Regional Energy Supply Planning: Chance Constraint Programming

Regional energy planning under uncertainty is an important concept in energy-economy models which makes the planning outcomes closer to reality and enables the decision maker to select the best decision. Reliability of local energy supply and the possibility of long-term access to resources and emissions reduction is an essential step. In this study, an urban energy demand which is supplied by electricity network is investigated with an optimal combination of alternative energy resources such as solar, wind and natural gas during the next 10 years. The optimal combination of fossil energy as well as renewable energies are determined by goal stochastic programming model. Isfahan province in Iran has been selected as a case study. Empirical results indicate that due to the importance of investment and operation costs, the dominant share of energy supply will belong to natural gas, while the shares of solar and wind energies remain constant in the next decade. In sum, the share of solar and wind energies increases by 8% in 10 years and therefore, it is not necessary to increase electricity supply by the network in order to meet annual increasing demand. CO 2 and NO x emissions will decrease significantly.


INTRODUCTION
Population growth and urban expansion, as well as increasing energy prices have exacerbated power shortage and changing climatic conditions within municipal energy supply systems. These issues are highly interrelated, not only among each other but also with a variety of social, economic, political, environmental and technical factors (Frei et al., 2003). In regional and urban planning, metropolitan areas must be considered as the regions that require energy planning severely. Dense and high population, goods and services dynamism and centralization of services, commercial and industrial activities have turned these areas to major centers of energy consumption.
Energy consumption has increased diffused pollution in large cities, so that their current development in the near future will be faced with productivity slowdown. With the intensification of pollution issue in large cities efforts for solving this problem were also intensified, which leads to the introduction of systems to replace with existing systems of energy supply in large cities. So energy resource allocation requires a multi-criteria decision approach. Nowadays power systems, regulated or deregulated, are exposed to ever more sources of uncertainty, such as fuel prices, demand fluctuations, as well as transmission constraints. This uncertainty and the increasing demand for power raise new challenges for utility planners, whose goal is to provide reliable power to consumers at the lowest possible cost. (Manickavasagam et al., 2015). Malik et al. (1994) offer an integer linear programming model to optimize the share of new and traditional technologies in the energy system. Ramanathan and Ganesh (1995) proposed an integrated model using multi-objective planning and hierarchical analysis process according to quantitative and qualitative criteria, which is developed for energy resources allocation. This model has been applied to the household sector in India which three scenarios have been developed and national grid only has been used for lighting. Groscurth et al. (1995) developed a model, in which urban and regional energy system was described as information flow networks. This model is a very flexible tool for minimization of dynamic demand, pollutants pollution and monetary costs in a stochastic framework. Bruckner et al. (1997) in a study about the competition among energy technologies in urban energy systems, proposed a dynamic optimization model to analyze the competition and different technologies for logical use of energy and renewable energies. Gas-fired, medium-size cogeneration units are found to be the best solution (30% primary energy and 2% cost savings) in a cost-benefit comparison with a system providing heat from conventional oil-fired boilers and receiving electricity from the inter-regional public grid. Dentcheva and Romisch (1998) optimized power generation operation under uncertainty using stochastic programming. This study determines least cost combination of local energy resources in order to supply consumers' demand. A dynamic model for the short-term operation and a power production planning model are used to introduce the cost-optimal generation of electric power under uncertain load. Lesourd (2001) gained the cost per unit of photovoltaic energy systems using life cycle cost analysis and compared the results with the cost per unit of energy in conventional power plants. He examined the advantages of photovoltaic power plant and concluded that photovoltaic power plant has a comparative advantage. Cormio et al. (2003) developed an energy planning model considering renewable energy resources and environmental constraints for a region in southern Italy. Their model includes some sections such as initial supply, electricity and heat generation and the ultimate consumer. The optimization process, aiming to reduce environmental impact and economic activities, provides feasible generation settlements that take into account the installation of combined cycle power plants, wind power, solidwaste, and biomass exploitation together with industrial combined heat and power (CHP) systems.
Another study has formulated an energy-economy planning model in Portugal using fuzzy multi-objective planning Borges and Antunes (2003). This approach is illustrated to tackle uncertainty and imprecision associated with the coefficients of an input-output energy-economy planning model, aimed at providing decision support to decision makers in the study of the interactions between Average solar radiation the energy system and the economy on a national level. Sadeghi et al. (2006) introduced fuzzy linear programming method for optimization of energy supply system in Iran to indicate the approach of application of FLP for optimization, then they figured out FLP is a flexible method that can be a great competitor for other confronting ways. A fuzzy-random interval programming is used by Cai et al. (2009) for the long-term planning of facility capacity. In this case study, multiple conventional (coal, refinery petroleum products, natural gas and nuclear) and renewable (solar radiation, wind, hydropower) energy resources were allocated to multiple end-users (municipal/commercial, industrial, transportation and agricultural sectors) through multiple facilities. A multi-objective linear programming model is proposed by Ren et al. (2010) in which optimal use of energy resources is specified. This model is developed to analyze the optimal operating strategy of a DER (distributed energy resource) system while combining the minimization of energy cost with the minimization of environmental impact which is assessed in terms of CO 2 emissions. Sensitivity analysis indicates that electricity buy-back, carbon tax, as well as fuel switching to biogas, has more or less effect on the operation of DER systems. Adeyefa and Luhandjula (2011) propose an up-to-date overview of how vital probability theory and multi-criteria decision analysis are to deal with situations that several objective functions and the stochastic nature of data are under one roof in a linear optimization context. The mathematical formulation of the problem and related solution have been developed by MOSLP model. Sampaio et al. (2013) presented a centralized power supply station in the city, which is a combination of possible technologies, including thermal power plants, hydroelectric power plant, wind systems, and photovoltaic systems with their relevant emission pattern. Koltsaklis et al. (2015) present a multi-regional, multi-period linear mixed-integer linear programming (MILP) model which combine optimization method with consideration of a Monte Carlo approach (MCA) and demand response. This paper indicates an optimizationbased method to address the generation expansion planning (GEP) problem of a large-scale, central power system in a highly uncertain and volatile electricity industry environment. The optimization goal considers the minimization of the total discounted cost by determining optimal power capacity additions per time interval and region, and the power generation mix per technology and time period. Saffari et al. (2016) presented a goal programming model, considering environmental and financial goals to introduce an optimal energy supply by renewable energy (wind, solar and natural gas) and nonrenewable energy in order to meet the electricity demands in Isfahan over 10 years. This model considered the annual electrical consumption, the potential of alternative technologies, the valuable and overnight cost of conventional and new power plants for determining the optimal portfolio of supplied energy. The thermal power plants in Isfahan have produced the sizable portion of energy which has been needed in this city, resulting in a major amount of air pollution (CO 2 and NO x ).

