Exercises: Electricity generation

Almost all power stations share three elements. Match each one with its function: (1) turbine, (2) alternator, and (3) transformer, with: (a) converts rotational motion into electrical energy; (b) receives the force of water, steam, or wind and converts it into rotation; (c) steps up the voltage to transmit energy with fewer losses.

Describe the chain of energy transformations in a coal or gas thermal power station, from fuel to the electricity grid: chemical energy, thermal energy of the steam, mechanical energy of the turbine, and electrical energy.

In a nuclear power station, heat is obtained from the fission of uranium atoms, without combustion. Explain what its process has in common with that of a thermal power station (everything that happens after high-pressure steam is generated) and what is different (how the heat is obtained and what waste is produced).

Explain what form of energy a reservoir hydroelectric power station uses (the potential energy of water stored at height) and write its chain of transformations up to the electrical grid, pointing out which component converts each form of energy into the next.

A wind turbine uses the kinetic energy of the wind. Write its chain of energy transformations and reason why, unlike a thermal power station, it requires neither a boiler nor steam: the wind directly turns the rotor.

Explain the difference between a photovoltaic solar plant (panels convert light directly into electricity) and a solar thermal power plant (mirrors concentrate sunlight to heat a fluid and drive a turbine). Which of the two is more similar in its process to a conventional thermal power station?

Explain which natural phenomenon each of these lesser-known renewable power plants exploits: (a) tidal power (the rise and fall of sea water with tides) and (b) geothermal power (the heat from the Earth's interior). State a geographical condition required by each.

At the alternator output, a transformer steps up the voltage to hundreds of thousands of volts for transmission. Reasoning with $P = V \cdot I$, explain why it is advantageous to transmit the same power at high voltage and low current (cable heating losses depend on the current).

Order the stages of a biomass thermal power plant, which are given out of order: (a) the high-pressure steam drives the turbine; (b) the transformer steps up the voltage; (c) the boiler burns the fuel and heats the water; (d) the alternator generates electricity. Write the complete sequence and name the form of energy at the beginning and end of each stage.

The figure shows the diagram of a reservoir hydroelectric power plant, with an element marked with a question mark, as shown in the figure.

(a) Identify the element marked with the question mark and explain what energy conversion it performs. (b) Write the complete chain of transformations of the plant, from the stored water to the power line. (c) Reason which two factors the power output of the plant depends on (the head height and the water flow rate).

Nuclear and thermal power are said to be 'baseload' energies and wind power is 'variable'. Explain what it means for a power plant to be dispatchable (it can decide when to generate) as opposed to variable (it depends on the wind or the sun) and reason why an electrical grid needs a combination of both types.

A power plant with $500\ \text{MW}$ operates at full power for a full day. (a) Calculate the energy generated in $\text{MW} \cdot \text{h}$, $E = P \cdot t$. (b) If a home consumes about $10\ \text{kW} \cdot \text{h}$ per day, calculate how many homes the plant's daily production is equivalent to (be careful with unit conversion). (c) Comment on the result: what does it tell you about the scale of a power plant?

A thermal power plant receives $1.000\ \text{MW}$ in the form of heat from the fuel and delivers $380\ \text{MW}$ of electrical energy. (a) Calculate its efficiency, $\eta = \frac{P_{electrica}}{P_{termica}} \cdot 100$. (b) Explain where most of the lost energy goes (heat transferred to the cooling water and to the atmosphere) and why those cooling towers are so visible.

A pumped-storage hydroelectric plant has two reservoirs at different heights: during the day it generates electricity as water falls, and at night, when cheap excess energy is available, it pumps the water back to the upper reservoir. Explain what function it serves in the electrical grid (it acts as a giant battery) and reason why, even though some energy is lost in the cycle, it is useful.

