write down a mathematical optimization model for case 1 The manager of a power g

Question

write down a mathematical optimization model for case 1

The manager of a power grid wants to buy energy from five different producers to meet the projected energy demand of 750 MWh for the next planning period. The producers are a nuclear power plant, a coal power plant, a wind turbine farm, a natural gas plant, and a solar photovoltaic plant. For each of these producers, Table 1 lists some key properties: the CO2 emissions in tonne/MWh, the maximum capacity (in MWh) of each producer for the planning period, and the cost (in Euro) that each producer charges per MWh of supplied energy. Environmental regulations dictate that at least 30% of the energy that the manager buys should come from CO2-free sources. Moreover, the total CO2 emissions during the planning period may not exceed 200 tonne, unless the manager buys carbon credits to compensate. Each carbon credit gives the right to emit one additional tonne of CO2 and costs Euro 30. (The manager may also buy fractions of carbon credits.) Formulate a linear optimization (LO) problem that the manager of the power grid may use to decide how much energy to buy from each producer and how many carbon credits to buy, in such a way that demand, capacity constraints and CO2 emission limits are all satisfied, and the total cost is minimized.

Explanation / Answer

Let the energy in MWh from the nuclear plant be = v

Let the energy in MWh from the coal plant be = w

Let the energy in MWh from the wind plant be = x

Let the energy in MWh from the natural gas plant be = y

Let the energy in MWh from the solar plant be = z

The constraints are :

The projected energy demand is = 750 MWh

atlease 30% of the energy needs to come form sources with 0 CO2 emission

=> v+x+z >= 30/100*750 or v + x + z >= 225

coal and natural gas plants produce carbon emission in tonne/MWh

1.022w + .516y <= 200

and there are limits to the energy being produced by each energy source and they are :

v <= 500

w <= 300

x <= 100

y <= 250

z <= 150

The objective fucntion is to minimize the cost :

Z = 150v + 80w + 190x + 75y + 240z

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