This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.
In this paper, the energy procurement for a distribution company (Disco) from the pool market, distributed generations (DGs) and contractual interruptible loads with a risk constraint stochastic programming framework is presented. We propose a formulation that considers uncertainty of some parameters such as pool price, end user demand and the items which are related to DGs so that maximize the expected profit of Disco. Also the revenue of not producing emission gas by utilizing clean energy is considered. A realistic case study illustrates the methodology proposed.
Keywords; Contractual interruptible load, CVaR, DG, pool market and uncertainty.
Traditionally, a Disco purchases energy from wholesale market, at a high voltage level, and then transfers this energy to final customers. Nevertheless, the restructuring process of the energy sector has stimulated the introduction of new agents and products, and the unbundling of traditional Disco into technical and commercial tasks, including the provision of ancillary services .
Distribution company planners continually endeavor to develop new planning strategies for their network in order to serve the load growth and provide their customers with a reliable electricity supply. In the present days, competitive electricity market forces drive the disco planners to investigate the economical and technical feasibility of new capacity expansion alternatives such as distributed generation (DG) .
Customers demand and pool price have fluctuation and are uncertain. Therefore, the distribution company must consider these uncertainties. In  and , for electricity procurement, risk is considered.  provides an overview of risk assessment tools in electricity markets, including appropriate tools to analyze the retailer perspective. Stochastic programming models  represent stochastic processes (e.g., electricity pool prices or electricity demands) via a set of scenarios, which are plausible realizations of the stochastic processes throughout the decision-making horizon.
Using risk management techniques for managing and controlling of such uncertainties is necessary. There are several instruments such as forward contracts, future contracts, call/put options, which can be used by the Disco for hedging against risks [7,8]. As risk hedging tool, Interruptible load programs are voluntary options, in which customers receive credits for permitting the utility to interrupt temporarily part or all of their loads during the period of the contract .
A DG investment planning from the perspective of a Disco that minimizes its investment and operation costs is proposed in . A static single-period energy acquisition market model with DGs and interruptible loads are presented in , while a multiperiod energy acquisition model in a day-ahead electricity market addressed in . In , a risk-constrained stochastic programming framework to decide which forward contracts the retailer should sign and at which price it must sell electricity so that its expected profit is maximized at a given risk level is modelled. A stochastic programming framework for electricity procurement of a large consumer from several alternatives (pool market, bilateral contracts and self-production) is addressed in [14, 15]. A mathematical method based on mixed-integer stochastic programming to determine the optimal sale price of electricity to customers and the electricity procurement policy of a retailer for a specified period is proposed in .
This paper provides a stochastic programming methodology with considering demand and pool price uncertainties, and utilizes DGs and contractual interruptible load as new sources for energy procurement with considering their uncertainties. This paper is organized as follows. Section II describes the problem formulation including DGs modelling, risk measure and objective function. Section III provides and analyzes results from a realistic case study. . Finally, section IV draws some relevant conclusions.
This section describes the main features of the proposed model.
The aim of this paper is to determine the best possible combination of electric energy sources for a distribution company in demand procurement. It is considered that the retail price to the consumers is yearly constant. In this model, a Disco can only purchase its electricity from pool market or procure it with self production. The planning horizon of the year has been divided into 12 monthly periods. It should be noted that the duration of each period is not the same. For example, January is 744 hours, while February is 672 hours. Hereinafter, the number of hours in period denoted by .
Typically, in a retail electricity market, there are different groups of customers such as residential, commercial and industrial. In this paper, one class of customers is considered. Disco has to face two major difficulties in demand acquisition. While purchasing electric energy, it must cope with uncertain pool prices. While selling electricity, it should handle the uncertainty of the end-user demand. This paper proposes using DGs for risk reduction in a stochastic programming.
For scenario generating of uncertainty parameters, forecasting of the history data or the average of history data is used. Demand and pool price parameters are forecasted with time series . The normal distribution with mean of forecasted curve and time dependent standard deviation is used for scenario generation to cover all plausible realizations of end user demand and pool price. This standard deviation logically indicates that the farther time periods are forecasted with less accuracy. The scenarios of gas price, wind speed and radiation are generated based on history data. In scenario generating method, we consider the average curve of these data as the mean, and the mean standard deviation of every month periods as curve variances; then utilize this mean and standard deviation for our normal distribution.
In this subsection, fixed and variable cost of DGs are modelled.
