Uncertainty and correlation modeling for load flow ...

Uncertainty and correlation modeling for load flow analysis of future electricity

distribution systemsProbabilistic modeling of low voltage networks with

residential photovoltaic generation and electric vehicle charging

Umar Hanif Ramadhani

AbstractThe penetration of photovoltaic (PV) and electric vehicles (EVs) continues to grow and is predicted to claim a vital share of the future energy mix. It poses new challenges in the built environment, as both PV systems and EVs are widely dispersed in the electricity distribution system. One of the vital tools for analyzing these challenges is load flow analysis, which provides insights on power system performance. Traditionally, for simplicity, load flow analy-sis utilizes deterministic approaches and neglecting correlation between units in the system. However, the growth of distributed PV systems and EVs increases the uncertainties and corre-lations in the power system and, hence, probabilistic methods are more appropriate.

This thesis contributes to the knowledge of how uncertainty and correlation models can im-prove the quality of load flow analysis for electricity distribution systems with large numbers of residential PV systems and EVs. The thesis starts with an introduction to probabilistic load flow analysis of future electricity distribution systems. Uncertainties and correlation models are explained, as well as two energy management system strategies: EV smart charging and PV curtailment. The probabilistic impact of these energy management systems in the electri-city distribution system has been assessed through a comparison of allocation methods and correlation analysis of the two technologies.

The results indicate that these energy management system schemes improve the electricity distribution system performance. Furthermore, an increase in correlations between nodes is also observed due to these schemes. The results also indicate that the concentrated allocation has more severe impacts, in particular at lower penetration levels. Combined PV-EV hosting capacity assessment shows that a combination of EV smart charging with PV curtailment in all buildings can further improve the voltage profile and increase the hosting capacity. The smart charging scheme also increased the PV hosting capacity slightly. The slight correlation between PV and EV hosting capacity shows that combined hosting capacity analysis of PV systems and EVs is beneficial and is suggested to be done in one framework. Overall, this thesis concludes that an improvement of uncertainty and correlation modeling is vital in pro-babilistic load flow analysis of future electricity distribution systems.

Keywords: Probabilistic load flow, Electricity distribution systems, Uncertainty model, Corre-lation model, Photovoltaics, Electric vehicle, Residential buildings, Hosting capacity

Umar Hanif Ramadhani, Department of Civil and Industrial Engineering, Civil Engineering and Built Environment, Uppsala University, SE-751 04 Uppsala, Sweden.

© Umar Hanif Ramadhani 2021

urn:nbn:se:uu:diva-434951 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-434951)

"Exploring the unknown requires tolerating uncertainty."Brian Greene

List of papers

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

I Ramadhani, U. H. , Shepero, M., Munkhammar, J., Widén, J.,Etherden, N. (2020)."Review of probabilistic load flow approaches forpower distribution systems with photovoltaic generation and electricvehicle charging", International Journal of Electrical Power & EnergySystems, Vol 120, 106003.

II Ramadhani, U. H., Fachrizal, R. , Shepero, M., Munkhammar, J.,Widén, J. "Probabilistic load flow analysis of electric vehicle smartcharging in unbalanced LV distribution systems with residentialphotovoltaic generation", Submitted to Sustainable Cities and Society.

III Fachrizal, R. , Ramadhani, U. H., Munkhammar, J., Widén, J. (2021)."Combined PV-EV hosting capacity assessment for a residential LVdistribution grid with smart EV charging and PV curtailment",Sustainable Energy, Grids and Networks, Vol 26, 100445.

Reprints were made with permission from the publishers.

Publications not included in the thesis

IV Shepero, M., Ramadhani, U. H., Munkhammar, J., J., Widén, J.(2019). "Estimating the impacts of single phase electric vehiclecharging and phtovoltaic installatios on an unbalanced 3-phasedistribution system". In proceedings of the 9th International Workshopon Integration of Solar into Power System, Dublin, Ireland, 15-16October 2019.

Notes on my contributionI contributed with the following in the appended papers:

Paper I, I surveyed the literature and wrote the paper.Paper II, I performed the simulations and wrote most of the paper.Paper III, I co-developed the simulations, co-wrote the discussion part, re-

viewed, and edited the paper.

Contents

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

2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Future electricity distribution systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Small-scale distributed generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 EV charging load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Energy management system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.4 Hosting capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Probabilistic load flow analysis process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Uncertainties in probabilistic load flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Aleatory uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Epistemic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Correlation in probabilistic load flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.1 Intra-variable correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2 Inter-variable correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Probabilistic load flow computation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Research gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Grid simulation and case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Distribution grid data and simulation tool . . . . . . . . . . . . . . . . . . . . . 193.1.2 Grid performance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.3 Energy management system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Modelling the uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 PV-EV penetration and allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 PV generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 EV charging model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.4 Residential building load model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Modelling the correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Probabilistic analysis of EV smart charging with different

allocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1 Voltage profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Phase unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.3 Peak loading and total losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1.4 Correlation between network nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Combined PV-EV hosting capacity with EV smart charging andPV curtailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Voltage profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.2 System losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.3 Combined PV-EV hosting capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Discussion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1. Introduction

Since the global Covid-19 pandemic started at the end of 2019, more than twomillion people have died, millions of people have lost jobs, and 130 millionpeople are estimated to live in extreme poverty if the crisis persists [1]. Eventhough the change forced by the pandemic was devastating, we still could tryto look at it from the bright side. In 2020, we saw the most dramatic drop inCO2 emissions, almost 8% lower than in 2019, mostly caused by the decline ofenergy demand due to lockdown policies [2]. It shows us that the war againstclimate change is possible to win if we put a massive effort into the way weuse our energy. However, the war is still far from over. The CO2 concentrationin the atmosphere, the result of cumulative past and current emissions, whichis what matters most for climate change, is still high [3]. Furthermore, theworld suffered from lockdowns. OECD stated that the annual GDP growthcould be 4-6 percentage points lower than it would have been with only threemonths of lockdown [4].

Instead, what we can do is to continue to do what we did in terms of cli-mate change mitigation before the pandemic, but at a higher pace than before.Several efforts in reducing the CO2 emissions before the pandemic have al-ready produced some promising results [2]. Changes in the power sector inadvanced economies have been proven to effectively flatten the CO2 emis-sions in 2019, before the lockdown and drops in energy demand occurred [2].The CO2 global emissions intensity from electricity generation was reducedby 6.5% in 2019 and the increase in renewable energy generation in advancedeconomies resulted in emissions cuts up to 130 Mt of CO2 emissions [5].).

Among the renewable generation technologies, solar photovoltaics (PV) hasthe highest expected growth potential, contributing to almost 60% of expectedgrowth [5]. The importance of renewable energy sources is even higher withthe increase of electric vehicles (EVs). It is also important that EV charging ismade with sustainable energy, since the emissions of EV charging in a powersystem with a high penetration of coal-based generation is similar to thoseof conventional combustion engine vehicles [6, 7]. However, with renewablesources EV charging can reduce emissions by up to 400 Mt, when comparedwith conventional combustion engine vehicles [8]. Currently, there are 7.2million EVs worldwide, with a 40% year-on-year increase in 2019, and thisfigure is expected to grow even higher in the future [9].

Interestingly, almost half of the expected growth of PV originates from dis-tributed solar PV, which is mainly located in the built environment [5]. Simi-larly, most EV charging takes place at home and work [9]. The growth of PV

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and EV then leads to significant changes in power management in the builtenvironment, such as in residential houses and offices. It changes both theconsumption and generation behavior and, consequently, changes the powerflow in the distribution system. This poses new challenges because tradition-ally, the distribution system was designed for a centralized power generationwithout distributed power sources such as PV systems and high loads such asEVs [10].

To some extent, it is similar to if we imagine that each house in our waterdistribution system is suddenly able to produce its own water and can sendtheir excess water to the neighborhood using the same pipes. This would forsure raise several concerns including the safety of the water supply, the ca-pacity of the pipes to receive backward supplies, the reliability of the waterdebit supply, and other operational problems would have to be ascertained.Similarly, the integration of PV systems and EVs into the grid can lead to sev-eral negative impacts. The impacts include voltage violation, a decrease inpower quality, equipment damage, power losses, reliability issues, and longerrestoration time from outages [11].

To solve or to mend the issue, several energy management systems (EMS)have been proposed. However, a proper assessment is required to quantify thereal impact of the proposed management system on the power system. Oneof the vital tools in solving these power system-related problems is load flowanalysis, as it provides insights on the power system performance at steady-state conditions [12].

Traditionally, for simplicity, load flow analysis approaches utilize determin-istic approaches and neglect correlation. However, the growth of distributedPV generation and EVs increase the uncertainties in the power system, bothspatially and temporally [13]. Hence, to increase the quality of the load flowanalysis, probabilistic methods are more appropriate.