DESCRIPTION OF CASE STUDY
This research is concentrated on 2010 statistical year. The electricity supply sector in Isfahan province has been composed of two major power plants, namely Montazeri and Isfahan, with 1600MW and 830 MW actual capacity, respectively as well as Hesa gas power plants with 69 MW capacity in 2010. The major fuel used Natural gas, Gasoil and fuel oil. The nominal capacity share of installed thermal power plants in Isfahan province is 8.6% which put the province in a third place however, the share of gross generation of thermal power plant was 11% that caused the first rank for Isfahan province in this year. Regarding total consumption in 2010, it was 12 TWh with overwhelming majority allocated to industrial sector which was 10745.4 GWh, 17% of total industrial electricity demand in Iran. Emission share of the majority of greenhouse gas such as SO 2 , NOx, CO 2 and name a few, was 10% in the mentioned year.
Regarding different dimensions of energy supply including economic costs and environmental dimensions, in this research a general and comprehensive regional energy supply framework has been developed with respect to renewable energy such as wind, solar accompanied by conventional power plants which has supplied the demand for energy in Isfahan. Uncertainty has been considered as a key factor in energy supply provision. The structure of the paper is as follows: In the next section, the details of mathematical model is discussed while in the third section data description is presented. Empirical results and main conclusions form fourth and fifth sections, respectively.

