The effect of time resolution on the modelling of domestic solar energy systems

. This paper investigates the modelling accuracy of small-scale solar energy systems, in particular the effect of changing the time resolution. Accurate modelling of solar energy systems is important in designing cost-effective solutions and in research into micro-grid functionality. Currently, most modelling uses a resolution of one hour when simulating the energy in small scale renewable energy systems. Within each hour, these models take an average of the load demand and solar generation values. An hourly time resolution is ignorant of the large variations in a daily load pro ﬁ le that result from high power appliances and also misses short-term variations in solar generation. Few researchers using hourly modelling consider the implications in terms of the simulation accuracy. In this paper, 3 load pro ﬁ les are modelled with grid connected solar systems, one with and one without a battery, totalling 6 models. Each model is simulated for 4 seasons of the year, modelled at hourly, 5 min and 1min time resolution. The results show that the hourly model was a poor predictor of battery behaviour, over-estimating the battery state of charge (SoC) by up to 10%. It is also shown that, for systems without battery, the quantity of energy exported and self-consumed changed by an average of 5% and 8% respectively. The study concludes that increasing time resolution from hourly to 5-min resolution in energy models would be advantageous, increasing their accuracy in terms of battery behaviour and predicted self-consumption of solar energy. In particular, modelling at 5-min instead of hourly resolution shows the full bene ﬁ ts of installing battery storage with solar systems.

1 Introduction and background 1

.1 Motivation
Global demand for energy is projected to continue to increase through the first half of this century. To meet these energy demands, new capacity must provide electricity while reducing carbon emissions to zero wherever possible. Renewable supplies such as solar will need to be deployed, often with battery storage to overcome the limits of intermittent generation. Some researchers predict growth of PV capacity from 0.73 terawatt (TW) in 2020 [1] to 30 TW by 2050 [2]. Despite current growth rates, renewable sources still account for around one quarter of global electricity generation and a much lower proportion of the primary energy supply. Most production is dominated by fossil fuels, which are responsible for providing over 60% of global electrical generation, emitting unsustainable quantities of greenhouse gases.
Relative to traditional sources, solar energy can provide a sustainable and cost-effective energy supply for many countries. Due to its distributed nature, using solar as a resource also offers a great way to tackle global energy poverty. Around 800 million people across the globe still lack access to electricity, and this limits opportunities for them to move out of poverty, as explained by other scholars [3,4]. Energy deprivation in the home deepens the divide between those of different social backgrounds, as the disadvantaged lose out on tools that can support economic growth and improve on quality of life.
In order to disseminate solar PV technologies to new markets, we need effective tools to model these systems. We also need to understand the factors that will limit the accuracy of results from typical simulations. A few researchers have identified issues with simulations that are based on hourly time resolution, especially Cao and Sirén [5]. A study by Meybodi et al. [6] looked at optimum time steps in modelling of Concentrated Solar Power (CSP) plants. It found that in many cases sub-hourly time resolution is necessary, although plant characteristics are quite different from single home PV systems.

