations 4.21 and 4.22)

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6 Simulation and results

This section presents different scenarios of the European Transmission System andthe results of the AC-DC optimal power flow. The proposed scenarios aim to deter-mine whether or not is better to have DC transmission lines in a large network likethe European System and what are the real benefits of the HVDC technology. Also,the computational performance is evaluated in the different scenarios, that are thefollowing:

• Scenario 1: Objective function is equal to zero, normal load flow procedure.

• Scenario 2: Power loss minimization of the base system

• Scenario 3: HVDC technology is used in all the cross-borders between countries

• Scenario 4: HVDC technology is used in the cross-borders with a capacityhigher than 1400 MVA

• Scenario 5: Minimum deviation from a voltage profile

6.1 Scenario 1

Firstly, the objective function of the system is set as zero in order to obtain the resultsof the load flow of the network. This is done to ensure that the optimization problemis being solved properly and that the results correspond to a ’normal’ performance ofa power system. The results cannot be compared with the ones obtained by Bialekin his research because he solved a DC power flow, which ignores the resistance andwhere the voltages are set to a predefined value. Thus, the optimization problemsolved in this section shows reactive power flow, voltages for each node and reactivepower losses at the different components of the network.

The following table shows the maximum and minimum limits for the different vari-ables in the system.

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6.1 Scenario 1

Table 6.1: Lower and upper limits of the system

Variable Maximum value Minimum valueDC voltage 1.1 0.9DC current 1 -1AC voltage 1.1 0.9

Generated power Different for each generatorVSC reactive power 1 -1

Voltage angle 0.5 -0.5Branch loading Different for each branch

Now, in GAMS the optimization problem is solved and the results are obtained.The simulation parameters are shown in the following table.

Table 6.2: Simulation parameters. Scenario 1

Solver Interior point method (IPOPT)Rows 8188

Columns 9148Non-zeroes 36168Iterations 37

Elapsed time 21,07 seconds

Having into account the dimension of the optimization problem, the problem issolved relatively fast because it only takes 37 iterations to get to the minimum value.However, it is not possible to conclude anything from the elapsed time because itdepends on the characteristics of the computer.Considering that the problem solves numerous equations and optimize more than5000 variables, just the most relevant results of the power flow are shown in thetables 6.3, 6.5 and 6.7.

Table 6.3: DC voltages and currents. Scenario 1

DC Node Voltage (p.u.) Current (p.u.)1 1,0003 0,01752 0,9998 -0,01753 1,0007 0,05644 0,9994 -0,05645 0,9997 -0,02026 1,0004 0,0202

47

Chapter 6 Simulation and results

7 0,9997 -0,07168 1,0002 0,07169 1,0231 0,789910 1,0083 -0,789911 1,0065 0,078212 0,9932 -0,078213 0,9999 0,014714 0,9994 -0,014715 0,9989 -0,063216 1,0011 0,063217 1,0000 -0,010118 1,0000 0,010119 1,0080 0,344620 0,9945 -0,3446

Table 6.5: VSC characteristics. Scenario 1

AC Node VSC losses (p.u.) Q VSC (p.u.) AC current VSC (p.u.)219 0,0005 -0,0705 -1,3295719 0,0006 -0,0691 -1,3257764 0,0006 -0,0604 -0,9581766 0,0007 -0,0459 -0,7674771 0,0016 -0,0361 -0,7674773 0,0016 -0,0243 -0,4184773 0,0016 -0,0215 -0,39111032 0,0018 -0,0190 -0,31731033 0,0020 -0,0164 -0,13671034 0,0027 -0,0124 0,11521035 0,0047 -0,0105 0,15501039 0,0049 0,0000 0,19901250 0,0058 0,0002 0,55341253 0,0060 0,0009 0,57501312 0,0060 0,0013 0,66921502 0,0082 0,0016 0,87071503 0,0091 0,0028 0,94001510 0,0200 0,0028 1,57531513 0,0217 0,0040 1,65711515 0,0244 0,0075 1,7805

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6.2 Scenario 2

Table 6.7: AC-DC OPF results. Scenario 1

Variable ValueVAC,MAX 1,0504 p.u.VAC,MIN 0,9117 p.u.VAC,MEAN 1,0005 p.u.PLOSS 3,6035 p.u.

PLOSS(%) 0,9209 %V SCsLOSS(%) 3,4127%

From the results presented in the table 6.3, it can be seen that almost every DCnode has a voltage level close to 1 p.u., which is expected for this power flow. Also,none of the DC branches transmit the maximum current allowed, in fact, just theline that connects the nodes 9 and 10 is transmiting more than 0,5 p.u.. The table6.5 shows the main characteristics of the converters at the AC side of the network,and it shows that the losses at the VSCs do not depend exclusively in the currentexchanged by the converter, other parameters may affect this value (AC and DCline parameters, generators, loads).

