Joakim Munkhammar

Markov-chain modeling of energyusers and electric vehicles

Applications to distributed photovoltaics

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AbstractTechnological improvements and falling prices on photovoltaic panels and electric vehiclessuggest that they might become more common in future households. The introduction of aphotovoltaic system and an electric vehicle has considerable impact on the energy balance of ahousehold.This licentiate thesis investigates the power consumption- and production patterns associatedwith the photovoltaic (PV) electricity production, household electricity consumption and homecharged plug-in electric vehicle (PEV) electricity consumption. This investigation is carried outon both an individual and aggregate household level.The methodology used in this thesis is interdisciplinary but the main contributions are math-ematical modeling and simulations of the three main components. Theoretical estimates ofelectricity consumption were constructed from extensions to the Widén Markov-chain modelfor generating synthetic household electricity use patterns. The main research contributionin this thesis is the development and analysis of two extensions of this Markov-chain model:(I) Electricity use from a home charged PEV, (II) Flexibility of end-user power use. Thesetwo extensions were used in studies regarding the coincidence - in particular the level of self-consumption - between PV electricity production and household electricity use. PV electricityproduction was modeled from high resolution solar irradiance data from the Ångström labora-tory.Results show that the home charged PEV load would increase the household load considerably.It was also shown that the level of correlation between PEV load and PV electricity productionwas low, but that to some extent the PEV load could help increase the self-consumption of PVpower, both on individual and aggregate household level.The modeling and simulations of end-user flexibility showed that the household load profilecould be altered to a certain degree. It was also shown that certain flexibility setup could im-prove the self-consumption of PV power production, more so than the introduction of a PEV.

Dedicated to my brother Johnny Munkhammar (1974-2012)who made all the difference by providing me with the guts

to follow my dreams and take the road less traveled by

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I P. Grahn, J. Munkhammar, J. Widén, K. Alvehag, L. Söder,"Plug-in-vehicle home charging model based on residential activitypatterns", submitted to IEEE Transactions on Power Systems (2012).

II J. Munkhammar, P. Grahn, J. Widén, "Quantifying self-consumptionof on-site photovoltaic power generation in households with electricvehicle home charging", Manuscript (2012).

III J. Munkhammar, J. Widén, "A flexible Markov-chain model forsimulating demand side management strategies - applications todistributed photovoltaics", in conference proceedings of WorldRenewable Energy Forum (WREF) Colorado, USA, May 13-17 (2012).

IV J. Widén, J. Munkhammar, "Widespread integration of distributedphotovoltaics at high latitudes: opportunities and challenges", inProceedings of the 26th European Photovoltaic Solar EnergyConference (EU-PVSEC), Hamburg, Germany, September 5-9 (2011).

Reprints were made with permission from the publishers.

Publications not included in the thesis

V J. Munkhammar, J. Widén, "A stochastic model for collective residentactivity patterns and energy use: preliminaries", In Sustainability in En-ergy and Buildings Vol. 1, Future Technology Press from Sustainabilityin Energy and Buildings (SEB’12) Stockholm, Sweden, September 3-5(2012).

VI L. Mattsson, A. C. Andersen, J. Munkhammar, "On the dust abun-dance gradients in late-type galaxies - I. Effects of destruction and growthof dust in the interstellar medium", Monthly Notices of the Royal Astro-nomical Society, Vol. 423, Issue 1, p.26-37 (2012).

VII M. Hellgren, P. Grahn, J. Munkhammar, "Photovoltaics, electric ve-ichles and energy users: A case study of the Royal Seaport - Visionsand energy user expectations", working paper no. 50 at Department ofTechnology and Social Change, Linköping University (2011).

Notes on my contribution

I contributed with the following in the appended papers:

Paper I, I co-developed the model, and assisted in writing it.

Paper II, I co-developed the model, did all simulations and wrote most of it.

Paper III, I did all simulations and wrote most of it.

Paper IV, I did literature survey and assisted in writing it.

Contents

Publications not included in the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vNotes on my contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Overview of thesis and appended papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Systems theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 System levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Introduction of distributed generation into the power system . . . . . . . 7

2.2.1 The Swedish power system and distribution grid . . . . . . . . . . . . 72.2.2 Challenges with distributed generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Renewable energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Integration of photovoltaics in the power system . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Properties of PV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 PV in the power system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 PV self-consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Electric vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.1 Electric and hybrid electric vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Electric vehicle car fleet in Sweden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Electric vehicle engine and battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.4 Electric vehicle and the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Demand-side management and end-user flexibility . . . . . . . . . . . . . . . . . . . . . 152.5.1 Measures of self-consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Modeling household electricity use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.1 Modeling behavior and electricity use . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.2 Time-use data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Methodology and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Model for household electricity consumption and production . . . 20

3.1.1 Household electricity consumption from activities . . . . . . . 213.1.2 The Widén Markov-chain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Self-consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Electric vehicle extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Plug-in electric vehicle electricity consumption model 273.2.2 Household electricity consumption from charging . . . . . . . 293.2.3 Limitations of the PEV extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Load flexibility extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 Formal setup of the flexibility extension . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Limitations of the flexibility extension . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Modeling PV electricity production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1 PEV home-charging model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 PEV extension and photovoltaic power . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Flexibility extension and photovoltaic electricity production . . . . . 374.3 Comparison of impacts on PV self-consumption . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1. Introduction

Reporter: "You’ve been called the Da Vinci of our times.What do you say to that?"

Tony Stark: "Absolutely ridiculous. I don’t paint."

Iron Man (2008)

During the time before the industrial revolution people had to deal withpoverty, starvation and generally bad living conditions. Life expectancy andworking conditions were immensely bad compared with today’s standards.Even the richest during those times - such as the kings - did not have accessto modern day dentistry or simple medications such as antibiotics, which ledto misery and suffering compared with even the most modest of today’s stan-dards in the western world. The industrial revolution brought on tremendousimprovements for people in the world by providing better food and curingdeceases. Technological advances enabled increased prosperity by more ef-fectively utilizing resources. The primary function of the revolution was nota redistribution of the wealth - but rather the enabling of the development ofspecializations and innovations which in turn led to more efficient and diverseproduction of goods and services by utilizing synergy. These processes ofindustrialization are today present - where not hindered - in developing coun-tries. The total GDP of the world increased nearly 50 times from 1820 to 1998[31, p.28] - the development of the world is dynamic and not a zero sum game.As a result of the industrial revolution societal systems in the industrialized na-tions grew in prosperity, size and complexity which required more energy tosustain its processes.

A key feature in upholding the structure and function of a society is a reli-able and abundant source of power. Today all energy sources - except nuclear,geothermal and tidal power - have their origin in solar radiation. The crucialproblem is how to utilize the solar irradiation energy. Coal and oil are solarradiation energy stored over long time during earlier epochs of earths history.Wind power obtains its energy from the weather system - which in turn isdriven by solar radiation. Hydro power is energy from reservoirs that has beenfilled by rain, biofuel is derived from plants that have absorbed solar radiationand via photosynthesis. Despite the most efficient modern photovoltaic (PV)cells having an average efficiency of about 16 percent they are still the mostefficient utilisers of the suns radiation per area [27, p.22], [34].

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Although coal and oil are naturally replenished energy sources they are notconsidered renewable due to the long time-scale they are replenished on. Theearth is radiated by about 800 million TWh annually which is 10000 timesmore than is necessary to match the global energy demand for a year [20,p.13]. In terms of the time-scale of renewability photovoltaic power is prac-tically immediately replenished. Despite these favorable features there arechallenges for the large scale development of photovoltaic electricity produc-tion in the world. In particular this is true at high latitude regions such asSweden. The price of PV cells is too high for large-scale production of PVpower in Sweden but it is economically viable or nearly economically viablefor households to reduce electricity consumption costs. Also there is no scale-dependence on the economy of a system, hence making large-scale centralizedPV power plants no more feasible than local distributed PV [59, p.32]. Thismakes the existing buildings - in particular residential buildings - interesting.

Residential buildings are interesting because of the proximity to the end-user and the total nation wide availability of roof areas. Installing PV moduleson a household level has become interesting from an economic perspective,especially when the PV system is connected to the grid and distribution isenabled. Indeed due to falling PV cell prices and the aid of governmental sub-sidies for installing PV-systems as well as generous feed-in tariffs - mostlyin Germany and Japan - the market for distributed photovoltaics (PV) has in-creased considerably since the turn of the millennium.

A large amount of installed PV as distributed generation has further pushedfor the need to rethink and redesign the power system into a smart grid whichcan handle intermittent distributed power sources such as distributed PV. Oneadvantage from power systems perspective for installing distributed PV is thatthe source of power is closer to the load. The injection of PV power into thegrid at the end-user should not surpasses the so-called hosting capacity of thegrid, that is when voltages or loading of components and losses in the gridreach unacceptable levels [6, p.89]. Hosting capacities of 60% PV generationas a fraction of the yearly load were recently determined for a representativerural distribution grid in Sweden [55]. Given no local energy storage in thehousehold the proximity of load and production is mostly beneficial if theload is matched with the production.

It is therefore interesting to investigate self-consumption of on-site PV asa way to handle locally high penetrations of PV. Electric vehicle batteries arespecial examples of possible PV power storages. It is interesting to investi-gate the coincidence between household electricity use and PV power when ahome-charged plug-in electric vehicle (PEV) load is added to the householdload, especially considering the expected increase of the number of electricvehicles in the future. Regarding household electricity use and the level ofself-consumption of PV power it is also interesting to investigate the demandresponse and end-user flexibility.

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1.1 Aim of the thesisThis licentiate thesis is a summary of the first part of a PhD project carried outwithin the Energy Systems Programme, a national research programme andgraduate school aimed at interdisciplinary research concerning energy systemsquestions.

The goal of the licentiate thesis is to investigate ways to increase self- con-sumption of on-site PV in residential buildings. The thesis is focused on threemain components: household electricity load, PEV electricity load and pho-tovoltaic electricity production. These three components are studied primar-ily on the single household level to determine the possibilities for PV self-consumption in individual households, but also on an aggregate of a numberof households is considered to indicate the impact on the distribution grid. Twoways to increase the self-consumption are studied in this thesis: introductionof PEVs and end-user demand flexibility. The first aim of the licentiate thesisis to investigate how introducing a PEV changes a household’s electricity loadpatterns and how these are matched by on-site PV. The second aim is to investi-gate how flexible households’ load profiles are and how they can be changed toincrease self-consumption of PV electricity. The thesis answers these aims bymodeling the three components by extending existing framework in two ways:

(I) A PEV load extension to a Markov-chain model for synthetic activitygeneration and energy use.

