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Risk-constrained framework for residential storage space heating load management
Electric Power Systems Research 119 (2015) 432–438
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Risk-constrained framework for residential storage space heating load management Mubbashir Ali ∗ , Amir Safdarian, Matti Lehtonen Department of Electrical Engineering and Automation, Aalto University, Espoo, Finland
a r t i c l e
i n f o
Article history: Received 14 May 2014 Received in revised form 26 September 2014 Accepted 26 October 2014 Keywords: Demand response Electric storage space heating Real time pricing Risk mitigation Smart grid Uncertainty
a b s t r a c t Residential demand response based on real time pricing provides a strong incentive for customers to reduce their energy payment. However, acting under an environment with time-varying prices will expose them to uncertain energy bills. This paper presents a risk-constrained framework for residential customers for scheduling the electric storage space heating load. The proposed decision framework attempts to accomplish desired settlement between expected cost minimization and cost deviation without altering the user’s thermal comfort. The price and load uncertainty are captured by a scenario based stochastic programming approach. The optimization model is solved using Genetic Algorithm and implemented using a moving-window procedure. The simulation results demonstrate that the proposed framework for scheduling the storage space heating load provides a method to selectively hedge against the price and load uncertainty risk. The optimal framework will result in an improved interaction between the electrical aggregator and its customers under the smart grid paradigm. © 2014 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation In today’s power system, safe integration of intermittent renewable energy resources is considered to be the most demanding challenge. Problems that further aggravated the situation include charging electric vehicles, aging infrastructure and institutional changes in electricity markets. In these circumstances, demand response is seen as one of the effective tools for alleviating these issues [1–3]. Demand response is a mechanism to motivate the customers, by either changing prices or giving incentives, to adjust the electricity consumption to bring favorable results for power system operation at different time-scales. Real time price (RTP) based demand response is considered to be the most effective mechanism for demand side management (DSM) [4]. Future smart grids can enable demand side management, integration of distributed energy resources and improve power system efficiency [5]. With the advent of smart grid technologies, the real time interaction
∗ Corresponding author at: Aalto University, Department of Electrical Engineering and Automation, Otakaari 5 A, P.O. Box 13000, Espoo FI-00076, Finland. Tel.: +358 50 4367307; fax: +358 9 47022991. E-mail addresses: mubbashir.ali@aalto.fi (M. Ali), amir.safdarian@aalto.fi (A. Safdarian), matti.lehtonen@aalto.fi (M. Lehtonen). http://dx.doi.org/10.1016/j.epsr.2014.10.024 0378-7796/© 2014 Elsevier B.V. All rights reserved.
between customers and electrical aggregators is becoming possible and a household customer can also participate in demand response programs implicitly optimizing the power system operations [6]. Amongst the household loads, the thermostatic controlled appliances (TCAs) such as electric space heaters, electric water heaters, fridge, etc are logically the most suited load for an effective demand response application. The time postponement and power-schedulable characteristic of the TCAs load qualifies it as a well-suited load for DSM programs from an electrical aggregator perspective. The flexibility of TCAs is enhanced if it is equipped with some degree of thermal energy storage. For instance, equipping space heating system with thermal storage (system commonly known as electric storage space heating, see Fig. 1) allows heating to uncouple from the system hence the power interruption can take place with almost no impact on customer’s thermal comfort and can be used for power system operation optimization. At present, many electric storage space heating consumers are charged according to the static Time of Day distribution tariffs despite considerable hourly variation in the wholesale market power price. The existing Time of Use (ToU) tariff help the household consumers to manage their heat consumption by charging the thermal storage during night time and then allowing the stored heat to coast through the rest of the day. Nowadays, with the advent of smart meters and the penetration of intermittent renewable generation, a contractual agreement between
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Nomenclature ˇr Ce Dl N Pch Pd Q qh,n S0t SLht t Smax ZN ˝(n) FR h n
risk coefficient electricity price (D /kWh) maximum power taken from the electric storage space heating system (kW) total number of scenarios considered. charging power (kWh/h) direct electric heating power (kWh/h) power taken from thermal storage (kWh/h) average heat demand (kWh) during hour h and scenario n. initial level of thermal storage (kWh) stored energy lost during hour h (kWh) net storage capacity (kWh) expected cost over N scenarios (D ) probability of scenario n storage loss coefficient financial risk index of hours index of scenarios
the end consumers and electrical aggregator can be made possible so that the customers can respond to more dynamic time varying prices. One of the serious implications of such a demand response program is that it could expose the end-user to the financial risk especially if the power prices are highly volatile [7,8]. A comprehensive report [9] by US department of energy on utility experience with real-time pricing suggest that RTP can be an attractive tariff however there are potential risk burdens associated with that which must be overcome in order to encourage the extent of customers participation in real-time demand response programs. Although a very attractive pricing scheme, the RTP prices can subject the consumers to a relatively high degree of risk by exposing them to stochastic power prices [9]. 1.2. Approach In an attempt to offset the customer’s potential risk that may be incurred due to the price volatility and load uncertainty as well as simultaneously reducing the energy payment, we present an optimal framework for scheduling the electric storage space heating load. The uncertainty is modeled using a scenario based stochastic programming approach. The proposed demand response model adequately considers the risk-level while reducing the expected energy cost of the customer. The risk is simply modeled by the minimum variance approach. Moreover, the customer’s thermal comfort is not compromised as the proposed demand response model manages the thermal storage operation without compromising on the energy demand during that hour. The decisions are
Fig. 1. Illustration of a typical electric storage space heating system. In this figure, the sum of Pch (charging power) and Pd (direct heating power) is the total thermal power output of the storage space heater.