The table 6.7 shows the most relevant results of the load flow and it is evident thatmost of the voltages in the AC side of the networks are close to 1 p.u. and themaximum and minimum values are far from the stipulated limits. The total lossesof the system are of 3603 MVA and this represents the 0,9209% of the generatedpower. Considering that in Europe the losses in transmission are between 0,5%and 3%[18], the value and the model obtained are trustworthy. The converters justrepresent the 3,41% of the total losses in the system.

6.2 Scenario 2

The second scenario solves the AC-DC optimal power flow of the European Trans-mission Network. Unlike the last scenario, in this case the problem has a objectivefunction different to zero. The algorithm minimizes the power losses in the powersystem and this is given by the equation 4.23. The boundaries are the ones in thetable 6.1. The system is the one described in the section 5 and the main parametersof the model in GAMS are shown in the next table.

49






The rows and the columns are the same as in the last problem because there’s nomodification in the variables or the constraints. However, the number on non-zerovalues and iterations increased as a result of the defined objective function whichmakes the problem more complex. The tables 6.9, 6.11 and 6.13 show the results ofthe optimal power flow for the AC-DC European Transmission System.


DC Node Voltage (p.u.) Current (p.u.)1 0,9291 1,00002 0,9000 -1,00003 0,9000 -1,00004 0,9228 1,00005 0,9609 -1,00006 0,9953 1,00007 1,0920 -1,00008 1,1000 1,00009 1,1000 1,000010 1,0813 -1,000011 1,0694 1,000012 0,9000 -1,000013 1,1000 1,000014 1,0694 -1,000015 0,9344 1,000016 0,9000 -1,000017 0,9000 -1,000018 0,9064 1,000019 0,9346 0,885020 0,9000 -0,8850


50

6.2 Scenario 2

AC Node VSC losses (p.u.) Q VSC (p.u.) AC current VSC (p.u.)219 0,0005 -0,2512 -1,6819719 0,0006 -0,1652 -0,7951764 0,0006 -0,1449 -0,7247766 0,0006 -0,0794 -0,5538771 0,0008 -0,0794 -0,5176773 0,0008 -0,0534 -0,4531773 0,0014 -0,0001 -0,45311032 0,0015 0,0000 -0,35881033 0,0015 0,0000 -0,32071034 0,0018 0,0000 0,09261035 0,0021 0,0000 0,09951039 0,0035 0,0000 0,21571250 0,0039 0,0000 0,41991253 0,0040 0,0000 0,46951312 0,0041 0,0000 0,48221502 0,0046 0,0000 0,49571503 0,0047 0,0000 0,54301510 0,0049 0,0000 0,55941513 0,0057 0,0009 0,57481515 0,0106 0,6081 0,6595



PLOSS(%) 0,2006 %V SCsLOSS(%) 7,4033%

The table 6.9 shows that the DC voltages are close to the minimum and maximumlimits (0,9 p.u. and 1,1 p.u.). Also, nine out of the ten HVDC transmission lines inthe network are fully charged, which means that they are transmiting their maximumcapacity. This occurs because the DC lines present a low resistance, which representslower losses compared to the ones of the AC grid. And since the objective functionis to minimize the losses, the OPF takes advantage of the HVDC line characteristicsto solve the problem.The voltage levels of the AC grid nodes are higher than the ones obtained in thefirst scenario, being the mean value 1,0896 p.u., which is almost the upper boundary.

51


The power losses obtained after performing the OPF are almost 4.5 times less thanin the case where the objective function is set to zero. The losses percentage is of0,2% (below the European levels) and the converters represent the 7,4% of the totallosses in the system as expected because more current is flowing through the DCtransmission lines.

6.3 Scenario 3

This scenario shows the situation where all the cross-border transmission lines arebased in HVDC technology, replacing the existent ones. The objective of this changeis to evaluate the impact of having numerous HVDC transmission lines all over theelectrical network. According to the Bialek data-set, there are 146 transnational andoverseas lines (including the 10 HVDC transmission lines presented in the case ofstudy) in the network, so the size of the DC network is bigger than in the previouscase. All the data processing performed in the section 5.3 is done again in orderto obtain the new conductance and susceptance matrices, the new interconnectionbetween AC and DC nodes, and the new constraints of the system. Is importantto state that the value of the resistance for the HVDC lines is the same as the oneselected in the section 5.2 for this type of lines. The objective function used is theone that minimizes the power losses of the system.After introducing the new data-set to GAMS, the model parameters are the follow-ing:





It can be seen that the size of the optimization problem is bigger as expected be-cause there are new constraints that represent the power balance at the nodes withconverters. Considering that the number of constraints and variables is higher, islogical that the number of iterations is bigger than in the other cases.Taking into account that the size of the AC and the DC networks are considerablybig, just the most important results of the optimal power flow are presented. Thefigure 6.1, allows to identify that there is not a clear pattern in the system aboutthe DC voltages. However, most of the nodes have voltages close to the upper limit(1,1 p.u.) and the mean value of these voltages is 1,0428 p.u.. Since the problem is

52

6.3 Scenario 3

minimizing the power losses in the system, the AC voltages rise up and this is thereason of the resulting mean value which is 1,0813 p.u..