Another important factor to increase the quality of the load flow analysis isthe correlation between input variables. The correlation occurs both as intra-variable correlation, i.e., between the same variables, and as inter-variable cor-relation, i.e., between different variables. The correlation arises, presumably,due to human routines and weather patterns [14]. The correlation happensand takes place both temporally and spatially. For example, a proper correla-tion model has been proved to successfully increase the accuracy of a spatio-temporal model of solar irradiance [15]. However, the inclusion of severalmanagement schemes for PV systems and EVs may also affect the correlationstructures between input variables.

To sum up, improved knowledge of the spatio-temporal uncertainties andthe correlation in the patterns of energy sources and energy use in the builtenvironment will potentially improve the modeling of input variables to loadflow analysis and help us to manage the future electricity distribution systembetter. It will provide more information on the impact of several new energytechnologies and management schemes. It will also, hopefully, allow the elec-

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tricity distribution system to host more green energy technologies by means ofapplying strategies which can mitigate CO2 emissions and reduce economiclosses.

1.1 Aims of the thesisThis thesis has the aim to contribute to the knowledge of spatio-temporal un-certainty and correlation modeling in order to improve the quality of load flowanalysis for electricity distribution systems with high penetration of residentialPV systems and EVs. In order to do so, three goals are formulated for this the-sis:

I Review the state of the art of probabilistic approaches in load flow analysisfor electricity distribution systems with PV generation and EV charging.

II Investigate the properties of spatio-temporal uncertainty and correlationsfor probabilistic load flow analysis of residential PV, EV charging, andload.

III Improve, building on aim II, spatio-temporal uncertainty and correlationmodeling for increasing levels of penetration and several energy manage-ment schemes.

1.2 Overview of the appended papersThe structure of the remainder of this licentiate thesis is as follows: Chapter 2presents the background of the research conducted in this thesis, includingsome basic explanations and classifications of probabilistic load flow (PLF),uncertainties, and correlation modeling in the PLF. Chapter 3 provides thedata, methodologies, parameters, and software tools used in this thesis. Chap-ter 4 presents the results from the appended papers. These results and futurework are then discussed in Chapter 5. Finally, the conclusions are drawn inChapter 6. The results of this thesis are based on these appended papers:

• Paper I provides a review of the state-of-the-art in PLF approaches. It fo-cuses on the application of PLF in electricity distribution systems with PVgeneration and EV charging consumption which has been identified to be avaluable addition in future power system planning and operation. The paperalso provides an overview of various methods that can be used to model theuncertainty and correlation structure in the PLF process.

• Paper II presents a PLF analysis of EV smart charging schemes in a elec-tricity distribution system. The paper provides a probabilistic model of PV,EV, and load with a smart charging scheme. The probabilistic impacts and

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correlation analysis were assessed in an unbalanced electricity distributionsystem. Two different spatial allocation methods of PV and EV were uti-lized, namely distributed and concentrated allocations, to observe the im-pact of handling these uncertainties.

• Paper III complements Paper II with a probabilistic impacts assessment oftwo different energy management system scenarios on combined PV-EVhosting capacity analysis with another allocation and penetration approach.The energy management, besides the smart charging, also includes PV cur-tailment. The paper also presents a study that assesses the correlation be-tween PV and EV hosting capacity.

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2. Background

This chapter provides a background of the thesis. Firstly, Section 2.1 describesthe possible changes in future electricity distribution systems and how thechanges are considered and not considered in this thesis. Then, Section 2.2explains the basic concept of PLF analysis. In Section 2.3 and Section 2.4,uncertainty and correlation modelling methods in PLF are explained, respec-tively. Section 2.5 introduces the load flow computation methods in PLF anal-ysis. Finally, the research gaps within this area are given in Section 2.6.

2.1 Future electricity distribution systemsElectricity distribution systems are evolving in several ways. Power gener-ation is shifted to small-scale distributed systems, EV charging is changingelectricity consumption patterns, and the interaction between local power gen-eration and load is becoming an increasingly important factor. To quantify theimpact of new generation units and loads that are integrated into electricitydistribution systems, the so-called hosting capacity has been introduced. Inthis section, some of these trends and concepts are discussed.

2.1.1 Small-scale distributed generationTraditionally, power is transmitted over long distances from centralized powergenerators to the customer side of the electricity distribution system. Thus,electricity distribution systems are usually considered as passive systems, whichmeans that they have the objective to distribute the power from centralizedgeneration to the customers only [16]. Currently, power generation is increas-ingly shifted to small-scale distributed generation at the distribution side of thepower system, which enables the customer to participate in the energy genera-tion mix [16]. The trend is motivated by the need for more sustainable energysupply and, therefore, the new distributed generators are mainly renewable.The world’s dominant distributed renewable generation mainly comes fromPV and is predicted to continue in the future [5].

Small-scale distributed PV systems are mainly connected using a single-phase connection because the customer musually only has access to a single-phase connection, with an exception of countries such as Sweden. Some coun-tries limit the size of single-phase PV inverters [17]. A study in [18] showedthe impact of single phase PV and EV charging installations and suggested fu-ture studies to simulate the load flow in their unbalance form. This motivatesall load flow analyses in this thesis to do the same.

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2.1.2 EV charging loadThere are also changes in the electricity consumption, where in particular elec-tric vehicle charging has an impact. Generally, electric devices are evolvingand follow the customer’s needs and behavior. Similarly as for PV, EVs areincreasing in numbers. The growth is motivated by the need for a cleaner al-ternative of transportation [9]. Therefore, the presence of EVs becomes oneof the main focuses of this thesis.

There are several types of EVs available in the market. Two of them canbe charged from the electricity grid, namely battery EVs and plug-in hybridEV. A battery EVs solely depends on the electric motor and is powered by abattery [19]. Plug-in hybrid EVs has both an internal combustion engine andan electric motor which is powered by a battery that can be plugged into theelectricity grid [19]. The EV batteries are usually charged directly from thedistribution power system, consisting of a mix of residential, workplace, andEV charging stations [20]. The simulations in this thesis, however, focus onresidential charging only.

2.1.3 Energy management systemThe presence of customers-owned PV systems increases energy exchanges andthe interaction between power generation and load in the electricity distribu-tion system [16]. It allows customers and power system planners to utilize theadvantages or to reduce the disadvantages of the distributed power generation,such as increasing the interaction and synergy between load and generation.Several technologies and schemes are available, including energy storage andEMS.

The basic idea of energy storage is to store the excess power generation anduse it later when the demand is higher than the generation. Some studies haveperformed load flow analyses featuring energy storage [21]. Energy storage,however, are generally not economically feasible [22]. Hence, several studiesemployed EMS without energy storage to improve the synergy between loadand generation. The EMS can be divided into supply-side management anddemand-side management.

At the supply side, the EMS aims to control the generation and adjust it tothe load demand. Basically, it is impossible for the PV systems without energystorage to adjust their power generation outputs to follow the load pattern atall times. The PV power generation, however, could be curtailed using a smartPV inverter which has been proved to be useful to increase the PV penetrationin the power system [23]. The PV inverter can also be equipped with reactivepower control. It should be noted that in the papers included in this thesis,however, the reactive power control was not considered.

Conversely, the demand side management aims to control the load for somepurpose, e.g. to adapt for generation or to generally flatten the peak load. For

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the case of EVs in the distribution power system, a scheme that shifts the EVcharging load to the non-peak load period and to periods of surplus PV gener-ations will be beneficial. The scheme, a so-called EV smart charging scheme,has been proven to be able to improve the synergy between PV generation andEV charging load [24].

2.1.4 Hosting capacityThe integration of PV systems and EVs into the future electricity distributionsystem leads to several impacts. The impacts include voltage violation, a de-crease in power quality, equipment damage, power losses, reliability issues,and longer restoration time from outages [11]. To avoid these issues, the im-pacts need to be quantified. Hence, the hosting capacity concept is introduced,defined as the maximum amount of the new generation or load that can beinstalled without violating the operational quality of the system [10].

There are several aspects that need to be defined in a hosting capacity calcu-lation. The first one is the performance index. Some phenomena considered inhosting capacity studies are: overvoltage, undervoltage, rapid voltage change,voltage unbalance, harmonics, thermal overload, and losses [25]. Secondly,the limit of the chosen performance index should be determined. The finalaspect is a calculation method for the performance index as a function of theamount of added generation or load.

2.2 Probabilistic load flow analysis processThe main objective of the PLF analysis is to study the uncertainties of theinputs in power flow calculations. These inputs are often represented by prob-ability density functions (PDFs), and the outputs are also often representedby PDFs of the intended parameters, such as voltages and phase unbalance[26,27]. How these PDFs are chosen will depend, however, on how the uncer-tainties are estimated as will be discussed deeply in the next chapter.

Figure 2.1 shows an illustration outlining the PLF process for an electricitydistribution system. Generally, the PLF process can be divided into three mainstages: input uncertainty modeling, PLF computation, and the PLF outputpresentation.

