Optimisation of Residential Solar PV System Rating for Minimum Payback Time Using Half-Hourly Profiling

Optimisation of Residential Solar PV System Rating for inimum Payback Time Using Half-Hourly Profiling HJ Vermeulen, T Nieuwoudt Abstract There is increasing interest in the residential load sector in the installation of grid-connected solar photovoltaic (PV) renewable energy sources as means to reduce the cost of electricity imported from the grid. The research presented here proposes a mathematical model used with a direct search methodology to optimize the rating of a solar PV system, with the view to minimize payback time, for a residential system with given load profile. Half-hourly average energy profiles are used to model the solar generation profile and load profile. Results are presented for two case studies, using a representative daily solar generation profile and daily residential load profile. The case studies include determining the optimal PV system rating, i.e. with minimum payback time, for a system with a fixed load profile and for a system where load scheduling optimization is applied. The results show, that in each case, the direct search optimization algorithm can quickly determine the PV system rating with the minimum payback time. The effects of the load profile and solar generation profile characteristics on the payback time are explored using the concept of a solar utilization factor. It is shown that the utilization factor plays a critical role in improving payback time. Index Terms Renewable energy, Solar energy, Residential load profile, Load scheduling, Energy system optimisation 1 INTRODUCTION Residential consumers are showing increasing interest in the installation of grid-connected solar photovoltaic (PV) renewable energy sources, as an option to reduce the cost of electricity imported from the grid. Installation of a PV system, however, represents a considerable capital investment cost. In order to maximize the financial benefit of a PV system and reduce the payback time, it is important to select the correct rating, especially in the absence of a grid in-feed incentive. This requires careful consideration of the daily solar generation profile and the load profile at the residence. For large PV systems where the solar yield exceeds the demand, it is usually possible to implement a load scheduling algorithm for schedulable loads to optimize local consumption [1, 2, 3]. The paper presents the results of an optimisation strategy to determine the optimum PV rating, i.e. with minimum payback time, for two case studies: For systems where the load profile is fixed, as well as for a system where load scheduling is applied to maximize local consumption of the generated solar power. A mathematical energy balance model is derived for a generic residential energy system that H.J Vermeulen, University of Stellenbosch, Faculty of Electric and Electronic Engineering, P O Box X1, Stellenbosch 7600, South Africa (email: vermeuln@sun.ac.za). T. Nieuwoudt, University of Stellenbosch,, Faculty of Electric and Electronic Engineering, P O Box X1, Stellenbosch, 7600, South Africa (email: tielman@sun.ac.za). is suitable for use with direct search optimization algorithms. A representative daily solar generation profile and daily residential load profile is used. Half-hourly average energy profiles are used to model the solar generation profile and load profile. The results show, that in each case, the direct search optimization algorithm can quickly determine the PV system rating with the minimum payback time, similar to what optimization software such as HOER would achieve [4, 5]. Further analysis of costrelated factors over a range of PV system ratings provide interesting findings on the relationship between PV system size and the load profile of the residence. 2 ATHEATIC ODELLING OF THE ENERGY SYSTE The energy balance model is implemented using the principle that electricity profiles can be represented by a number of averaging intervals of equal duration. Using halfhourly intervals, a daily profile is thus represented by 48 average values of power or energy consumption. This halfhourly averaging interval works well with standard metering systems and is compatible with Time of Use (TOU) tariff systems, whilst having a resolution that is high enough to represent the temporal properties of the underlying profiles. 