(II) An end-user load flexibility extension to the Markov-chain model forsynthetic activity generation and electricity use.

The setup on the photovoltaic simulations is for high latitudes in order toaddress these questions for the location of Sweden.

1.2 Overview of thesis and appended papersThis thesis is composed of papers based on interdisciplinary research collab-orations. My main contributions to the papers is in part the mathematicalmodeling and simulations, and in the cases where I am head author most ofthe writing. The results in the thesis are based on the following appended pa-pers:

• Paper I presents a stochastic home-charging model for PEVs based onthe stochastic model for activities and electricity use developed by Widén[57, 58]. The model produces high resolution charging patterns overtime and is used to study how load patterns change with PEV introduc-tion.

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• Paper II is a study on the application of the stochastic model in paper Ito a household with photovoltaic electricity production. It investigatesthe impact on the electricity consumption/production balance for house-holds with PV installed when introducing a PEV.

• Paper III deals with an extension on the stochastic model developedby Widén that simulates flexible behavior in household electricity con-sumption. The flexibility extension is then used to study increased self-consumption of PV electricity in connexion with flexible behavior.

• Paper IV provides a summary of previous research on systems questionsfor distributed PV in Sweden and gives a background and motivation forpapers (I)-(III).

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2. Background

The general background for the research is presented in this chapter.

2.1 Systems theoryThere are many types of systems such as for example physical systems, bio-logical systems, social systems and power systems. Despite having differentcontent systems often have many aspects in common, such as for examplebasic underlying structures, and rules for over time evolution. There existmodels, principles and laws which are universal and can be found in manysystems. These laws are often formulated by mathematics, and one exampleof this is exponential growth which is a phenomenon found in many differenttypes of systems. Bertalanffy defined General Systems Theory as a subjectmatter of formulation and derivation of principles which are valid for systemsin general [4, p.31]. In that sense systems theory can be seen as an interdisci-plinary theory of how systems work in general. In many ways a system can beseen as a series of components which together constitutes a complex structure[23]. It is possible to arrange systems in a classification scheme which re-lates to their complexity. In [7, p.18-30] Boulding provided such an orderingof systems regarding their complexity, see Figure 2.1. Boulding showed howthe world can be investigated from each type of system level [7]. Informationand communication are important topics in systems theory where for examplefeed-back loops allows self-regulation in a system.

2.1.1 System levelsThe largest system is perhaps all possible configurations of the entire threedimensional universe over all time. Given an observed history or assuming adeterministic view of the universe the largest system is the three dimensionaluniverse over all time. It contains hundreds of billions of galaxies, each galaxywith different characteristics and history at each time. Lets assume that thesystem is limited in time-span to only a few years. If one then takes thisuniverse and zooms in on a single galaxy trillions of stars in various stages ofevolution emerge along with interstellar gas and dust. If zooming in furtherthe next typical system level might be a planetary system with a central star,orbiting planets and asteroids. If one zooms in on a single special planet one

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Social systemsHuman systems

Evolutionary systemsEcological systems

Demographic systems

Creodic systemsPositive feedback systems

Cybernetic systems

Reproductive systems

Mechanical systems

IncreasingComplexity

Figure 2.1. Different levels of system complexity according to Boulding [7].

may see water, mountains and a hint of an atmosphere. Further zoom perhapsreveals a city, with buildings, large-scale infrastructure and vital componentssuch as power plants. Zooming in further to a neighborhood individuals withcomplex actions and interactions emerge. The next level is a single householdwith its inhabitants and their activities, electricity consumption and possiblyphotovoltaic electricity production. The final level is an individual in whosemind a thought of the entire universe as a system is present.

Each system level has its own characteristic dynamics; the universe as awhole contains galaxies much like particles in a gas. Each galaxy is composedof stars - which on the galaxy level can be approximated as heavy particlesmoving with gas and dust. Planetary systems can usually be described asa central mass with orbiting particles. A planet has overall texture such asmountains, sea and atmosphere. A city has buildings and infrastructure. Eachhousehold has inhabitants and electricity consumption - possibly production.A mind has thoughts. Every system level has its characteristic level of detailsand its own dynamics - which is manifested with input and output from thesystem boundaries.

In practice system theory - at least when implemented in the research car-ried out for this thesis - is mainly used as a means to set limits for the re-search. In this research project the system limit is mainly of a single house-hold - whether it be a detached house or an apartment - and the energy-relatedpractices of the inhabitants within. Also the system level of an aggregate ofhouseholds is considered. Here it is important to quantify the processes ofinput and output of a system in the typical system limit set for the research.For example the energy input and output of the particular household that is in-vestigated. This might then be used in other studies of for example aggregatesof households by widening the system limit of the initial study. Also otherstudies can use the input and output of a system so that different systems caninteract.

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Figure 2.2. A system-based schematic illustration of a house, an inhabitant, an electricvehicle and a PV module, which in essence is the main system investigated in thisthesis.

2.2 Introduction of distributed generation into the powersystem

In this section we will review the Swedish power system and distributed PV.

2.2.1 The Swedish power system and distribution gridPower systems have been described as the most complex systems ever cre-ated and operated by humans [46, p.139]. Historically in Sweden - in thelate 1800’s - the electricity production was local. Often the transition wasmade from old hydro-powered mills to hydro power plants in Sweden [25]. Inthose places where a hydro power plant could not be constructed a coal powerplant was setup instead. As the technology for transmission and distribution ofelectric power was developed the possibility for establishing centralized powerwas enabled. The use of alternating current made it possible to first constructregional grids and eventually a national one [25]. The current energy mix inthe grid consists of mostly hydro power and nuclear power, but with a growingpart of renewable energy supply. This renewable energy part - other than hy-dro power - mostly consists of wind power. PV provides a small contributionwith an installed power of 15.8 MWp in 2011 [54, p.5].

This has further pushed the need for smart-grids in the Swedish power sys-tem. The Swedish distribution grid can be divided into three major levels.The power output from large-scale power generation such as hydro power isat the high-voltage level 400kV . This is distributed to a major transformerstation where it is transformed to the mid-voltage level 130kV . This is thendistributed to a regional transformer which transforms it to the low-voltagelevel 40− 70kV [52]. The power is then transmitted to a distribution stationwhich transforms the power to 10kV and the eventually a substation trans-

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forms it to 400V which is distributed at the household level [52]. Distributedgeneration may be injected at any level, but the PV we consider in this thesisis injected at the household level.

2.2.2 Challenges with distributed generationUnlike centralized electricity production PV is often situated at the end-userwhich makes it part of so-called distributed generation. An increased introduc-tion of distributed generation in the power system increases the complexity ofthe system. There are basically two different challenges with distributed andvariable generation like PV. On the one hand, with large amounts of variablegeneration the need for power system balancing to keep the system frequencywithin limits may change. On the other hand, which is most relevant for thisthesis, the local distribution grid may experience new, reversed, power flows,voltage rises and component overloading. Voltage disturbances includes re-duction of the lifetime of equipment, erroneous tripping of the equipment, anddamage to equipment [6, p.92]. This calls for advanced power system controlwhen - often intermittent - power sources such as PV are connected at the enduser [46, p.139].

The introduction of distributed generation will impact the power systemperformance. This impact is harder to estimate if the distributed power sourceis intermittent. The problem for distributed generation is that the distributiongrid has limits regarding the amount of power which may be injected at theend-user site. One may define hosting capacity as the amount of distributedgeneration for which the performance becomes unacceptable [6, p.89]. It ismeasured as a fraction of injected power compared with the load on an annualbasis [55]. It is a performance index which is suitable to use as power qualityindicator regarding issues such as voltage rise, overloading and harmonics. Itwas determined that the hosting capacity in Sweden was 60% for rural andsuburban grids while for a city grid it could be as high as 325% [55].

2.2.3 Renewable energyDistributed generation such as PV is often from renewable energy sources. Fora modern society the production of electric power is necessary for providinga decent life for any citizen. If many options for production are available thenthere is usually a preference relation among the options. In face of possibleabundance limitations of fossil fuel combined with potential environmentalproblems related to their combustion has generated a considerable amount ofattention to the field of renewable energy sources [42]. The definition of re-newable energy sources is the following according to encyclopedia Britannica[10]:

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Renewable energy is usable energy derived from replenishable sources such asthe sun (solar energy), wind (wind power), rivers (hydroelectric power), hotsprings (geothermal energy), tides (tidal power), and biomass (biofuels).

Today most electricity production in the world comes from non-renewableenergy sources [42, p.16]. A graphical representation of the distribution ofdifferent kinds of energy sources in the world 2011 is shown in Figure 2.3.Among the renewable energy sources PV has expanded considerably duringthe last decade. Most of the installed PV power is in the International EnergyAgency Photovoltaic Power System Programme (IEA-PVPS) countries wherethe biggest producers are Germany and Italy [59, p.25]. The total amount ofinstalled PV power for the IEA-PVPS countries is shown in Figure 2.4.

Figure 2.3. The primary energy in the world 2011 [42, p.16].

The majority of the installed PV-systems are grid-connected and connectedat the end-users which is located at the very end of the distribution grid [54,p.4].

Figure 2.4. Total PV peak power installed in the IEA-PVPS countries between 2000and the end of 2011 [54].

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2.3 Integration of photovoltaics in the power systemIn this section we review photovoltaic technology, its applications and chal-lenges with its integration in the power system.