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made for next 24 h but executed and revised on an hourly level (the approach commonly known as a moving window procedure) as the new information on prices and demands are updated on an hourly basis. 1.3. Related work and contribution A significant amount of research studies have concentrated on the residential demand response. The most relevant works are reviewed in the following to demonstrate the contribution of this paper. The authors of [10] presented an optimal operation of major residential load under a smart grid scenario. The work in [11] presented an optimal energy scheduling framework for an automatically operating appliance. A co-evolutionary particle swarm optimization (PSO) based decision support tool for optimally scheduling distributed energy sources at household level is developed by Angelo et al. [12]. Rad et al. [13] formulated an optimal household load control under a real-time pricing and inclining block rate environment to achieve a tradeoff between energy payment and waiting time. Peizhong et al. [14] proposed a scheduling scheme based on optimal stopping rules to achieve the same objective. A robust optimization model to schedule the hourly-level consumption is developed by Conejo et al. [15]. In [16], an appliance commitment algorithm that schedules thermostatically controlled appliances considering user comfort under the price and demand uncertainty situation. A Markov decision framework to schedule the power consumption based on future price forecasts was discussed by Kim and Poor [17]. An efficient scheduling algorithm to optimize the household appliance operation considering demand and intermittent renewable uncertainty was proposed in [18]. In [19], a study has been performed to assess the domestic demand response potential of responsive appliances. The work [20] proposed a distributed algorithm solution for unleashing domestic demand response in smart grid environment. The research in [21] coordinated the load management of heating load and electric vehicles to bring customer economic savings. The demand response model evaluates the cost of discomfort and temperature preferences however the framework does not account the uncertainty and risk issues. The work [22] demonstrated the performance of dynamic controller for residential building demand response application. Some studies have principally focused on the demand response control of a particular flexible appliance such as water heater [23], electric vehicle [24], and freezer [25] to name just a few. Most of the above reviewed work focused on customer load scheduling in order to minimize the user energy payment under dynamic prices. However, despite its importance, the customer’s financial risk associated with electricity purchasing under RTP were not adequately discussed in the literature. Nonetheless, there exist a few papers that addressed the problem from the perspective of a risk-averse energy retailer [26–28]. A few risk measures were described and used in these papers which can be useful in the current study as well as the future works, however, more comprehensive models are required to be provided to achieve the ultimate benefits of demand response potentials. In this regard, this paper develops a risk constrained model to minimize the expense of heating demand for a residential customer. The core contributions of this paper may be summarized as follows: • It presents a user-eccentric tool for customers to optimize their space heating load under uncertainty environment without deteriorating the customer’s thermal comfort. • To minimize customer’s cost at a given risk level, a non-linear optimization problem is formulated to obtain the optimal amount
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of charging power and direct electric heating power to meet the energy demand. • Genetic Algorithm (GA) is used to solved the demand response model due to its simplicity and robustness to handle non-linear objective function. • The influence of thermal storage capacity on the customer’s cost and financial risk is critically analyzed.
1.4. Paper organization The remainder of the paper is organized as follows. In Section 2, some theoretical basics are briefly described. The risk-considered demand response framework for electric storage heating is presented in Section 3. Simulation results are presented and discussed in Section 4. Finally, Section 5 concludes the paper.
2. Theoretical basis
2.3. GA-based solution technique Owing to the non-linear objective function, we selected GA for solving the problem. Although the model can be approximated to a linear one which can be then solved using efficient linear programming solvers, however then the accuracy of the results may be compromised. GA is a random search algorithm inspired by the natural evolution process. GA employs probabilistic search rules. It considers a population of chromosomes (set of genes) which encode the possible solutions. The combination of individuals take place on the basis of fitness function to create individuals that are more fit [31]. Unlike classical optimization methods, such as interior point method, the GA does not require the information of derivative of the objective function and take the least computational resources and thus can be easily embedded into the home energy management system with minimal infrastructure cost. It is worth mentioning that GA is selected as a tool to illustrate the proposed framework effectiveness and the proposed problem can be solved by any other heuristic techniques or classical optimization methods.
2.1. Uncertainty description The uncertainty in future power prices and load poses challenges in scheduling the electric space heating load. Both of the uncertainty sources negatively affect customers enrolled in RTP based demand response programs. Due to the uncertain future heating load the power scheduled may not match the realized heating demand. This would consequently affect the customer’s thermal comfort as well as the energy expense. In such an energy mismatch case, the heating power is to be then supplied from direct electric heating in the case of deficit. While the heating power is to be stopped if the required heat demand is lower than the scheduled power. The operation of storage space heating needs to be efficiently managed so that the expected electricity payment can be minimized under the dynamic power prices. To handle the price and load uncertainty, we employ a scenario based stochastic programming approach [29] to generate the load and price scenarios. A set of most likely scenarios are generated considering the expected value and time-varying standard deviation. It is supposed that the heating load and power prices can be projected with good accuracy for the upcoming hour while the standard deviation linearly increases as the prediction time increases. The probability of happening of each scenario is captured by normal probability distribution function. Then the problem is solved over all the scenarios. The goal is to minimize the expected cost value given the price and load scenarios. The constraints are taken to be same in all of the considered scenarios. Although the greater number of scenario increases the accuracy of solution but it happens at the cost of complexity of the model. To balance the tradeoff between complexity and accuracy, scenario reduction methods can be used to lessen the number of scenarios without compromising on accuracy.
3. Proposed framework for heating load scheduling We consider a smart grid environment in which real-time prices are offered to the residential customers and they respond to the time-varying prices to minimize their energy payment while limiting the financial risk due to uncertainties to a pre-defined level. We consider that a smart home load management system is integrated in the household and an adequate decision framework is required for load management. The proposed decision support tool for home load management system is presented in the following subsection. 3.1. Problem formulation The proposed decision support tool, based on the future heating load and power prices, will help the customers to decide the optimal share of direct heating power and storage charging power to meet the heating energy requirement. The objective of the framework is to optimize the operation of space heating load in order to minimize the user expected payment while the financial risk imposed by uncertain power prices and stochastic heat demand is restricted to a certain level. The objective function can be mathematically written as follows: minimize
Z N + ˇr (FR)
where, ZN =
N
24
˝(n)
n=1
2.2. Risk measure Consider a consumer involved in the RTP-based demand response program. The uncertain future power prices will render the customer to uncertain energy payment especially if the power prices are volatile. To tackle this inconvenience, a naive risk measure is integrated in the decision-model. The risk management is imposed through the mean-variance approach [30]. In general, it is simply the volatility (standard deviation) of the cost at different scenarios. The minimum variance approach has only the objective of mitigating the risk rather and leads to a noticeable concentration in low volatility costs at different scenarios.
(1)
e d Ch,n (Phch + Pn,h )
(2)
h=1
⎡ 24 2 ⎤ N e (P ch + P d ) − Z N ⎦ FR = ⎣ ˝(n) Ch,n h n,h n=1
(3)
h=1
In the objective function (1), the coefficient ˇr principally models the customer’s risk attitudes. The higher values of risk coefficient (ˇr > 10) represent the risk-averse customers whereas the lower values (0 < ˇr ≤ 10) represent the risk-seeking customers. ˇr would be zero if and only if the risk is ignored and it corresponds to the case of an extreme risk affine customers. The detailed subject of setting the risk-coefficient parameter is not the focus of this paper.
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Fig. 2. A moving window procedure for optimization.
