A FLEXIBLE MARKOV-CHAIN MODEL FOR SIMULATING DEMAND SIDE MANAGEMENT STRATEGIES WITH APPLICATIONS TO DISTRIBUTED PHOTOVOLTAICS

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1 A FLEXIBLE MARKOV-CHAIN MODEL FOR SIMULATING DEMAND SIDE MANAGEMENT STRATEGIES WITH APPLICATIONS TO DISTRIBUTED PHOTOVOLTAICS Joakim Munkhammar, Joakim Widén Uppsala University, Sweden Address: P.O. Box 534, SE-75, Uppsala, Sweden ABSTRACT In this paper a stochastic model for load shifting was utilized for the purpose of investigating the potential for increased self-consumption of photovoltaic (PV) generation in households. We show the results in terms of power consumption, PV power production and solar fraction from a number of scenarios involving end-user flexibility on the order of a few percent. Simulations are performed on both an individual and an aggregate level. Results indicate that the solar fraction is only improved by a few percent both on an aggregate and individual level even for the most extreme scenarios of load shifting. The lack of substantial increase in solar fraction from imposed flexibility can partly be attributed to complementary energy use; when certain energy-demanding activities are downshifted in probability other activities are up-shifted. Another explanation for the lack of increased solar fraction is the total available fraction of flexible activities at a certain time..introduction Theoretical high-resolution load profiles for domestic power consumption are useful for the setup and energy optimization of various kinds of systems (4),(8). In particular they are useful in studies of distributed and variable power sources - such as distributed photovoltaics - which are often introduced into the grid at the end-user. When there is a lack of substantial feed-in tariffs the selfuse of PV electricity becomes particularly important since the electricity generated is normally worth more if used on site rather than sold (). At least for high penetration levels of photovoltaics increased self-use is also a possible way to accommodate more photovoltaics in distribution grids. The household power demand is usually not ideally matched by photovoltaic power production since most power-consuming activities are performed during late evenings (9). One possibility to increase the self-use fraction is to implement demand side management strategies (DSM) in particular load shifting (8). This strategy aims at shifting certain power-consuming activities such as for example washing and dishwashing from a certain time to another time for example from onpeak periods to off-peak periods (8). Electricity demand is largely uncontrollable and varies over season and day, with a typical minimum demand on summer nights (6). The variability of power use requires regulating power and puts stress on the distribution grid (8). Demand side management strategies aim to alter the use of energy at the end-user in order to fulfill some set of goals. The specific concept of load management strategies aims to lower the demand during periods of on-peak and increase it during off-peak periods (8). One possible way to do this is to introduce a time-differentiated price tariff and thus produce price-incentives for altered electricity use (8). The approach in this paper assumes that the optimal utilization of the self-produced photovoltaic power is self-use. This then acts as a form of price-incentive for shifting loads from night-time to day-time. In this paper we have utilized a flexible discrete-time Markov-chain model for activities in order to simulate the effects of flexibility on the power consumption and on self-use of PV power in a number of scenarios. The model generates synthetic sequences of activity data (9),(0). Each activity is associated with a certain energy use, thus the model generates typical household electric energy consumption patterns over time. The model was calibrated with a large set of activity data from Statistics Sweden (SCB) (9). This approach is stochastic but is not dependent on external variations such as electricity price or the benefit of increased solar fraction. The flexibility extension to this model that is used in this paper is designed so that the transition probabilities of the Markov-