Example simulations
As funding is directed towards sustainable development projects, the need to effectively model new energy systems is becoming greater. One example of modelling software is HOMER. The supplier's website states that "HOMER Grid serves the grid-connected market with cutting edge algorithms for optimizing solar, storage, and more to reduce your overall energy costs" [7]. It has over 200,000 users in 193 countries, making it a leader for the design and optimization of micro-grids. The software makes energy balance calculations based on a predicted load profile and the available energy resource defined in the model. It can simulate what happens over the course of a year, providing time steps of up to one minute. However, the load profiles available in HOMER are only modifiable hour-by-hour.
A good example of HOMER in use is described by Amutha and Rajini [8]. They explore electrification of a rural community in India by comparing the economics of extending the national grid infrastructure against building a hybrid renewable energy system. The load profile of the village is constructed from state government records detailing what buildings, industries and institutions are present. The domestic electricity demand is from appliances such as fans, TVs and refrigerators, whilst the commercial/industrial demand is from activities such as schools, shops, streetlights, milk processing plants, agriculture etc. There is also a telecom tower. This load profile is displayed in Figure 1 [8]. According to the simulation results, the designed hybrid renewable energy system in combination with energy storage is capable of meeting the load demands and is cheaper than extending the grid.
The software is also used for the assessment of sustainable projects in the developed world. A study by Kästel et al. [9] uses HOMER to assess the influence of selfconsumption on the economics of electricity prosumers in the UK. It uses two load profiles generated by Elexon Ltd, one for a UK domestic load and the other for a small UK non-domestic load. They have been adjusted for the seasons and have a time granularity of 30 min. An example of this load data is shown in Figure 2 [9], which represents the domestic load profile during the winter season for Wednesday, Saturday and Sunday. The results conclude that when the levelized cost of solar electricity is lower than Fig. 1. Hourly load profile as input for renewable energy system model [8]. the grid supply price, high self-consumption levels provide cheap electricity and protect the consumer against energy price inflation.
The load profiles in both papers are ignorant to large short-term load spikes produced from higher power appliances, as can be seen in Figures 1 and 2. It is very common for simulations to operate at no higher than a halfhourly time granularity, as can be seen in many other studies [10][11][12][13]. In reality, a daily load profile has a lot more diversity in power demand throughout the day. Several common household appliances have power ratings of 2-3 kW, which cause large spikes that may last for only a few minutes. This does not have much impact at the network level, where household demands are aggregated, but will have an impact at the individual household level.
One solution to introduce solar energy into a grid network is for a household to install its own photovoltaic (PV) array as part of a larger smart grid system. Smart grids can offer greater reliability and are more sustainable compared with the conventional centralised power grids operating today [14]. The PV array acts as the primary energy source for the building, exporting any surplus electricity back into the grid, whilst the grid provides any energy that cannot be generated by solar. With the addition of energy storage, these systems are capable of supplying power to commercial or residential dwellings and relieving stress from the electric transmission and distribution system [15]. A recent paper by Ahmed [16] explains how multiple homes could be connected to an energy storage "community". In this case, modelling has been done at 1 min resolution to accurately simulate the battery performance. Another study from Canada, looking at what would be needed for a community of retrofit homes to reach "Net Zero" has used 5-minute time steps for PV and loads [17]. In each case, energy storage is installed to alleviate intermittency issues that arise from renewable sources.

Published analysis of time resolution for PV and load simulation
The paper by Cao and Sirén [5], gives a comprehensive overview of the issue of modelling accuracy with varying time resolution. It uses data from a UK database which is accessible through the CREST institute at Loughborough University [18] and also uses data from a house in Finland where PV output and load demand were monitored each minute through the day.
The conclusion reached in this paper is that significant error can occur when 1-hour time resolution is used to model domestic PV systems. In all cases the use of hourly averaged data caused an overstatement of the selfconsumption from the solar system. A theoretical analysis explains why this is the case, based on assuming that peak loads are a factor of 8 times the base load. Within the current research we present further analysis for varying values of peak load.
The study of Cao and Sirén appears to be the only one that analyses the effect of using different time resolutions. The study also has some limitations, which are recognized by the authors. Their research only looks at individual days, in Summer and in Winter, which limits the possibility of drawing more general conclusions. There is no estimate of what the error in modelling would be over a complete year. Most of the analysis is for systems without battery, but in the future it is likely that battery storage will become commonplace for houses with PV systems. One system with battery is analysed but uses an ideal battery model with zero energy loss and no current limits.
Other published research stresses the importance of choice of time resolution but does not carry out an analysis to evaluate the effects of using different resolutions. Typical load profiles at shorter than hourly resolution contain large variations, as seen in the paper by Murray and Stankovic [19] which comments on the importance of time resolution for UK households with and without solar PV. Alrawi et al. show how 5-minute resolution highlights variations in load and also in PV output for homes in Qatar [20]. A review paper by Luthander [21] also comments on the fact that use of hourly averaged values causes errors in system modelling, especially an overestimation of selfconsumption by a household with its own solar PV system. However, the review does not evaluate the overestimation. Another review paper by Hoevenars and Crawford [22] covers a wide range of research into renewable energy systems with various time-steps used in modelling, but without a systematic comparison using the same data with different time resolutions they were unable to draw definitive overall conclusions or make a recommendation for the best time step to use.
The current paper therefore sets out to take this investigation further, using reliable household and solar data and analysing for four seasons of the year and over consecutive days, with and without battery storage, modelled with three different time resolutions. It also looks at the case of solar home systems in a tropical country, which has not been investigated elsewhere.