A $400\ \text{W}$ photovoltaic panel receives the equivalent of $5\ \text{h}$ of peak sun per day. (a) Calculate the energy it generates per day, $E = P \cdot t$, in $\text{kW} \cdot \text{h}$. (b) Calculate how many panels would be needed to cover $10\ \text{kW} \cdot \text{h}$ per day. (c) Reason why the calculation uses 'peak sun hours' and not the total hours of daylight.

Match each power plant to its most characteristic impact, briefly justifying your answer: (a) coal-fired thermal ($\text{CO}_2$ emissions), (b) nuclear (long-lived radioactive waste), (c) reservoir hydroelectric (flooded valley and altered river ecosystems), (d) wind (landscape impact and effect on birds) and (e) photovoltaic (land use and panel recycling).

The figure shows the basic operating diagram of a nuclear power plant.

During an energy audit, the following plant data are recorded: thermal power released in the reactor = $3.000\ \text{MW}_t$; net electrical power injected into the grid = $1.020\ \text{MW}_e$.

(a) A popular science article claims that the equipment marked with "?" performs very poorly because it "wastes almost two thirds of the energy produced by the reactor". Refute this statement technically: identify the actual function of "?", its typical electromechanical efficiency, and explain in which component of the closed circuit (missing from the block diagram) most of that energy is actually rejected, and by which physical law this is unavoidable.

(b) During a heatwave, the temperature of the river cooling the external circuit rises from $18\ ^\circ\text{C}$ to $29\ ^\circ\text{C}$. Predict with reasoning how this warming will affect the maximum theoretical thermal efficiency (Carnot cycle) of the plant and the electrical power it can deliver.

(c) If the nuclear reactor were replaced by a coal-fired or natural gas boiler while maintaining the same thermal input power, explain why a gas-fired thermal power plant can achieve higher overall efficiencies ($> 50\,\%$) than a nuclear plant ($33\,\%\text{-}35\,\%$), taking into account the temperature and pressure limits of steam in both types of reactor/boiler.

The table shows the expected average electricity demand profile in an isolated system during a winter day:

The grid operator has the following generation fleet available:

• Nuclear: $10.000\ \text{MW}$ installed (variable cost $14\ \text{€/MWh}$; start-up time over 24 hours; zero flexibility to modulate power rapidly).

• Combined-cycle gas turbines: $14.000\ \text{MW}$ installed (variable cost $68\ \text{€/MWh}$; start-up in $1\text{-}2\ \text{h}$).

• Reservoir hydroelectric plants: $6.000\ \text{MW}$ maximum turbining capacity (stored energy in water limited to a maximum of $24.000\ \text{MWh}$ per day; start-up under 3 minutes).

• Pumped-storage plant: pumping capacity $2.500\ \text{MW}$ and round-trip storage cycle efficiency of $72\,\%$.

(a) Design the hourly dispatch plan for each time slot (specifying how much power each technology should provide), minimising the total economic cost while respecting the water availability and flexibility limits of each plant.

(b) Demonstrate with an economic balance whether it is cost-effective to use the pumped-storage plant during the early morning to turbine that water during the evening peak, replacing natural gas.

(c) At 15:00, demand unexpectedly rises by $2.000\ \text{MW}$ in under 5 minutes. Justify which technology must immediately take on this power step and why it is not technically feasible to assign it to the nuclear power plant.

A wind farm developer must select wind turbines for a coastal site where the average wind speed is $7{,}5\ \text{m/s}$ (air density $\rho = 1{,}225\ \text{kg/m}^3$). Two technical bids are available:

• Wind turbine A: rotor diameter $D_A = 90\ \text{m}$; rated power $P_{n,A} = 3\ \text{MW}$ (reached at a rated wind speed of $13\ \text{m/s}$).

• Wind turbine B: rotor diameter $D_B = 120\ \text{m}$; rated power $P_{n,B} = 3\ \text{MW}$ (reached at a rated wind speed of $10\ \text{m/s}$).

(a) Calculate the incident kinetic wind power passing through the swept area of each wind turbine at the average speed of $7{,}5\ \text{m/s}$. From this result, determine the theoretical maximum mechanical power that each model could extract according to the Betz limit.