2 chain model are re-weighted based on a synthetic timedifferentiated tariff with the intent of increasing the solar fraction (8). This is in practice a re-weighting of the transition-matrices under the assumption that the net value of the self-produced photovoltaic electricity is highest if self-consumed. The disposition of the paper is as follows. First we give some methodology of the model and the flexibility extension. Then we show some results regarding flexibility and energy use and continue to show results from self-use of PV electricity. Finally we discuss the results and further research.. METHODOLOGY AND DATA. The Markov-chain model In (9), (0) a discrete time Markov-chain model for generating synthetic activity patterns and consequent energy use was constructed. This model here called the Widén model is stochastic and calibrated with a large dataset of time-use data (TUD) (9), (0). The model was based on the assumption that each resident occupies one state at each time. The canonical set of activities chosen in (9) are presented in Table. TABLE : LIST OF ACTIVITIES Code Activity. Away. Sleeping 3. Cooking 4. Washing 5. Dishwashing 6. Drying 7. TV/VCR/DVD 8. Computer 9. Audio 0. Other The model was based on the probability of transition from one state of activity to another in this fixed set of states. The probability of occupying any of the states of activity at any time step was set equal to unity. The model was also designed to satisfy the Markov property by only being dependent on the previous time step (), (9). Since there are X possible states the model was based on an X X - sized transition-matrix for each time-step. The model had minute-based resolution. Each transition-matrix was defined as p where i, [, X ]. In general there is one i transition matrix for each time-step t [,440] since there are T=440 minutes per day. The model has different matrices for weekend-day and weekday, and for apartments and detached houses as well as it can allow any number of individuals per household. The model was calibrated and verified with electricity-use measurements (9).. Flexibility extension The Widén-model was designed to be independent of external variations in for example electricity price or potential interest for self-use of photovoltaic production. Instead a flexibility extension to the Widén-model was proposed in (8), that is developed further and evaluated in this paper. In order for the flexibility model to be consistent as a stochastic load shifting model it has to fulfill two criteria: (I) (II) The probability at each time-step for transition to any state in the next time-step has to be unity. The total amount of probability (the sum over probabilities) for a certain activity over a 4 hour period has to be the same for the standard case as for the flexibility case. The first criterion is a mere requirement from probability theory that the total probability for transitioning to any of the predefined states equals one. The second criterion regards the model consistency of load-shifting that the load is actually shifted but conserved - otherwise the model is simply a valley filling or peak clipping method (8). The flexibility model focuses on simulating the shift of activity patterns for a specified amount of residents in a population. The particular approach in this paper aims to up-shift certain activities during daytime and down-shift them during night time. These activities belong to the set U. Since the model is a closed system of probabilities for occupying any activity at a certain time-step there has to be a complement of activities which are down-shifted during daytime and up-shifted during night time, these activities belong to set D. In this approach there are activities which are left unchanged for all time-steps, and they belong to set N. The set of all activities is then A U D N. The day-time is defined as the set T which contains minute time-steps from 07:00-:00 and the night-time is defined by the set T and which contains the minutes during :00-07:00. Both sets make out the full 4 hours of a day. Mathematically the framework for flexibility works as follows (8). From the Markov matrix

3 p we can determine the probability for the occurrence of i an activity at a particular time-step. The probability for the up-shifted states U at time t T are: p ( t) p ( t) (.) i A i which holds for i U. Here we introduce the flexibility coefficient - which is the fraction of probability that is shifted. In practice the fraction is equivalent to the fraction of people who perform this shifting. We can define the down-shifted probabilities for t T as: pˆ ( t) ( ) p ( t) (.) i i which hold for i D, A. In order to fulfill flexibility criterion (I) the sum of down-shifted probabilities: p( t) p ( t) (.3) i D i There is an inherent freedom of choice regarding the distribution of the sum of down-shifted probabilities onto the up-shifted category of probabilities, but in general the up-shifted probabilities may be defined as: pˆ ( t) ( p( t) ) p ( t) (.4) i i For t T, i U and Awhere the coefficients distribute the down-shifted probabilities at the same time step. The coefficients also need to be normalized via U. This approach is by default a peak clipping or a valley filling strategy and does not automatically meet flexibility criterion (II) by being a model that truly shifts load from one time-step to another. However one may adopt an approximate flexibility strategy with a specific setup on as was done for the model in (8). In order to approximately fulfill flexibility criterion (II) then and consequently equation (.3) is defined as: p pˆ ( t) p( t) p ( t) i i p (.5) for t T, i Aand U where p for is the average probability over T of residents involved in the up-shifted activities during that period: p T p () t (.6) T t and the sum of those probabilities is then: p p (.7) U where U. The same procedure is then performed during night time T except reversed the up-shifted during day-time is down-shifted during night-time and vice versa..3 Photovoltaic model In order to estimate the power production from a residential-scale PV system we utilized the PV model previously developed and used in (). This model is based on Swedish irradiation and temperature data from the year 99 obtained from the Swedish Meteorological and Hydrological Institute (SMHI) (5). The dataset contained data from (5), and the first year in the data-set was chosen for the PV-simulations. The output data from the model is calculated from insolation data, array size, location, orientation, and temperature dependence..4 Simulations The model is setup according to the following for the simulations. The assumptions were zero azimuth angle, 30 degree tilt angle and irradiation data from Stockholm, photovoltaic size of 45m and an efficiency of 4 percent. The size of the photovoltaic panel for each household was estimated so that the household would be a net-zero energy building (NZEB), or ust above net-zero, making it a plus energy building. The total peak power of the PVsystem was 4.kW, and the total PV power production P was 5.50 MWh/yr compared with the household power consumption at 5.8 MWh/yr making it a plus energy building for the standard scenario. As regards the standard setup from which scenarios are constructed it was the same as used in (9). 3