In the input uncertainty modeling stage, the uncertainty of all loads andgenerations are modeled. The uncertainties can be categorized as aleatoryand epistemic uncertainties. The uncertainty modeling stage, including thecomparison between the two categories, is explained in Section 2.3. Currentstudies have shown the importance of representing the correlations betweenthe input variables at this stage [28]. The correlations occur both betweenthe same variables (intra-variables) e.g., between PV systems, and between

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Figure 2.1. Illustration of the PLF process for an example distribution network.

different variables (inter-variables) e.g., between PV system and load. Thecorrelation modeling is explained more deeply in Section 2.4.

In the PLF computation stage, the load flow problem is solved based onthe input uncertainty and correlation models. The PLF analysis can be doneusing numerical methods, analytical methods, and approximation methods.The basic concept and the differences between these methods are explainedin Section 2.5, including how the third stage, the PLF output presentation, isexecuted for the different methods, which is strongly related to the choice ofcomputation methods.

2.3 Uncertainties in probabilistic load flow analysisIn our daily life, our judgments and decisions entail uncertainties. Whetherwe are estimating the weather, choosing a new car, or having a marriage, weusually do not know in advance how it will turn out precisely. However, thequality of the uncertainties could be significantly different. For example, theuncertainty raised by the question about which number will be shown by arolling dice and about whether Stockholm is bigger than Uppsala, are dis-tinctly different.

In the first question, the uncertainties come from the stochastic outcome ofthrowing a dice. Meanwhile, the second question can be answered accuratelywith proper knowledge. These two different uncertainties can be categorizedas aleatory and epistemic uncertainties, respectively. To answer the first ques-

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Figure 2.2. Illustration of the categorization of uncertainties for three different ques-tions.

tion, knowledge about the best probability outcome is still needed to give thebest approximative answer. The uncertainty, however, cannot be further re-duced.

A pure epistemic uncertainty would be related to a question that can bejudged as true or false. The second question, "Is Stockholm bigger than Up-psala?", is a proper example of pure epistemic uncertainty. With completeknowledge, this question can be answered clearly. There could still be un-certainty in answering this question if the knowledge or information is notcomplete; for example, when the only available data is the number of individ-uals in the population. The question can be approached based on this data, butof course, it contains high uncertainties.

The definition of aleatory and epistemic uncertainties has been discussedin psychology and the linguistic area, as can be seen in [29]. In the afore-mentioned study, the aleatory uncertainties, also known as random or statisti-cal uncertainties, can formally be categorized as necessitating an assessmentof stochasticity that arises from natural stochastic behavior. Aleatory uncer-tainty is represented with a class of possible outcomes and basically difficultto reduce [29]. In contrast, epistemic uncertainty, also known as knowableuncertainties, is formally categorized as necessitating missing knowledge orexpertise about an event that is, in principle, knowable [29].

It is important to note that the categorization is not strict and still debatable[29]. One of the examples is the question about which public travel mode willtake us to Stockholm faster. This question, even though people with properknowledge can answer it confidently, may still contain some uncertainties,e.g., from a traffic accident or the departure time. Hence, the question can beanswered both directly with our knowledge or can be represented with a classof possible outcomes. A comparison of this categorization compared to theprevious two questions is given in Figure 2.2.

The choice of how the uncertainties are handled is often on the people whoanswer them. Hence, the way we perceive the uncertainties can be different,

9

and consequently, the treatment and its effects will be different as well. There-fore, it is vital for a study in PLF with uncertainties to explain how the studytreats each uncertainty.

In load flow analysis, these uncertainties have been proven to be predomi-nant in determining the hosting capacity. A study in [30], for example, showsthat the different treatments of these uncertainties directly leads to differenthosting capacity estimates. In this section, some categorization of aleatoryand epistemic uncertainties of household load consumption, residential PVgeneration, and EV charging load for load flow analysis will be given basedon the literature review. How to model these uncertainties in the appendedpapers is introduced in the Methodology section in Section 3.2.

2.3.1 Aleatory uncertaintiesAleatory uncertainties in load flow analysis originate from variables that stronglyrelate to stochastic behavior and chaotic systems that are difficult to reduce.This type of uncertainty is mainly represented directly using PDFs. Therefore,the choice of PDFs is vital to handle this uncertainty. Different input vari-able PDFs could lead to significant changes in load flow analysis results [30].Probabilistic modelling of inputs is explained din detail in Paper I, Section3. Alternatively, the uncertainties can be modeled using historical data and atime-series stochastic model.

Using historical data is perhaps arguably closest to reality, but ideally, thedata should be sufficiently long, which is rare, to include wider phenomena[31]. The time-series model is computationally expensive and also prone tothe risk of losing the generality [31]. However, the time-series model is oftenrequired to model the impact of new technologies or EMS schemes that requiretime-series-based optimization.

The household power consumption in the grid is a variable that is com-plex to quantify and highly stochastic [32]. The uncertainties of load con-sumption arise from customer behaviors, electrical equipment variations, andclimate conditions that makes it a challenge to generate a perfectly accurateload consumption model. Hence, the load consumption uncertainties are of-ten considered as aleatory. With a well studied activity-generating model ofindividual household members, it is possible to produce a realistic time seriesmodel and probability distributions of load consumption [32]. An individualhousehold model also can be generated from historical smart meter data asproposed in [33].

Alternatively, a wide range of PDFs have been utilized to directly representthe household uncertainty. The distribution modeling can be classified withregard to load aggregation. Most studies, especially when dealing with a big-ger power system, considered the aggregated power system load, while othersmodeled a more specific model for residential loads. In the studies with aggre-

10

gated power system load, the normal distribution is often utilized [34–39]. Incontrast, if the study focuses on the residential loads only, the normal PDF israrely employed. One of the main benefits of not using a normal distributionis avoiding negative power consumption. The study in [40] recommends thegamma distribution as the most suitable distribution to represent the electricityuse of 100 residential customers. Several non-normal distributions also gaveacceptable results, including beta, inverse normal, lognormal, and Weibull dis-tributions.

PV generation contains some uncertainties that come from the natural stochas-tic behavior as well. For PV generation, the main source of the aleatory uncer-tainties is the absorption and scattering of solar radiation in the atmosphere,in particular from clouds [41]. Hence, the uncertainties from PV generationoutput are often considered as aleatory. However, some fluctuations in PVgeneration are predictable, such as seasonal variations. Hence, a proper PDFis important in modeling PV generation uncertainties.

Historical data of solar irradiance is more widely available than PV outputgeneration data. Hence, a PV generation time-series model is often generatedbased on recorded irradiance data. Similarly, modeling the PV generation us-ing probability distributions can be approached directly with a distribution ofPV generation and indirectly with a distribution of the solar irradiance or clear-sky index, subsequently fed into a PV system model. The clear-sky index is ameasure defined as the ratio of global irradiance to the responding theoreticalclear-sky irradiance at the same time and location [41].

For modeling the solar irradiance, normal, beta, uniform, and empirical dis-tributions have been used [34–37, 42–44]. For modeling the clear-sky index,a wider range of PDFs has been utilized, typically a mixture of distributionssuch as normal, lognormal, polynormal [45, 46]. Some studies chose singlepeak distributions while other used bimodal distributions, representing cloudyand clear skies. Some studies also consider the PV temperature uncertaintiesas aleatory and combine them in an indirect approach [34–36]. Meanwhile,the direct approach does not typically perform the output simulation from so-lar distribution. Direct PV power distributions have been modeled using ex-ponential, Weibull, Beta, and normal [47, 48].

As in the case of household load consumption, the aleatory uncertainties ofthe EV charging model mainly come from human behavior. Consequently,some variables in an EV charging model are often considered as aleatoryuncertainties, including trip mileage, battery capacity, departure time, arrivaltime, and charging time [26]. These variables are used to generate a bottom-upmodel for EV charging. Another option is to model the overall EV chargingdemand using a PDF.

It is important to note that the time range of load flow simulations differsbetween studies. Some of them consider a wide time range and others consid-ers only the ’worst case’ time. The different time ranges may lead to differentdistribution models for these aleatory uncertainties. The aleatory uncertainty

11

model of the input variables also rarely considers the EMS, neither for thesupply nor the demand side, which presumably would change the probabilitydistribution parameters.

Another variable that is often considered as aleatory is the allocation ofboth PV and EV, i.e., which houses would have PV systems and EVs when thepenetration is less than 100%. This variable, however, can also be modeledas more of an epistemic uncertainty based on a PV and EV adoption model.Several studies show that PV adoption is impacted by both socio-economicbackground as well as the peer effect [49–51]. The impact of this choice,especially in the distribution system, is still rarely discussed.

Lastly, another important variable that can be modeled as aleatory is thebackground voltage variations. In [52], the distributions of the backgroundvoltage, based on measurements, differed even between individual substations.The distributions that fitted well were generalized wxtreme value (GEV) andWeibull.