METHODOLOGY
Goal stochastic programming approach is one of the most interesting approaches which has been developed in order to include uncertainty as the probable stochastic variables that may change in a predefined domain. The Probable distribution function of uncertain parameters can be different in various circumstances.
In order to solve the proposed multi-objective model initially, it is necessary to convert it to a single objective model by using goal programming technique. After allocation of related weight and goal, this technique will be implemented by a decision maker.
Goal programming is one of the available techniques for solving a multi-objective problem. In this method, objectives are transformed into goals by establishing associated targets and are then ranked according to their importance. In the sequence, the goals are transformed into deviations, i.e., variables that represent the distance between the target and the actual attainment of the goal. The preference over the goals may be expressed by deviations in priority levels, often referred to as a preemptive formulation or a "utility function." In a goal programming model, the original objective functions are taken to the constraints and a new objective function written in terms of under-and over-achievement of the considered objectives is minimized. As Sampaio et al. (2013) noted that it is not possible for a goal to be positively and negatively deviated at the same time, d − and d + must be zero, or d − .d + = 0.
A general goal programming function that weights (g i ) the under and over achievement of objectives (d + and d − ) is described by Eq. (1).
, , ,..., m x, , Where g 1 : Related weight for positive deviation from environmental goal which is equal to US$ 0.3925 that is the cost of reducing one kilogram CO 2 . g 2 : Related weight for positive deviation from environmental goals which is equal to US$ 5.325 that is the cost of reducing one kilogram NO x .
As it can be seen, positive deviations from environmental goals are the main weights, since due to the expansion of environmental issues v j − and f j − are considered as negative environmental effects which lead to destruction and degradation as well as social costs, thus they are defined and minimized in target function. In addition, to create a homogeneous weight of these two variables in target function it has been multiplied by one kg CO 2 and NO x emission costs, respectively, to compare it easily with other variables and minimize according to currency value. In goal programming, objective function minimizes inappropriate deviations. In this study, economic and environmental goals for Isfahan province in Iran is defined in order to suggest new technology systems and obtain an appropriate portfolio of electricity generation from renewable and non-renewable energy resources to fulfill the required demand. In fact, each of the goals show the current status of power supply system in terms of economic costs and pollutants diffused from power plants in Isfahan province.
Eq. (3) shows an economic goal for E kj , annual generated energy for the kth electric generating at the jth year with Variable costs (c kj ) and investment cost (I kj ) to meet the incremental yearly energy consumption D j (kWh/year) at a public electric power grid variable cost, C grid , and overnight capital cost, a grid , repectively.
Over-achievement (m + ) and under-achievement (m − ) are the economic targets for the problem.
Eq. (4) and (5)   . , , , , Eq. (6) states that total annual energy produced from new technologies (E kj ) must be consistent with total annual energy demand (D j ). To satisfy each of the uncertain limitations in this model with a predetermined probability (e.g. 1 − p i ), p i ∈ [0,1].
It must be noted that all stochastic variables are considered as normal independent stochastic variables. For example, assume that the mean and standard deviation of stochastic variable  D is equal to D and σ(D), respectively. We consider 1-φ parameter as standard normal cumulative distribution function which its value can be obtained according to corresponding statistical tables. The parameter p i is determined according to decision maker opinion and is related to satisfaction degree of each constraint in the model. In this research, (1 -p i ) is assumed 95%, energy demand and generation are determined (E wind,avg , E pv,avg ) as the normal stochastic variable with mean and standard deviation (using the standard deviation of 10%, 20% and 30%, respectively).
Eq. (7) to (17) are considered technical, functional and environmental constraints related to corresponding technologies to produce energy. Eq. (7) computes average generation of photovoltaic energy for a solar panel that is equal to 40W power multiplied by the average amount of solar radiation per day (I avg (j)). Average generation of wind power is calculated by Eq.  (14), cost coefficient of overnight capital cost in photovoltaic and wind systems (a kj ) is multiplied by the capacity of power plants. Gas turbine and other steam technologies overnight capital costs for their capacity (P k,j ) are calculated using Eq. (14). , , ≥ ( ) Eq. (15)