4 TABLE : FLEXIBILITY SETUP Activities Shifting 3-9 Up during day,0 Down during day Neutral If up-shifted during day-time it is down-shifted during night-time, and vice versa. The shifting applies to both week-days and weekend-days. 3. RESULTS 3. Household electricity use In figure the change in probability over the course of 4 hours is given for a weekday. The probabilities for an activity to be occupied at a given time is calculated from the model. Certain activities are shifted towards day-time compared with others that are shifted towards night-time. As the flexibility coefficient increases from left to right, from up to down, the difference between the two flexibility periods becomes more acute. Fig. : Probability distributions of weekday activities. Standard setup (upper left) followed by 5% (upper right), 5% (lower left) and 50% (lower right) flexibility. The shifting is according to the setup in table. 4

5 Fig. : Average electric energy use for each activity per day for a 360-day sample simulated by the extended stochastic model. Standard setup (upper left) followed by 5% (upper right), 5% (lower left) and 50% (lower right) flexibility coefficient. 3. Photovoltaics and load matching In this section we show the results from the simulations of household electricity use and PV power production according to the setup in section.4. Fig. 3: Average photovoltaic production from the 45m PV-panel along with averaged aggregate load profiles for the standard setup and from the top: 5%, 5% (lower left) and 50% (lower right) flexibility coefficient. The aggregate scenario consists of 360 households, each containing four residents. The data from the simulations are found in Table. 5

6 TABLE 3: PV POWER, POWER USE AND SOLAR FRACTIONS Level of flexibility Average PV KWh /day Household KWh/day Solar Fraction (SF) Aggregate SF 0% (Standard) 5% % % In Table 3 the data from simulations of household energy consumption and PV production are given for a series of scenarios. The data include average PV power production, household load along with individual household solar fraction and 360 households aggregate one day. All scenarios of flexibility had higher percentage of solar fraction both on an individual household level and on an aggregate level. The highest percentage was reached by the 50% flexibility-coefficient scenario. However the level of solar fraction increase was modest even for the most extreme scenario which is a fundamental problem of flexibility that will be discussed in next section. Another issue regards the alteration in household power demand from one scenario to another. A model that perfectly shifts the load from one time to another thus fulfilling flexibility criterion (II) above should ensure that the total energy use is kept constant. However this is not possible since the total flexible probability during daytime and night-time might often not be the same and hence the flexibility is limited. This problem is not exclusively a model-related problem, since it is reasonable that in real life the total amount of flexible loads will vary over time and certain activities are less flexible at certain times. 3. Flexibility and complementary activities Flexible behavior means in practice shifting an activity from one time to another usually based on some incentive. Since the total probability of performing any activity at a particular instance of time equals unity there has to be a corresponding complementary shifting of one or more other activities at the same time. Most activities use energy, but the degree of energy consumption differs. In Table 3 the total daily energy use associated with each activity is shown. Some activities are perhaps even unreasonable to shift, such as Away, which was assumed in the simulations in this paper. TABLE 4: DIURNAL ENERGY USE Time for activity (min/person /day) Appliance runtime (min/day) Appliance energy use (Wh/day) Away Sleeping Cooking Dishwashing 0, Washing, TV Computer Audio Other Given the results from Table 4 we may construct an extreme scenario of shifting only the two most energy consuming activities for the other activities, while assuming that activities Sleeping and Away are inflexible. The two most energy-intense activities are Cooking and Computer, see Table 4. The setup of this special flexibility model is given in Table 5. TABLE 5: SPECIAL FLEXIBILITY SETUP Activities Shifting 3,8 Up during day 4,5,6,7,9,0 Down during day, Neutral The result from a 5% flexibility factor for this scenario is shown in Table 6 and Figures 4-6. TABLE 6: SPECIAL FLEXIBILITY RESULTS Level of flexibility 0% (Standard) 5% Flex. (special) Average PV KWh /day Household KWh/day Solar Fraction (SF) Aggregate SF