The objective function (1) is subjected to the following operational constraints. The constraint (4) ascertains the quality of service by not altering the minimum heating requirement: d Pn,h + Qn,h = qn,h ,
∨ h, n
(4)
The rated power of direct electric heating Pd and charging Pch is enforced by constraints (5) and (6), respectively: Fig. 3. Flow chart illustrating the demand response model implementation. ch 0 ≤ P ch ≤ Pmax , d 0 ≤ P d ≤ Pmax ,
∨h
(5)
∨ h, n
(6)
The following constraint bounds the storage discharging power limits: 0 ≤ Q ≤ Qmax ,
∨h
(7)
The constraint (8) models the dynamics of thermal storage. t Sh+1 = Sht + (Phch − Qh )h − SLht ,
∨h
(8)
The storage losses are modeled in (9). An additional constraint formulated in (10) that guarantees that total power taken from the heating system shall not exceed a pre-defined maximum power limit: t SLh+1 = Sht ,
P ch + P d ≤ Dl ,
∨ h, n ∨ h, n
(9) (10)
3.2. Demand response model implementation The optimal demand response framework of electric storage space heating is developed in the Section 1.4. In this section, the demand response framework implementation is discussed. The implementation of this proposed decision model does not require any costly hardware setup. A two-way communication is needed between the utility and end-user. The communication infrastructure is required to be integrated with the smart home energy management system. The utility announces the timevarying prices, and expected outside temperature profile for the following hours to the customer. The proposed demand response model is optimized using the GA algorithm technique for a moderately short-term horizon [h, h + 24] and implemented using a moving window procedure. The moving window procedure as illustrated in Fig. 2 basically optimizes the thermal storage operation for a finite optimization horizon while considering the future heating energy demand and power prices. Only the first step of the optimization result is implemented and then the inputs are sampled again. The optimization routine is repeated resulting in a new
predicted thermal storage charging/discharging profile. The sample time is taken as 1 h, whereas, the scheduling horizon of 24 h is chosen. The flow chart of moving window algorithm for solving the demand response framework is shown in Fig. 3. First off, the heat demand and electricity prices are forecasted and announced for h + 24 h. Next, the demand response framework is solved using a GA approach. The gradient-free feature and the ability to reach a near-optimal solution for non-linear problem in quick time make GA a suitable tool for the demand response applications of electric storage space heating at the household level. The inputs of the optimization model are power price scenarios, stochastic heat demand, customer risk preferences and thermal storage size. The decision (output) variables are amount of direct electric heating power and charging power. The purpose of the GA-based demand response optimization is to search for an optimal proportion of direct electric heating power and charging power to meet the current hour heating load while deciding on charging the thermal storage tank to use it for later hours considering the future costs and heating load. The proposed demand response framework is solved and the output charging power and direct electric heating profile is obtained for the next 24 h, however, only the decisions are made for the first hour (h + 1) and then the price and heating data is updated. This moving window approach makes the thermal storage scheduling immune from the price and load forecasting error to some extent. It would also tackle the problem of empty storage level by the end of the optimization period, which is the case if the scheduling horizon is static. This hourly moving window algorithm is suitable to handle the dynamics of a relatively massive-structured (high thermal constant) building to be heated.
4. Case study In order to show the usefulness of the proposed demand response decision framework, it is applied to a case study. A typical medium massive structure Finnish detached house equipped
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Expected Cost (€ )
(Euro/MWh)
60 50 40
1.6
β =50
1.55
r
β =25
1.5
r
β =15
1.45
r
β =12
1.4
r
β =10 β r=5
1.35
30
1.9
20 0
4
8
12 (Hours))
16
20
2.1
24
2.3 2.5 2.7 Cost standard deviation (%)
r
β =0
2.9
3.1
Fig. 6. Evolution of the expected cost versus cost standard deviation. Fig. 4. Power price scenarios (The expected hourly power prices are shown with dashed line).
with storage space heating system is considered for the analysis. The obtained results are reported and discussed in this section. 4.1. Data To model the uncertainty on hourly power prices, Nordic electricity market data [32] is used. Without loss of generality, the power price uncertainty is modeled through a set of 7 possible scenarios. The real-time price deviation linearly increases with time and it is set to be 10% at 24 h later. The probability of each scenario is assigned considering the well-known 7-step approximation of the normal distribution function. Fig. 4 illustrates the price scenarios considering the expected price and time-varying standard deviation. The hourly heating demand is constructed using 1-capacity building model as described in [33]. The model takes into account building thermal dynamics and ambient temperature. To tackle the heat load variations, the stochastic hourly heat demand for a typical winter day is modeled as function of different probable scenarios. The heat load scenarios are based on the expected demand and time-varying standard deviation and are shown in Fig. 5. For the sake of simplicity, it is to be noted that the customer’s heat demand is assumed to be deterministic in the simulations. It is a reasonable assumption if and only if the building under study has a slow moving heat dynamics due to high level of thermal insulation and only space heating load is considered, as is the case in Nordic countries. The thermal energy storage charging profile would be same for all price scenarios; however, the direct electric heating demand matrix is updated on an hourly basis to moderately compensate the future forecast errors. In addition, the thermal energy storage is considered to be an ideal as the standby storage losses of commercially available heaters is about 0.5% per h at full storage [34]. 4.2. Simulation results The proposed demand response-decision framework is optimized using the described GA-based technique using MATLAB
Heat demand Scenarios [kW]
2
1,5
1 0
3
6
9 12 15 Time of Day (hours)
18
21
Table 1 Scenario of energy payment associated with different risk levels. Risk-level (ˇr )
0 5 10 12 15 25 50
Possible future costs (D ) Expected value
Optimistic case
Pessimistic case
1.35 1.36 1.37 1.4 1.45 1.52 1.58
1.21 1.23 1.24 1.29 1.34 1.43 1.5
1.48 1.49 1.5 1.52 1.55 1.61 1.67
optimization toolbox. The problem is elucidated for the customers equipped with heating system having 50% thermal storage1 and having different risk preferences ˇr . In Fig. 6, the expected costs associated with different risk levels are depicted. The results present an expected-cost/deviation-cost tradeoff for customers with different risk taking abilities. For risk-seeking (0 < ˇr ≤ 10) customers, the expected cost would be lower but the cost deviation would be higher as compared to risk-averse (ˇr > 10) customers whose expected costs would be much higher but with reduced risk. For an extreme risk-seeking (ˇr = 0) customer, the cost standard deviation could be significantly lowered (14%) if it is willing to accept a slight increase in the expected cost (2.9%). Nevertheless, there is no risk-free zone; there is a lower bound to the cost standard deviation and beyond that value there is no noteworthy reduction in risk. In the simulated case, the minimum cost deviation that could be achieved is 1.9% of the customer’s expected energy payment. Table 1 lists the possible future energy payments given different risk levels. The energy cost could soar up to 9.6% more than the expected cost in the case of a risk-seeking customer. However, there is a likelihood of a drastic cost reduction as well (10.4% declination from expected cost). On the other hand, for the extreme risk-averse customers, the best and the worst-case scenarios are insignificant as the total payment is not vulnerable to power price uncertainty. The deviation cost of this category of customers is about 35% less than an extreme risk-affine customer. Fig. 7 provides a comparison of heat load management approach between risk-averse and risk-seeker customers. It shows the storage charging profile for customers having different risk-attitude. The risk level has a strong influence on charging pattern, for instance, a risk-seeking (ˇr = 10) customer did not charge the thermal storage until 07:00 h, and meet the minimum energy requirement through direct electric heating, in the hope that power prices would get cheaper. On the contrary, the customers with the high-risk taking attitude resorted to the storage charging during early hours marked by high prices and thus mitigated the financial risk at the expense of a higher expected cost.