2.3.2 Epistemic uncertaintiesEpistemic uncertainties in load flow analysis mainly come from the lack ofknowledge or information regarding variables related to the choice and instal-lation of technologies. These uncertainties can be reduced with more infor-mation [29]. In previous studies, epistemic uncertainties are sometimes notexplained as uncertainties. Instead, they are sometimes simply included asdata.

In terms of household electricity consumption, most of the epistemic un-certainties come from the lack of knowledge about the installment of tech-nologies that will be applied by the household. Some examples are: i) Thetype of heating technologies used in the house. ii) Type of phase connectionused in the house i.e., will it be three-phase or single-phase? iii) If it utilizes asingle-phase connection, which phase will it be [30, 31]?

The heating load model depends on the heating system utilized in the neigh-borhood. A community could use district heating, electric heating, or the com-bination of these technologies [53]. To model electric heating loads, a separatemodel from other electricity loads could be utilized.

If the load flow analysis is applied to an existing system, these questions canbe answered easily because the phase connection is less likely to be changed inthe future. However, PLF is often simulated in a prototype environment or forthe system where PV systems and EVs are not yet installed, usually for scien-tific studies [25]. Hence, educated guesses or stochastic approaches are oftenused [31]. Several test grids have been developed to provide representative in-formation on phase connections that can be used in simulations. Alternatively,the phase connection can be chosen randomly which means it is treated as analeatory uncertainty.

12

PV generation models contain further epistemic uncertainties. The uncer-tainties mainly occur from the lack of knowledge about the modeled PV in-stallations. Some of them are: i) The properties of the PV system, e.g., PVpanel efficiency and performance. ii) The direction and the tilt of the panels.iii) Type of phase connection used in the installation and which connection itwill be for a single-phase connection. iv) The size of the PV systems. v) EMSapplied in the system. These questions are difficult to answer because usually,the PV generation is the variable that is assessed in PLF analysis [30, 31].

The properties of the system can be approached using historical data, but itneglects the learning curve of PV development. The direction and the tilt of thepanels also require building data. The properties of the system, the direction,and the tilt information can be neglected if the PV output uncertainties utilizedirect PDF modeling as discussed in the previous subsection.

The phase connection, similarly to the load, can be modeled based on his-torical data or is treated as an aleatory uncertainty. The size of the PV system,if treated as an epistemic uncertainty, can be approached based on historicaldata or can be modeled based on the load data by scaling it according to theratio of annual PV power production to load data. Alternatively, the study mayconsider the PV system size as an aleatory uncertainty. For the EMS, previousstudies usually neglected it, because most of the systems do not have EMS. Ifthe EMS is taken into account, simulations are needed.

Epistemic uncertainties in EV charging load mainly come from the knowl-edge gap regarding future EV operational parameters and charging infrastruc-ture. The epistemic uncertainties include: i) The type of EVs, i.e., whetherit is plug-in hybrid EV or battery E.V ii) Charging power rate. iii) The sizeof the battery. iv) The charging algorithm. These uncertainties rely on thedevelopment of EVs.

To model a plug-in hybrid EV, the uncertainties of the operating status needto be taken into account [54]. For charging power and battery capacities, anassumption about technology should be made, e.g., based on historical data.Advanced charging algorithms, as a part of a demand-side EMS, are rarelymodeled. If the charging algorithm is taken into account, a charging simula-tion is needed.

2.4 Correlation in probabilistic load flow analysisThe correlation, as explained in the previous section, can be categorized as aspecial case of the uncertainties because basically we do not know in advancewhat the correlation between variables is. The correlation model in a PLFanalysis aims to characterize the dependencies between inputs that containuncertainties. In the power system, input variables are to a varying degreecorrelated with each other. As mentioned, the correlation can be divided intointra-variables correlation, e.g., PV-PV, and inter-variables, e.g., PV-EV.

13

Generally, the joint variability of two random variables can be measured bythe covariance [55]. A positive covariance means that higher values of onevariable coincide with higher values of the other variable, and vice versa [55].Hence, the covariance measures the strength of a linear relationship betweentwo variables. For two random variables X and Y, both with a sample size ofN, the covariance of X and Y is:

Cov(X ,Y ) =N

∑i=1

(xi − x)(yi − y)N

, (2.1)

where x and y are the means of X and Y respectively. This value, however, isoften difficult to interpret due to its dependence on the magnitudes of the vari-ations in X and Y. Therefore, the correlation coefficient (CC), the normalizedcovariance, is often used:

Corr(X ,Y ) =Cov(X ,Y )

σx.σy, (2.2)

where σx and σy are the standard deviations of X and Y respectively.The CC, however, assumes that a change in one variable is corresponding

to a proportional change in another variable. In some cases, two variablesare better represented as having a monotonic relationship (either increasing ordecreasing), but not necessarily at a constant rate. In this case, the numericalvalues could be transformed to their rank on the sorted data, called ranking instatistics [56].

2.4.1 Intra-variable correlationsWithin the variables with the same input data type in distribution power sys-tems, correlation occurs between load-load, PV-PV, and EV-EV. Correlationsbetween residential loads, for example, occur due to human routines and sim-ilar responses to the weather [57]. At night, for example, people tend to turnon the light and when it is getting colder, people tend to turn on the heater.In previous studies, correlations between loads have been modeled using lin-ear dependence, Gaussian copulas, and Cholesky decomposition [34, 58, 59].The diversity between households, however, is also important when creating arealistic synthetic household demand model because a perfect correlation be-tween households is almost impossible due to the stochastic nature of humanbehavior [32].

Meanwhile, the spatial correlations between PV systems occur due to thephysics of the sun and clouds [14]. The spatial correlations between PV sys-tems have been modeled using copulas and joint cumulants. It is importantto note that assuming a perfect correlation between PV systems could over-estimate the negative impacts of the PV system, especially in a larger powersystem due to the dispersion-smoothing effect [15].

14

Similarly, correlations between EV charging loads at dispersed locationshave also been observed [60]. The correlations were caused by human activ-ities, charging infrastructure, and, presumably, the charging algorithm. Thecorrelations between EV charging at different locations, however, have notbeen studied to the same extent as correlations between loads and betweenPV systems. Some studies have modeled the correlations between EVs usingGaussian copulas and linear dependence [57, 61].

2.4.2 Inter-variable correlationsCorrelations between different variables, even though not discussed exten-sively in the literature, could impact the overall system uncertainties. A studyin [35] shows that the standard deviation of the voltage was significantly higherin a correlated case compared to an uncorrelated one.

Small positive correlations between PV and load within 24 hours were iden-tified in [62], but the correlation was much more significant when the time wassplit into separate periods of morning, afternoon, and night. Residential loadand EV charging are positively correlated due to the presence of the user athome. EV charging have also been found to have a mismatch with PV gener-ation. In previous studies, inter-variables correlations have been modeled us-ing Gaussian copula and Cholesky decomposition for numerical methods, andCholesky decomposition, unscented transformation, and linear dependence forother methods [34, 37, 39, 62, 63].

A positive correlation between generations and load variables in buildings,particularly residential houses, would benefit the buildings and help to achievea so called net-zero energy balance [24]. One of the methods is to utilizesmart charging and storage. The impact of these methods in the inter-variablescorrelation, however, is rarely discussed. Achieving net-zero energy buildings(NZEB) also will help to add more PV systems and EVs into the system.

The previous studies usually determine the hosting capacity of PV systemsand EVs separately. The synergy between PV systems and EVs theoreticallywill improve the hosting capacity, depending on how intensive the synergy willbe. The impact of one technology on the other’s hosting capacity, however,was rarely included in the previous hosting capacity studies.

2.5 Probabilistic load flow computation methodsThe PLF computation stage aims to characterize the PDFs of the intendedoutput variables, e.g., voltages and phase unbalance, based on the given sta-tistical information in the input variables. There are several methods that havebeen used for PLF computations. As seen in Figure 2.3 the methods can becategorized into three main groups: numerical, analytical, and approximationmethods. Paper I, Section 5 discusses these methods in depth.

15

Figure 2.3. Classification of PLF computational methods [27].

Most of these probabilistic methods still require performing a determinis-tic load flow simulation as one of its steps. The numerical methods performa deterministic power flow calculation on a set of new samples repeatedly.The analytical methods usually do the deterministic calculation to generatethe sensitivity matrices, which is one of the important steps in the cumulant-based analytical method to represent the relationship between input and outputvariables. The approximation method utilizes the deterministic calculation tocalculate the output of every sample point to produce the output’s raw mo-ments [28].

There are several available load flow iteration methods to solve determin-istic load flow. One of the most popular methods is Newton-Raphson iter-ation, which is often used in load flow calculations for transmission systems[64]. The distribution system, however, has different characteristics with a lowreactance-to-resistance (X/R) ratio. According to [65], which compared theperformance of the Newton-Raphson method and forward/backward sweep it-eration methods, found that the latter is more suitable for distribution systems.OpenDSS, a power system simulation tool used in this thesis, includes a fixed-point iterative method which is referred to as the normal mode in the softwareto solve power system simulation in the distribution system [66].