(b) A sales representative claims that, since both models have the same rated power ($3\ \text{MW}$), they will produce exactly the same electrical energy over the year at that site. Refute this claim by interpreting the behaviour of both rotors under moderate winds, and predict which one will achieve a higher annual capacity factor.

(c) If the selected model ultimately generates $8.400\ \text{MWh}$ annually per turbine, calculate its capacity factor (equivalent full-load hours per year and percentage of the $8.760\ \text{h}$ of the year). Knowing that it displaces a thermal power plant that emits $0{,}42\ \text{t}$ of $\text{CO}_2$ per $\text{MWh}$ and that each tonne of emissions costs $70\ \text{€}$, calculate the annual economic emissions savings attributable to each wind turbine.

A tidal barrage power plant is planned in an estuary with a floodable surface area of $22\ \text{km}^2$. On this coastline, the average tidal range (the difference in level between high tide and low tide) is $H = 7{,}0\ \text{m}$, with two complete filling and emptying cycles every day ($24\ \text{h}$).

(a) When the reservoir is emptied during low tide, the center of gravity of the water mass drops by an average height $h = H/2$. Deduce and calculate the theoretical gravitational potential energy available in each tidal cycle ($E = m \cdot g \cdot \frac{H}{2}$, assuming $\rho = 1.025\ \text{kg/m}^3$ and $g = 9{,}8\ \text{m/s}^2$). Express the result in $\text{MWh}$.

(b) If the combined efficiency of the hydraulic turbines and alternators is $30\,\%$, calculate the electrical energy generated per day and the equivalent average power that the plant delivers over the 24 hours.

(c) A city council on the Mediterranean coast proposes building a similar tidal power plant in a bay with tides of average range $H = 0{,}4\ \text{m}$. Analytically demonstrate why the energy obtained per square meter would be reduced by a factor greater than 300 compared to the oceanic estuary, and justify two technical and environmental reasons why tidal energy has such a limited worldwide deployment compared to offshore wind energy.

The figure shows the diagram of a central receiver solar thermal tower power plant.

(a) A designer proposes replacing the thermal receiver, turbine, and equipment “?” with a conventional high-power photovoltaic panel placed at the top of the tower to generate electricity directly and eliminate the thermal loop. Justify two physical and technological limitations (related to temperature and irradiance concentration) explaining why this approach would destroy or severely degrade the silicon cells.

(b) The tower receiver receives a net concentrated radiant power of $90\ \text{MW}_t$. Molten salts with a specific heat capacity $c = 1{,}55\ \text{kJ/(kg}\cdot\text{K)}$ are used as the heat transfer fluid. If the salts enter at $290\ ^\circ\text{C}$ and must exit at $565\ ^\circ\text{C}$, calculate the mass flow rate ($q_m$, in $\text{kg/s}$) that the pump must circulate through the receiver.

(c) The plant has two molten-salt thermal storage tanks with a usable capacity of $6.000\ \text{t}$. Knowing that the efficiency of the power block (turbine + alternator “?”) is $38\,\%$, calculate how many hours the plant can operate at its rated power of $45\ \text{MW}_e$ during the night using exclusively the thermal energy stored between $565\ ^\circ\text{C}$ and $290\ ^\circ\text{C}$.

An isolated island grid has an hourly demand that fluctuates between a night valley of $3\ \text{MW}$ and a daytime peak of $7\ \text{MW}$. Currently, it is supplied entirely by diesel generators (average specific fuel consumption of $0{,}24\ \text{liters/kWh}$; fuel cost of $1{,}25\ \text{€/liter}$). To reduce emissions, a project is approved to install $9\ \text{MW}$ of solar photovoltaics, $6\ \text{MW}$ of wind power, and a lithium battery energy storage system (BESS).

