Objective function

The objective function in this multi-objective optimization model is to minimize the weighted sum of overall expected cost of scenarios and the worst (highest) scenario cost.

The expected cost is defined as in the stochastic model, but now it has to work with the influence of the other robustness component – worst-case cost. As mentioned before worst-case cost and partial mean of costs are two popular measures broadly used in robust optimization. The study of Kang et al. indicated that worst-case cost would be the better choice as the economic robustness measure because this approach avoids the problem of choosing target value in partial mean of costs and gives more allowance to for the model to consider technical robustness. In the current modeling, worst-case cost is adopted as the economic robustness measure, as well as one of the two objectives.

The weights of expected cost and worst-case cost are controlled by the model users with the parameter L, as shown in (1).

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

Min   (1 ) (1)

The expected cost of scenarios is defined as the summation of all individual scenario costs multiplied by their corresponding probabilities, as shown in (2)

s N

s

s Cost

p

Expected 



The worst-case cost is defined as the highest scenario cost among all individual scenario costs, as shown in (3).

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

Cwmax _s 1,..., (3)

The individual scenario cost is composed of a number of cost items, including cost of annual capital investments, electricity purchase from the national grid, fuel consumption

for energy generation, system operation and maintenance, and carbon tax imposed based on greenhouse gas emissions, as well as cost of starts and stops of equipment. The revenue obtained from sales of electricity to the national grid is presented as a deduction from the total cost.

s CInv CEbuyN CFuel COM CCtax CSS CSal

Cost        (4)

The cost of annual capital investments is calculated as the present value of the amortized amount of initial invested capital, which is derived from the summation of the unit fixed capital cost multiplied by the planned capacity for each DER technology selected. The amortization of the fixed cost of each kind of DER equipment is considered over the estimated life time of each kind with respect to a given interest rate. The calculation can be expressed in (5). The cost of buying electricity from the national grid is represented by (6). The cost

structure consists of two parts, demand charge and mobile electricity charge. Demand charge is determined by the regulated demand charge rate of electricity multiplied by the peak electricity demand in one certain month, where the peak electricity demand is estimated as the average electric power provided by the national grid and for all kinds of usage (i.e. power, heating, and cooling) in one month multiplied by an assumed factor C, as shown in (7). On the other hand, mobile electricity charge is calculated as the actual amount of electricity consumed in one month multiplied by the utility electricity tariff rate.

The cost of fuel consumption can be broken down into two parts as shown in (8). The first part accounts for the direct fuel consumption other than DER usage for heating and

cooling purposes, and the second part accounts for the fuel consumed by different DER technologies for power generation. They are all determined by the cumulative amount of fuel usage multiplied by unit fuel charge, with respect to each kind of fuel. It should be noted that the relationship between the electricity produced (for all kinds of ultilization including meeting demands, sales to the national grid, and storage) and the fueled consumed is governed by the distinct efficiency of each kind of DER technology with respect to its corresponding fuel type.

fs

The cost of system operation and maintenance is constituted by the fixed cost and the variable cost of the DER equipment, as described in (9). The fixed cost of the equipment can be calculated by the summation of the unit fixed operation and maintenance cost of all DER technologies multiplied by their respective planned capacities, while the variable cost is obtained from the summation of the amount of electricity production by different DER sources multiplied by their unit variable operation and maintenance cost.

 

s Efrom ESal Etostore OMv Cap OMf

COM (9)

The cost of carbon tax, which is illustrated in (10), considers the total carbon credits accumulated by direct fuel consumption for non-DER use and by distributed power generation with relation to the carbon intensity of each kind of fuel used. The carbon tax cost of purchasing electricity from the national grid, on the other hand, is calculated by the multiplication of unit carbon tax rate, carbon intensity of electricity provided by the national grid, and the cumulative amount of electricity bought from the national grid.

fs

As the operation planning of microgrids is included in this model, the cost of starts and stops of different DER equipments should be taken into account. It is well known that frequent starts and stops may cause damage to equipment and increase of cost of maintenance. With only a few expections, most of the DER technologies may incur additional cost when they start up or shut down. The following equation (11) addresses this issue by assuming a linear relationship between the cost of start and stop and the frequency of start and stop for each DER technology, with a fixed unit start and stop cost given as a parameter.

When the excess electricity is sold to the national grid, the microgrid system will receive an income, which can be expressed in (12). The revenue from the sales of electricity equals to the summation of selling price of electricity (can be uniform or different among different power sources) multiplied by the amount of electricity sold.

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The primary constraints in this microgrid model include demand-supply relationships, energy balance, and the operation characteristics of the microgrid components, etc.

1. Demand-supply relationships

A fundamental principle under this model is that all forms of local energy demand (including electricity, heating, and cooling loads) must be satisfied in every time period, as illustrated in (13). The energy can be supplied from one or more of the following sources:

a. Electricity produced from one or more DER sources for different end use b. Electricity bought from the national grid for different end use

c. Heating and cooling output transferred from direct fuel consumption via boilers and absorption chillers

d. Heating and cooling output transferred from recovered waste heat during power generation by certain CCHP DER technologies

e. Electricity, heating, and cooling output supported by energy outflow from electricity batteries and thermal storage.