Numerical methods utilize numerical estimations to solve mathematicalproblems. Monte Carlo simulation, one of the numerical methods, is oftenused in PLF computations. The Monte Carlo method estimates the outputprobability distributions by repeatedly taking samples from the input proba-bility distributions and solving the power flow problem from these samples.Several sampling methods have been introduced to increase the computationalspeed, such as the Latin hypercube and uniform design sampling [59, 67, 68].

The drawn samples, which depend on the type of variables, are sometimesused directly as the input for load flow calculations, and are sometimes used

16

to generate a time-series model first before solving the power flow calculationfor every generated time step. This method is referred to as the time-seriesmethod and the Monte Carlo method without time-series model is sometimesreferred to as the probability distribution method [31]. The output of numericalmethods can be represented directly as the distribution of simulation results.

The analytical methods solve the PLF computation by performing arith-metic operations on the input PDFs. The older studies in analytical-based PLFutilized a conventional convolution method which requires a burden convo-lution calculation [69]. To avoid the expensive computation, the cumulantsmethod was introduced. The idea is to replace the moments of the PDFs withthe cumulants and performing the computation based on these cumulants [28].

After that, the input-output sensitivity of the system is generated, usuallybased on several deterministic load flow calculations to calculate the cumu-lants of the output. To determine the shape of the output distribution fromthe cumulants of the output, several types of expansion series are commonlyused, such as the Cornish-Fisher, Gram-Charlier, and Edgeworth expansions[33, 54, 70, 71].

The last group is called approximation methods. The basic idea is to useapproximations based on a small number of sample points of the input PDFsand their weights to estimate the statistical properties of the output PDFs. Themain goal of these methods is to use a minimum number of samples from theinput variables to generate sufficient information about the PDFs of the inputs.

The most popular methods are based on point estimation. The point estima-tion method utilizes statistical properties of the input variables, such as meanand variance, and does not require complete information about the distribu-tion that makes it computationally efficient. The drawback of this method isthe low accuracy when dealing with higher statistical moments [37].

2.6 Research gapsProbabilistic load flow analysis in distribution systems is a rapidly developingresearch area. However, the implementation of PLF analysis for distributionsystems with PV generation and EV charging load still has a great potentialfor improvement. The improvement could come from combining the currentstate-of-the-art in this area and utilizing applicable knowledge from other ar-eas. Several uncertainty variables and correlation models have also not beenexplored in previous studies, including how to best represent them, and theireffects on PLF analysis. The following research gaps have been identified inthe appended papers:

• Probabilistic models and impact analysis of PV systems and EVs in electric-ity distribution system rarely include the future EMS scenarios. The simula-tion of these scenarios becomes increasingly important to achieve the NZEB

17

in the future. Paper II attempts to address this gap by modeling the prob-abilistic analysis and simulating the impact of PV systems and EVs withsmart charging scenarios. Paper III elaborates this study with the additionof PV curtailment scenarios.

• The impact of different PV and EV allocation as one of the uncertaintysources in load flow analysis has rarely been studied. The allocation isusually treated randomly as aleatory uncertainties or assumed to be evenlydistributed. Paper II addresses this issue by performing a probabilistic im-pact analysis of PV systems and EVs for concentrated allocation methods.Hence, the impact of distributed and concentrated allocations are compared.In Paper III, an evenly distributed allocation method is utilized.

• The impact of EMS on intra-variables and inter-variables correlations inpower distribution grids with PV systems and EV charging has rarely beendiscussed in the literature before. Paper II compares the spatial correlationsin the distribution grid with and without smart charging schemes. Paper IIIassesses the impact of one technology on the other’s hosting capacity withinseveral EMS scenarios.

18

3. Methodology

This chapter describes the data, case studies, tools, and the methodologiesfor how uncertainties and correlations are treated in Paper II and Paper III.In Section 3.1, the distribution grid data, grid performance parameters, andthe EMS used in the appended papers are presented Section 3.2 explains howthe appended papers treat the uncertainties of each input variable. Section 3.3explains how several correlations are modeled in the appended papers.

3.1 Grid simulation and case studiesIn this section, the information regarding the grid data, grid simulation tools,grid performance parameters, and the case studies in Paper II and Paper IIIare presented. This thesis focuses on the electricity distribution system in aresidential area with building load, PV generation, and EV charging. All of thePV systems and EVs are considered single-phase connected as an asumption.Two supply-side and demand-side management schemes are considered in thesimulations in this thesis: EV smart charging and PV curtailment. Energystorage, reactive power control, and any further control algorithms, however,are not simulated yet in this thesis. Other energy system technologies suchas combined heat and power (often referred to as CHP) and small scale windturbines are excluded as well.

3.1.1 Distribution grid data and simulation toolThe result of load flow analysis is usually valid only for the specific studiedgrid. However, the method, in general, should apply to other grid environ-ments. The included studies in this thesis aim to illustrate the method to assessthe impact of different EMS and allocation methods with uncertainties for loadflow analysis in the electricity distribution system. The grid operator shoulduse their own data to assess the impact on their system.

In this thesis, the load flow analyses were performed on the IEEE EuropeanLV test feeder [72]. It represents a single feeder from a real LV grid in UK, andhas been used before in, e.g., [73,74]. It consists of 55 customers with a single-phase connection. Note that it is different with Swedish distribution grids thathave three-phase connection all the way down to the residential customers.The grid is supplied by an 800 kVA 11 kV/0.416 kV transformer. The grid

19

208

344783

7473

178

225264

248 249

327

SubstationPhase 1Phase 2Phase 3

320314

276289

387342388

337349

629563

611502

562

676682

899

619639

886906

898

780835

813755

896900

702

406

70

701778

861817 860688

614785

522539

556458

Figure 3.1. Topology of the IEEE European LV test feeder [72].

topology and phase connection are shown in Figure 3.1. The voltage on thelow voltage side was set to 1.04 pu at all times.

The unbalanced load flow analysis in this thesis was simulated in OpenDSSusing the fixed-point iterative method known as the normal mode in OpenDSS[66].

3.1.2 Grid performance parametersAs explained in Section 2.1.4, it is important to decide the grid performanceparameters should be used in load flow analysis as this is a central part of thehosting capacity analysis. In the appended papers, several parameters describ-ing the operational quality of the grid are utilized. In Paper II, voltage profiles,phase unbalance, peak loading, and total losses are assessed. In Paper III, thephase unbalance is not assessed, but the hosting capacity is studied based onvoltage violation and loses.

Voltage violation is often considered to be the main obstacle to high pene-tration of PV generation and EV charging. According to the European stan-dard [75], rms voltages in the electricity distribution system should be withinthe range of 0.90 - 1.10 pu during 95% of the time on a weekly basis. This

20

percentile, however, depends on the tolerable amount of risk that will be takenby the stakeholders [31].

There are several definitions of phase unbalance. In Paper II, the phase un-balance is defined as the maximum deviation, among the three phases, betweenphase and average line voltage. The definition and a comparison of several dif-ferent definitions are given in [76]. The safe limit of phase unbalance basedon this definition is 3% [77].

Peak loading is simply the highest demand in the system. Meanwhile, thetotal losses represent the energy efficiency of the system. The Council of Euro-pean Energy Regulator reported that the losses in the power system are higherin the distribution part of the system, which makes it an important parameter tostudy [78]. In the appended papers, the total losses in the distribution systeminclude both line losses and transformer losses.

3.1.3 Energy management systemThere are two EMS scenarios simulated in the appended papers in this thesis:distributed smart charging and PV curtailment. In general, smart or controlledcharging aims to control the EV charging demand. The distributed term meansthat the control is done at the individual building level rather than centrally fora whole area. Meanwhile, PV curtailment limits the PV generation output.

The distributed smart charging scheme utilized in this thesis has as its mainobjective to reduce the net-load variability. With this scheme, flatter andsmoother net-load profiles are expected, and hence, the scheme is expectedto increase the self-consumption and decrease the peak loads. The variabilityis measured by the variance and, therefore, the net-load variability is repre-sented by a variance equation. In the appended papers, the smart charging isbased on the study in [79].

The optimization problem is formulated with a quadratic programming ap-proach considering building load, PV generation, the targeted SoC, and thegiven future departure time. The optimization problem of the smart chargingscheme can be written as

mintdep

∑t=tarr

(xt + lt − st −μt park)2, (3.1)

s.t. ηx

tdep

∑t=tarr

xt ·Δt = SoCtarget −SoCarr,0 ≤ xt ≤ xmax, (3.2)

where tarr and tdep are the arrival and departure times of the car respectively,xt is the charging power rate at time t, lt is the household load at time t, stis the solar power production at time t, μt park is the mean net-load during theparking period including EV charging load. In the constraint, ηx is the charg-ing efficiency, Δt is the time step, which in this case is 15 minutes, SoCtarget is

21

the targeted state of charge (kWh) of the battery, SoCarr is the state of charge(kWh) of the battery on arrival and xmax is the maximum charging power rate.