2.3.1 Properties of PVThe common consensus when speaking of active solar energy is that it regardsdirect conversion from solar radiation into heat and/or electricity [46, p.61]. Inthis category PV is a special case since it directly converts the energy from in-coming solar photons into a current using semiconductor materials [22, p.4].In terms of physics solar cell technology is based on the photovoltaic effectfirst reported by Bequerel in 1839 [22]. The function of each photovoltaic cellis to absorb the incoming photons which then frees electrons in the semicon-ductor material [46, p.62]. This in turn generates a DC-current. The mostcommon PV technologies which are available on the market are silicon celltechnology but there also exist emerging technologies such as organic PV cellswhich encompasses advanced thin film technology [27, p.7]. Technically themain component of a PV system is a PV panel which is made up of single solarcell modules [56]. Depending on size and efficiency these modules typicallyrange from a few watts up to some 100 watts, hence PV systems have a widerange of applications from pocket calculators to large fields of PV plants. Cur-rent PV modules have an average efficiency of about 16 % [27, p.22]. A 10square meter PV module with efficiency of 16% has a peak power of 1.6kWp.

Figure 2.5. PV modules at the Ångstrom Laboratory (Uppsala University). Photo:Joakim Munkhammar.

10

The output of power from photovoltaic systems depends on the setup of theparticular photovoltaic system used. Latitude and tilting of planes are exam-ples of important variables for the setup of a photovoltaic system. In Figure2.6 the daily average PV production over a year for a 25m2 PV array located atUppsala is given along with an example of one year averaged household load.

2.3.2 PV in the power systemAn overview of the challenges and opportunities of distributed PV at highlatitudes was done in appended paper (IV). A main conclusion was that theSwedish power system seems to allow for a high penetration level of PV,but that grid-connected PV power at the end-user site was in need of fur-ther research, in particular in connection with for example PEV load. TheIEA-PVPS defines a classification of systems according to [54, p.3]: (a) off-grid domestic, (b) off-grid non-domestic, (c) grid-connected centralized and(d) grid-connected distributed. Among these categories of systems it is thegrid-connected systems (c) and (d) which are the most prominent today inthe world [27]. Recent years the markets for PV systems have progressedfaster than even the most optimistic predictions [67, p.4]. The total installedgrid-connected PV power in the world rose by 28 GWp during 2011 makingthe total installed peak power over 62 GWp in the world [54, p.4]. PV is anintermittent power source with daily- and seasonal variations. A substantialamount of distributed PV injected at the end-user site in the grid needs a suf-ficient hosting capacity in the grid. There are essentially three main methodsfor increasing the hosting capacity for PV: Adjusting tap changer settings atthe transformer substation, PV inverter active power curtailment and reactivepower control. It was shown that the most effective options for dealing withover voltages during limited time intervals and narrow control ranges was re-active power control and curtailment [55]. During times of high-load adjustingtap changers lowered all voltages, not just the critical ones [55, p.7]. Anotherway to circumvent the problem of insufficient hosting capacity is to increasethe self-consumption of the PV power to lower the power injected into thegrid. We will go into this in detail in the next section.

2.3.3 PV self-consumptionThe coincidence between PV electricity production and household electricityload is generally suboptimal. In particular for countries at high latitudes, suchas Sweden, where household electricity demand and PV electricity productionare negatively correlated both on annual and diurnal basis [60, p.1953]. Onemay for example compare a simulated average load profile and an average PVelectricity production profile in Figure 2.6.

11

00:00 12:00 24:000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (Hours)

Pow

er (

kW)

Figure 2.6. This figure represents PV electricity production (dotted) and householdload (solid). The household load is represented by one day year average synthetic loadfor two inhabitants in a detached house. The PV electricity production is year averageoutput over a day on minute-based resolution from a 25m2 PV array located at Uppsalawhich was modeled from irradiance data from a panel of 45 degrees tilt facing southat Uppsala Sweden. Peak power of the PV system was 4.3 kWp if assuming a 17%module efficiency. Since the data is averaged over a year both load and production arenearly smooth. The setup in this Figure is net-zero energy.

The importance of increasing the hosting capacity by increasing the level ofself-consumption of PV power was mentioned previously. There are emergingtechnologies that integrate PV systems with energy storage systems - mainlyas a means to increase the level of self-consumption [28, 50]. In particularself-consumption is interesting in Germany due to its high levels of distributedPV, where even favorable tariffs for self-consumption have been implemented[8]. Studies on battery storage as a means to increase self-consumption alsopoint to the need for improved technology and demand-side management [8,11, 28]. There are new technological solutions which help to improve theself-consumption, such as the "Sunny home manager" from SMA [47]. Self-consumption of PV has been investigated in connexion with PV and net zeroenergy buildings in [62]. A study shows the impacts of various options forobtaining a lower mismatch between production and consumption [60]. Thatstudy focused on the following three possible options: PV array orientation,demand side management and electrical storage. The PV array orientation ismerely the azimuth and tilt angles that a photovoltaic array are aligned [60,p.10]. Electrical storages such as for example batteries were concluded in [60]to be the most effective option for increasing self-consumption, at least for alarger penetration level. An emergent field of potential PV power storage isthe PEV battery storage [28].

Research indicates the need for further research regarding demand-sidemanagement and energy storage possibilities in relation to PV electricity pro-duction. In general demand-side management - or load management - can be

12

done in several ways which is described in section 2.5. Despite economic in-citements for load shifting the problem of shifting load is not obvious sinceperhaps only a few appliances are shiftable [60, p.13]. Regardless such eco-nomic incitements might perhaps more effectively be utilized with an hourlyprice on the electricity consumption [62]. This thesis deals with how self-consumption is altered from the introduction of PEVs and end-user flexibility.A background to these options for altering the self-consumption is provided inthe following sections.

2.4 Electric vehiclesThe first electric vehicle (EV) was built by Gustave Trouvé from France in1881 [14, p.12]. It was a tricycle powered by a 0.1 hp EV motor fed by lead-acid batteries. As gasoline powered vehicles started to become more powerful,flexible, and easier to handle the EV started to disappear from the market[14]. The efficiency related to the power regulation of the electric vehiclewas improved in the 1940’s with the invention of the transistor. The mostprominent electric vehicle that used this updated technology was perhaps theextraterrestrial Lunar Roving Vehicle, which the Apollo astronauts used onthe Moon [35]. Despite advances in battery technology and power electronicsin the 1960’s and 1970’s the range EV’s was still a significant obstacle [14].During the 1980’s a number of electric vehicles from major car manufacturerswere released, for example the EV1 from GM. Throughout EV history themain problem has been battery capacity, and above all the amount of energyit can store per kilogram weight. Despite intense research in the 2000’s notmuch progress has been made [14]. Instead electric vehicles have diversifiedinto several sub-types of electric and hybrid electric vehicles which have eitherlonger range because they contain an internal combustion engine or they havebeen adapted for more short range purposes.

2.4.1 Electric and hybrid electric vehiclesToday the electric vehicle concept covers a wide range of different types ofvehicles. The plug-in electric vehicles (PEVs) have an electric motor and thepossibility to be charged from the grid. The Tesla Roadster and Nissan Leafare examples of PEVs. There are also hybrid electric vehicles (HEVs) whichhave both an electric engine and an internal combustion engine. The most fa-mous example in this category is perhaps the Toyota Prius. In addition to thesecategories there is also a combination of PEV- and HEV technology in plug-inhybrid electric vehicles (PHEVs) which have an electric engine with the possi-bility to plug it into the grid and an internal combustion engine. Chevrolet Voltand Fisker Karma are examples of this type. There also exist electric vehicleswith on-board PV modules for charging the battery, examples of this type of

13

Figure 2.7. A Chevrolet Volt being charged in Denver Colorado (USA). Photo: JoakimMunkhammar.

vehicle is the concept car Koenigsegg Quant. These main types of electricvehicles mainly cover electric vehicles that in scale and function aim to mimicordinary vehicles with internal combustion engines. There are other types aswell such as two-wheeled the self-balancing Segway PT and electric bicycles(e-bikes). The electric vehicles which are considered in this thesis are mainlyPEVs and PHEVs .

2.4.2 Electric vehicle car fleet in SwedenEVs in style and function similar to ordinary cars have been produced in smallnumbers in Europe, Japan and the United States since the 1980’s [10]. Al-though the rise of EVs in the world has been slow the increased researchand development recently within the major automobile companies indicatesa rapid expansion [30]. Many automotive manufacturers have released severalEVs models in recent years, including Nissan Leaf and Chevrolet Volt. TheSwedish car fleet had about 4.4 million cars registered for use in 2011 [45].Out of those 366 were EVs. But a total of about 21 thousand were HEVs [45].Modern customers demand the vehicles to have attributes such as reliability,good performance, in principle infinite range, good comfort, high safety andfair price [30, p.21]. The technology that is available today has limitations byexpensive manufacturing costs and the perhaps most prominent limitation isthe battery capacity and thus the EV’s practical range of use [30]. The rangeis of course improved in PHEVs, but at the extra cost of adding an internalcombustion engine to the construction [30]. We will go into detail regardingEV batteries in section 2.4.3 below.

14

2.4.3 Electric vehicle engine and batteryThe electric motor has an efficiency of about 80% compared with the effi-ciency of an internal combustion engine which has the efficiency of around30% [30]. While the internal combustion engine typically runs on gasoline,diesel or biofuel the electric engine usually runs on power from a battery. Inthe PEV the energy in the battery is supplied by power from the grid - whichhas its origin in the current energy mix. Apart from the cost one importantaspect of a battery is its ability to contain stored energy per kilogram - the so-called specific power of the battery [13, 21]. The most common battery typeused in EVs is the Lithium-ion battery. These batteries have relatively highspecific energy, specific power, energy efficiency and cycle lifetime in com-parison with other battery technologies [21]. The lifetime and performanceof a battery is reduced with deep discharges and affected by external temper-atures [32]. The battery on board EVs are currently expensive and have lowspecific energy compared to fuel for ordinary combustion engines [32].

2.4.4 Electric vehicle and the gridWhen a PEV or PHEV is plugged into the grid there are several possibilitiesfor charging. One option is to use the regular one phase outlets which wouldallow charging at a power of 2.3 kW in Sweden (230V, 10A). Another optionis to use faster charging using the three-phase charging which could charge at6.9 kW. The most common way of charging is with conductive technology, butinduction solutions are also being considered [29]. Another possibility is tohave replaceable batteries and battery swapping stations [68]. We have mainlydescribed the way in which PEVs and PHEVs can be charged from the grid -that is power flow in one direction only - but it is in practice possible to reversethe flow and enable the PEV to deliver electric power to the grid. A collectivename for these technologies is known as Vehicle to Grid (V2G). In generalsuch a setup is more complex and requires feedback.