Capital Cost and Demand Forecasting
In order to find the optimal generation, overnight and variable costs in the future years must be predicted for each of the alternative technologies (solar, wind, gas). In addition, given that power plants use subsidized fuel and also according to dollar fluctuations, in order to determine real costs of network, capital cost estimation coefficient is used for steam power plants. In order to estimate the capital used in power plant (fossil and non-fossil), the nominal capacity of power plant can be multiplied by the overnight capital coefficient. Obviously, these coefficients change each year, and in the power plant a different coefficient is considered depending on the type of used technology. According to estimated data by US Energy Information Administration (EIA, 2016), capital cost coefficient for solar and wind technologies with decreasing growth rate in future years are estimated −0.022 and −0.0673 and for steam and gas power plants are estimated 0.02 and −0.0231, respectively ( Table 1).
In order to estimate demand parameter, electricity consumption in previous years has been assessed and using these predictions future years are predicted based on Growth formula by Eq. 18. Where y is demand level in year t, y 0 is demand level in the base year, g is the growth rate of demand and t is time. Obviously, this form of scenario is only used to obtain supply level in future years so that it is not considered in terms of prediction accuracy. Since the aim is to provide a reliable prediction, first it is necessary to consider different elements of demand such as price, consumer preferences and substitute goods in calculation then predict by presenting models, which are based on demand theories. Therefore, in order to predict consumption in Isfahan within the next 10 years we use costumers' data and average annual consumption ranging

Solar Radiation and Wind Forecasting Models
For estimating the potential of photovoltaic and wind energy generation in Isfahan from 2015 to 2025, a time series simulation method is proposed. Time series method is the most commonly used renewable energy system optimization routine, in which time series meteorological station data is processed for developing the feasibility study of hybrid systems. The solar radiation quarterly and wind speed quarterly series are analyzed with ARIMA model ranging from 2007 to 2012 with an adjustment to the first order autoregressive process, as suggested by the auto-correlation function. As prescribed by the ARIMA method, the relevant data are used for model construction and 2013 data are selected for model validation. The time series model and forecast are adjusted using Eviews software.
In order to choose an appropriate model, it is necessary to use some of the tests used in time series analysis. The Dickey-Fuller unit root test for solar radiation and wind speed is performed to determine the stationarity of these variables. As shown in Table 3 for each quarter solar radiation series has a unit root with first-order differencing stationary data.
When applied to the data available for Isfahan from 2007 to 2012, the solar radiation ARIMA models for first to fourth quarter are expressed, respectively, by Eq. (19) to (22) The unit root test for wind speed series is reported in Table 4 which is integrated by first-order differencing.
According to the estimated model, Eq. (23) to (26) describe long-term quarterly model wind speed for the first quarter to the fourth quarter.    Based on the collected data for Isfahan, the electricity demand is predicted until 2025. After obtaining the annual quarterly series, the annual average solar radiation and annual wind speed, the electrical energy produced is estimated by each solar panel or each wind turbine using Eq. (7) and Eq. (8).

EMPIRICAL RESULTS AND DISCUSSION
In this section, optimal results of planning are addressed considering stochastic parameters in the first, the second, and the third scenarios with 10%, 20% and 30% standard deviation, respectively. It must be noted that in Tables 6-8 given that variables related to fossil fuel in Isfahan power network are always zero, thus this variable was reported regardless of value, and also considering constant value of wind and solar energy generation in some scenarios, it is addressed in first column of Table 1 to prevent repetitive numbers.

Demand Sensitivity Analysis
The results of energy planning model for each of the power generation technologies at 95% significance level are presented in Table 5. Obviously, in scenarios 1 to 3 gas energy generation increases and solar and wind energy values remain constant. It can be said that its reason is the high cost of wind and solar energies compared to natural gas. In other words, in proposed model production costs are main drivers that play an important role in energy portfolio selection. The results indicate that with demand uncertainty, the share of natural gas in energy generation exceeds solar and wind energies.

Sensitivity Analysis of Solar Power Plant Capacity
Average generation of energy produced by solar panels is also considered as the stochastic variable, since solar irradiation at any time of the day has a different value. It can be said that generation  efficiency of solar panels is variable with solar irradiation change.
Obviously, in scenarios 1 to 3 energy generation by photovoltaic panels decreases and generation of energy by natural gas power plants increases, while wind energy generation remains constant because of increasing standard deviations of solar irradiation parameters ( Table 6). It is noteworthy that the amount of electricity produced by the network will be zero.

Sensitivity Analysis of Wind Power Plant Capacity
Regarding Eq. (8), generation of wind energy by each of the wind turbines depends on a cubic value of wind so average energy produced can be considered as the stochastic variable. According to each of the three scenarios, it is clear that as long as wind energy production standard deviations increase, generation of wind energy decreases and generation of electricity by gas increases (Table 7), It definitely by changing scenario the risk of wind energy generation is increasing which is due to capacity changes of wind power plant so the amount of electricity produced from wind power plant are decreasing.