With this charging scheme, the user needs to state the expected departuretime with a risk of having the EV not fully charged if the EV is used earlierthan the given time. Furthermore, residential load and PV forecasts are neededas well.

The PV generation curtailment is used in Paper III, and that study assessesthe PV and EV hosting capacity in one framework. In this paper, initially, fullcurtailment is shown to illustrate the best scenario in terms of voltage violationand PV hosting capacity enhancement. Later, it is compared to the partialcurtailment to find the trade-off between voltage profile and PV electricityutilization. The result of this comparison, however, is not included in thisthesis but can be found in the appended papers.

In the full curtailment method, the building is not allowed to transfer anyexcess PV generation into the grid. Hence, with this method, the maximumvoltages on the customer side is expected to remain the same. In partial cur-tailment, transfer of excess generation is allowed, but is limited based on theupper voltage limit of the system.

The generated PV power after full curtailment scurt,t can be defined as

scurt,t = min(st , lt + xt). (3.3)

3.2 Modelling the uncertaintyThis section explains how the appended papers treat the uncertainties of eachinput variable. It includes the categorization of the uncertainty, i.e., whether itis aleatory or epistemic and how exactly the uncertainty is modeled. It shouldbe noted that some variables are modeled differently in Paper II and Paper III.

3.2.1 PV-EV penetration and allocationThe appended papers aim to assess the impact of new technologies at severallevels of PV and EV penetration. Here, the PV and EV penetration levels areconsidered as epistemic uncertainties. In Paper II, where smart charging isthe only utilized EMS, ten different penetration levels are considered. It isimportant to note that in Paper II, every house with an installed PV system isassumed to have exactly one EV. The penetration is based on the number ofhouses that have both PV and EV.

One of the contributions of Paper II is the assessment of two different PVallocation methods: concentrated and randomly distributed. These alloca-tions are the consequence of considering the allocation as an epistemic and analeatory uncertainty, respectively. For each method, five different sets of allo-cation scenarios are used for every penetration level. For the concentrated dis-tribution, the five sets of allocation scenarios are concentrated on five equally

22

distributed locations, i.e., one set is concentrated at the end of the network, an-other one is set near the substation, and the other five are concentrated betweenthem.

In Paper III, except in the 0% PV scenario, all houses are assumed to havea PV system. In this paper, the penetration is represented by the PV and EVshare, which is related to the PV system installed capacity and number of EVsat each house. The installed capacity of the PV system is adjusted to eachPV share level. For EV allocation, one EV annual charging demand equals50% of the annual electricity consumption of one household. Hence, a 25%share means that there is one EV load per two households and a 75% EV sharemeans three EVs per two households. The allocation for Paper III is evenlydistributed.

3.2.2 PV generationPaper II and Paper III both use a PV power generation model based on histori-cal irradiance data. The irradiance time-series data is from 2018, measured inStockholm, Sweden, by the Swedish Meteorological and Hydrological Insti-tute (SMHI) [80].

The PV system size in these studies was scaled relative to the intended totalyearly electricity demand, i.e., P/L ratios, which in the appended papers aretreated as epistemic. The P/L ratio distribution, however, is different betweenthe two studies. In Paper II, the P/L ratios are assumed to be the same asthe P/L ratios of buildings in [81]. In Paper III, five different P/L ratios weresimulated: 0%, 25%, 50%, 75%, and 100%.

To generate the PV power generation time-series, the irradiance time-seriesare multiplied with a factor that makes the yearly PV generation satisfy theP/L ratios. The PV power generation st at time t can be expressed as

st = ηpv × It , (3.4)

where ηpv is the PV system efficiency factor times area in m2, and It is the in-cident solar radiation, which in this case is the GHI. GHI data are used as theincident solar radiation profile for simplicity, which means the roof tilt uncer-tainties are neglected. The PV system area is not defined explicitly, becausehere we are only interested in the relative size of the PV system and not whichexact area it would correspond to. The PV system efficiency factor times areaηpv can be written as

ηpv =∑35040

t=1 ltΔt ×P/L(∑35040

t=1 ItΔt), (3.5)

where lt is the electric load at time t, Δt is the time step, which in this caseis 15 minutes, and P/L is the ratio of yearly electricity production to yearlyelectricity consumption. The total number of timesteps used is 35,040, which

23

00.00 06.00 12.00 18.00 24.00Time of the day

2

4

6

8

10

12Pe

rcen

tage

(%)

(a)

departurearrival

(b)

06.00 12.00 18.00 24.00Time of the day

0

20

40

60

80

100

Frac

tion

of ve

hicles

(%)

weekdayweekend

Figure 3.2. User mobility statistics: (a) home-arrival and home-departure times and(b) mean daily fraction of vehicles parked at residential areas. In (b), the light greenarea represents the weekday fraction, the light red area the weekend fraction, and thebrown area just the intersection area between the two fractions.

corresponds to four timesteps within every hour. In Paper II, the electricitydemand included both the household load and the EV charging demand. ThePV power generation was assumed to have a constant power factor of 1.

3.2.3 EV charging modelThe mobility and the charging demand were modeled using a bottom-up ap-proach. The uncertainties of the arrival and departure times, the origin and thedestination locations of the trips, and the distance traveled within the trips weremodeled based on the historical Swedish travel survey data from 2006 [82].

The arrival and departure times were randomly sampled. The daily chargingdemands were calculated as

E = η ×D, (3.6)

where η is the specific consumption of EVs (kWh/km) and D is the dailydriving distance (km). η was set to 0.15 kWh/km in Paper II and 0.16 kWh/kmin Paper III. These consumption rates conform to the usual EV consumptionrates as given in [83].

Meanwhile, D was calculated by doubling a randomly sampled distancefrom the travel survey data. The reason for this is that the distance repre-sented the round-trips from and to home, assuming that each EV travels intwo equally long trips a day, such as from home to workplace and back tohome. The maximum usable energy in the battery was set to 30 kWh. Thiswas assuming that the battery could provide sufficient energy for the trips byEVs within a city. The charging efficiency was set to 90%, which is the av-erage efficiency of Level 2 chargers [84]. The maximum charging power wasset to 3.7 kW which is a typical home-charging power [85].

24

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Figure 3.3. Mean daily load profile with (a) the uncontrolled charging scheme and (b)the smart charging scheme. The figures were generated for a case of a house with aP/L = 0.5.

Home-work-home mobility patterns were used to define the arrival and de-parture times and the charging demands on weekdays, while home-other-homemobility patterns were used for weekends. See [86] for information aboutwhich trips were considered ending at the categories work and other. The usermobility statistics that was used in this study is illustrated in Figure 3.2.

3.2.4 Residential building load modelSynthetic household electricity time-series data for Paper II and Paper III weregenerated with the Widén model [32]. This is a Markov chain-based modelthat generates electricity profiles based on Swedish occupant activity profiles.First, the model generates synthetic activity patterns. Then, the patterns areconverted into power demand. The input, for modeling lighting demand, isthe daylight data that is converted from solar irradiance. The parameters areMarkov-chain transition probabilities which makes the model able to create alarge random set with realistic random diversity.

In Paper II and Paper III, the model was used to generate electricity use pro-files without electric heating for detached houses. In these papers, all housesare assumed to have two adult inhabitants per household. The expected meandaily residential load model, alongside daily PV generation and two differentcharging profiles, are shown in Figure 3.3.

3.3 Modelling the correlationThis section explains how several correlations are modeled in the appendedpapers. Some other correlations are not modeled but observed after the imple-

25

mentation of the EMS, and will be discussed in the Chapter 4. In addition, theinformation on the excluded correlations from the appended papers is given aswell.

The relationship between household loads in Paper II and Paper III is theresult of the Markov-chain model being used in both papers [32]. The cor-relation is not quantified explicitly but the model has been proved to producea realistic diversity between households even with a small household data set(the diversity between only 14 household sets was tested).

The correlation between PV systems in Paper II and Paper III is one, i.e.,a perfect correlation is assumed. It is of course not 100% accurate and isprone to overestimate the PV impact as studied before for a larger system [15].However, the correlation of PV in a small system and its impact on the gridhas not been studied before, which makes the exact accuracy of the assumptiondifficult to quantify.

Meanwhile, the variation in the EV mobility data in both Paper II and PaperIII comes solely from variability inherent in the historical travel survey data.The sampling of arrival and departure times are not correlated. In Paper II,the average correlation of EV charging load between nodes in an uncontrolledscheme is calculated to be 0.103.

In general, Paper II and Paper III do not model the inter-variable correla-tions. Because the correlation between household load and EV is not modeled,it is possible that the model generates a mismatch scenario. For example, thehousehold load may assume that all household members are present in thehouse while the mobility model states that the EV is still used. This is a limi-tation of the model, and improvements on this is left for future work.