Fixed investment cost on DGs is apportioned and paid monthly; but in our problem it is considered at the end of the year.
Let be the annual flow of revenues required to recover the investment, defined as :
Where is the initial investment, r is the discount rate, and t is the life time of DG units.
Gas Fired Distributed Generations
One of popular technologies of distributed generation is gas fired generators. Variable cost of this type of generators is modeled as:
Where is variable cost of gas fired DG, is active power generation of DG, , and are gas price, and operation and maintenance (O&M) cost in t-th period and scenario s respectively. is duration of period t. The uncertain parameter of gas price is its major part. is used, whereas the power balance constraint should be satisfied in each period and with every scenario. It appears in our model in order to consider its cost; however our aim is not the accurate determination of this variable.
Wind Turbine Generators
The main element in this type of generators is that the wind speed for planning is not definite. In the other hand, the power generated by a wind turbine depends on uncertain parameter of wind speed. For simplicity, it is considered that the generator output only depends on wind speed, not its direction. For modeling, some kinds of wind turbines are chosen and their curves are accessed linearly. Then they are applied to objective function for optimal value determination.
In catalogues, a minimum speed (a) for turbine start up, and a nominal speed (b) in which turbine reaches its maximum output power are considered. Between these speeds, linear curve can be used truly. The output will be approximately set on zero under (a) speed, and maximum value over (b) speed until a threshold speed. After threshold speed, generator has no output. Considering the above assessments for every selected turbine, it is modeled as:
Where is the generated power of i-th wind turbine, is the capacity of i-th turbine and is the wind speed.
One example of linear approximated curve of wind turbine is illustrated in Fig. 1.
The operational power of i-th wind turbine considering above linear approximation is:
Linear curve of wind turbine output
Where is the operational power of i-th turbine and is wind speed in s-th scenario and t-th period.
The solar cell output power depends on solar radiation. The obtained power of each photovoltaic (PV) panel is related to some effective factors as below :
Where is the solar radiation in with s-th scenario in t-th period, is angle of incidence, is the efficiency of received radiation, is area of PV panel in and is the efficiency of PV panel.
PV panels contain some solar cells to make the facing area larger. for similar panels linear approximated relation between panel area and capacity can be considered. If Disco utilizes some specific available technologies of this kind for investment, the former approximation will be acceptable. Consequently, the operational power of PV panel is:
Where indicates the relation between the j-th panel capacity and its area. Also is capacity of j-th panel.
In this problem, we consider 5 uncertain parameters. Target is maximizing Disco expected profit (profit=revenue-cost).
Where is probability of -th scenario and is Disco annual profit in scenario that is modelled as follow:
Where is duration of period t that related to average hours with solar radiation. is total end user demand, is power purchased from pool market in t-th period and scenario s. is capacity of f-th DG technology. is installation cost of f-th DG technology, is selling price settled by Disco to consumer in , is price which is paid to clean energy for not producing emission gas, is price of electricity in the pool during period t with scenario s in . is quantity of load that the Disco decide to interrupt at price in cent/kWh.
Equation (9) states that for demand procurement, the supplied power of both pool and self-production must be equal to the demand in each period and scenario. Constraints (10) state the power generated from DG must be less than the DG capacity in each period and scenario. Expression (11) is used for constraint on demand procurement from system because of substation capacity limit.
The proposed objective function includes both revenues and costs. The revenue contains two parts: the revenue of selling electricity to the customers, and the revenue of not producing pollutant gas with utilizing clean energy of wind and solar technology. The cost includes 3 parts: the cost of electricity purchasing from pool for demand procurement, the installation cost of DGs, and variable cost.
The variable cost of DGs operation is considered at final part of objective function. This cost for gas fired generators depends on uncertain parameter of gas price. Wind and solar systems are dependent on uncertain parameters of wind speed and radiation.
The risk measure used in this work is the conditional value-at-risk at Î± confidence level (Î±-CVaR). Î±-CVaR is equal to the solution of the following optimization problem :
The optimal value of , , represents the greatest value of the profit not exceeded by any profit outcome with a probability equal to . is known as the value-at-risk (VaR). Furthermore, is the difference between VaR and the profit of scenario .
The risk-constrained formulation of the problem faced by the Disco is summarized below:
The tradeoff between expected profit and risk is enforced through the weighting factor . If risk is not considered (risk-neutral Disco), the value of is set to 0. The higher the value of , the more risk averse the Disco. For example, a risky Disco prefers maximizing its expected profit assuming a high risk of profit volatility. Therefore, this retailer will select a factor Î² close to 0. On the other hand, a conservative Disco is not willing to assume a high risk and accepts a lower profit by selecting a larger value of Î².