Research objectives
The aim of this research is to assess how the time resolution for the model of a small-scale solar energy system effects the accuracy of predicted behaviour. With that in mind, the objectives are as follows: review existing research on modelling of domestic solar systems and the theoretical analysis relating to impact of time-step for simulation of operating behaviour; produce a model for a domestic solar energy system, with and without battery storage, which can be used with realistic load profiles; gather or generate load and generation data for households with PV in the UK and in India suitable for simulating to a resolution of 1-minute; simulate each system, with and without battery, operating over an appropriate sample period of each of four seasons throughout the year; compare the simulation results using time steps of one hour, five minutes and one minute to measure how time resolution effects the performance and annual outputs; compare the results with those of other researchers, where appropriate, and draw conclusions on the implications of the results.
2 Method of analysis and theoretical basis 2.1 Resources

CREST demand model
The Centre for Renewable Energy Systems Technology (CREST) [18] at Loughborough University has designed a model capable of producing one minute resolution demand data of a dwelling. It is an open-source development in Excel VBA that is free to access from the CREST web page. As the software is capable of producing a load profile for residential buildings in the UK and India, it was decided to model 3 load profiles; UK, Rural India and Urban India. The appliances used in each dwelling have been listed below. For the UK, this is randomly selected by the CREST demand model, however for the Indian profiles, the simulation was run multiple times to find comparable appliances to that described in case study papers [21,23].
Appliances À United Kingdom Appliances À Urban India [21] Appliances À Rural India [11] When looking at the appliances, what is important to note is the quantity of appliances in the UK compared to India, as well as the use of items such as electric oven, kettle, washing machine, dishwasher, shower, vacuum and microwave. These are all high-power devices that are responsible for the large load spikes. The CREST Demand Model assumes that the lighting load demand in India is 27.5% of that in the UK. Note that the rural India profile uses fans, whilst the urban India house uses air conditioning for cooling, which will generate some larger load spikes. Further corroboration of appliances used in rural households can be found in a paper by UNDP and ETH Zurich [24] in which "typical" rural household electricity use for India or Kenya is given as 3 lights (each 6 W), mobile phone charger (5 W) and fan (10 W); every 5 th house has a TV (60 W). Actual load data from India presented by Bhatti and Williams [25] shows a similar pattern, while Barman et al. [26] note household wishes (fans, mobile phone charging) when power is available for more than lighting.
Another important variable that can be selected in the CREST model is the building index. This is important as it defines the thermal properties of the dwelling, which in turn impacts how the cooling systems operate. The average house in the UK is a semi-detached house [27] with an average occupancy of 3 [28]. The urban India dwelling is equivalent to a large flat, and the rural India dwelling is equivalent to a small brick bungalow. The average occupancy is 4 and 5 respectively [29]. It has been decided to bulk the Indian load profiles to represent 6 households together, inspired by the modelling from studies previously cited. Due to the lower load demands, a single battery is often used for several houses to reduce costs.

Generation of daily profiles
An example of a load profile generated by the CREST Demand Model is shown in Figure 3 [18]. This is a highresolution model of domestic electricity that uses daily activity profiles and domicile active occupancy patterns to create a simulation of appliance use. The variation in load is much greater than for the hourly profiles (Figs. 1 and 2), with noticeable spikes in the one-minute Air Conditioning TV Lights Fans TV Lights resolution graph that do not appear with longer time resolution. This means that there is a disparity between using software such as HOMER and the actual operation of the system. One of the limitations of the CREST demand model is that it produces unique daily profiles by randomly selecting which appliances are in use each day, meaning there can be large inconsistency between the load profiles for the same house from one day to the next. Also, it is not possible to model a continuous week of operation, varying the occupancy activity at the weekend. To avoid this issue, single daily load profiles were repeated over the course of 3 days and used for every season. It is important to model more than one consecutive day, so that the impact of battery storage is carried forward from one day to the next. Other issues with the demand profile are that it under-represents seasonal variation and nightly consumption, as stated in the original report [18].

Solar data
To match the high time resolution of the load data, it was decided to use high resolution solar data. It is difficult to find solar data that is of a higher time granularity than hour-by-hour, however Solcast [30] can provide solar data to a precision as high as every 5 minutes. Normally, the data is available for purchase, but they offer free credits for student research projects. This was enough to retrieve one week for each season of high precision solar data, set in Leeds at the co-ordinates (-1.6653, 53.86181). The azimuth angle was set to face south at 180 degrees, and the surface tilt angle was set to 43.2 degrees.
Once it was decided to complete models for locations in India, solar radiation data had to be retrieved from another source as there were no credits remaining for Solcast. Hourly solar data was then retrieved from the EU PV Geographical Information System (EUPVGIS) [31] for 3 days of each season, centred around the summer and winter solstice. The co-ordinates used are (28.6, 77.2), which is New Delhi in India. The tilt angle is 31 degrees and the azimuth is 180 degrees facing south. As the data was recorded in a different time zone to that of New Dehli, all data points had to be moved forward by 5 hours.
It was required to convert the hourly data to 5-minute resolution, and the 5 minute resolution data to one-minute resolution.
Based on a moving average, the first conversion used the formula: where Hour A is solar radiation for an initial hour, and Hour B is radiation for the hour after. N is the value from 1 to 12, representing each 5 min interval between the hours. This assumes that in India the solar radiation changes gradually from one hour to the next. The same process was carried forward when converting the 5 minute data to every minute using a similar formula: where 5 min A and 5min B represent the 5 min intervals, and m is the value from 1 to 5 representing each minute interval. This process assumes that changes in solar radiation are gradual, with no short-term cloud cover, which may reduce the impact of changing time resolution.