2.5 Demand-side management and end-user flexibilityDemand-side management (DSM) is a method for altering the use of energyat the end-user site in order to fulfil some goal [1, 5]. Demand response islargely used as a synonym for DSM - which covers a number of technical,economical and legal aspects [3]. Household electricity demand is largely un-controllable and varies daily and seasonally typically with a minimum duringsummer nights [64]. Among DSM strategies load management is a specialcase which aims at lowering demand during on-peak periods and increasing itduring off-peak periods. The goal of load management is to reduce the loadon the distribution grid and reduce the need for regulating power. An inves-

15

tigation of DSM strategies might also be an investigation of the result of al-ternative behavior - as opposed to common behavior - and thus alleviating thediscussion regarding the forcing from demand policy. It is possible to identifysix load management strategies [1, 26]:

1. Peak clipping - reduction of load during short usage peaks.

2. Valley filling - load buildup during off-peak periods.

3. Load shifting - a combination of peak clipping and valley filling strate-gies by conserving the load and shifting it from one time to another.

4. Strategic conservation - decreasing the overall load demand by increas-ing the efficiency of energy use.

5. Strategic load growth - increased electric energy use either by replacinginefficient fossil-fuel equipment or improve consumer productivity andquality of life.

6. Flexible load shape - specific contracts and tariffs with the possibility toenable flexibility control of consumers’ equipment.

The third type of load management - the load shifting strategy - is the pri-mary one used in this thesis, in appended paper (III). In practice load manage-ment strategies can be divided into direct and indirect. Direct load manage-ment is defined by the direct switching on and off different equipment whereasindirect load management is through regulations and economic incitements[1]. A time-differentiated tariff is an example of an economic incitement. Es-pecially a time-differentiated tariff with different buying prices for electricityduring periods of low load and high load [64, p.38].

2.5.1 Measures of self-consumptionThere are two prominent measures for self-consumption that is used in thisthesis: Solar fraction (SF) of the load and load fraction (LF) of the PV powerproduction.

SF is defined as the fraction of load that is matched with PV power. Math-ematically from the illustration in Figure 2.8 we get:

SF =B

A+B+C. (2.1)

LF is defined as the fraction of PV power which is matched by load. Math-ematically from the illustration in Figure 2.8 we get:

16

Power

Time

A BC

D

Figure 2.8. A schematic illustration of a load curve (solid) and production curve(dashed) with appropriate labels for energy that is used to estimate solar fraction andload fraction.

LF =B

B+D. (2.2)

These measures will be essential in the evaluation of the self-consumptionof the flexibility extension developed in this thesis. It should be noted that fornet-zero energy buildings the solar fraction equals the load fraction since theparts A+B+C and B+D are equal. A more detailed mathematical treatmentof SF and LF is given in Section 3.1.3.

2.6 Modeling household electricity useIn this section we review the background for modeling household electricityuse.

2.6.1 Modeling behavior and electricity useHousehold electricity use is about 20% of Sweden’s total energy use [19].Thus quantifying human household activities can contribute significantly tothe understanding and predictions regarding Swedish electricity use over time.The complexity of predicting human activities over time suggests an interdis-ciplinary approach where both quantitative and qualitative research is neces-sary in order to properly quantify domestic electricity use [56, p.57]. Thereare many possible approaches to model activity patterns and consequent elec-tricity use based on time-use data. One way is to use a deterministic approach

17

Time-useData

DeterministicModel

StochasticModel

SyntheticHouseholdElectricity

Use

Figure 2.9. Deterministic and stochastic models of household electricity use based ontime-use data.

where a model is created directly on the basis of the data, and a determin-istic model is defined as always giving the same output given the same in-put. In contrast to a deterministic model a stochastic model is calibrated withinput data and produces synthetic output and does not give the same outputfor the same input [12]. A deterministic model can for example be used togive a direct connection between activity data and electricity consumption. Astochastic model is used to produce synthetic activity patterns which can thenfor example be used in power system calculations. Both deterministic andstochastic models of energy use may be based on time use data, see Figure2.9.

Examples of deterministic approaches to estimating household energy useare [41, 66]. Some stochastic approaches to modeling household electricityuse are [36, 38, 39, 49, 57, 58]. The output from the models is useful for forexample gaining understanding of electricity variations over time which mayin turn be useful in various power system calculations [56]. It can also be usedto for example estimate household energy use flexibility patterns [64], whichwas further investigated in appended paper (III).

2.6.2 Time-use dataActivity patterns can be used to estimate electricity use in households. Thereare many approaches to quantifying household activity patterns [17]. Oneof these approaches is the method involving time-use data, in particular viaso-called time diaries (TUD). The concept involves collecting data about theparticular states of activity for inhabitants in households [17]. These time di-aries are distributed to inhabitants with the intent that they write down theiractivities over time [16, 17]. One assumption in the time diary approach isthat each inhabitant only occupies one state at the time, and may at any timeswitch from one state of activity to another [15, 57, 58]. In order to properlyobtain information regarding inhabitant activities the time diaries should bedesigned to obtain the following information [16, p.2]:

18

• At which time the activity starts (which also gives the time when theprevious activity ended)

• What activity is performed.

• Where the activity is performed.

• If the activity is performed together with someone else and then withwho.

Major time-use surveys have been conducted by Statistics Sweden overtime in Sweden. The data has been gathered over intervals and each inter-val is presented in a report, currently there are data for 1990/1991, 2000/2001and 2010/2011 [43, 44]. The general premise for modeling energy use fromtime-use data is the concept that activities consume electricity. Time-use datacould be used as a direct means to estimate - in a deterministic fashion - theelectricity consumption associated with the set of activities that was observed[66]. It could also for example be used to calibrate a stochastic model whichin turn produces synthetic electricity consumption patterns [38, 39, 57, 58].This will be discussed in more detail in section 3.1.

A related concept is time geography which can be a useful tool in estimat-ing electricity use in households based on household activities. The conceptof time geography was developed in the late 1960’s by Hägerstrand [24] andit aims to provide a basis for analyzing activities in which the the basic di-mensions of space and time are central [18]. The essence can be traced togeography where the physical world and the human utilization of it is in focus[18, p.40]. Time geographic representations can be useful in energy research,in quantifying which activities were performed at what times [56]. They canalso be used in visualizing the connection between activities and energy use[17, 56]. Because of these useful properties time geography can also be usedin interdisciplinary energy research by bridging the gap between quantitativeand qualitative studies.

19

3. Methodology and data

In this section we present the methodology and the data that was used in theappended papers.

3.1 Model for household electricity consumption andproduction

The aim of this thesis is to investigate ways to improve self-consumption of PVin households which includes PEV load. In order to investigate this a generalmodel for a household including PEV load and PV electricity production hasbeen developed. The research presented in this thesis is based on modeling twosystem levels of electricity consumption/production: (I) A single householdwith one or a few inhabitants, one PEV and a PV module, (II) An aggregateof single households each equivalent to system level (I)-type of households.The modeling carried out within the papers of this thesis is based on the singlehousehold system level (I) and then the scaling to a larger aggregate of Msystems is performed via simulating M different systems and summing up theload profiles. Since the modeling is stochastic each household most likelyhas unique behavior. The single household level is modeled according to thefollowing setup. The setup of the household contains three main contributionsto the total balance of power for this system. First off we have the photovoltaicproduction of electric power PPV (t) where t is the index for time step t. Weshall assume that there is a fixed time step length of typically 1 minute. Secondwe have the household electricity consumption from activities over time t,denoted PHousehold(t). Third we have the electricity consumption from thePEV over time when connected PPEV (t). Then the discrete time evolution ofthe electricity production of the entire system S of the household is given bythe following equation:

P(t) = PPV (t)−PHousehold(t)−PPEV (t). (3.1)

In this equation negative electricity production means in practice electricityconsumption. For simplicity we assume that t is a natural number (t ∈ N).This makes P a T ×1 matrix where T is the total number of time steps giventhat t = 1, ...,T . The equation (3.1) thus has three terms - each correspondingto a component of the electricity consumption/production - which have to bequantified in order to obtain the complete electricity consumption/production

20

pattern of the particular household. The setup of the model is initially neitherdeterministic nor stochastic, but the stochastic model will be described furtherin detail in the next section.

(t)

Time use data of

activities

Stochastic model

Electricity consumption

PV model Electricity production

Stochastic model

Electricity consumption

Model for PEV use

P

P

P

P

PV

Household

PEV

Time use data of

activities

Total electricity production

Solar irradiation data (t)

(t)

(t)

Figure 3.1. A schematic illustration of the model for a single household over theelectricity consumption and production.

3.1.1 Household electricity consumption from activitiesThe household electricity consumption component of (3.1) is based on thepremise that inhabitant activities generate load. Then in order to estimate thehousehold electricity consumption from activities we have to estimate the elec-tricity consumption associated with each activity. Based on the time-use datain section 3.1 we may calculate the electricity consumption associated witheach activity. Assume that the set of activities C(t) ∈ Activities for each timestep t corresponds to electricity consumption over time PHousehold(t) ∈ R viathe function:

C(t)→ PHousehold(t) (3.2)

We shall consider that there is no electricity production from household ac-tivities; PHousehold(t)≥ 0. In order to properly quantify the function (3.2) letsadopt some simplifying assumptions:

(I) Each individual may only occupy one activity at each time step.

(II) Each individual may only occupy a state among the predefined activitiesat any given time.

21

(III) To each activity there is a predefined electricity consumption associatedwith it.

Lets assume that there are N activities defined in the set of activities. Thislets us setup the electricity consumption of these activities according to thefollowing equation:

PHousehold(t) =N

∑i=1

AitLi +PSpecial(t)+PBase(t). (3.3)

In this setup we have divided the household electricity consumption PHousehold(t)from activities into three terms. The first term represents those activities thatconsume a particular fixed amount of power when active (eg TV) and the sec-ond represents those appliances which produce more complex electricity con-sumption patterns over time once activated (eg washer), see Section 3.1.2. Letus review these terms. The first term consists of two factors, one covariantvector Li and a matrix Ait . The vector Li - which is of size 1×N - containsthe instantaneous electricity consumption from the appliances that are runningfor each activity. Each activity thus has a corresponding entry in Li, and thenotation Li is the transposed version of Li (in order to make the multiplicationcorrect). The second factor is the matrix Ait which is a boolean matrix with en-tries defined by time t and activity i. The size of the matrix A is N×T whereN is the size of vector L and T is the total number of time steps. A typicalsetup on the activity matrix A is as follows:

A =

1 0 · · · 11 1 · · · 0...