The correlation between load and PV is not modeled either. The syntheticload model, however, takes into account the daylight which is converted fromthe irradiance data from the same location. The correlation between EV charg-ing load and PV generation is not modeled, but is determined to be almostzero for the uncontrolled scheme in Paper II. As explained in Section 2.4.2,the synergy between PV systems and EVs due to the introduction of an EMSis likely improving the hosting capacity of both technologies. The impact ofone technology on the other’s hosting capacity is assessed in Paper III.

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4. Results

This chapter summarizes the main results from the appended papers. The sec-tions are ordered based on the results in the papers. Section 4.1 summarizes theresults of Paper II, and present the probabilistic analysis of EV smart charg-ing with increasing penetration levels. Two allocation methods are compared:distributed and concentrated allocation. The correlations between nodes wereobserved as well. Section 4.2 present the results of Paper III regarding thecombined PV-EV hosting capacity. Two impacts of two different EMS arecompared: EV smart charging and PV curtailment. The impact of one tech-nology on the other’s hosting capacity is assessed as well.

4.1 Probabilistic analysis of EV smart charging withdifferent allocation methods

This section summarizes the results from Paper II. It consists of voltage pro-files, phase unbalance, peak loading, total losses, and the correlation betweennetwork nodes.

4.1.1 Voltage profilesThe voltage profiles in the distribution grid obtained when the PV and EV pen-etration level was increased in Paper II are shown in Figure 4.1. The voltagedeviation increased across all phases when the penetration of PV and EV wasincreased and the voltage deviation and range of uncontrolled charging werehigher than those of smart charging.

The impact of different allocation methods on the voltage deviation washigher for the lower penetration levels. When PV and EV penetration wereless than 55%, the voltage deviation for concentrated allocation was higherthan for distributed allocation. Concentrated allocation also gave higher maxi-mum voltages for the lower penetration levels. The maximum voltages startedto exceed the allowed level of 1.1 pu when the penetration was equal to 40buildings in distributed allocation. Meanwhile, in the case of concentrated al-location, it exceeded 1.1 pu when the penetration was equal to 25 buildings,i.e., 3 penetration levels earlier.

The minimum voltage in the uncontrolled charging cases was always lowerthan that in the smart charging cases and the difference was increasing with

27

Figure 4.1. Comparison of violin plots of the phase voltages for different chargingschemes (smart and uncontrolled charging) and different allocation methods (concen-trated and distributed). Five out of ten penetration levels were presented for a simplerpresentation (even n only), with n being the number of houses with PV and EV. Thehorizontal marks of each violin plot represent the minimum and maximum values.

increasing PV and EV penetration. In addition, the minimum voltage in thecase of uncontrolled charging with concentrated allocation was always lowerthan with distributed allocation. The minimum voltages, however, were stillabove the allowed level of 0.9 pu even at the highest penetration level.

During most of the time in the higher penetration cases, the voltages werestill within ±5% of the rated value. For smart charging, the 5% percentileswere 1.025 with both allocation methods and the 95% percentiles were 1.057pu with distributed allocation and 1.058 with concentrated allocation. For un-controlled charging, the 5% percentiles were 1.019 with both allocation meth-ods and the 95% percentiles were 1.057 pu with distributed allocation and1.057 with concentrated allocation.

4.1.2 Phase unbalanceThe phase unbalances with respect to increasing PV and EV penetration onthe tested system are shown in Figure 4.2. The mean values of the phaseunbalance are given in Table 4.1. It can be observed that increasing the PVand EV penetration amplified the phase unbalance, and the phase unbalancewith concentrated allocation was higher than with distributed allocation. Inaddition, smart charging helped reduce the phase unbalance.

In all PV and EV penetration scenarios, smart charging helped reduce thephase unbalance. Smart charging also significantly decreased the maximumphase unbalance and the number of events in which the phase unbalance washigher than 3%, which is considered as the safe limit [77]. This safe limit,

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Figure 4.2. Comparison of violin plots of the phase unbalance for different chargingschemes: (smart and uncontrolled charging) and different allocation methods (con-centrated and distributed). The left and right figures represent distributed and con-centrated allocation, respectively. Ten penetration levels were considered. Red andblue represent smart charging and uncontrolled charging, respectively. The horizontalmark of each violin plot represents the maximum value.

Table 4.1. Mean values in % of the phase unbalance for different charging schemes(smart and uncontrolled charging) and different allocation methods (concentrated anddistributed). Ten penetration levels were considered.

Houses with PV 5 10 15 20 25 30 35 40 45 50

Uncontrolled, distributed 0.32 0.35 0.39 0.43 0.49 0.50 0.51 0.58 0.61 0.60Smart, distributed 0.30 0.31 0.34 0.37 0.42 0.41 0.41 0.47 0.48 0.44Uncontrolled, concentrated 0.35 0.39 0.45 0.47 0.51 0.53 0.53 0.58 0.57 0.59Smart, concentrated 0.34 0.36 0.41 0.41 0.43 0.45 0.42 0.46 0.43 0.43

however, was exceeded more often with concentrated allocation than with dis-tributed allocation.

During most of the time, the phase unbalance was lower than 3%. The 95%quantile of the phase unbalance of uncontrolled charging ranged from 0.87%for the lowest penetration level to 1.70% for the highest penetration level.

4.1.3 Peak loading and total lossesThe peak loading and losses for all cases and penetrations are presented inFigure 4.3. As expected, the loading and losses increased with increasing PVand EV penetration.

As regards the peak loading, the smart charging impact was insignificant atlower penetration levels, but the impact of smart charging was increasing withincreasing penetration level. At the maximum penetration level, the peak loadwas reduced up to 57%. Similarly, the reduction of losses due to the smartcharging scheme is less significant in the low penetration scenarios comparedto the higher penetration scenarios. This is basically due to fewer flexibleloads, i.e., EVs, involved in the lower penetration scenarios than in the higherpenetration scenarios. For the active power, the loss reduction ranged from 59

29

Figure 4.3. Peak load and losses for different charging schemes (smart and uncon-trolled charging) and different allocation methods (concentrated and distributed). Tenpenetration levels were considered.

kWh for the lowest penetration level to 795 kWh for the maximum penetrationlevel.

In terms of allocation methods, the concentrated allocation scenarios havehigher losses and peak load in all uncontrolled charging scenarios and higherlosses in the smart charging scenarios. For the peak load in the smart charg-ing scenarios, however, the differences were insignificant. This implies thatthe reduction of peak load due to the smart charging scheme is higher withconcentrated allocation.

4.1.4 Correlation between network nodesThe average values of the correlation coefficients for EV charging load andvoltages, respectively, between all pairs of bus nodes, are given in Table 4.2for uncontrolled charging and smart charging. It can be observed that smartcharging slightly increases the correlation between nodes for both EV charg-ing load and voltages, even though there is no communication between nodesapplied, as in the case of centralized smart charging. This is due to the PVpower generation being completely correlated between buildings, so control-ling the charging to match the PV generation leads to more correlated chargingprofiles. The correlation between the EV charging load and the PV generationis given in Table 4.3. Similarly, the correlation is higher in the smart chargingscheme than in the case of uncontrolled charging.

30

Table 4.2. Average values of the correlation coefficient for EV charging load andvoltages, respectively, evaluated and averaged over all pairs of the 55 network nodesfor both uncontrolled charging and smart charging

Variables Uncontrolled charging Smart charging

EV charging load 0.103 0.164Voltages 0.987 0.995

Table 4.3. Correlation between EV charging load and PV generation for uncontrolledcharging and smart charging

Uncontrolled charging Smart charging

0.007 0.141

4.2 Combined PV-EV hosting capacity with EV smartcharging and PV curtailment

This section summarizes the results from Paper III. It consists of voltage pro-files, system losses, and the combined PV-EV hosting capacity. Some resultsfrom the paper are not included but can be seen in the appended paper.

4.2.1 Voltage profilesFigure 4.4 shows the voltage probability distribution for a variety of EMS andseveral shares levels of PV systems and EVs. It can be seen that even withoutEMS, there were no voltage violations if both the PV and EV share were below75%.

The smart charging scenario was able to avoid the undervoltage problem inthe scenarios of 100% EV share. Smart charging could also lower the maxi-mum voltage if the PV share was low, as can be seen if we compare the 25%PV share without EMS and with smart charging only scenarios. With PV cur-tailment, the overvoltage problem in the scenarios of a 100% PV share can beavoided.