2.1 Energy, Average Power and Energy Cost Profiles The solar generation and load profiles are represented by sets of timestamps and values. Using set theory notation, the timestamps are expressed as T = {t k } k = 1, 2, 3 N (1) where N defines the total number of timestamps. The timestamps give rise to a set of averaging intervals T given by where ΔT = {Δt k } k = 1, 2, 3 N 1 (2) Δt k = t k+1 t k (3) and t k and t k+1 denote the beginning and end respectively of the k th averaging interval. A daily power profile P(t) can be expressed as a set of average power values P, such that P = {P k} k = 1, 2, 3 N 1 (4) where P k denotes the average power for the k th averaging interval and is given by the relationship t k+1 P(t)dt P k = t k k = 1, 2, 3 N 1 (5) t k+1 t k The energy E k associated with the k th averaging interval is related to P k through the relationship

E k = P k (t k+1 t k ) (6) The associated energy profile E is given by E = {E k } k = 1, 2, 3 N 1 (7) The cost C associated with a given energy profile can be expressed as where C = {c k } k = 1, 2, 3 N 1 (8) c k = R k E k (9) and c k and R k denote the energy cost and energy cost rate respectively for the k th averaging interval. The total energy cost C for the period represented by the profile is C = N 1 k=1 c k (10) 2.2 Energy Balance odel Fig. 1 shows the topology of a generic energy system consisting of a total of energy subsystems connected to a common energy bus, where P i (t) denotes the instantaneous power flow from the i th subsystem to the bus. where j denotes the grid interface subsystem and R k denotes the energy tariff in R/kWh for the k th averaging interval. 3 RESIDENTIAL ENERGY SYSTE TOPOLOGY The generic energy system topology and mathematical model given above can be applied to optimize the residential system considered in this investigation. 3.1 Residential Energy System Topology The topology of the residential energy system considered in this investigation is shown in Fig. 2. The system consists of four subsystems, namely a grid connection, PV energy source, a set of non-schedulable loads and a set of schedulable loads. The non-schedulable loads include loads with fixed operating schedules such as security lighting, cooking appliances, etc. The schedulable loads include loads for which the operating schedules can be scheduled by an optimisation process, e.g. geysers, swimming pool pumps, etc. Fig. 2. Residential energy system topology used in the case studies Fig. 1. Energy system topology for a residential system Conservation of energy requires that the instantaneous power flowing into the system summates to zero, i.e. i=1 P i (t) = 0 i = 1, 2, 3 (11) This also holds true for the average power P ik and energy E ik of the i th subsystem in the k th averaging interval, giving rise to and i=1 P ik = 0 i = 1, 2, 3 (12) i=1 E ik = 0 i = 1, 2, 3 (13) The above relationship gives rise to the following expression for the energy E jk associated with the j th subsystem in the k th averaging interval E jk = i=1 E ik i j i = 1, 2, 3 k = 1, 2, 3 N 1 (14) The relationships given above represent a convenient mathematical model for optimizing energy flow in the generic system, especially where the objective function can be defined in terms of the energy flow or average power profile of one of the subsystems. The cost of energy imported from the grid C gives rise to an objective function of the form N 1 k=1 C = i=1 R k E ik i j i = 1. 2. 3 k = 1, 2, 3 N 1 (15) 3.2 Power Flow Constraints for each Subsystem The power flow constraints apply for the various subsystems can be summarized as follows: Grid connection: The power flow can be bidirectional, with P G max P G (t) P G max (16) where P G max denotes the power rating of the grid connection. PV source: The power flow is unidirectional, with 0 P PV (t) P PV max (17), where P PV max denotes the maximum power rating of the PV system. Non-schedulable loads: The power flow is unidirectional, with P LU max P LU (t) 0 (18) where P LU max denotes the sum of the rated values of the non-schedulable loads. Schedulable loads: The power flow is unidirectional, with P LC max P LC (t) 0 (19) where P LC max denotes the sum of the rated values of the schedulable loads. For the purpose of optimisation, the subsystems shown in Fig. 2 are characterized in terms of the associated average energy profiles. The power flow constraints listed above translate to constraints for the half-hourly energy profile values in accordance with (5) and (6). For a half-hourly averaging interval, with energy measured in kwh, this yields the constraints summarized in Table I.