.... . .

...0 0 · · · 1

(3.4)

In this particular example A21 = 0 and A12 = 1 which means that activity 1is off at time step 2 and that activity 2 at time step 1 is on. The second termof PHousehold(t) is the special activity power vector PSpecial(t) which in simi-larity with P(t) has size 1×N. The vector PSpecial(t) contains the electricityconsumption from the complex activity patterns such as washing and drying.The total electricity production containing all terms becomes:

P(t) = PPV (t)−N

∑i=1

AitLi−PSpecial(t)−PBase(t)−PPEV (t) (3.5)

The third term PBase(t) in PHousehold(t) is just a term for basic backgroundappliances such as for example standby and cold appliances.

22

3.1.2 The Widén Markov-chain modelThe setup of the power consumption model in the previous section was generalbut without input data regarding actual activities. This data can be providedby empirics (typically time-use data) or by a stochastic model. The modelfor household activities and electricity consumption which is the basis for thisthesis is the Widén model - which is a discrete time Markov-chain model forproducing synthetic activity data and consequent electricity use. This modelis then in appended paper (I) extended with PEV load and in appended paper(III) extended with a flexibility extension. We shall begin by reviewing theWidén model.

State

Time123456789

t t+1 t+2 t+3 t+4 t+5

Figure 3.2. A schematic illustration of a Markov-chain process of entering one stateat a time. See Table 3.1 for legend on the states of activity.

Generally a discrete-time Markov chain S(t) is a discrete stochastic processbased on the premise that each time step t is occupied by one state Eµ in anumber of states defined by E1, ...,EN . Each state is defined stochasticallyon the basis of the previous state only which is the so-called Markov property.The number of time steps is defined via t = 1, ...,T and µ = 1, ...,N is the indexdetermining which state it is in. The probability that the process occupies aparticular state µ at time step t is [12]:

pµ(t) = Prob(Xt = Eµ). (3.6)

Since the system is closed with respect to the fixed number of states theprobability for an individual to occupy any state in the predefined set of statesat a given time t is unity:

N

∑µ=1

pµ(t) = 1. (3.7)

23

As the process evolves from one time step t to the next t + 1 the state of theprocess at time step t+1 is determined from the state at time t via the transitionprobabilities:

Pµν(t)≡ Prob(Xt+1 = Eν |Xt = Eµ). (3.8)

Here µ = 1, ...,N and ν = 1, ...,N since (3.8) is a square matrix of sizeN×N. We can see that (3.8) satisfies the Markov property: the state at timet +1 is only dependent on the state at time t. The transition matrix Pµν(t) canthen be written as [56, p.50]:

Pµν(t) =

P11(t) P12(t) · · · P1N(t)P21(t) P22(t) · · · P2N(t)

......

. . ....

PN1(t) PN2(t) · · · PNN(t)

. (3.9)

The probability that a state is occupied can be computed according to thefollowing equation [56, p.50]:

pµ(t) = pµ(1)t−1

∏τ=1

Pµν(τ), (3.10)

where pµ(t) and its covariant version pµ(t) represents the probability ofoccupancy in time step t.

Estimation of transition probabilitiesA Markov chain model has to be calibrated with data. This "calibration" isin practice determining the transition matrix Pµν(t) for each time step withsome form of data [56, p.50]. Given a set of empirical data it is a straightforward task to estimate the transition probabilities from there according tothe following. Suppose that we have a series of data sµ(t) representing theactivity for µ = 1, ...,N number of individuals and t = 1, ...,T time steps. Letsassume that from time step t to time step t +1 the total number of transitionsnµν(t) from µ to ν are summed up over all individuals. This gives the totalnumber of transitions kµ(t) from state µ at time t:

kµ(t) =N

∑ν=1

nµν(t), (3.11)

from which we get the transition probability estimates:

Pµν(t) =nµν(t)kµ(t)

. (3.12)

The transition probability matrix Pµν is hourly averaged [56, p.50].

24

Activities in the Widén modelIn this section we describe the activities in the Widén model. The model isbuilt up on transition matrices which are calibrated with time-use data, thisadds a component in the function from time-use data to electricity consump-tion (3.2). We assume that the set of activities C(t) ∈ Activities for each timestep t is defined by the discrete time Markov-chain process S(t) which in turnis modeled to household electricity consumption over time PHousehold(t) ∈ Rvia the function (For a detailed description of PHousehold(t) see section 3.1.1):

S(t)→C(t)→ PHousehold(t). (3.13)

For more detailed information regarding the Markov-chain process S(t) see[12, p.106]. The transition probabilities Pµν(t) used in the Widén model arederived from the equations in section 3.1.2 based on time-use data. The statesof the model are shown in Table 3.1.

Table 3.1. Activity states of the Markov-chain model.

Code Activity

1 Away2 Sleeping3 Cooking4 Dishwashing5 Washing6 TV7 Computer8 Audio9 Other

The activity states in Table 3.1 each contain complex electricity consump-tion processes such as for example washing programs. Each activity has to beidentified in the activity data.

3.1.3 Self-consumptionThere are different measures of PV self-consumption, and in this thesis weshall use solar fraction (SF) of load and load fraction (LF) of PV power, seesection 2.5.1 for more information on that. In this section we provide a math-ematical tool for calculating both SF and LF from the setup of the householddefined in section 3.1. In order to simplify the notation we assume that theelectric electricity consumption of the PEV PPEV (t) is included in the house-hold electricity consumption from activities PHousehold(t). This gives the fol-lowing total production of electricity for the household:

25

P(t) = PPV (t)−PHousehold(t), (3.14)

where PPV (t) is the photovoltaic electricity production like in previous setup.Solar fraction SF(t) is the fraction of load PHousehold(t) which is matchedby PV electricity production PPV (t). An assumption which is made, but canbe derived from the definition is that if production exceeds the load (PPV,i >PHousehold(t)) then the fraction SF(t) does not exceed 1. Lets define PLack(t) asthe amount of power at a particular time that is not matched by PPV (t). Thenwe have that PLack(t)/PHousehold(t) is a measure of how "unmatched" PPV (t)and PHousehold(t) are. According to the normalization criterion the solar frac-tion SF(t) at time step t is defined as:

SF(t)≡ 1− PLack(t)PHousehold(t)

. (3.15)

We have the following identity for the "unmatched ratio":

PLack(t) =|P(t)|−P(t)

2(3.16)

which can be proven by the following arguments. If P(t)≥ 0 then PLack(t)=0 and conversely if P(t) ≤ 0 then PLack(t) = |P(t)|. This brings the followingexpression for the solar fraction SF(t) (3.15):

SF(t) = 1− |P(t)|−P(t)2PHousehold(t)

(3.17)

If this is summed up over time and divided by the number of time steps onegets the SF over that particular period of time:

SF = 1− ∑Tt (|P(t)|−P(t))

2∑Tt PHousehold(t)

(3.18)

If the solar fraction is to only be calculated between the PV electricity pro-duction and household power consumption then one could simply excludePEV load PPEV (t) from PHousehold(t) in this special setup of meshing all loadsin PHousehold(t). Conversely if to calculate the solar fraction for only thePEV load, then the household load PHousehold(t) would simply be replacedby PPEV (t). It should be emphasized that the solar fraction only measuresthe percentage of the electricity consumption that has been matched by pho-tovoltaic production. The complementary measure to solar fraction is loadfraction which is a measure of the amount of PV electricity production whichis matched by household load. According to (2.2) and (3.18) we get:

LF = 1− ∑Tt (|P(t)|+P(t))

2∑Tt PPV (t)

. (3.19)

26

Similar to the solar fraction it is possible to measure the load fraction of justthe household electricity consumption or just the PEV electricity consumptionby setting up PHousehold(t) to include only the desired load that is to be mea-sured against PV electricity production.

3.2 Electric vehicle extensionA PEV extension to the Widén model was developed in paper (I) and used tostudy self-consumption of PV-electricity in appended paper (II). In this sectionwe shall review this model. The PEV extension was implemented in MATLABfor the simulations in appended papers.

3.2.1 Plug-in electric vehicle electricity consumption modelBased on the Widén model the PEV extension for estimating the electricityconsumption from a home-charged PEV was developed. The PEV extensionenables the Widén model to estimate electricity consumption associated withthe use of a home-charged PEV. This concept aims to as accurately as possiblemimic the actual behavior and consequent energy use associated with the useof a PEV which is plugged in and charged at home. Although the extension isdesigned to fit with the Widén model it is general and built on the followingprinciples:

(I) The PEV is used by one fixed inhabitant in the household during a cer-tain percentage of the states "Away" for that individual.

(II) The choice for the fixed inhabitant to take the PEV is with the probabil-ity in (I) made each time the state changes to state "Away". This choiceis kept for every time step after this which is not changed from state"Away".

(III) The PEV electricity consumption is proportional to its time away un-til the state of charge (SOC(t)) reaches the minimum state of chargeSOCmin.

(IV) If the PEV is away for a longer time than is possible without breachingthe minimum state of charge (SOCmin) then the PEV has either pausedfor some time during the excursion or run on some other fuel during theremaining trip.

(V) The PEV is only charged when at home and not fully charged.

27

Lets go into mathematical detail based on these principles. The mathemat-ical description of this model follows the mathematical structure for describ-ing the Markov-chain model in section 3.1.1. Since the Markov-chain modelgenerates synthetic activity patterns and consequent electricity use this PEV-extension is also based on the same premise and produces synthetic PEV usepatterns and consequent electricity use over time. According to principles (I)and (II) - in the PEV-extension - there is a certain probability PPEV that thePEV is used by the inhabitant when the inhabitants state is changed to the state"Away", see Table 3.1 for a list of states in the Widén model. Mathematically,this means that for time steps t = 1, ...,T if A1t = 0 and A1t+1 = 1 then thereis a probability PPEV that the PEV is taken during the time interval for whichthe model enters state "Away". According to principle (III) the PEV consumesa constant amount of electric power during each time step when away, but ac-cording to principle (IV) this is only true for as long as the minimum state ofcharge SOCmin is not reached. For trips that are longer time than there is bat-tery energy for - according to this setup - there is an assumption that the PEVstopped during the trip and depleted the state of charge upon arrival home.Another interpretation is that the PEV has some other on-board engine (forexample a combustion engine) which provides the energy for the remainder ofthe trip, for a thorough discussion on this see [21, p.44].