24
Fig. 5. Probable scenarios of hourly heat demand for a typical medium massive structure house.
1 Capacity of storage is expressed as a percentage of daily heat demand of a typical winter day.
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1.6 Expected Cost (€ )
Storage level (p.u)
r
β =0
1
r
β = 10
0.8
r
β = 25 r
β = 50
0.6 0.4 0.2 0
3
6
9 12 15 Time of day (hours)
18
21
24
Fig. 7. Thermal storage charging profile of different risk-attitude customers.
0.8 r
Probability
β =0 r
β =12
0.6
r
β =50
0.4 0.2 0
1.2-1.3
1.3-1.4
1.4-1.5 Cost (€ )
1.5-1.6
1.6-1.7
Fig. 8. Histogram of the total energy payment.
Expected Cost (€ )
1.7
r
β =50
100%
50%
25%
10%
r
β =50
1.6
r
β =50
r
β =50 r
1.5
r
β =15
β =15
r
r
β =15
1.4
r
r
β =0
β =0
β =0 r
1.3
β =0
1.5
2
2.5 3 3.5 Cost standard deviation (%)
4
4.5
437
r
σ decreased by 1%
β =50
r
β =50 βr=50
1.5
r
β =15
base case ( σ = 10%) σ increased by 1%
r
β =15
r
β =15
1.4
r
β =0 r
β =0
1.3
2
r
β =0
2.5 3 3.5 Cost standard deviation (%)
4
Fig. 10. Consequences of the price uncertainty on the demand response optimality.
reduces the energy payment by storing the heat during the cheaper hours, but also hedges against the volatile prices to a certain degree. The greater the storage capacity the lesser the deviation cost for a given risk level. For instance, in the case of 100% storage, the risk reduces to 50% as compared to partial storages (≤25%) for the same expected cost. Also, for the full storage (100%) case, the deviation cost of a risk-averse customer is reduced to less than half of the expected energy payment of the risk-affine customer. On the contrary, the partial storages (storage size less than 30%) have the lowest potential for off-setting the financial risk. For example, in the case of 10% storage, the range of deviation cost lies within 1% of the expected cost and hence not much prospect to equipoise the risk. The practicality of the proposed demand response framework, like any other decision framework, depends on the forecasting accuracy of input parameters, e.g., power price. In Fig. 10, the consequences of power price uncertainty on the expected cost/deviation cost tradeoff are illustrated. The plot gives a comparison between price forecasting extreme cases for different risk levels. As anticipated, the price uncertainty would change the results significantly. As an example, in the case of an extreme risk-seeker (ˇr = 0), even 1% change in price deviation from the base case (Price standard deviation is assumed to be 10%) would lead to an increase of the customer’s energy cost deviation by 8.5%.
Fig. 9. Effect of storage capacities on the expected cost/cost deviation tradeoff.
5. Conclusion For the extreme case when the financial risk is not considered (ˇr = 0), the storage is only charged during the cheapest hours 09:00–12:00 h to reduce the expected energy payment to minimum. It is worth observing here that the storage charging profile is dictated by power price scenarios and risk preferences. This is the reason the customer with (ˇr = 25), who is in between the extremes of risk-affine and risk considered attitudes would also like to reduce energy cost in addition to the reduction of cost deviation, hence the discharging trend can be seen during early hours of the day when the power prices were relatively higher. In Fig. 8, the distribution of energy payment occurs in various scenarios for three different risk levels are illustrated. It can be seen from the figure that as the risk coefficient increases, the average customer’s energy cost is shifted to the right side of the plot, representing an increase in the expected energy cost. The average energy payment increases by 17% as the ˇr increases from 0 to 50. While, the cost variance decreases with decrease in the values of ˇr . 4.3. Sensitivity analysis A set of sensitivity analyses is performed to explore (a) dependence of thermal storage size on customer’s energy payment and risk (b) price uncertainty on the customers’ expected cost given a risk level. Fig. 9 shows the effect of storage capacity on the tradeoff between the expected cost and risk. The thermal storage not only
This paper presented an optimal decision model for the household heat load management under a RTP environment; the aim is to minimize the customer’s expected cost given a certain risk level without deteriorating the user’s thermal comfort. The uncertainty associated with customers’ heat demand and future prices was taken into account using a scenario-based stochastic programming approach. The optimal demand response framework efficiently considered the risk-attitude of customer and quality of service constraint to optimize the thermal energy storage scheduling. The problem was solved using a GA-based technique on an hourly basis using a moving window approach. From the simulation results, it is clear that as the consumer concern on the risk increases, the expected cost increases as well. The deviation cost for a risk-affine customer can fluctuate significantly from the expected cost. In addition, it is evident from the results that the thermal energy storage can be an effective tool for risk-management under uncertain environments. The future work would focus on large scale integration of residential storage space heating demand response in distribution networks with the special attention to the customer preferences and accurate heating load and thermal energy storage modeling. Acknowledgements This work is supported by Aalto Energy Efficiency (AEF) Research Program through Smart Control Architecture for Smart Grids (SAGA) project.