4.2.2 System lossesSimilar to the results in Paper II, the losses increased with increasing EV shareand the smart charging once again decreased the grid losses. However, it isinteresting to see that for PV without curtailment, losses first decrease to aminimum, then start increasing again as shown in Figure 4.5. This was notfound in Paper II because Paper II assumes that the shares of PV and EVare increasing together. This shows that the PV power injected into the gridreduces the losses if the PV size is optimal. If the PV production is too high,

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Figure 4.4. Voltage probability distributions in different EMS scenarios and with com-bined PV-EV shares.

the losses then start increasing again. The PV curtailment reduced the losses,but the losses did not include the curtailed electricity.

4.2.3 Combined PV-EV hosting capacityThe combined PV-EV hosting capacity estimation with different EMS scenar-ios is shown in Figure 4.6. Please note that the x-axis and y-axis are limitedin the figures for a more focused analysis. It can be seen that the smart charg-ing scheme not only increased the EV hosting capacity significantly but alsoincreased the PV hosting capacity slightly. Meanwhile, the PV curtailmentincreased the PV hosting capacity but did not increase the EV hosting capac-ity. There is a slight correlation between PV and EV hosting capacity, whichmeans that the increased PV share improved the EV hosting capacity and theincreased EV share improved the PV hosting capacity.

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5. Discussion and future work

This chapter firstly discusses the contribution of this thesis. Then, furthersuggestions on future work in this area are presented.

5.1 DiscussionPV systems and EVs are rapidly penetrating electricity distribution systemsaround the world. To allow more of these technologies in our power systemsafely and efficiently, studies in PLF analysis of future systems are vital forgrid operators, infrastructure planning, and policymakers.

In this thesis, PLF approaches in the analysis of power distribution systemswith PV systems and EVs are reviewed in Paper I, including the uncertaintyand correlation models. Paper II performs a PLF analysis of EV smart charg-ing in a electricity distribution system with PV and compares the impact ofallocation methods. Paper III complements the previous research with PV cur-tailment methods, another allocation and penetration approach, and introducesa combined PV-EV approach.

Paper I provides an overview of the state of the art in PLF approaches forelectricity distribution systems with PV systems and EVs. Improving the inputdistribution modeling, developing an effective spatio-temporal model, and theneed for more data are among the research gaps identified in this paper.

Paper II shows the probabilistic impacts of smart charging and spatial al-locations. According to the result of this paper, the smart charging scheme,which has as its main objective to reduce the net-load variability, improves theLV distribution system performance without energy storage by decreasing thevoltage deviation, reducing the phase unbalance, cutting the peak loading, andlowering the total losses.

In this paper, the voltage profile is closer to the upper safe limit than to thelower safe limit. With smart charging, the system will be able to lower thedefault voltage value which was set at 1.04 pu to lower the risk of overvolt-age. An increase in correlations between network nodes is also observed dueto the smart charging scheme, which is important for PLF studies with theprobability distribution method.

Regarding the spatial allocation, the results show that concentrated alloca-tion has more severe impacts, in particular at lower penetration levels. Hence,it is suggested for future studies to consider the spatial allocation as an epis-temic uncertainty instead of aleatory, especially if the hosting capacity of thenetwork is low.

35

Paper III shows that for the tested grid, the presence of two EVs and 100%share PV system at every house is still able to be hosted with EV smart charg-ing and PV curtailment. The number can be further increased but more simu-lations are needed.

This study also shows that smart charging increases the PV hosting capacityas well, but PV curtailment only allows more PV. A correlation between PVand EV hosting capacity was found. Hence, probabilistic impact studies aresuggested to assess the impacts of both PV and EV in one framework insteadof assessing the impacts independently, especially with the presence of severalkinds of EMS that increase the interaction between variables.

5.2 Future workThis thesis shows that many aspects of uncertainties and correlation in PLFanalysis are still largely unexplored. These research gaps come from the needto include other variables, comprehensive sensitivity analysis of the variableuncertainties, further comparison of different methods, integration of moretechnologies, improvements regarding the accuracy of the model, and com-parison with real grid performance data.

Several uncertainties in the system were not modeled in this thesis. Furtherstudies that include more variables would be beneficial. Some of the uncer-tainty variables include the roof condition (orientation, shading, etc), the PVinverter parameters, EV charger parameters, and background voltage.

It is difficult to choose which uncertainties to add and which to not includebecause the impact of all uncertainties is not yet known. Hence, one of thevital suggested possible future works is a comprehensive sensitivity analysisof as many input variables as possible in one test framework. Then, this studycan suggest which uncertainties have a significant effect and need further de-velopment, and which uncertainties can be neglected in PLF analysis.

Two distinct methods used in load flow studies considering uncertainties aretime-series and probability methods. The time series method generates time-series using synthetic data for PLF input, while the probability distributionmethod directly sample input data from PDFs, usually considering only a fewworst times. A comparison of these two methods in PLF is also beneficial.However, a fair comparison is difficult to carry out due to the lack of data fromactual physical systems. This comparison would also aid the development ofa method to include EMS in the probability distribution method.

Integration of more technologies in these studies would also interesting.Some possible technologies that could be included are energy storage, com-bined heat and power systems, EV smart charging with different objectives,and reactive power control.

To increase the accuracy of the uncertainty and correlation modeling, someimprovements can be applied: i) The use of a spatial adoption model of PV

36

systems and EVs will help to reduce the allocation uncertainties that are provenin this thesis to have a significant impact on the grid. ii) The spatial correla-tion of PV systems in a small system and its impact on the grid has not beenstudied before, but it requires several measurements in one small distributionnetwork. iii) Integration of PLF analysis with geographic information systems(GIS) will help in power network planning and can include more technologiessuch as EV public charging and more complex load at the city level. iv) Syn-chronization of EV and load models in a residential building, e.g., whetherthe occupant is present at home or not, will also improve the quality of thetime-series model.

The availability of real data for input variables such as solar PV, EV charg-ing, mobility patterns, and load, in general, is crucial to improve the quality ofuncertainty and correlation modeling for PLF analysis.

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6. Conclusion

This thesis contributes to the knowledge regarding the state-of-the-art of prob-abilistic approaches in load flow analysis for electricity distribution systemswith PV generation and EV charging. The growth of distributed PV genera-tion and EVs increases the uncertainties in the electricity distribution system,and hence, probabilistic methods are more appropriate. Another important fac-tor to increase the quality of the load flow analysis is the correlation betweeninput variables.

In this thesis, properties of spatio-temporal uncertainty and correlations forPLF analysis have been investigated. The uncertainties from PV-EV alloca-tion, PV generation, EV charging, and residential building load have beenmodeled. The results indicate that concentrated allocation of PV systems andEVs has more severe impacts, in particular at lower penetration levels. Therewas no correlation between EV charging load and PV generation in the modelwhen there was no EMS applied. The slight correlation between PV and EVhosting capacity showed that hosting capacity analysis of EVs and PV benefitsfrom being done in one single framework.

Probabilistic impact analysis of EV smart charging with an objective ofminimizing the net-load variability has been simulated in this thesis for in-creasing levels of penetration. The results show that this smart charging schemeimproves the electricity distribution system performance. Furthermore, an in-crease in correlations between nodes is also observed due to this smart charg-ing scheme.

Combined PV-EV hosting capacity assessment shows that a combination oftwo EMS strategies, EV smart charging and PV curtailment, in all buildingscan further improve the voltage profile and increase the hosting capacity. Thesmart charging scheme also increased the PV hosting capacity slightly.

Overall, this thesis concludes that an improvement of uncertainty and corre-lation modeling is vital in probabilistic load flow analysis of future electricitydistribution systems. There is a lot of room for improvement in PLF anal-ysis, but it mainly requires actual physical system measurement data for thedevelopment and improvement of the probabilistic model. Further research issuggested, such as integrating more uncertainty variables, conducting compre-hensive sensitivity analysis of input uncertainties, comparing different meth-ods, including more technologies, improving the accuracy of the model, andcomparing the results with real output data.

38

Acknowledgments

Alhamdulillah, all praises to Allah, the Omniscient, and the Absolute Truth.

I want to thank my supervisor Joakim Widén, and my co-supervisor JoakimMunkhammar, for their outstanding help, support, guidance, and leadershipduring this process. Your ability as supervisor, teacher, researcher, scientificwriter, and organizer really amaze me. I also wish to thank my co-authors-beside my supervisors- Mahmoud Shepero and Reza Fachrizal for their valu-able contribution to my work. Thanks to all of my colleagues at the Divisionof Civil Engineering and Built Environment, for the great working atmosphereand fun after-work activities, even during the pandemic!

I would like to thank Swedish Centre for Smart Grids and Energy Stor-age (SweGRIDS) for funding this PhD project. I would like to also thankmy industrial supervisor, collaborator, and co-author in this project, NicholasEtherden from Vattenfall Research and Development.

Special thanks to my family in Indonesia, especially my mom, for all prayers,unconditional love, and everlasting trust. Thanks for believing in me, evenmore than myself.

Lastly, for my dearest little family. My life-long learning and adventurepartners, from 42◦C in Canberra to -24◦C here in Uppsala. Thank you somuch for everything. I am grateful to have you in my life.

39

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