Lets define the energy level of the battery - the state of charge - at a giventime t as SOC(t). When fully charged the state of charge is at a maximum levelSOC(t) = SOCmax. When the PEV is used then according to principle (III) thestate of charge is decreased linearly with an electricity consumption of CPEV

times the seasonal coefficient S(t) - until it is depleted to the minimum depthof discharge, which is here modeled as a minimum state of charge SOCmin - orreturned home (as the state has changed from "Away" to some other). Uponarrival at home the PEV is plugged in immediately and according to principle(V) it is charged when plugged in. The PEV is charged with charging powerCCharge, which in practice increases SOC(t) until some time step τ for whichSOC(τ) equals SOCmax. This can be expressed via the following equation:

SOC(t +1) =

SOC(t)−CPEV S(t)∆t if consuming,SOC(t)+CCharge∆t if charging,SOC(t) else.

(3.20)

The level of charge CCharge is dependent on which type of charging is used.The PEV extension also has to make sure that the state of charge is kept withinthe accepted domain:

SOCmin < SOC(t)≤ SOCmax (3.21)

The parameters and nomenclature is listed in Table 3.2. The effect of thePEV-charging on the household energy balance is given in section 3.2.2.

28

Table 3.2. Parameters for the PEV extension.

Parameter Symbol

State of charge at time t [kWh] SOC(t)Maximum state of charge [kWh] SOCmaxMinimum state of charge [kWh] SOCminCharging power [kW] CCharge

Electricity consumption [kW] CPEV

Vehicle load [kW] PPEV (t)Probability to take the PEV [%/100] PPEVSeasonal coefficient [%/100] S(t)

3.2.2 Household electricity consumption from chargingWhen plugged in to the household the PEV is charged according to the prin-ciples in section 3.2.1, also see nomenclature list in Table 3.2. Upon arrivalthe PEV is connected to the grid and charged with CCharge until the the maxi-mum state of charge SOCmax has been reached. This creates a household loadPPEV (t) at time t which can be expressed according to the equation:

PPEV (t) ={

CCharge if charging,0 else. (3.22)

where K ∈U(0,1) is a stochastic variable which is uniformly distributedbetween 0 and 1, randomized each time the state is changed to state "Away".The seasonal variation in fuel consumption for the standard setup was assumedto be a factor S(t) times the average fuel consumption. For the different sea-sons the assumption in appended paper (I) was according to the following.Winter: S(t) = 1.2, spring: S(t) = 1, summer: S(t) = 0.8, fall: S(t) = 1.0.The season coefficient was inspired from [40], and it adjusts the variability ofload from vehicle use from seasonal conditions from for example heating. ThePEV load PPEV (t) adds to the total power balance of the household at time taccording to:

P(t) = PPV (t)−PHousehold(t)−PPEV (t). (3.23)

The general setup of this extension does not allow for using the PEV forelectricity production - instead of just load - during for example householdpeak load. Thus PPEV (t) ≥ 0, however allowing for PPEV (t) < 0 would be aninteresting but in need of an appropriate a smart-grid setup.

3.2.3 Limitations of the PEV extensionThe PEV extension is general, but based on a number of delimiting assump-tions. The Markov-chain model can produce synthetic activity patterns and

29

Table 3.3. Seasonal coefficient adjusting for the seasonal effect on average electricityconsumption of the PEV.

Season Seasonal coefficient, S(t)

Jun-Aug 0.8Sep-Nov, Mar-May 1.0Dec-Feb 1.2

energy use for any number of household inhabitants. The PEV-extension islimited to only one inhabitant using it, at least in the setup of this thesis. An-other delimitation is the charging, the inhabitant is only expected to charge thePEV at home which is perhaps not entirely realistic. If for example the PEVis taken to work it could perhaps reasonably be charged there. Also the useof the PEV battery as a power source could prove to be interesting, but thecurrent setup of the extension does not allow for that. Future improvements ofthe model on the aforementioned shortcomings could prove to be interesting.

3.3 Load flexibility extensionIn this section we review the end-user activity load flexibility extension to theWidén model which was investigated in appended paper (III). The flexibil-ity extension was implemented in MATLAB for the simulations in appendedpapers.

3.3.1 Formal setup of the flexibility extensionOne of the goals of this thesis was to investigate the potential for increasedself-consumption of PV power. In this section we describe an extension tothe Widén model which enables the study of the effect on self-consumptionfrom alternative behavior in terms of altered probabilities for performing cer-tain activities over certain periods of time. This flexibility extension was ini-tially developed in [64]. This extension was later used in appended paper (IV)to study self-consumption of local PV electricity production. Formally it isbased on the Markov-chain model (the Widén model) for household activi-ties and energy use. The Widén model is by design independent of responsefrom external influence from for example inhabitants. Thus such a model isnot susceptible to for example economic incitements such as fixed time-of-usetariff or a varying electricity price following market prices. This was the mainreason for the development of the flexibility extension [64]. The extension isbased on the following principles (See appended paper III):

30

(I) The probability at each time-step for the transition to any state in thenext time-step has to be unity.

(II) The total amount of probability (the sum over probabilities) for a certainactivity over a 24 hour period has to be the same for the standard case asfor the flexibility case.

Criterion (I) is a necessity from the theory of Markov-chains [12]. Thetotal probability of transition to any state has to equal unity. Criterion (II) isused as a consistency measure of load-shifting - that load is shifted and notjust increased at some point in time and not compensated by shifting down atanother point. Without this criterion the model would simply be a valley fillingor peak clipping model [64].

The idea of the model is to up-shift the probability for certain activities dur-ing a certain time interval and down-shift the probabilities for these activitiesduring a different time interval (of equal duration). Let these activities belongto the set U . Correspondingly, and according to criteria (II) of the flexibil-ity model there has to be "complementary" activities which are down-shiftedduring the time the other activities are up-shifted and vice versa. Let theseactivities belong to the set D. It is also possible that there is a set of activitieswhich are left unchanged during all time, let those activities belong to the setN. Thus we have the total set of activities as A =U ∪D∪N. For simplicity theday is divided into two time-intervals ∆T1 which runs from 07 : 00 to 21 : 00and ∆T2 which runs from 21 : 00 to 07 : 00 and these are the time-intervals usedfor up-shifting and down-shifting respectively. This setup of periods aims atmaximizing utilization of PV electricity production. The mathematical for-mulation of the model follows the following prescription. The Markov chaintransition matrix Pµν(t) of the Widén model determines the probability fortransition from one state of activity µ to another ν at time step t, see section3.1.2. Let us define the up-shifted states U for t ∈ ∆T1 as:

p̃µ(t) = ∑ν∈U

Pµν(t) (3.24)

Here we introduce the flexibility coefficient α which is a measure of theamount of probability which is shifted. It changes the probabilities in theMarkov-chain model either up or down with α . This coefficient is definedas the fraction of inhabitants that changes activities according to the setup onperiods with up-,down- and neutral activities [64]. We may define the down-shifted probabilities during time t ∈ ∆T1 for µ ∈ D,ν ∈ A as:

P̂µν(t) = (1−α)Pµν(t). (3.25)

In order for the flexibility criterion (II) to be fulfilled we have to setup thefollowing sum of the down-shifted probabilities from (3.24):

31

p̃(t) = ∑ν∈D

p̃ν (3.26)

From the onset of the principles there is an inherent freedom of choiceregarding the distribution of the sum of down-shifted probabilities onto theup-shifted ones. This may generally be formulated as:

P̂µν = (1+α p̃(t)λν)Pµν(t) (3.27)

for t ∈ ∆T1, µ ∈ U and ν ∈ A where the coefficients λi redistributes thedown-shifted probabilities at the same time step [64]. The coefficients λi dis-tributed also need to be normalized:

∑j∈U

λ j = 1. (3.28)

This approach is by default a peak clipping or a valley filling strategy sincethere is no regulation on the compensation for the up-shifting/down-shifting.In order to satisfy flexibility criterion (II) a compensating mechanism has tobe established. In general this is a formidable task because of a few challengesinvolved in this. In general the main problem in up-shifting and then findinga perfect compensation with down-shifting of other probabilities at anothertime is the available amount of flexible probabilities. This available amountvaries over time. One possibility to still solve this problem - even if onlyapproximately - is to setup the following strategy, which was formulated in[64]. Redefine λν , and (3.27) to:

P̂µν(t) =

(1+α p̃(t)

p̄ν

p̄

)Pµν(t) (3.29)

for t ∈ ∆T1, µ ∈ A and ν ∈U where p̄µ is the average probability over ∆T1of residents involved in the up-shifted activities during that period:

p̄µ =1T

T

∑t=1

pµ(t) (3.30)

and in order to satisfy criteria (I) for the model (the unity criteria fromprobability theory) we then have:

p̄ = ∑µ∈U

p̄µ . (3.31)

This shifting is then repeated for ∆T2 except reversed, that is the up-shiftedduring day-time is down-shifted during night-time and vice versa.

32

3.3.2 Limitations of the flexibility extensionThe modeling of flexible end-user behavior, at least in the current setup byaltering probabilities in a Markov-chain model over activities, is a challenge.There is an upper limitation on the amount of flexibility that is possible toperform since the total amount (in terms of probability) flexible behavior isnot constant throughout the day. This causes problems for the model since itattempts to compensate for the flexibility during certain times by adding "com-plementary" flexibility at other times. The model also has to assure that theprobability is unitary for being in any of the predefined states. Together theseissues form a mathematical modeling problem that was not solved completelyby this model. Therefore the higher the flexibility level the less the model islikely to accurately move load from one time to another and instead act as aform of valley filling or peak clipping model.