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[18] C. Xiaodao, W. Tongquan, H. Shiyan, Uncertainty-aware household appliance scheduling considering dynamic electricity pricing in smart home, IEEE Trans. Smart Grid 4 (June (2)) (2013) 1–10. [19] V. Hamidi, F. Li, F. Robinson, Demand response in the UK’s sector, Electr. Power Syst. Res. 79 (12) (2009) 1722–1726. [20] A. Safdarian, M. Fotuhi-Firuzabad, M. Lehtonen, A distributed algorithm for managing residential demand response in smart grids, IEEE Trans. Ind. Inform. 10 (4) (2014) 2385–2393. [21] D.T. Nguyen, L. Le, Joint optimization of electric vehicle and home energy scheduling considering user comfort preference, IEEE Trans. Smart Grid 5 (January (1)) (2014) 188–199. [22] J.H. Yoon, R. Baldick, A. Novoselac, Dynamic demand response controller based on real-time retail price for residential buildings, IEEE Trans. Smart Grids 5 (1) (2014) 121–129. [23] S.A. Pourmousavi, S.N. Patrick, M.H. Nehrir, Real-time demand response through aggregate electric water heaters for load Shifting and balancing wind generation, IEEE Trans. Smart Grid 5 (March (2)) (2014) 769–778. [24] R.M. Oviedo, Z. Fan, S. Gormus, P. Kulkarni, A residential PHEV load coordination mechanism with renewable sources in smart grids, Int. J. Electr. Power Energy Syst. 55 (2014) 511–521. [25] N. Baghina, B. Ioannis Lampropoulos, W.L. Asare-Bediako, P.F. Kling, Ribeiro, Predictive control of a domestic freezer for real-time demand response applications, in: Proc. Innovative Smart Grid Technologies (ISGT Europe), October 2012, pp. 1–8. [26] A. Safdarian, M. Fotuhi-Firuzabad, M. Lehtonen, Impacts of time-varying electricity rates on forward contracts scheduling of DisCos, IEEE Trans. Power Deliv. 29 (April (2)) (2014) 733–741. [27] M. Carrión, A.J. Conejo, J.M. Arroyo, Forward contracting and selling price determination for a retailer, IEEE Trans. Power Syst. 22 (November (4)) (2007) 2105–2114. [28] A. Ayed, K. Bhattacharya, A generic operations framework for discos in retail electricity markets, IEEE Trans. Power Syst. 24 (February (1)) (2009) 356–367. [29] J.R. Birge, F. Louveaux, Introduction to Stochastic Programming, SpringerVerlag, NY, USA, 1997. [30] X.Y. Zhou, D. Li, Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Appl. Math. Optim. 42 (2000) 19–33. [31] Coley, A. David, An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific, 1999. [32] Nordpool. Available: http://www.nordpoolspot.com [33] M. Ali, J. Jokisalo, K. Siren, M. Lehtonen, Combining the demand response of direct electric space heating and partial thermal storage using LP optimization, Electr. Power Syst. Res. 106 (January) (2014) 160–167. [34] K. Vanthournout, R. D’hulst, D. Geysen, G. Jacobs, A smart domestic hot water buffer, IEEE Trans. Smart Grid 3 (December (4)) (2012) 1–7.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Risk-constrained framework for residential storage space heating load management Mubbashir Ali ∗ , Amir Safdarian, Matti Lehtonen Department of Electrical Engineering and Automation, Aalto University, Espoo, Finland
a r t i c l e
i n f o
Article history: Received 14 May 2014 Received in revised form 26 September 2014 Accepted 26 October 2014 Keywords: Demand response Electric storage space heating Real time pricing Risk mitigation Smart grid Uncertainty
a b s t r a c t Residential demand response based on real time pricing provides a strong incentive for customers to reduce their energy payment. However, acting under an environment with time-varying prices will expose them to uncertain energy bills. This paper presents a risk-constrained framework for residential customers for scheduling the electric storage space heating load. The proposed decision framework attempts to accomplish desired settlement between expected cost minimization and cost deviation without altering the user’s thermal comfort. The price and load uncertainty are captured by a scenario based stochastic programming approach. The optimization model is solved using Genetic Algorithm and implemented using a moving-window procedure. The simulation results demonstrate that the proposed framework for scheduling the storage space heating load provides a method to selectively hedge against the price and load uncertainty risk. The optimal framework will result in an improved interaction between the electrical aggregator and its customers under the smart grid paradigm. © 2014 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation In today’s power system, safe integration of intermittent renewable energy resources is considered to be the most demanding challenge. Problems that further aggravated the situation include charging electric vehicles, aging infrastructure and institutional changes in electricity markets. In these circumstances, demand response is seen as one of the effective tools for alleviating these issues [1–3]. Demand response is a mechanism to motivate the customers, by either changing prices or giving incentives, to adjust the electricity consumption to bring favorable results for power system operation at different time-scales. Real time price (RTP) based demand response is considered to be the most effective mechanism for demand side management (DSM) [4]. Future smart grids can enable demand side management, integration of distributed energy resources and improve power system efficiency [5]. With the advent of smart grid technologies, the real time interaction
∗ Corresponding author at: Aalto University, Department of Electrical Engineering and Automation, Otakaari 5 A, P.O. Box 13000, Espoo FI-00076, Finland. Tel.: +358 50 4367307; fax: +358 9 47022991. E-mail addresses: mubbashir.ali@aalto.fi (M. Ali), amir.safdarian@aalto.fi (A. Safdarian), matti.lehtonen@aalto.fi (M. Lehtonen). http://dx.doi.org/10.1016/j.epsr.2014.10.024 0378-7796/© 2014 Elsevier B.V. All rights reserved.
between customers and electrical aggregators is becoming possible and a household customer can also participate in demand response programs implicitly optimizing the power system operations [6]. Amongst the household loads, the thermostatic controlled appliances (TCAs) such as electric space heaters, electric water heaters, fridge, etc are logically the most suited load for an effective demand response application. The time postponement and power-schedulable characteristic of the TCAs load qualifies it as a well-suited load for DSM programs from an electrical aggregator perspective. The flexibility of TCAs is enhanced if it is equipped with some degree of thermal energy storage. For instance, equipping space heating system with thermal storage (system commonly known as electric storage space heating, see Fig. 1) allows heating to uncouple from the system hence the power interruption can take place with almost no impact on customer’s thermal comfort and can be used for power system operation optimization. At present, many electric storage space heating consumers are charged according to the static Time of Day distribution tariffs despite considerable hourly variation in the wholesale market power price. The existing Time of Use (ToU) tariff help the household consumers to manage their heat consumption by charging the thermal storage during night time and then allowing the stored heat to coast through the rest of the day. Nowadays, with the advent of smart meters and the penetration of intermittent renewable generation, a contractual agreement between
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Nomenclature ˇr Ce Dl N Pch Pd Q qh,n S0t SLht t Smax ZN ˝(n) FR h n
risk coefficient electricity price (D /kWh) maximum power taken from the electric storage space heating system (kW) total number of scenarios considered. charging power (kWh/h) direct electric heating power (kWh/h) power taken from thermal storage (kWh/h) average heat demand (kWh) during hour h and scenario n. initial level of thermal storage (kWh) stored energy lost during hour h (kWh) net storage capacity (kWh) expected cost over N scenarios (D ) probability of scenario n storage loss coefficient financial risk index of hours index of scenarios
the end consumers and electrical aggregator can be made possible so that the customers can respond to more dynamic time varying prices. One of the serious implications of such a demand response program is that it could expose the end-user to the financial risk especially if the power prices are highly volatile [7,8]. A comprehensive report [9] by US department of energy on utility experience with real-time pricing suggest that RTP can be an attractive tariff however there are potential risk burdens associated with that which must be overcome in order to encourage the extent of customers participation in real-time demand response programs. Although a very attractive pricing scheme, the RTP prices can subject the consumers to a relatively high degree of risk by exposing them to stochastic power prices [9]. 1.2. Approach In an attempt to offset the customer’s potential risk that may be incurred due to the price volatility and load uncertainty as well as simultaneously reducing the energy payment, we present an optimal framework for scheduling the electric storage space heating load. The uncertainty is modeled using a scenario based stochastic programming approach. The proposed demand response model adequately considers the risk-level while reducing the expected energy cost of the customer. The risk is simply modeled by the minimum variance approach. Moreover, the customer’s thermal comfort is not compromised as the proposed demand response model manages the thermal storage operation without compromising on the energy demand during that hour. The decisions are
Fig. 1. Illustration of a typical electric storage space heating system. In this figure, the sum of Pch (charging power) and Pd (direct heating power) is the total thermal power output of the storage space heater.