Table I. Energy profile constraints for the residential energy system model Subsystem Grid connection PV source Energy profile constraints [kwh] 0.5P G max E Gk 0.5P G max 0 E PVk 0.5P PV max Non-schedulable loads 0.5P LU max E LUk 0 Schedulable loads 0.5P LC max E LCk 0 3.3 Schedulable Loads Subsystem Energy Consumption Constraints The average power and energy profiles of the schedulable loads subsystem must satisfy the energy consumption constraints imposed by the operating conditions required for the individual loads. The constraint for the energy consumption profile is the total energy consumption E LC required by the set of schedulable loads over the profile timeline, i.e. 24 hours. This gives rise to the relationship E LC = N 1 k=1 E k LC (20) where E k LC denotes the energy consumption of the schedulable loads energy profile for the k th averaging interval. The value of E LC is derived from the relationship N m E LC = m=1 P m LC T m LC (21) where P m LC and T m LC denote the power rating and total operating time respectively for the m th load component and N m denotes the total number of loads comprising the schedulable loads subsystem. The load power rating P m LC reflects an average value that incorporates the effects of duty cycle, etc. 3.4 Payback time The simple payback period [6] is defined as the time required to recover the capital investment costs of an investment through the income or savings produced by the investment. For the purpose of this investigation, the financial savings per day, C s, is defined as the reduction in the cost of energy imported from the grid as a result of the energy supplied by the PV system, giving rise to the relationship C s = C old C new (22) where C old and C new denote the cost of energy imported from the grid with no solar generation and with solar generation respectively. The simple payback time T pb can be expressed as T pb = C C C S (23) where C C denotes the capital cost of the PV system and T pb is measured in days. Other economic indicators, including Levelized Cost of Electricity (LCOE), Capital Recovery Factor (CRF) and Internal Rate of Return (IRR) can be considered, but require additional factors, such as predicted interest rate. In view of the complexities and uncertainties introduced by these factors, only the simple payback period will be considered. 4 CASE STUDIES 4.1 Overview Results are presented for two case studies aimed at determining the optimal system rating to minimize the payback time of a residential rooftop PV system. The two case studies consider the case where no load scheduling is used and the case where the load schedules of schedulable loads are optimized to reduce payback time. 4.2 Case study parameters The generic load profile used in the case studies is shown in Fig. 3, and represents a typical residential load profile in South Africa [7]. The load profile consists of loads that cannot be scheduled, such as lights, televisions, etc. and schedulable loads, represented by a geyser and a pool pump. Fig. 3. Generic residential load profile used in the case studies The half-hourly solar energy yield profile used in the case studies is given in Fig. 4. It represents the electricity generation of a 1 kw PV rooftop system for a clear day in the Gauteng for the month of October, i.e. approximately midway between the winter and summer months [8]. The total amount of energy generated throughout the day is 6535 Wh. Fig. 4. Normalised solar profile representing the solar electricity generation for a 1 kw PV system Fig. 5 shows the relationship between the normalized capital cost in R/W and system rating in Watt used in the case studies. This relationship reflects the 2015 industry prices for the installation of solar systems, based on the installation costs of actual systems rated at 1,3 kw, 3 kw and 5 kw approximately. These costs include the PV panels, grid-tied inverter, roof mounts and accessories such as cables and connecters, but exclude the labor costs. Actual installation costs are highly dependent on the location, provider and environment of the installation; therefore this approximation is only relevant for the current parameters. The optimisation algorithm uses a linear approximation

derived from the three system ratings shown in Fig. 5, giving rise to the relationship C cu = ( 2,09. 10 3 P s + 31,47) (24) where C cu denotes the normalized capital cost in R/W and P s denotes the rating of the solar PV system. The capital cost C c is then given by the relationship C c = ( 2,09. 10 3 P s + 31,47)P s. (25) F u = Solar energy absorbed by the load Solar energy produced by the PV installation. 