Systems
The usual form of energy storage for solar PV is the use of batteries. They store energy within an electro-chemical structure through charge transfer reactions, which occur at the positive and negative electrodes of each cell. An electrolyte enables ion transfer between the electrodes and an external circuit carries the liberated electrons back to complete the process [32]. The proportion of energy left within a battery is called the State of Charge (SoC). Accurate estimation of battery SoC remains a challenge in battery research.
There are multiple ways of estimating battery SoC. Detailed knowledge of battery characteristics and physical properties would be needed when using methods such as coulomb counting or electrochemical impedance spectroscopy [33]. A basic battery model with ideal voltage source and constant internal resistance [34] is not useful as these parameters could not be reliably found. For this reason, it was decided to use a simpler battery model. Just as in HOMER, the SoC is calculated by summing the electrical energy (Wh) of the system over a given time interval, and assuming that the battery operates with constant efficiency.
Another issue with modelling of battery charging is the variable loss in the converter, as highlighted by Panagiotou [35]. However, for most systems the converter loss is less than the battery loss and has therefore been integrated as a constant efficiency in the model used here. In [35] the measured internal resistance of the battery has been used for R b , which neglects the main battery loss due to electrochemical reactions and therefore overstates the relative significance of the converter loss, especially for lead-acid batteries.
The three load profiles are applied to two variations of the same system. The models are a simulation of a PV array connected to a larger grid network. In one system, there is energy storage via batteries and in the other there is none. This is shown in Figures 4 and 5.
The basic operation of the system with battery is that any energy produced by the PV array is prioritised to be consumed by the load. Should there be any surplus energy, it is then directed to charge the battery. If either there is no battery or the battery is fully charged, the surplus energy is instead exported to the grid. When the PV output is not able to meet that load demand, the battery will then discharge to provide the required power. In the situations where there is no battery or the battery is at minimum capacity, electricity is then imported from the grid to power the load. Often in the UK, the battery is installed on the load side of the inverter. However, to enable consistency in modelling, the UK load profile was simulated with the system described in Figure 5.
The system specifications varied for each profile and are detailed in Table 1.
The size of the solar array for the UK is based on the average roof area for a semi-detached house (20m 2 ) [36]. The same area was used for the Urban India array (Table 2), which was then reduced for the Rural India array (Table 3) by the same factor as the reduction of power in the load profile.
One important difference to highlight between the UK and Indian systems is the different battery type. As leadacid are cheaper, they are often the chosen form of energy storage in developing countries. They have a lower efficiency than lithium-ion batteries [37] and different charging characteristics. For Li-Ion, the battery should have a minimum SoC of 20% and a maximum SoC of 80% [37]. Lead acid batteries on the other hand should have a minimum SoC of 50% and should be charged to full capacity [38].

Model equations
The equations detailed in this section solve for key outputs: -How much electricity is imported from the grid? -How much electricity is exported to the grid? -What is the battery state of charge at any given time? -What is the self-consumption of the system?
The energy in Wh was calculated at different locations in each system for each time interval. The first point to solve for was the output of the PV array, using the equation from Nelson et al. [39]: where G is the solar irradiance (Wh/m 2 ), A is the area (m 2 ) of the array, h PV is the efficiency of the solar module and h CR is the efficiency of the charge regulator. It has been assumed that the output of the array is not dependant on temperature.
The Net Power (Wh) on either side of the inverter needs to be solved. The net power PV side is found from: where P L (Wh) is the load demand during the given time interval, and h In is the inverter efficiency. The net power on the load side is found from:

Equations À with battery
The first set of equations below detail the modelling of the battery connected system. The battery will discharge if NP PVS is less than 0 and the SoC is above minimum.
IF [(NP PVSn < 0) AND (SoC nÀ1 > SoC min )], then SoC n = SoC nÀ1 À NP PVSn The battery will charge if the NP PVS is greater than 0 and the SoC is below maximum.
then SoC n = SoC nÀ1 + NP PVSn Â h Batt where SoC n is the SoC at time interval n and h Batt is the battery round trip efficiency. Only equation (7) is multiplied by h Batt as it is defined as "the ratio of energy required for charging to the discharging energy needed to regain original capacity" [33].
Electricity is imported from the grid when the system is not capable of meeting the load demand on its own. This is when the NP LS is less than 0 and the battery SoC is at its minimum.