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made for next 24 h but executed and revised on an hourly level (the approach commonly known as a moving window procedure) as the new information on prices and demands are updated on an hourly basis. 1.3. Related work and contribution A significant amount of research studies have concentrated on the residential demand response. The most relevant works are reviewed in the following to demonstrate the contribution of this paper. The authors of [10] presented an optimal operation of major residential load under a smart grid scenario. The work in [11] presented an optimal energy scheduling framework for an automatically operating appliance. A co-evolutionary particle swarm optimization (PSO) based decision support tool for optimally scheduling distributed energy sources at household level is developed by Angelo et al. [12]. Rad et al. [13] formulated an optimal household load control under a real-time pricing and inclining block rate environment to achieve a tradeoff between energy payment and waiting time. Peizhong et al. [14] proposed a scheduling scheme based on optimal stopping rules to achieve the same objective. A robust optimization model to schedule the hourly-level consumption is developed by Conejo et al. [15]. In [16], an appliance commitment algorithm that schedules thermostatically controlled appliances considering user comfort under the price and demand uncertainty situation. A Markov decision framework to schedule the power consumption based on future price forecasts was discussed by Kim and Poor [17]. An efficient scheduling algorithm to optimize the household appliance operation considering demand and intermittent renewable uncertainty was proposed in [18]. In [19], a study has been performed to assess the domestic demand response potential of responsive appliances. The work [20] proposed a distributed algorithm solution for unleashing domestic demand response in smart grid environment. The research in [21] coordinated the load management of heating load and electric vehicles to bring customer economic savings. The demand response model evaluates the cost of discomfort and temperature preferences however the framework does not account the uncertainty and risk issues. The work [22] demonstrated the performance of dynamic controller for residential building demand response application. Some studies have principally focused on the demand response control of a particular flexible appliance such as water heater [23], electric vehicle [24], and freezer [25] to name just a few. Most of the above reviewed work focused on customer load scheduling in order to minimize the user energy payment under dynamic prices. However, despite its importance, the customer’s financial risk associated with electricity purchasing under RTP were not adequately discussed in the literature. Nonetheless, there exist a few papers that addressed the problem from the perspective of a risk-averse energy retailer [26–28]. A few risk measures were described and used in these papers which can be useful in the current study as well as the future works, however, more comprehensive models are required to be provided to achieve the ultimate benefits of demand response potentials. In this regard, this paper develops a risk constrained model to minimize the expense of heating demand for a residential customer. The core contributions of this paper may be summarized as follows: • It presents a user-eccentric tool for customers to optimize their space heating load under uncertainty environment without deteriorating the customer’s thermal comfort. • To minimize customer’s cost at a given risk level, a non-linear optimization problem is formulated to obtain the optimal amount
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of charging power and direct electric heating power to meet the energy demand. • Genetic Algorithm (GA) is used to solved the demand response model due to its simplicity and robustness to handle non-linear objective function. • The influence of thermal storage capacity on the customer’s cost and financial risk is critically analyzed.
1.4. Paper organization The remainder of the paper is organized as follows. In Section 2, some theoretical basics are briefly described. The risk-considered demand response framework for electric storage heating is presented in Section 3. Simulation results are presented and discussed in Section 4. Finally, Section 5 concludes the paper.
2. Theoretical basis
2.3. GA-based solution technique Owing to the non-linear objective function, we selected GA for solving the problem. Although the model can be approximated to a linear one which can be then solved using efficient linear programming solvers, however then the accuracy of the results may be compromised. GA is a random search algorithm inspired by the natural evolution process. GA employs probabilistic search rules. It considers a population of chromosomes (set of genes) which encode the possible solutions. The combination of individuals take place on the basis of fitness function to create individuals that are more fit [31]. Unlike classical optimization methods, such as interior point method, the GA does not require the information of derivative of the objective function and take the least computational resources and thus can be easily embedded into the home energy management system with minimal infrastructure cost. It is worth mentioning that GA is selected as a tool to illustrate the proposed framework effectiveness and the proposed problem can be solved by any other heuristic techniques or classical optimization methods.
2.1. Uncertainty description The uncertainty in future power prices and load poses challenges in scheduling the electric space heating load. Both of the uncertainty sources negatively affect customers enrolled in RTP based demand response programs. Due to the uncertain future heating load the power scheduled may not match the realized heating demand. This would consequently affect the customer’s thermal comfort as well as the energy expense. In such an energy mismatch case, the heating power is to be then supplied from direct electric heating in the case of deficit. While the heating power is to be stopped if the required heat demand is lower than the scheduled power. The operation of storage space heating needs to be efficiently managed so that the expected electricity payment can be minimized under the dynamic power prices. To handle the price and load uncertainty, we employ a scenario based stochastic programming approach [29] to generate the load and price scenarios. A set of most likely scenarios are generated considering the expected value and time-varying standard deviation. It is supposed that the heating load and power prices can be projected with good accuracy for the upcoming hour while the standard deviation linearly increases as the prediction time increases. The probability of happening of each scenario is captured by normal probability distribution function. Then the problem is solved over all the scenarios. The goal is to minimize the expected cost value given the price and load scenarios. The constraints are taken to be same in all of the considered scenarios. Although the greater number of scenario increases the accuracy of solution but it happens at the cost of complexity of the model. To balance the tradeoff between complexity and accuracy, scenario reduction methods can be used to lessen the number of scenarios without compromising on accuracy.