100 (26) Fig. 7 shows the solar utilization factor as a function of solar yield for the fixed load profile shown in Fig. 3. The relationship shows that the utilization factor decreases as the solar yield increases past a defined minimum value. Fig. 5. Unit capital costs of a solar PV system as a function of system rating. The electricity cost model applied in the case studies reflects the typical electricity tariff scenario currently in use for residential electricity costs in South Africa. Electricity is purchased from the grid at a flat rate of R1/kWh and no remuneration is received for electricity exported to the grid. 4.3 Solar energy utilization Fig. 6 shows the solar generation profile and load profile relationships for various solar generation profiles. The load profile is fixed as no scheduling optimisation is performed for the schedulable loads, i.e. the geyser and pool pump. Fig. 6. Solar electricity generation and load profiles as a function of solar yield rating. Fig. 6 shows that for smaller PV systems, e.g. the 500 W PV installation, 100% of the generated solar energy is used by the load. As the PV system rating is increased, as shown for the 1 kw and 3 kw systems, the solar yield exceed the load of the residence for one or more of the half-hour averaging intervals. As the solar yield increases, a decreasing percentage of the solar electricity is used by the load, reducing the cost benefit. In the absence of an in-feed tariff incentive, this affects the payback time negatively. The utilization of solar electricity for a given solar yield profile and a given load profile can be expressed in terms of a solar utilization factor F u, where Fig. 7. Solar utilization factor as a function of PV system rating for a fixed load profile 4.4 Optimisation of PV System Rating to minimize payback time Having defined the normalized capital cost as C cu in (24) and the solar utilization factor F u in (26), an alternative calculation is presented for the daily financial savings as opposed to (15) and (22). This allows (23) to more clearly show the relationship between cost of purchase and the amount of electricity from PV utilized. 4.4.1 Fixed load profile The daily financial savings C s is now expressed as C s = K. P s. a. t. z (27) where K is the sum of energy from the normalized 1 kw PV energy vector, P s is the solar system rating, a is a factor used so that P s. a provides a scaling factor that scales K to give the daily generated solar electricity. Factor t is the cost of purchasing energy, and z represents the utilization factor as given in Fig. 7. The parameters as used here are given in Table II. C s is accordingly then expressed as C s = 6,535.10 3 P s. z (28) With these formulas as defined, some simplification can be done so that (23) can be rewritten as T pb = 2,09.10 3 P s +31,47 6,535.10 3 z Table II. Parameters for the case study Parameter Value Dimension K 6353 Wh/day a 1.10 3 1/W t 1.10 3 R/Wh PS variable W z variable none (29) A plot for (29) is shown in Fig. 8, indicating the payback time as a function of PV system rating, calculated for 50 W intervals, for the fixed load profile given in Fig. 3. The graph indicates that a minimum payback time of 12.61 years

is achieved for a solar system rating of 700 W. The initial downwards trend represents the region where a solar utilization factor of 100% applies and is due to the downward price curve shown in Fig. 5. When the utilization factor decreases below 100%, the savings decrease and payback time increases. Fig. 8. Payback time as a function of PV system rating for a fixed load profile The optimum PV system size can be obtained using a direct search optimisation methodology. This was implemented using the Pattern Search algorithm available in atlab [9]. The methodology implements the mathematical model introduced in Section 2 and the constraints and system boundaries identified for the residential system topology defined in Section 3. The objective function was taken as payback time (23). Fig 9 shows the iterations produced by the search algorithm, using an initial solar rating of 3.25 kw, to find the solar system rating with minimum payback time. As expected, the payback curve in Fig 9 follows the curve in Fig. 8. The optimal solar system rating is 676 W, with a payback time of 12.60 years. Fig. 10. Comparison of optimized load profiles for different PV system ratings Comparison of the geyser and pool pump load profiles in Fig. 10 and Fig. 3 shows that in each case the schedules of the schedulable loads have been adapted to make as much use of the solar energy as possible. Fig. 11 shows the solar utilization factor as a function of PV system rating when load schedule optimisation is applied. Comparison between Fig. 7 and Fig. 11 show that with load schedule optimisation a 100% utilization factor can be achieved for a wider range of PV system ratings. The deviations from the linear trend to 3 kw and the saw tooth effect for system ratings close approaching 6 kw is due to the optimisation algorithm wrongly identifying a local minimum rather than the absolute minimum point as the optimal solution for the load profile at these points. Fig 9. Iteration steps followed by the direct search optimisation to find the optimal solar system rating for minimum payback time with a fixed load profile 4.4.2 Optimized Load Profile By optimizing the load profile of schedulable loads, the solar utilization factor can be improved for higher solar system ratings. For this case, the schedulable loads, i.e. geyser and pool pump, are scheduled by the optimisation algorithm to maximize the energy cost savings achieved by the solar PV system. The load profile characteristics produced by the optimisation algorithm depend on the solar system rating, as shown by the load profiles given in Fig. 10. Fig. 11. Solar utilization factor as a function of PV system rating for an optimised load profile The improved utilization factor gives rise to the more favorable payback time graph shown in Fig 12. The optimal payback time of the system is reduced to 10.56 years for a 3.05 kw system. The variations in the trend line in Fig 12 are directly related to those found in Fig. 11.