IF NP LSn
Sometimes there will be enough energy in the battery to supplement part of the load demand, and the rest is supported by the grid.
Electricity is exported to the grid when the system is producing more energy than is needed from the load demand. This is when NP LS is greater than 0 and the battery SoC is at its maximum.
Sometimes the excess will be greater than what is needed to fully charge the battery, and the excess would be exported to the grid.
The self-consumption (SC) of a system is the amount of energy produced and utilized without any import from an external source. In this case the external connection is the grid, and the energy produced from the PV array is utilized when supplying the load or charging the battery. The first check is to see if the NP LS is positive, meaning the PV output can supplement the load, and that the battery is not fully charged. If this is the case, then the self-consumption is the load, plus however much the battery is charged by.
In equation (12), if the battery is fully charged, then SoC n À SoC n-1 = 0. If NP LS is negative, then the selfconsumption is simply the PV Output.

Equations À no battery
Below are the equations for modelling systems without a battery. Continuing with self-consumption, equation (13) is also applicable when there is no battery. When the NP LS is positive, the self-consumption is simply just the load.
When there is no battery, any excess electricity is exported to the grid.
Any energy for the load that cannot be met by the PV array, is supplemented by the grid.  Excel was used to compile the data into tables and graphs.

Key assumptions
Within this paper, various assumptions have been made when creating the models. The first is that the PV array performance is assumed to be independent of the temperature. In reality, as the temperature increases, efficiency falls, meaning the PV array's efficiency would increase in winter. Additionally, it has been stated in Section 2.2 that the batteries are assumed to have a constant efficiency and it also assumed that the inverter behaves the same. In practice the inverter efficiency increases with increasing current. This is opposite to battery behaviour, therefore balancing against each other and minimizing the impact.

Theoretical consideration of averaging error
The theoretical basis for the errors caused by hourly averaging is well explained in the paper by Cao and Sirén [5]. They show the error that occurs when there is a load pulse of peak value 8 times the base load. Figure 6 illustrates this case, when there is a step change in load within an hour, with a pulsed load having a duration of 15 minutes and constant generation. As explained in [5], the worst error occurs when the generation is equal to the average load, which in this case is 2.75 times base load. If this were modelled using the average hourly load, all of the load could be met by on-site generation, which implies 100% self-consumption.
In reality, the fraction of the pulse above mean power (5.25 times base load) cannot be supplied during the 15 minutes duration of the peak. The actual value of selfconsumption is therefore only 5.75/11 = 52% of the total load. This represents an error of 48/52 = 91%.
Cao and Sirén define two parameters for which they compare the errors in simulation relative to a time resolution of one minute. These parameters are called on-site energy fraction (OEF) and on-site energy matching (OEM), which are defined as: OEF = Self-consumption/Household load demand OEM = Self-consumption/Total PV generation In the worst case for error, when PV generation is equal to average load, these two parameters have the same value and the error in each is also the same. As can be seen from the example above, this error can, in theory, be large. Note that the load could be split into several shorter pulses of the same total length to create the same error. To check the worst theoretical values, the maximum error has been plotted against the ratio of pulse peak to base load for different pulse durations. The results are shown in Figure 7.
From this theoretical approach, it can be seen that multiple short pulses with high power values are likely to cause the greatest error when hourly averaging is used instead of 1 min time steps.