3.4 Modeling PV electricity productionPhotovoltaic data was used in the simulations in appended papers (II) and(III). Two different kinds of modeled photovoltaic data was used in the ap-pended papers. For paper (III) the following setup was used. In order toestimate the data for PV electricity production Swedish irradiation and tem-perature data 1992-1999 from the Swedish Meteorological and HydrologicalInstitute (SMHI) was used [48]. This data was inserted in a model for PVsystems developed by Widén [65]. The output was estimated from incidentsolar radiation and temperature data, panel size, location, tilt and azimuth onan hourly basis during 1992.

For paper (II) high-resolution insolation data from the PV-group at theÅngström laboratory was used. The data which was used was minute-basedand for the entire of year 2011. The output was then calculated from the for-mula:

PPV (t) = η×A×G(t) (3.32)

where PPV (t) is the power output over time from the PV panel, A is the PVarea (m2), η is the efficiency of the PV system and G(t) is the incident solarradiation (W/m2). The system efficiency was set to η = 13%. The incidentsolar radiation data G(t) was 2-second resolution and measured in a planetilted 45◦ with a pyranometer at the Ångstrom laboratory at Uppsala Sweden(59◦50′19” N 17◦38′50” E) during January 1 to December 31 2011. The datawas averaged to 1 minute resolution in order to match the resolution of thehousehold- and PEV load from the stochastic model. The data was adjustedfor day light savings time.

33

4. Results

In this section we review the main results from the appended papers. Thesections regard the results from the different model extensions to the Markov-chain model in connection with local PV electricity production. Results fromthe Markov-chain model (without extensions) will not be reviewed in detailhere since that research was carried out previously by Widén and collaborators[57, 58].

4.1 PEV home-charging modelThe PEV home-charging extension to the Markov-chain model is describedin detail in section 3.2.1. The model was developed in appended paper (I)and investigated in connection with PV electricity production in paper (II). Apreliminary study of the model was carried out in not appended paper (VII).The PEV home-charging extension contained a set of input parameters, seesection 3.2.1, but a standard scenario was devised according to Table 4.1.

Table 4.1. Standard setup on the PEV-model.

CCharge CPEV PPEV SOCmin SOCmax

2.3 kW 8.4 kW 0.2 21 kWh 35 kWh

In Table 4.1 CCharge is the charging power when the PEV is charged. PPEVis the probability to take a car when leaving the house/apartment, SOCmin isthe minimum state of charge and SOCmax is the maximum state of charge (hereequivalent to the total battery storage capacity). CPEV is the average powerconsumption when the PEV is used. These free parameters are then addedto the free parameters of the Widén model. Thus for example the choice ofapartment or detached house are parameters in the Markov-chain model whichapplies to the PEV extension as well. For more detailed information regardingthe calculations see appended paper (I). As an example output the one yeardaily average electricity use for a single detached house containing two in-habitants with one inhabitant only using the electric car is shown in Figure4.1.

The corresponding electricity use for each appliance of the household (wherePEV charging is considered an appliance) is shown in Table 4.2.

34

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (Hours)

Pow

er (

kW)

A: Cold AppliancesB: CookingC: WashingD: DishwashingE: TelevisionF: ComputerG: AudioH: LightingI: Additional AppliancesJ: Electric Vehicle

Figure 4.1. A stacked bar plot over the activities and their energy use averaged over ayear. The corresponding electricity use from each activity is given in Table 4.2.

Table 4.2. Output (MWh/year) from the Markov-chain model with the PEV-extension.A detailed plot and nomenclature list is given in Figure 4.1.

A B C D E F G H I J

0.72 0.33 0.12 0.18 0.29 0.38 0.06 1.14 0.93 1.55

One of the results is that the electricity consumption from the home-chargedPEV is a considerable amount of the total household load. This was shown tobe true for both the single household over time and an aggregate of households.

Table 4.3. Standard deviation (W) of the load profiles from the Markov-chain modelwith the PEV-extension. A nomenclature list is given in Figure 4.1.

A B C D E F G H I J

7 37 13 24 18 4 0.3 54 0 125

Standard deviation of the different appliances is shown in Table 4.3. ThePEV charging is strongly intermittent and high peak, which results in highstandard deviation.

35

4.1.1 PEV extension and photovoltaic powerOne of the main research questions that motivated appended paper (II) re-garded the level of coincidence between local PV electricity production andhome-charged PEV load. The main result was that the introduction of a PEVincreased the self-consumption of PV electricity production, but that in gen-eral the coincidence between load and production was low. This was true bothon a single household- as well all aggregate level, see Figure 4.2 for a compar-ison of the load with and without PEV load as well as two different PV-sizes.The self-consumption on the single household level - measured as load frac-tion - was increased from about 31 % to about 34 % for the 25m2 PV arraysize when the PEV was introduced. For the larger 34 m2 size the increase inload fraction when introducing the PEV load was from 25% to 28%, see Table4.4 on load fractions and solar fractions for the single household scenarios.For the aggregate scenarios the PEV introduction gave a load fraction increasefrom 34% to 40% for the 25 m2 PV array size and a 27% to 32% increase forthe 34m2 PV array size, see Table 4.5 for load fractions and solar fractions forthe aggregate scenarios. It might be concluded that the self-consumption wasboth initially higher and increased more on the aggregate level when introduc-ing the PEV.

Table 4.4. Solar fraction and load fraction for the individual household scenariosfor various setup. PV1 = PV electricity production from 25m2, PV2= PV electricityproduction from 34m2, H = Household load, PEV = PEV load.

Load/ Solar Loadproduction fraction fraction

unit of load (%) of solar (%)PV1-H 31.6 31.3

PV1-H-PEV 25.5 34.4PV2-H 34.4 25.0

PV2-H-PEV 28.0 28.0

See also the duration graph of both individual household level and aggregatelevel in Figure 4.3. It shows how the peak electricity consumption is lower onthe aggregate level, an artefact of the averaging of the stochastic behavior -and a main contributor to the reason for the higher load fractions and solarfractions in the aggregate scenarios.

The PV electricity production was constructed from incident solar radiationdata from Uppsala with the model described in section 3.4.

The introduction of a home-charged PEV increases the household load from4.17MWh/year to 5.7MWh/year which is an increase of 37 percent. The in-troduction of a PEV to a household which has net-zero energy makes the solarfraction drop by 20 percent while the load fraction increases 10 percent. If ahousehold without PEV has net-zero energy setup and a PEV and additional

36

Table 4.5. Solar fraction and load fraction for the aggregate scenarios with 1000households for different type of setup. PV1 = PV electricity production from 25m2,PV2= PV electricity production from 34m2, H = Household load, PEV = PEV load.

Load/ Solar Loadproduction fraction fraction

unit of load (%) of solar (%)PV1-H 34.2 33.7

PV1-H-PEV 29.1 40.2PV2-H 36.8 26.7

PV2-H-PEV 31.7 32.2

00:00 12:00 24:000

0.5

1

1.5

2

Individual household

Time (Hours)

Pow

er (

kW)

00:00 12:00 24:000

0.5

1

1.5

2

Aggregate of households

Time (Hours)

Pow

er (

kW)

EV+HouseholdHousehold

34m2 PV

25m2 PV

Figure 4.2. Year-average daily electricity use and production. On the left hand sidethe single household with and without PEV load for the two options of PV-setup of25m2 and 34m2 is shown. On the right hand side the year average of an aggregate of1000 households is presented for both the scenario with and without a PEV.

PV area is added to the house so that the house once again is net-zero energythen the solar fraction increases with 13 percent and the load fraction decreasesby 10 percent. While having about equal LF and SF in each of both net-energyscenarios their magnitude is lower for the net-zero energy household with PEVdue to the higher mismatch between load and production.

4.2 Flexibility extension and photovoltaic electricityproduction

This section will show some results from the Markov-chain model with theflexibility extension. The model is discussed in detail in appended paper (III)and section 3.3. One of the main research questions that motivated researchinto developing a flexibility extension was whether or not flexible behaviorcould increase self-consumption of PV electricity production. The extensionwas based on changing the probabilities for performing certain activities at

37

0 2000 4000 6000 8000−5

0

5

Hours

Pow

er (

kW)

Aggregate of households

Household+EV−PV2Household+EV−PV1Household−PV2Household−PV1

0 2000 4000 6000 8000−5

0

5

Hours

Pow

er (

kW)

Aggregate of households

Household+EV−PV2Household+EV−PV1Household−PV2Household−PV1

0 2000 4000 6000 8000−5

0

5

Hours

Pow

er (

kW)

Individual household

PV2PV1HouseholdHousehold+EV

0 2000 4000 6000 8000−5

0

5

Hours

Pow

er (

kW)

Aggregate of households

PV2PV1HouseholdHousehold+EV

Figure 4.3. This figure represents a duration plot over household load with and withoutPEV load and the two scenarios with PV production: PV1=25 m2 and PV2=34 m2. Onthe left hand side the duration graphs of single households over a year are presented,on the right hand side duration graphs for average households in an aggregate of 1000households is presented (also over a year). The result above zero is the magnitudeof electricity consumption, the result below zero is a measure of the magnitude ofelectricity production.

certain times in the Markov-chain model. The general premise was that theextension would up-shift the probability for a certain set of activities duringa certain time interval and down-shift them during another time interval. Inorder to keep the probability for occupying any state at a given time (see Sec-tion 3.3) there had to be complementary activities which were down-shiftedin probability when the other activities were up-shifted and vice versa. Theinitial flexibility setup of the extension in the study was according to Table4.6.

Table 4.6. Flexibility setup, nomenclature for activities is found in Table 3.1.

Activities Shifting

3-9 Up-shifted during day-time

2,10 Down-shifted during day-time

1 Neutral

38

The setup shown here (which was also used in the study) was a detachedhousehold with four inhabitants. In the study the shifting was done for a se-ries of different scenarios with different levels of shifting. Also an alternativeversion of the shifting-scheme in Table 4.6 was utilized in the study but notinvestigated here. The result from 10% flexibility compared to the standardsetup in terms of probability of using certain appliances and electricity use isgiven in Figures 4.4 and 4.5.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time (Hours)

Pro

babi

lity

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time (Hours)

Pro

babi

lity

AwaySleepingCookingDishwashingWashingDryingtv/vcr/dvdcomputeraudioother

Figure 4.4. The difference in probabilities for certain activities with no flexibility onthe left hand side and 10% flexibility on the right hand side according to the setup inTable 4.6.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

Time (Hours)

Pow

er (

kW)

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

Time (Hours)

Pow

er (

kW)

Cold AppliancesCookingWashingDishwashingTelevisionComputerAudioLightingAdditional Appliances

Figure 4.5. The difference in electricity consumption for certain activities with noflexibility on the left hand side and 10% flexibility on the right hand side according tothe setup in Table 4.6

The flexibility extension was investigated in connection with PV electricityproduction. The investigation primarily focused on the change in correlationbetween household electricity use and PV electricity production given flexiblebehavior according to the flexibility extension. A plot over the load curves

39

from the standard scenario household electricity consumption and the loadcurve produced with 10% flexibility from the flexibility extension is showntogether with the average PV electricity production curve in Figure 4.6.