The most important advantage of changing the transmission lines at the intercon-nections for HVDC technology is the reduction in the total power losses, whichrepresents just the 0,1028% of the generated power in the system. This value islower than the losses percentage in Finland, which is the country in Europe withless losses due to transmission[18]. Also, it is important to identify that the 90% ofthe losses is because of the converters.

Figure 6.1: Voltage profile of DC network. Scenario 3


Variable ValueVAC,MAX 1,1 p.u.VAC,MIN 0,9 p.u.VAC,MEAN 1,0813 p.u.VDC,MAX 1,1 p.u.VDC,MIN 0,9 p.u.VDC,MEAN 1,0428 p.u.PLOSS 0,3839 p.u.

PLOSS(%) 0,1028 %V SCsLOSS(%) 90,96%

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6.4 Scenario 4

This scenario is similar to the third one, with the difference that the DC networkis smaller. For this case, only the cross-border transmission lines with a capacityhigher than 1400 MVA are taken into consideration for the AC-DC optimal powerflow. Thus, the total number of HVDC transmission lines is 48, including the tenHVDC links explained in the section 5.2. The same procedure as in the scenario 4is performed and the new conductance and susceptance matrices are obtained. Theobjective function to minimize is the total power loss in the system.

The following table presents the main parameters of the simulation in GAMS.





The size of the problem is smaller than the one in the scenario 3 but bigger than theother ones because there are more constraints due to the AC-DC converter stations.Also, the number of iterations to get to the minimum is considerable if comparedwith the first two scenarios, but smaller than the third one.

The figure 6.2 and the table 6.17 show the main results of the optimal power flow.Analyzing the chart presented, most of the nodes of the DC network have a voltageof 0,9 p.u., which is the lower boundary, and just three nodes have voltages greaterthan 1 p.u.. On the other hand, the mean value of the AC grid is 1,0891 p.u. andmost of the voltages are above this value. Then, when a AC-DC OPF is solved, theDC voltages tend to go down and the AC ones go up.

Furthermore, the total power losses in the grid represent the 0,1758% of the gener-ated power, which means that having HVDC technology instead of HVAC systems,is beneficial for the power network. The losses in the converters represent more thanhalf of the total losses of the system mainly because of the increased number of DClines.

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6.5 Scenario 5

Figure 6.2: Voltage profile of DC network. Scenario 4


Variable ValueVAC,MAX 1,1 p.u.VAC,MIN 0,9 p.u.VAC,MEAN 1,0891 p.u.VDC,MAX 1,1 p.u.VDC,MIN 0,9 p.u.VDC,MEAN 0,9195 p.u.PLOSS 0,6828 p.u.

PLOSS(%) 0,1758 %V SCsLOSS(%) 55,014%

6.5 Scenario 5

This final scenario is based on the original system exposed in the section 5 but theobjective function changes. Now the goal is to minimize the voltage deviation inthe AC nodes of the power system and the new objective function is given by theequation 4.26. For this problem, the set voltage is 1 p.u. and the parameters of thesimulation are seen in the next table.

55






The number of rows is exactly the same as in the two first cases. However, thecolumns, the non-zero elements and the number of iterations are bigger than thescenarios one and two. This is mainly because the problem is using different variablesto minimize the objective function. The principal results of the simulation are shownin the tables 6.19, 6.21 and 6.23. These results show that the voltage level of theDC nodes is between 0,9 p.u. and 1,1 p.u., while the currents flowing through thebranches is almost 1 p.u. for three lines and zero for the rest. The mean voltage ofthe AC system is 1,0001 p.u. which is desirable because the problem is minimizingthe voltage deviation of the nodes taking as reference a voltage equal to 1 p.u. andthe objective function at the minimum point is equal to 0,0518.