Results
A total of 6 cases were modelled, with 3 load profiles each simulated in 2 systems. Each system is simulated over 3 days of each season of the year, and each season is simulated with three time resolutions. As an example to show how the results have been compiled and analysed, the results for the UK battery system operating in the month of December are illustrated in the graphs below. Figures 8-10 all show simulations of the system operating over 3 days. On the left y-axis is the energy for a given time interval in watt-hours. This axis is shared by the PV Output (blue) and the Load Through Inverter (green). The right y-axis is the Battery state of charge (SoC À orange) as a percentage. The x-axis is the time over the course of the 3 days, starting at midnight. Figure 8 shows the activity over the 3 days at a time resolution of 1 hour. The time resolution is increased to 5 minutes in Figure 9, and then increased to every minute in Figure 10. Figure 11 shows how the battery SoC changes over the course of 3 days for all 3 of the time resolutions, overlaid on one graph. This graph shows how the modelled battery SoC changes with different time resolutions, effectively summarising the results. In systems without a battery, the net power on the load side is used to summarise instead.
There are key differences in the graphs when increasing time resolution from hourly to 1-minute. Firstly, note that the scale of the y-axis (Wh) decreases. As the time intervals get shorter, the power is multiplied by a smaller value when calculating the energy in Wh. For example, in the hourly graph the peak load is about 2000 Wh, equivalent to a power of 2 kW over the hour period, whereas in the 5 minute resolution the peak is roughly 450 Wh, which is equivalent to 5.4 kW over 5 minutes. At 1 minute  resolution, the peak is around 180Wh, which represents a power of 10.8 kW for a duration of 1 minute. Likewise, the difference between the peaks and troughs gets relatively larger with increased time resolution. With hourly time resolution, large load spikes are averaged out, so that Figure 8 displays fewer details than the higher resolutions, and thus a less complex behaviour. The annual performances of each system have been compiled into Table 4. It shows how each system's output variables change with time resolution. The rows labelled 'Mean' show the average % change of all systems, with and without battery. For systems with battery, the changes in export and self-consumption are sometimes positive and sometimes negative, so the mean values of the changes are small. Based on the magnitudes, hourly to 5-minute mean change of export is 1.8% and 5 to 1 minute mean change is 1.0% for the battery systems. Likewise, the mean magnitude of change in self-consumption is 1.9% for hourly to 5 minute and 0.3% for 5 to 1 minute.
Detailed data for the UK household with battery are shown in Tables 5 and 6. Table 5 shows how much energy in kWh has been imported, exported and self-consumed for 3 days of every season. It then shows what the change in kWh is when improving time resolution; an increase with  shorter time resolution is shown by a positive value. The change is then shown as a percentage of the value at lower resolution. Table 6 uses the results from 3 days, multiplied by 30.42 (average days per calendar month) to give values for 3 months to represent each season. Summing these values gives the annual performance of each system, as detailed in the final row.