3. Proposed framework for heating load scheduling We consider a smart grid environment in which real-time prices are offered to the residential customers and they respond to the time-varying prices to minimize their energy payment while limiting the financial risk due to uncertainties to a pre-defined level. We consider that a smart home load management system is integrated in the household and an adequate decision framework is required for load management. The proposed decision support tool for home load management system is presented in the following subsection. 3.1. Problem formulation The proposed decision support tool, based on the future heating load and power prices, will help the customers to decide the optimal share of direct heating power and storage charging power to meet the heating energy requirement. The objective of the framework is to optimize the operation of space heating load in order to minimize the user expected payment while the financial risk imposed by uncertain power prices and stochastic heat demand is restricted to a certain level. The objective function can be mathematically written as follows: minimize
Z N + ˇr (FR)
where, ZN =
N
24
˝(n)
n=1
2.2. Risk measure Consider a consumer involved in the RTP-based demand response program. The uncertain future power prices will render the customer to uncertain energy payment especially if the power prices are volatile. To tackle this inconvenience, a naive risk measure is integrated in the decision-model. The risk management is imposed through the mean-variance approach [30]. In general, it is simply the volatility (standard deviation) of the cost at different scenarios. The minimum variance approach has only the objective of mitigating the risk rather and leads to a noticeable concentration in low volatility costs at different scenarios.
(1)
e d Ch,n (Phch + Pn,h )
(2)
h=1
⎡ 24 2 ⎤ N e (P ch + P d ) − Z N ⎦ FR = ⎣ ˝(n) Ch,n h n,h n=1
(3)
h=1
In the objective function (1), the coefficient ˇr principally models the customer’s risk attitudes. The higher values of risk coefficient (ˇr > 10) represent the risk-averse customers whereas the lower values (0 < ˇr ≤ 10) represent the risk-seeking customers. ˇr would be zero if and only if the risk is ignored and it corresponds to the case of an extreme risk affine customers. The detailed subject of setting the risk-coefficient parameter is not the focus of this paper.
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Fig. 2. A moving window procedure for optimization.
The objective function (1) is subjected to the following operational constraints. The constraint (4) ascertains the quality of service by not altering the minimum heating requirement: d Pn,h + Qn,h = qn,h ,
∨ h, n
(4)
The rated power of direct electric heating Pd and charging Pch is enforced by constraints (5) and (6), respectively: Fig. 3. Flow chart illustrating the demand response model implementation. ch 0 ≤ P ch ≤ Pmax , d 0 ≤ P d ≤ Pmax ,
∨h
(5)
∨ h, n
(6)
The following constraint bounds the storage discharging power limits: 0 ≤ Q ≤ Qmax ,
∨h
(7)
The constraint (8) models the dynamics of thermal storage. t Sh+1 = Sht + (Phch − Qh )h − SLht ,
∨h
(8)
The storage losses are modeled in (9). An additional constraint formulated in (10) that guarantees that total power taken from the heating system shall not exceed a pre-defined maximum power limit: t SLh+1 = Sht ,
P ch + P d ≤ Dl ,
∨ h, n ∨ h, n
(9) (10)
3.2. Demand response model implementation The optimal demand response framework of electric storage space heating is developed in the Section 1.4. In this section, the demand response framework implementation is discussed. The implementation of this proposed decision model does not require any costly hardware setup. A two-way communication is needed between the utility and end-user. The communication infrastructure is required to be integrated with the smart home energy management system. The utility announces the timevarying prices, and expected outside temperature profile for the following hours to the customer. The proposed demand response model is optimized using the GA algorithm technique for a moderately short-term horizon [h, h + 24] and implemented using a moving window procedure. The moving window procedure as illustrated in Fig. 2 basically optimizes the thermal storage operation for a finite optimization horizon while considering the future heating energy demand and power prices. Only the first step of the optimization result is implemented and then the inputs are sampled again. The optimization routine is repeated resulting in a new
predicted thermal storage charging/discharging profile. The sample time is taken as 1 h, whereas, the scheduling horizon of 24 h is chosen. The flow chart of moving window algorithm for solving the demand response framework is shown in Fig. 3. First off, the heat demand and electricity prices are forecasted and announced for h + 24 h. Next, the demand response framework is solved using a GA approach. The gradient-free feature and the ability to reach a near-optimal solution for non-linear problem in quick time make GA a suitable tool for the demand response applications of electric storage space heating at the household level. The inputs of the optimization model are power price scenarios, stochastic heat demand, customer risk preferences and thermal storage size. The decision (output) variables are amount of direct electric heating power and charging power. The purpose of the GA-based demand response optimization is to search for an optimal proportion of direct electric heating power and charging power to meet the current hour heating load while deciding on charging the thermal storage tank to use it for later hours considering the future costs and heating load. The proposed demand response framework is solved and the output charging power and direct electric heating profile is obtained for the next 24 h, however, only the decisions are made for the first hour (h + 1) and then the price and heating data is updated. This moving window approach makes the thermal storage scheduling immune from the price and load forecasting error to some extent. It would also tackle the problem of empty storage level by the end of the optimization period, which is the case if the scheduling horizon is static. This hourly moving window algorithm is suitable to handle the dynamics of a relatively massive-structured (high thermal constant) building to be heated.
4. Case study In order to show the usefulness of the proposed demand response decision framework, it is applied to a case study. A typical medium massive structure Finnish detached house equipped
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Fig. 6. Evolution of the expected cost versus cost standard deviation. Fig. 4. Power price scenarios (The expected hourly power prices are shown with dashed line).