0 5 10 15 200

0.5

1

1.5

2

Time (Hours)

Pow

er (

kW)

FlexibleStandardPV

Figure 4.6. This plot shows the two load curves from the standard scenario and for the10% flexibility scenario and a curve over the PV-production. See Table 4.6 for setupon flexibility.

In this simulation a 45m2 PV array with a 45 degree tilt angle facing southwas setup in the Widén model for PV electricity production described in sec-tion 3.4.The setting was for Uppsala, Sweden. The solar fractions for thestandard setup with different levels of flexibility are found in Table 4.7. Theincrease in solar fraction is most notable in the lower percentages of flexibilityfraction.

Table 4.7. Solar fraction (SF) for the different levels of flexibility for the standardsetup.

Flexibility level SF Aggregate SF0 % 33 % 48 %5 % 36 % 53 %

15 % 37 % 55 %50 % 38 % 57 %

The overall higher solar fraction in the aggregate scenarios is related to theaveraging of each households intermittent energy use along with estimatingthe solar fraction from this with an average of PV electricity production. Thiswas done in an approximative sense, a more accurate way would have beento take each day PV electricity production instead of the average, this wouldpresumably have lowered the solar fraction. If each house had a unique PVprofile related to the geographical distribution of households then the accuracyof the aggregate solar fraction calculations could have improved further.

40

In conclusion the result - in terms of coincidence between household elec-tricity use and photovoltaic electricity production - shows that although thedifference between the standard scenario and the flexible scenario is small itis possible to change the solar fraction with flexible behavior.

4.3 Comparison of impacts on PV self-consumptionThe previous sections have regarded the results from the PEV extension andthe flexibility extension. In particular the focus has been on quantifying theimprovement on the level of self-consumption of PV power. In this section wecompare the PEV extension with the flexibility extension in terms of self-consumption of PV Power. The investigations of the PEV extension andthe flexibility extension in relation to self-consumption was made in differ-ent projects with different setup of the model and input PV data for the PVmodeling. This sets a limit on the level comparison that is feasible betweenthe two sets of results, and a more detailed study of this relation would be in-teresting future work. Despite this shortcoming we will here do a more overallcomparison of the results. The PEV model was setup with two inhabitants andone PEV along with 1-minute resolution data from the Ångström laboratory.The flexibility model was setup with four inhabitants and the PV-data was pro-vided by the model developed by Widén which was based on hourly averagedSMHI data. Self-consumption can be measured with either solar fraction orload fraction. In the Tables 4.9 and 4.8 the solar fractions and load fractionsare given for the PEV extension and the flexibility extension. Both extension-simulations regard households in detached houses.

Table 4.8. Solar fraction (SF) and load fraction (LF) for the different levels of flexi-bility for the standard setup in the flexibility extension for a single household.

Flexibility level (%) SF (%) LF (%)0 33 325 36 34

15 37 3450 38 35

For the flexibility extension it is clear that both solar fraction and load frac-tion increases as the flexibility percentage increases. In comparison with thePEV extension it is clear that although the load fraction is increased when thePEV is introduced it also yields a decreased solar fraction. This is an artefactof the fact that PEV charging occurs mostly during times when PV electricityproduction is low and that the total load is increased mostly during evenings,night time and mornings. The total fraction of load that is matched by PV elec-tricity production is lower, hence solar fraction is lower. In this sense the flex-

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Table 4.9. Solar fraction and load fraction for the individual household scenarios fordifferent setup on the PEV extension. PV1 = PV electricity production from 25m2,PV2= PV electricity production from 34m2, H = Household load, PEV = PEV load.

Setup SF (%) LF (%)PV1-H 32 31

PV1-H-PEV 26 34PV2-H 34 25

PV2-H-PEV 28 28

ibility model is better for matching PV electricity production with load. How-ever the PEV load could in principle increase the load fraction of PV powereven to 100 percent - given small enough PV setup - but the total electric-ity consumption would increase and the solar fraction between the householdelectricity consumption and PV electricity production would decrease. Givena household without PEV load and net-zero energy setup on the PV system. Ifintroducing a PEV load into this system then increasing the PV area to onceagain reach net-zero energy setup then the self-consumption is lower for thesystem containing PEV compared to the one without. A fruitful comparisonbetween the PEV extension and the flexibility extension in terms of aggregateself-consumption was not really possible due to the approximation of the PVmodel in the aggregate scenario. In general such a comparison would be bestgiven the same PV model, something that is left for future work. A conclusionfrom comparing the extensions is that introducing PEVs with default setup oncharging patterns does not result in increased self-consumption if the setup ofthe household already is net-zero energy. Thus load shifting is more promis-ing than the home-charged PEV for increasing the self-consumption, at leastin this setup.

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5. Discussion

In this section we discuss the thesis and elaborate on future work to be carriedout within the PhD-program.

The intersection between household electricity use from activities, PEV useand local distributed PV electricity production is an interesting research topicfor a number of reasons. Due to technological improvements as well as fallingprices for PV technology and PEVs the constellation of having a PEV and aPV module locally connected to the household is likely to become more com-mon in the future. There are many open issues related to this setup. What isthe grid interaction? What is the economic feasibility of that setup? How is theself-consumption of PV electricity production maximized? In face of differ-ent possible configurations regarding this setup there is a need for theoreticalinvestigation of the interrelations between the components prior implementa-tion.

The work in this thesis addressed this aspect to some extent. The investi-gations carried out within the appended papers primarily regarded two systemlevels: single household level and aggregate household level. In these scenar-ios each household - detached house or apartment - typically had between oneand four inhabitants, one PEV and one PV system on the roof of the house.The investigations in the appended papers have focused on certain parts of thisconstruction. One part is modeling of household load from activities - whichwas done with the Widén Markov-chain model. This model was endowedwith a PEV extension and an end-user flexibility extension. The PV electricityproduction was modeled directly from empirical irradiance data. The flexibil-ity model showed that if the probability for performing certain activities wasincreased - at the cost of decreasing the probability for other activities - thenelectric load could indeed be shifted. It was shown that self-consumption ofPV power could indeed be increased with a flexible strategy - although onlyto a limited degree because of both theoretical and practical problems inherentto flexible behavior. Flexible behavior demands that an inhabitant changes acertain activity for another at a given time but in reality most household ac-tivities are power consuming, making the effect only proportional to the netdifference in electricity consumption between the shifted and compensated ac-tivities. The PEV extension proved to be a realistic model for representing ahome-charged PEV. The model showed that for a household of two inhabitantsand one PEV the PEV load was a considerable amount of the household load.

The PEV extension had the limitation of only one inhabitant per householdbeing able to use the PEV. Also the flexibility model had a limited range of

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flexibility due to the problem of variable total amount of flexible activitiesduring a day. These are issues which are in need of improvement.

5.1 Future workDuring the remainder of the PhD-program the research will continue to focuson investigating household electricity consumption including PEV load andhow it can be related to PV electricity production. In many cases the method-ology will be similar, with the continued development of mathematical mod-els for representing the dynamics of these three components. As an examplea Monte-Carlo model calibrated with data on electricity consumption fromthe Swedish Energy Agency is currently being developed. The intention is todevelop this model so that it could be used for both power flow calculationsand estimations of photovoltaic self-consumption. In the not appended paper(V) the preliminaries for how a stochastic model of collective inhabitant be-havior and consequent energy use is described. Future work might be carriedout on this topic - in particular in connection with time geography. Anotherinteresting research topic that will be investigated is the complex smart gridconstruction of having a battery connected to the household that could storeenergy during certain times and deliver energy at other times in order to forexample increase the amount of self-consumption of PV electricity produc-tion. Instead of having flexible energy users to maximize self-consumption -like in appended paper (III) - this would allow for an automated improvementof self-consumption.

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6. Conclusions

Self-consumption of PV power by means of introducing PEV home-chargingand flexible behavior was investigated in this thesis. Results show that whenintroducing a PEV load to the household load the load fraction increases butthat the solar fraction decreases. This occurs since the PEV load causes the to-tal load to become less matched with PV electricity production. The flexibilityextension on the other hand shows that it is possible to increase both solar frac-tion and load fraction with flexible behavior. On the other hand the flexibilityextension shows limits in the total fraction of flexible behavior which meansthat a perfect match between household load and PV electricity productionmight not be possible to achieve using such a flexibility setup.

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7. Acknowledgements

This thesis was made possible with the support of a number of people. Firstoff I would like to thank my main supervisor Joakim Widén for his excellentapproach to supervision. My co-supervisor Ewa Wäckelgård for her supportand for providing insight into the infrastructure of power systems research.

The members of the Program Energy Systems who set the stage for inter-esting collaborations, courses and research debates. I thank Pia Grahn andMattias Hellgren for good collaborations on courses and research. A specialthanks also goes to my co-authors Karin Alvehag and Lennart Söder fromKTH.

Thanks goes Clas-Göran Granqvist and to the Division of Solid State Physicsat Uppsala University, especially the rest of the Build Environment EnergySystems Group (BEESG) members: Magnus Åberg and Annica Nilsson. Thanksalso goes to Fredrik Wallin and Javier Campillo at Mälardalens Högskola(MdH) for research collaborations. A special thanks goes to Uwe Zimmermanfor providing highly resolved solar irradiance data from the very PV panels Iwatch every day from my office.

I would like to thank my partner Anna Celander for her joyful attitude andstrong forcing for directions such as heavy training and enjoying life. Lastbut not least I would like to thank my family and friends for the support andencouragement throughout my time as a PhD-student.

You ain’t seen nothing yet

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