Discussion
This section covers two aspects. Firstly, there is a discussion of the key results and the impact of modelling at different time resolutions on overall performance parameters. Secondly there is a discussion of the  implications of these changes in terms of financial returns and choice of system configuration for domestic solar energy.
To see the overall impact of changing time resolution, the results from Table 4 have been analysed. Table 4 shows that there is a much larger change in all variables when increasing time resolution to 5-minute resolution from hourly, than there is increasing from 5-minute resolution to one-minute resolution. Where errors are sometimes positive and sometimes negative it is important to use means of magnitudes rather than arithmetical means to see the pattern in the results. Taking an average for all parameters, and using means of magnitudes where appropriate, changing the resolution from hourly to 5minute resolution (a 12 times shorter resolution) produces a change of 3.8%, whereas from 5-minute to one-minute resolution (a 5 times shorter resolution) gives a mean change of only 0.6%. There is therefore a proportionally greater impact for the initial improvement in resolution from hourly to 5-minute data. This is further re-enforced by the fact that every initial percentage change is either greater than or similar to the second percentage change. Looking at the load patterns it can be seen that most short load spikes initially become visible in the graphs after the first time resolution transition À i.e. they are 5-minutes or longer in duration. The largest errors due to hourly time resolution are seen in export and in self-consumption for the UK no battery and Urban India no battery systems. The rural India systems show much lower error because the loads (lighting, fan, mobile phone charging) do not have large short-term load spikes (see Figs. 19-22 in the Appendix). On average, systems without battery export 5.63% more electricity to the grid than what is predicted from a traditional hourly analysis. Hourly modelling predicts lower imports for systems without battery, which again leads to an overestimation of self-consumption. The impact of these two effects means that hourly modelling consistently overestimates self-consumption relative to real operation. The average error is 8% for the initial change in time resolution, whereas it is only 1% when changing from 5 to 1 minute resolution. The same pattern, in relation to the changes in resolution, is seen for the systems with battery, though here even the largest changes (UK household) are much less (4.4% and 0.8%) and the self-consumption is underestimated when using hourly resolution. The reason for the opposite change in self-consumption in the battery systems is that hourly averaging underestimates the short-term flows of charge into and out of the battery to supply peaky loads or absorb short bursts of solar on mainly cloudy days. In contrast, the pattern for the Urban Indian household with battery shows an overestimate of self-consumption when using hourly resolution. For this case there is a higher percentage increase in imports when modelling at higher resolution, which is probably due to the use of a lead-acid battery which has relatively larger losses than the Li-ion battery modelled for the UK case.
The households with battery generally show smaller changes when modelled at a shorter time resolution. This is mainly because time averaging tends to underestimate opportunities for the battery to discharge (supplying power to house loads) and recharge. The increases in import and export are therefore outweighed by additional losses in use of the battery and inverter. In the case of the September model of the UK battery system, the actual value of export is very small, so a small difference represents a large percentage change. When we look at the raw values, we can however see that the losses from exported electricity are similar to the gains in self-consumption. This behaviour can be seen in the UK battery system for the month of June, looking at the battery SoC graph shown in Figure 12. At the top where the batteries are at maximum charge (80%), during the hourly (blue) simulation the battery remains fully charged, represented by the flat top. The 5-minute resolution (orange) and minute resolution (green) simulations behave differently. The battery can be seen discharging and recharging, represented by the dips at the top. The most significant difference appears during the third day at around 2.30pm, when the battery discharges by about 8% before recharging to full shortly afterwards. A similar behaviour can be seen in Figure 11, showing the UK battery SoC for the month of December. The hourly simulation does not show the battery charging except for during the second day. However, during the 5-minute resolution and minute resolution simulations, the battery SoC spikes by more than 2%.
One of the key differences between the low-and highresolution load profiles, is that higher resolution brings out visibility of the full variation in load demand over the day. This can be seen in Figure 13, which shows the net power on the load side of the inverter for the Urban India system with no battery in March. Negative power values show net consumption and positive values net generation. This graph is similar to the example of Figure 10, in that it compares the behaviour of the system simulations at all time resolutions. We can see in Figure 13 how improving the resolution brings out load fluctuations produced by an appliance. CREST does not detail a single appliance load profile, but this is likely to be a refrigerator as it follows a typical on-off cycle occurring approximately each hour.
With both five-minute and one-minute resolution, the spikes have significantly higher maximum power. Looking shortly after 12:00:00 on the 3rd day, the hourly spike is À2000W, whilst the five-minute resolution spike is close to À7000W and the one-minute spike slightly larger. This is a significance difference of 5000W. This helps to explain why, in the systems without a battery, there tends to be a negative change in self-consumption, as the PV array is not capable of producing this much power. We can also see that the spike is a lot thinner for the higher resolutions. This means that the net power is below 0 for a shorter amount of time, and that PV will actually export to the grid for a longer period of time. The percentage exported therefore increases with time resolution for systems without battery storage.
To see the impact of the time resolution on prediction of financial parameters we need to take into account the incentives for solar generation. Many governments promote the growth of solar energy through feed-in-tariffs or net-metering. In the UK for example, there is a Smart Export Guarantee that has an average price around 5.50 p/ kWh [40]. The hourly simulation predicts the no battery system exports 155 kWh less a year than the one-minute resolution and also imports 155 kWh less. This equates to £8.50 less export payments per household per year, but more significantly an additional £50 paid by each household for imports, based on Autumn 2022 rates of 32p/kWh. The error for the system with battery is in comparison very small, as the largest use of power is in winter when the battery absorbs all of the electricity generated by the solar PV and delivers it back to household loads. Changes to time resolution of the model do not have much impact on this behaviour.
As a solar installation costs a few thousand pounds, an error of £50 per year in imported electricity cost will have a small effect on the payback time. However, a more significant impact of this error, which would result from using hourly data, is that the benefit of adding battery storage to a solar system is underestimated. For the system without battery, hourly averaging overestimates self-consumption by over 12% but for the system with battery it underestimates self-consumption by 5.6%. Currently the installation of battery storage is faced with large economic challenges, with a 10 kWh Li-Ion battery costing more than £3,500 [41]. If the benefit of having batteries with solar is underestimated due to inaccurate modelling, this may reduce take-up of cost-saving storage technology.
There are 28 million households in the UK [42], with rising interest in domestic solar installations. Aggregating the error when using hourly time-resolution, which underestimates solar exports from installations without battery, could impact the modelling of voltage variations in distribution networks. Voltage fluctuations can be reduced significantly if energy storage continues to become cheaper and becomes a more consistent component in domestic renewable energy systems. On the contrary, if batteries do not further penetrate the domestic solar market, variable export of power will increase.
There may be further implications to the errors in battery modelling. Firstly, in relation to battery management. If the software is set up based on hourly modelling, it will not use the battery storage with full efficacy. In future systems where EV batteries are used in grid support mode, this could have significant implications for the effective charging of the EV. Another issue is that a battery is designed to operate for a specific amount of charge cycles. Each time the battery charges and discharges, it undergoes some form of degradation. In the case of a Li-Ion battery, each charge cycle develops the Solid Electrolyte Interphase (SEI) on the negative electrode [43]. The SEI induces loss of continuous lithium-ions and electrolyte decomposition. Similarly, aging occurs with each cycle in lead-acid batteries due to positive active mass degradation, where discharging and recharging morphs its shape [44]. Additional charging and discharging, although relatively small, may reduce expected battery lifetime.