with storage space heating system is considered for the analysis. The obtained results are reported and discussed in this section. 4.1. Data To model the uncertainty on hourly power prices, Nordic electricity market data [32] is used. Without loss of generality, the power price uncertainty is modeled through a set of 7 possible scenarios. The real-time price deviation linearly increases with time and it is set to be 10% at 24 h later. The probability of each scenario is assigned considering the well-known 7-step approximation of the normal distribution function. Fig. 4 illustrates the price scenarios considering the expected price and time-varying standard deviation. The hourly heating demand is constructed using 1-capacity building model as described in [33]. The model takes into account building thermal dynamics and ambient temperature. To tackle the heat load variations, the stochastic hourly heat demand for a typical winter day is modeled as function of different probable scenarios. The heat load scenarios are based on the expected demand and time-varying standard deviation and are shown in Fig. 5. For the sake of simplicity, it is to be noted that the customer’s heat demand is assumed to be deterministic in the simulations. It is a reasonable assumption if and only if the building under study has a slow moving heat dynamics due to high level of thermal insulation and only space heating load is considered, as is the case in Nordic countries. The thermal energy storage charging profile would be same for all price scenarios; however, the direct electric heating demand matrix is updated on an hourly basis to moderately compensate the future forecast errors. In addition, the thermal energy storage is considered to be an ideal as the standby storage losses of commercially available heaters is about 0.5% per h at full storage [34]. 4.2. Simulation results The proposed demand response-decision framework is optimized using the described GA-based technique using MATLAB
Heat demand Scenarios [kW]
2
1,5
1 0
3
6
9 12 15 Time of Day (hours)
18
21
Table 1 Scenario of energy payment associated with different risk levels. Risk-level (ˇr )
0 5 10 12 15 25 50
Possible future costs (D ) Expected value
Optimistic case
Pessimistic case
1.35 1.36 1.37 1.4 1.45 1.52 1.58
1.21 1.23 1.24 1.29 1.34 1.43 1.5
1.48 1.49 1.5 1.52 1.55 1.61 1.67
optimization toolbox. The problem is elucidated for the customers equipped with heating system having 50% thermal storage1 and having different risk preferences ˇr . In Fig. 6, the expected costs associated with different risk levels are depicted. The results present an expected-cost/deviation-cost tradeoff for customers with different risk taking abilities. For risk-seeking (0 < ˇr ≤ 10) customers, the expected cost would be lower but the cost deviation would be higher as compared to risk-averse (ˇr > 10) customers whose expected costs would be much higher but with reduced risk. For an extreme risk-seeking (ˇr = 0) customer, the cost standard deviation could be significantly lowered (14%) if it is willing to accept a slight increase in the expected cost (2.9%). Nevertheless, there is no risk-free zone; there is a lower bound to the cost standard deviation and beyond that value there is no noteworthy reduction in risk. In the simulated case, the minimum cost deviation that could be achieved is 1.9% of the customer’s expected energy payment. Table 1 lists the possible future energy payments given different risk levels. The energy cost could soar up to 9.6% more than the expected cost in the case of a risk-seeking customer. However, there is a likelihood of a drastic cost reduction as well (10.4% declination from expected cost). On the other hand, for the extreme risk-averse customers, the best and the worst-case scenarios are insignificant as the total payment is not vulnerable to power price uncertainty. The deviation cost of this category of customers is about 35% less than an extreme risk-affine customer. Fig. 7 provides a comparison of heat load management approach between risk-averse and risk-seeker customers. It shows the storage charging profile for customers having different risk-attitude. The risk level has a strong influence on charging pattern, for instance, a risk-seeking (ˇr = 10) customer did not charge the thermal storage until 07:00 h, and meet the minimum energy requirement through direct electric heating, in the hope that power prices would get cheaper. On the contrary, the customers with the high-risk taking attitude resorted to the storage charging during early hours marked by high prices and thus mitigated the financial risk at the expense of a higher expected cost.
24
Fig. 5. Probable scenarios of hourly heat demand for a typical medium massive structure house.
1 Capacity of storage is expressed as a percentage of daily heat demand of a typical winter day.
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Fig. 7. Thermal storage charging profile of different risk-attitude customers.
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Fig. 8. Histogram of the total energy payment.
Expected Cost (€ )
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r
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4
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σ decreased by 1%
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4
Fig. 10. Consequences of the price uncertainty on the demand response optimality.
reduces the energy payment by storing the heat during the cheaper hours, but also hedges against the volatile prices to a certain degree. The greater the storage capacity the lesser the deviation cost for a given risk level. For instance, in the case of 100% storage, the risk reduces to 50% as compared to partial storages (≤25%) for the same expected cost. Also, for the full storage (100%) case, the deviation cost of a risk-averse customer is reduced to less than half of the expected energy payment of the risk-affine customer. On the contrary, the partial storages (storage size less than 30%) have the lowest potential for off-setting the financial risk. For example, in the case of 10% storage, the range of deviation cost lies within 1% of the expected cost and hence not much prospect to equipoise the risk. The practicality of the proposed demand response framework, like any other decision framework, depends on the forecasting accuracy of input parameters, e.g., power price. In Fig. 10, the consequences of power price uncertainty on the expected cost/deviation cost tradeoff are illustrated. The plot gives a comparison between price forecasting extreme cases for different risk levels. As anticipated, the price uncertainty would change the results significantly. As an example, in the case of an extreme risk-seeker (ˇr = 0), even 1% change in price deviation from the base case (Price standard deviation is assumed to be 10%) would lead to an increase of the customer’s energy cost deviation by 8.5%.
Fig. 9. Effect of storage capacities on the expected cost/cost deviation tradeoff.
5. Conclusion For the extreme case when the financial risk is not considered (ˇr = 0), the storage is only charged during the cheapest hours 09:00–12:00 h to reduce the expected energy payment to minimum. It is worth observing here that the storage charging profile is dictated by power price scenarios and risk preferences. This is the reason the customer with (ˇr = 25), who is in between the extremes of risk-affine and risk considered attitudes would also like to reduce energy cost in addition to the reduction of cost deviation, hence the discharging trend can be seen during early hours of the day when the power prices were relatively higher. In Fig. 8, the distribution of energy payment occurs in various scenarios for three different risk levels are illustrated. It can be seen from the figure that as the risk coefficient increases, the average customer’s energy cost is shifted to the right side of the plot, representing an increase in the expected energy cost. The average energy payment increases by 17% as the ˇr increases from 0 to 50. While, the cost variance decreases with decrease in the values of ˇr . 4.3. Sensitivity analysis A set of sensitivity analyses is performed to explore (a) dependence of thermal storage size on customer’s energy payment and risk (b) price uncertainty on the customers’ expected cost given a risk level. Fig. 9 shows the effect of storage capacity on the tradeoff between the expected cost and risk. The thermal storage not only
This paper presented an optimal decision model for the household heat load management under a RTP environment; the aim is to minimize the customer’s expected cost given a certain risk level without deteriorating the user’s thermal comfort. The uncertainty associated with customers’ heat demand and future prices was taken into account using a scenario-based stochastic programming approach. The optimal demand response framework efficiently considered the risk-attitude of customer and quality of service constraint to optimize the thermal energy storage scheduling. The problem was solved using a GA-based technique on an hourly basis using a moving window approach. From the simulation results, it is clear that as the consumer concern on the risk increases, the expected cost increases as well. The deviation cost for a risk-affine customer can fluctuate significantly from the expected cost. In addition, it is evident from the results that the thermal energy storage can be an effective tool for risk-management under uncertain environments. The future work would focus on large scale integration of residential storage space heating demand response in distribution networks with the special attention to the customer preferences and accurate heating load and thermal energy storage modeling. Acknowledgements This work is supported by Aalto Energy Efficiency (AEF) Research Program through Smart Control Architecture for Smart Grids (SAGA) project.
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