DC Node Voltage (p.u.) Current (p.u.)1 0,9381 0,99642 0,9090 -0,99643 1,0000 0,00094 1,0000 -0,00095 1,0000 0,00266 1,0000 -0,00267 1,0916 -0,99958 1,0996 0,99959 0,9993 -0,077010 1,0008 0,077011 1,0968 0,990612 0,9290 -0,990613 1,0001 0,004614 0,9999 -0,004615 1,0016 0,094216 0,9983 -0,094217 1,0000 -0,001118 1,0000 0,0011

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6.5 Scenario 5

19 1,0065 0,286020 0,9953 -0,2860


AC Node VSC losses (p.u.) Q VSC (p.u.) AC current VSC (p.u.)219 0,0006 -0,4766 -1,3379719 0,0006 -0,4366 -1,3305764 0,0007 -0,3768 -0,8471766 0,0009 -0,1781 -0,8133771 0,0016 -0,1781 -0,8133773 0,0018 -0,0985 -0,4813773 0,0019 -0,0954 -0,41191032 0,0019 -0,0002 -0,38461033 0,0020 0,0000 -0,08231034 0,0020 0,0000 0,12611035 0,0038 0,0010 0,16541039 0,0045 0,0010 0,19101250 0,0046 0,0028 0,45581253 0,0055 0,0060 0,52971312 0,0061 0,0100 0,54361502 0,0061 0,0169 0,63901503 0,0073 0,0210 0,79711510 0,0074 0,0983 0,80441513 0,0111 0,2081 1,08041515 0,0219 0,9992 1,6668



PLOSS(%) 0,8489 %V SCsLOSS(%) 2,7196%

z 0,0518

57


6.6 Discussion

Having solved the optimal power flow for all the scenarios proposed, it is evident thatthe utilization of HVDC transmission lines decrease the total losses in the Europeannetwork, which is desirable for the system. This happens because the resistance perkilometer of the DC conductors is lower than the AC resistance, and due to thetransmission capacity of this type of technology. The effect of the AC-DC OPF isthat there’s more current flowing through the DC lines than in the normal scenarioand that is why the conductors are almost fully charged as it can be seen in thefigure6.3. This figure shows that when the objective function is the minimizationof the power losses, 9 out of 10 transmission lines are fully charged, while when thefunction is set as zero, none of the lines is working at the full capacity.

Figure 6.3: Percentage of current flowing through the HVDC lines

In addition to the reduction of the power losses in the system, when implementingthe AC-DC OPF adding new HVDC lines to the network as in the scenarios 3 and4, the total power losses decrease and the percentage of losses due to the convertersincrease. This means that when the HVDC technology replaces the HVAC trans-mission lines, the power tend to flow through the DC links and the losses in thesystem decrease. The figure6.4 shows that for the scenarios with more HVDC lines,the percentage of losses because of the converters is bigger.On the other hand, the voltages are also affected when the objective function ischanged in the optimization problem. The figure 6.5 shows that the voltages getcloser to the upper boundary when the problem is minimizing the power losses inthe system, while this levels are almost 1 p.u. when there is no optimization. Also,for the case of the minimization in the voltage error, the figure presents that thevoltage level for almost every node of the AC network is 1p.u.. The figure 6.6 shows

58

6.6 Discussion

Figure 6.4: Proportion of losses for the five scenarios

the angles for all of the 1515 nodes in the system in the three different scenarioswhere the original system is analyzed, and it is clear that when the power loss isminimized the angle of the nodes is almost the same, while in the other cases, theseangles are different for each node.

59


Figure 6.5: Voltage profile for different scenarios

Figure 6.6: Voltage angle for different scenarios

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7 Conclusions

The project purpose was to perform an optimal power flow of the hybrid ACDCEuropean Transmission Network. After processing the data, coding the solution andidentifying the different scenarios, the results were obtained, leading to interestingconclusions about the nature of the problem.Firstly, when solving the ACDC OPF, the power flowing through the convertersincreased leading to a power loss reduction. Setting the objective function as theminimization of the losses in the network led to a change in the voltage profile ofthe system and the path followed by the current, obtaining as a final result a valueof 0,2% of total losses in the system, which is lower than the European average.However, it is important to consider that the dataset used to solve the problem, wasbuilt based on assumptions, so the final results are not totally accurate.Moreover, according to the results obtained by changing the cross-border intercon-nections for HVDC links, there is a reduction in the power losses. This phenomenawas observed in two of the proposed scenarios and the losses could even decreasemore if the converters are improved because the HVDC system efficiency relies onthese elements. Nevertheless, the problem is more complex to solve and it requiresmore computational capabilities.Despite of the technical advantages that suppose the implementation of HVDCtransmission lines in the cross-border countries, an economical approach is neededto determine the real benefit of the introduction of this new technology. This projectprovides an important tool to begin this analysis because it gives the power savingswhen using HVDC at a certain moment depending on the generation and demandof the system. Thus, the electrical energy savings for a given lifespan could bedetermined and the economical feasibility too.Finally, if the information (demand, line parameters, generator capacities) of theEuropean Transmission Network is available, a more accurate optimal power flowcould be performed. Then, the results would be relevant to a future expansion planof the European power system, considering that the number of offshore wind powerplants is increasing all over the continent and the HVDC tranmission links are abetter solution to connect them to the grid as discussed in the document. Also,this technology play an important role in the development of a supergrid in Europe,taking into account the future demand scenarios.

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