Pollutants Emission
In order to determine the levels of emissions, the total amount of CO 2 as well as NO x emissions derived from an optimal portfolio of certain and uncertain are presented in Figures 1 and 2, respectively. As it is shown CO 2 and NO x emissions are derived for both deterministic and stochastic cases and indicates that in the uncertain conditions pollutant emissions are less than optimal portfolio. It must be noted that NO x emission decreases significantly, since emission coefficient for this pollutant is more than emission coefficient for the former pollutant ( Figure 2).

CONCLUSION
Regarding local energy planning issue, a stochastic goal programming model has been used. The empirical results indicate that gas turbine, wind, and solar power technologies will be beneficial alternatives for conventional steam power plants and must be considered as priorities in investment policies. It will initially improve optimal utilization of gas turbine power plants and consequently the optimal share of wind energy is expected to be more than solar energy, and meanwhile steam turbine power plants have no share to fulfill demand. It can be said that advantage of probability distribution assurance method of a random variable is to consider a variable risk in different scenarios. In addition to converting a continuous variable to some discrete parameters, converts model from functional form to parametric form. This method converts the stochastic programming model to a parametric programming model. In this study, the amount of demand, solar radiation, and wind speed are investigated in different scenarios with standard deviations of 10%, 20%, and 30%, respectively. The main findings indicate that: 1. Based on environmental constraints, policy makers should allocate more production between wind and gas power plants when demand is considered as a stochastic variable. As demand grows over the next 10 years, contribution of solar power plant decrease while the amount of wind and gas power plants production increase in the optimal energy production, although in the first scenario the overwhelming allocation goes to gas power plant, a mere 10% and 4% portions are allocated to wind and solar energy, respectively. By changing the scenario of demand from 10% to 20% and 30%, solar and wind power plant production level off while production of gas power plant increases due to the importance of cost. 2. Considering solar power plant capacity as a stochastic variable, its production reaches the lowest point due to an increasing amount of standard deviations of solar radiation parameters and consequently increasing risk of produced photovoltaic energy (Table 6). Consequently, more optimal production is devoted to wind and gas turbine power plants in the first scenario, given the fact that natural gas reserves are an undoubted part in optimal generation basket. It is noteworthy that in the alternative scenarios (increased standard deviation of solar power plant capacity), production of this power plant will continue to decrease, in contrast, the contribution of energy production of gas turbine power plants will increase to meet the demand. In addition, wind power plant remains constant in the alternative scenarios. 3. Similarly, when the capacity of wind power plant is stochastic  (Table 7), in the first scenario, its production will slow down to its lowest level resulting in the concentration of gas and solar power plants generation. Importantly, the order of production contribution by policy maker will be the same as two aforementioned modes with the largest share will be devoted to gas and wind respectively and the smallest share will be allocated to solar energy. It should be noted that in the alternative scenarios, wind power plant production will decrease because of risk increment in the wind energy. Consequently to fulfill demand, gas turbine power plant generation will compensate for the supply and production of solar power plant remains constant due to cost constraints.
It should be noted that with changing from the first scenario to second and third scenarios, the generation of wind and solar power plants stay constant or decline (because of high investment costs) due to risk reduction. In return, gas power plant production share increases. Thus, it can be concluded that in this model, policy makers increase the capacity of solar and wind power plants up to the maximum standard deviation of 10% (first scenario) and by increasing the standard deviation, energy supply will be mostly provided by natural gas power plants. So, cost constraint has a major significant contribution to determine energy production in each power plant compared to the environmental constraint.
Low pollutant emission, lower investment cost and abundance of natural gas are the merits of gas power plants. Regarding wind power plants, it can be said that energy utilization is lower compared to gas power plants but at the same time, it is higher than photovoltaic power plant portion. Regarding photovoltaic power plant, it can be said that although Isfahan province enjoys relatively high radiation, allocating a low portion because of high investment costs. In fact, reduction of CO 2 and NO x emissions up to 87% show that incremental demand can be met annually by new energy supply system. It can be concluded that despite certain budget and environmental constraint, investment can be directed to renewable energy technologies (Tables 5-8).