7 Fig. 4: Probability distribution over activities with 5% special flexibility setup. Fig. 5: Power consumption with 5% special flexibility setup. fact that flexibility for a limited number of activities leads to complementary energy consumption. This is an artifact of the model, but the model as an approximation of real life is based on the notion that one activity is substituted for another, which also typically also demand energy. Another reason for the lack in increased solar fraction from even the most extreme flexibility scenarios is the time-varying total amount of flexible probability. This sets a limit on the amount of probability for activities that are actually flexible. In this model this causes a total energy demand difference between the standard scenario and the flexible scenario. There is also a fundamental limit to maximizing the solar-fraction on a daily load shifting basis due to seasonal variation in PV power production, in particular at high latitudes (). In (3) the maximum daily optimal match was on the order of 65%, which is then the upper limit of the over-time solar fraction. The annual optimal match was estimated at 95% and is thus in practice the limit of the aggregate solar fraction. There exist other measures than solar-fraction, for example focusing on the net power generation and load profiles. Another important factor in flexibility regards the economy of the system, which will be investigated in further research. It is also possible to extend this flexibility model to investigate the effect of flexibility on electric vehicle charging patterns. That could in turn also be used to study a potential increase in solar fraction when including electric vehicle charging. ACKNOWLEDEMENTS Fig. 6: Standard versus shifted load curve in the special flexibility scenario, see Tables 4 and 5. It is clear that in the extreme scenario a higher solar fraction is achieved, but the magnitude is still only on the order of a few percent. The problem of altered total energy use in the flexibility extension is even more prominent in this extreme or special flexibility scenario. 4. DISCUSSION With the lack of substantial feed-in tariffs the optimal use of photovoltaic power installed at the end-user is selfconsumption. One of the main aims of this paper was to investigate how the demand side management strategy of load shifting could aid the self-use solar fraction. The conclusion from the model is that the solar fraction is only increased by a few percent even in the most extreme scenario of flexibility. This problem stems in part from the This work has been carried out under the auspices of the Energy Systems Programme, which is primarily financed by the Swedish Energy Agency. REFERENCES () E. Cinlar, Introduction to stochastic processes, Prentice-Hall International, inc, 975. () S. Conti, S. Raiti, Probabilistic load flow using Monte Carlo techniques for distribution networks with photovoltaic generators, Sol. Energy 8, 007. (3) P.Grahn, M.Hellgren, J.Munkhammar, Photovoltaics, electric vehicles and energy users: A case study of the Royal Seaport Visions and energy user expectations, working paper 50 Linköping University, 0. (4) J.V.Paatero, P.D.Lund, A model for generating household electricity load profiles, Int. J. Energy Res. 30, 006; 30 (5) T.Persson, Measurements of solar radiation in Sweden , In SMHI, 000. (6) G.Strbac, Demand side management: Benefits and challenges, Energy Policy (7) M. Thomson, D.G.Infeld, Impact of widespread photvoltaics on distribution systems, IET Ren. Power Gen (8) J.Widén, A. Molin, K. Ellegård, Models 7

8 of domestic occupancy, activities and energy use based on time-use data: deterministic and stochastic approaches with application to various building-related simulations, J. Building Performance Simulation 5 (0). (9) J. Widén & E. Wäckelgård (00), A high-resolution stochastic model of domestic activity pattern and electricity demand, Applied Energy 87, 00. (0) J. Widén A.Nilsson, E.Wäckelgård, A combined Markov-chain and bottom-up approach to modelling of domestic lighting demand, Energy and Buildings 4, 009. () J.Widén, Correlations between large-scale solar and wind power in a future scenario for Sweden, IEEE Transactions on Sustainable Energy, Vol., 0. () J.Widén, B.Karlsson, End-user value of on-site domestic photovoltaic generation with different metering options in Sweden, Proceedings of EuroSun Graz, 00. (3) J.Widén, E.Wäckelgård, P.D.Lund, Options for improving the load matching capability of distributed photovoltaics: Methodology and application to high-latitude data, Solar Energy 89,