In this section, the performance of the proposed model is illustrated through an implementation on realistic data with numerical result.
A time series of six years from 2003 to 2008 is used to characterize the demand and pool price of the electricity market of mainland Spain .
The uncertainty of the end user demands is modeled through a set of five scenarios. Demand scenarios are generated by adding a random term to the expected demand of end user. The standard deviation of the random term increases with the time period. Fig. 2 depicts the demand for the five scenarios considered.
Pool price uncertainty is modeled through a 7 scenario set. Fig. 2 shows the pool prices in all of the 7 scenarios for the 12 periods considered. Likewise pool price scenarios are generated with a increasing random term.
Henry-Hub Index is used for natural gas spot price data of six years from 2003 to 2008 . The averages of daily price data are considered in each month and the averages of these prices in similar months are computed during these years. The computed data with their standard deviation in a normal distribution are used to model gas price through a set of three scenarios. Fig. 2 shows the considered three scenarios of gas price.
Data history for wind speed of a wind farm in Spain is available in . These data are in detail and have been classified hourly for five years from 2004 to 2008. Here the monthly averages of these years are computed. Based on these averages, considering their standard deviations with a normal distribution, the wind speed is modeled through a set of three scenarios. Fig. 2 illustrates the wind speed for the three scenarios considered.
The data of solar radiation are hardly available.  shows some estimated data of solar radiation in different geographic areas. For considering the uncertainty of this parameter the estimated data are used. A normal distribution is used for scenario generating with the averages of these data and the standard deviations that are considered 0.2 of the averages. A set of three scenarios for solar radiation is depicted in fig. 2.
A joint scenario tree of 945 scenarios is generated by taking into account each of 5 demand scenarios, 7 pool price scenarios and 3 scenarios for each of gas price, wind speed and solar radiation. Probability of each scenario for 5 uncertain parameters is provided in TABLE I. Note that the probability of each resulting scenario is equal to multiplication of demand, pool price, gas price, wind speed and solar radiation scenarios. In this way, the sum of the probabilities over all scenarios of the joint tree is equal to 1.
Demand, Pool price, Gas price, Wind speed and Radiation scenarios.
Economical characteristics of DGs are detailed in TABLE II. Selling price to end users is set in 0.5 $/kWh, the price of not producing emission gas is set in 15 $/co2-tone. Substation capacity limit is considered 280 MVA. Discount rate in this work, is supposed to be 10%.
For analyzing the risk that a Disco is encountered, three cases are considered as follow:
Case 1: purchasing from pool market and DGs.
Case 2: purchasing from pool market with DR contracts.
Case 3: purchasing from pool market and DGs with DR contracts.
Fig. 3-5 depicts the probability distribution function of expected profit in three cases above at confidence level of 95% with. The price for interrupting end user demand is considered 150 cent/kWh.
Since is shown in Fig. 3-5, the risk on profit variability refers to the volatility of that probability distribution profit. In case 1, the Disco has the highest quantity of risk, because selling from pool and installing DGs is crossed with uncertainties. Case 2 has the least quantity of risk, but the expected profit in this case is the least. Case 3 that combine each three options for energy procurement has expected profit and risk between two cases former cases. This result is also valid for other Î² values, but to save space, the plot is not shown.
The result (number of scenarios determined) can be verified using the central limit theorem (CLT). Based on CLT, if the number of scenarios is sufficiently large, then the sampling distribution of sample means is approximated by a normal distribution . As shown in Fig. 3-5, the probability density function is approximately close to normal distribution. Therefore, the number of scenarios (samples) is sufficiently large.
Fig. 5 shows that with increasing risk averse parameter, the expected profit decrease. On the other hand, whatever the Disco wants to avoid the risk, he should earn the lower expected profit.
probability Distribution Function in Case 1
Probability Distribution Function in Case 2
Probability Distribution Function in Case 3
Expected Profit Versus Risk Parameter
This paper provides a risk constraint stochastic programming methodology with considering demand and pool price uncertainties that allows a distribution company to engage in medium-term. The proposed linear formulation considers options such as pool market, DG installation and contractual interruptible load for energy procurement in one year planning horizon. Numerical results with realistic data show the capability of this method. In addition, it was shown the risk aversion parameter, affects the selected power supply options.