Conclusion
This paper aimed to investigate the impact of time resolution when modelling domestic solar PV systems. The results show that choice of time-step is important when modelling these systems, due to averaging errors. Theoretical analysis was extended and indicates that predicting the ability of PV systems to meet household load demands can have significant errors when an hourly time resolution is used. A detailed study was completed using real solar data and modelled household electricity demand, based on the best freely available data sources. Analysis of the results shows that modelling with an hourly time resolution misses crucial details and overestimates self-consumption. In theory, errors of up to 90% in calculation can occur in the short-term. However, using realistic 3-day cycles, a maximum error of 15% was found over the whole year.
Whereas 1-minute time resolution gives the most accurate results, there is little improvement relative to 5-minute resolution. For most practical cases it is recommended to use 5-minute resolution in preference to hourly as it makes visible the fluctuations in a daily load profile and portrays more accurately the maximum power of the load spikes. These spikes, in systems without a battery, require more power than available from the PV. Further increase to 1-minute resolution would require greater time to process and space for data storage, without significant increase in accuracy.
Modelling with an hourly time resolution consistently understates both export and import for households with solar systems but no battery. The estimated self-consumption is therefore overstated when modelling at hourly time resolution, by up to 12% over a full year relative to 5-minute modelling. This could impact predicted financial benefits for prosumers, since each household is a separate prosumer. If only hourly simulation is available, an allowance for modelling error needs to be made when calculating selfconsumption, based on the values shown in this paper.
When modelling systems with batteries, the changes due to increasing time resolution are generally smaller and of a more variable nature, as the underestimation of battery and inverter losses counter the averaging effect when using hourly time resolution. The result is that hourly modelling understates the increase in self-consumption that can result from battery storage, which may reduce its appeal as a cost-effective technology at household level. The only cases where use of hourly data was found to give acceptable results were the rural households in India, which did not have the types of appliances, e.g., refrigerators, that have short-term load surges.
For systems with a battery, the hourly model misses short-term changes in the state of charge (SoC) of the battery. This could have a significant impact on the design of a smart grid, which will aim to use energy storage over different timescales to optimize power delivery. In addition, it would incorrectly estimate charging of an EV battery from the smart grid. Again, if only hourly simulation is available, allowance should be made when selecting battery size for such systems, as hourly simulations will tend to underestimate the levels of charging and discharging required.
There are also implications for the prediction of longterm health of batteries. Higher resolution simulations show that the batteries charge and discharge more often than is seen with hourly simulations. With both lithium ion and lead acid batteries, this increased cycling of the battery will accelerate aging and reduce battery life.
There are some limitations recognized in the current study, which could be topics for further research. Firstly, there are some limitations to the CREST demand model. It would be beneficial for studies such as this, if the appliances could be selected, instead of being randomly generated. This would enable a consistent, multi-day load profile to be produced and used.
Although the study was only for solar PV generation, the same issues are likely to occur for small-scale wind power generation, especially as there is greater variation in power output at a time-scale of less than one hour. Few households have their own wind generator, but wind power is often a component within a micro-grid, where timeresolution is a relevant factor for accurate modelling. Another area for further research would be the impact of time resolution on the choice of optimum battery storage size and type. For that research, the use of improved battery models is recommended to ensure accurate assessment of system performance.

Implications and influences
The investigation carried out shows that care needs to be taken when choosing the time resolution for modelling domestic solar energy systems. Data for actual systems is analysed to show that modelling using hourly resolution, which is commonly used, may significantly reduce the accuracy of results, especially where there is no energy storage available. In these cases, modelling at 5-minute resolution is recommended.