(a) A local political party proposes completely decommissioning and dismantling all diesel generator sets on the island, arguing that the $15\ \text{MW}$ of renewables easily exceeds the peak demand of $7\ \text{MW}$. Technically refute this measure by justifying the need to maintain firm synchronous power in an isolated grid against frequency variations and during prolonged calm weather conditions with overcast skies.

(b) At 13:00 on a spring day, the island's demand is $5\ \text{MW}$, the PV generates $7\ \text{MW}$, and the wind turbines generate $3\ \text{MW}$. The diesel generators cannot be shut down entirely and must operate at a minimum technical safety load of $1\ \text{MW}$. Determine the net power that the batteries must absorb at that moment to avoid curtailment (forced disconnection) of renewable energy.

(c) The battery bank has a cumulative usable capacity of $18\ \text{MWh}$ and a round-trip efficiency of $85\,\%$. If the batteries are discharged to cover a constant demand of $3\ \text{MW}$ during the night without diesel support until their usable energy is depleted, calculate how many hours they will cover and the direct economic savings in fuel achieved that night.

A $15\ \text{MW}_e$ thermal power plant burns forestry wood chips with a lower heating value (LHV) of $13\ \text{MJ/kg}$ (commercial moisture content of $25\,\%$) and an overall plant efficiency of $28\,\%$. The carbon content of the biomass used is $48\,\%$ by mass relative to the biomass as burned.

(a) Calculate the plant's hourly biomass consumption at full load (in $\text{t/h}$) and determine the mass of $\text{CO}_2$ released into the atmosphere for each hour of continuous operation (molar masses: $\text{C} = 12\ \text{g/mol}$, $\text{O} = 16\ \text{g/mol}$).

(b) The supplier company claims that the plant is “carbon neutral from the very first minute of combustion because wood is renewable”. Refute this claim by introducing and analyzing the concepts of “carbon debt” and “carbon payback time”, explaining why felling mature live trees produces a vastly different net climate impact compared to using agricultural pruning residues or forest thinning.

(c) Biomass is transported to the plant by trucks with a $22\ \text{t}$ payload that consume $34\ \text{liters of diesel}$ per $100\ \text{km}$ (emissions: $2{,}64\ \text{kg of CO}_2$ per liter consumed). Determine the maximum road supply radius (round-trip distance: laden outbound and unladen return) so that transport emissions do not exceed $4\,\%$ of the total $\text{CO}_2$ emitted by burning that biomass in the boiler.

A power plant generates an active power of $150\ \text{MW}$ that must be supplied to a distribution center located $90\ \text{km}$ away. The transmission line is a two-wire line, and each of the two conductors has a linear ohmic resistance of $0{,}075\ \Omega/\text{km}$ (total round-trip resistance: $R_T = 2 \cdot 90 \cdot 0{,}075 = 13{,}5\ \Omega$). Assume a unity power factor ($\cos \varphi = 1$).

(a) If you tried to transmit this power directly at the output voltage of the plant's alternators, $V_1 = 15\ \text{kV}$:

1. Calculate the current that would have to flow through the line ($I_1 = P / V_1$).

2. Calculate the power that would be dissipated by the Joule effect ($P_{\text{pérdidas}} = I_1^2 \cdot R_T$).

3. Physically interpret the result obtained and justify why it is impossible to carry out transmission at that voltage.

(b) To solve the problem, a step-up transformer substation is installed to raise the line voltage to $V_2 = 400\ \text{kV}$:

1. Calculate the new current ($I_2$) and the power dissipated by the Joule effect in the line.

2. Determine the percentage of power lost in transmission relative to the injected power.

3. Calculate the voltage drop along the line ($\Delta V = I_2 \cdot R_T$) and the actual voltage arriving at the receiving substation.

(c) A student suggests: “To reduce losses even further, we should always transmit at voltages of around $1.500\ \text{kV}$ or higher on all transmission and distribution lines”. Technically refute their proposal by analyzing the dielectric breakdown limits of air (corona effect), the costs and footprint/volume of insulators and transformers, and the required safety clearances.