Additional advantageous can be pointed out for indicators such as IRR, but this falls outside the scope of this paper. Fig 12. Payback time as a function of PV system rating for an optimized load profile 4.4.3 Finding the minimum payback time through optimisation As in the case for the fixed load profile, the optimum PV system rating for minimum payback time can also be derived using the direct search optimisation methodology applied for the fixed load profile case. In this case the methodology includes a load profile optimisation loop for each PV system rating. Fig. 13 shows the iterations of the search algorithm, using an initial PV system rating of 200 W. As expected, the payback curve in Fig. 13 follows the curve in in Fig 12. The optimal solar system rating is 3.07 kw, with a payback time of 10.55 years. 100%, the payback time increases due to not using the optimal amount of energy from the solar system. The case study results show that load profile optimisation, whereby the load profile of schedulable loads such as geysers and pool pumps are optimized to make use of the available solar energy, has a marked effect on the optimum solar system rating. Depending on the characteristics of the initial fixed load profile, the payback time can be reduced by increasing the solar system rating up to a point where the solar utilization factor falls below 100%. The optimal solar system rating for the residential network, using payback time as the main criteria, is highly dependent on factors such as the grid electricity tariffs, the per-unit capital cost trend of PV installations, load characteristics, solar yield profile, etc. Furthermore, the introduction of tariff schemes such as Time of Use (TOU) tariffs, grid in-feed incentives and capital investment rebates can have major impacts on capital payback times and the financial feasibility of solar PV installations in the residential sector. The modeling and optimisation strategy proposed in this paper is suitable for determining optimal PV system sizing in these complex scenarios. The overall accuracy of payback time versus solar PV rating results can be improved by performing the optimisation exercises for representative solar generation and load profiles spanning a longer period of time, e.g. a calendar year that includes the effects of seasonal trends. The PV system unit cost approximation curve in Fig. 5 can be a higher order function to more closely approximate industry prices. From a financial perspective, further research is required to extend the results to include economic indicators such as LCOE, CRF and IRR. REFERENCES Fig. 13. Optimisation iterations to find the optimal solar PV rating for minimum payback time with an optimized load profile 5 CONCLUSION The results presented in this paper shows that the payback time of a solar PV system in a residential network can be conveniently calculated, and an optimum PV system rating with a minimum payback time determined, using halfhourly load profiles and the concept of a solar utilization factor. It is shown that the same result can be achieved using a direct search method such as a pattern search algorithm. A given load profile and solar yield profile gives rise to solar energy utilization factor, which plays a crucial role in minimizing payback time of the solar PV installation. Where a solar utilization factor of 100% applies, the payback time decreases with increasing solar system rating due to the decreasing normalized capital costs in R/W of the solar installation. As the solar utilization factor drops below [1] T. Yu, D. S. Kim and S.-Y. Son, "Home Appliance Scheduling Optimization with Time Varying Electricity Price and Peak Load Limitation," Information Science And Technology, vol. 23, pp. 196-199, 2013. [2] T. Yu, S. D. Kim and S.-Y. Son, "Optimization of Scheduling for Home Appliances in Conjunction with Renewable And Energy Storage Resources," International Journal of Smart Home, vol. 7, pp. 261-271, 2013. [3] K. C. Sou, J. Weimer, H. Sandberg and K. H. Johansson, "Scheduling Smart Home Appliances Using ixed Integer Linear Programming," in IEEE Conference on Decision and Control, Orlando, FL, USA, 2011. [4]. A. Salam, A. Aziz, A. H. A. Alwaeli and H. A. Kazem, "Optimal sizing of photovoltaic systems using HOER for Sohar, Oman," International Journal of Renewable Energy Research, vol. 3, pp. 301-306, 2013. [5] G. okhtari, G. Nourbakhsh and A. Gosh, "Optimal Sizing of Combined PV-Energy Storage for a Grid-Connected Residential Building," Advances in Energy Engineering, vol. 1, no. 3, pp. 53-65, 2013. [6] Ray H Garrison, "Capital Budgeting Decisions in anagerial Accounting, 6 th ed, D van Bakel, USA, Von Hoffman Press, 1991, ch 14, pp. 615-621. [7] S. Davis,. Prier, B. Cohen, A. Hughes and K. Nyatsanza, "easuring the rebound effect of energy efficiency initiatives for the future", Energy Research Centre, University of Cape Town, Project Number EU0607-122, 2010. [8] SineTech, "eteocontrol Energy and Weather Services," SineTech, [Online]. Available: http://www.meteocontrol.com/en/. [Accessed 10 December 2014]. [9] ATLAB, "Patternsearch," atlab, 2014. [Online]. Available: http://www.mathworks.com/help/gads/patternsearch.html? [Accessed 10 December 2014].

AUTHORS Hendrik J Vermeulen received B.Eng,.Eng, and Ph.D Eng degrees in electrical engineering and an.phil (usic Technology) degree from the University of Stellenbosch. His work experience includes three years with Eskom as a research engineer before joining the University of Stellenbosch, where he is an Associate Professor in Electrical Engineering focusing on power system dynamics, high voltage engineering, energy management and renewable energy. He is a member of IEEE, AEE, SAEE and CVPSA and Certified easurement and Verification (&V) Professional. Tielman Nieuwoudt received his BIng degrees in Electric and Electronic engineering from the University of Stellenbosch, in 2013. From 2014 he continued his studies as a fulltime asters student at the University of Stellenbosch. Presenting author: The paper will be presented by Tielman Nieuwoudt