Green energy for developing nations

Paradoxically, those nations which are most vulnerable to climate change’s ill effects also require significant energy investment. Yet, emerging economies such as China’s and India’s cannot grow whilst still relying on coal-fired electricity and oil for transport. The consequences for the planet and human lives would be catastrophic. It’s clear that developing countries must leapfrog current technologies in favour of low or zero-carbon energy sources.

This may seem an unfair burden to impose on less prosperous nations. Yet, solar power is becoming financially attractive, in addition to it’s green credentials. The levelised cost of electricity, or the minimum price for which electricity must be sold so that a power plant breaks-even, shows solar power converging on gas and coal. Such gains were driven by significant cost reductions in the manufacturing of solar panels since 2010.

Long-term contracts to purchase solar power in developing countries including South Africa, the United Arab Emirates, Peru and Mexico support such analysis. The Economist cites an example earlier this year: Enel Green Power, an Italian power company, won a tender to provide Peru with 20 years of PV solar power at a rate of less than $48/MWh. Soon after, Mexico also awarded the company a long-term contract to generate solar power at a price of about $40 per MWh. Bloomberg New Energy Finance describes these contracts, and another awarded to ACWA in Dubai in January, as the lowest subsidy-free solar contracts seen so far. 

Large grid-connected solar projects in China and India accounted for most of the global growth in solar capacity additions last year. China’s biggest project yet – a 200 MW solar power plant  in the Gobi desert – is now under construction. It could eventually power up to a million homes. The Indian government is flirting with offering 2-4 GW tenders for solar power plants. Solar power is central to both the Chinese and Indian governments’ plans for economic growth and reducing greenhouse gas emissions. 

Off-grid solar in Africa

Grid-connected, large-scale solar does not suit developing countries currently lacking in grid infrastructure though. Further, difficult terrain, a significant rural population or remote communities present a challenge to electrification. M-Kopa is an innovative company currently bringing cheap, off-grid solar electricity to more than 200, 000 households across Kenya, Uganda and Tanzania – reaching places that landlines and power lines do not. Customers pay 35 dollars upfront for a solar panel, LED bulbs and a flashlight, a radio and cellphones chargers. The package would normally cost around 200 dollars. This is paid off, via a mobile banking service, in installments proportional to the amount of energy consumed. Once their initial loan is paid the electricity is free.

By M-Kopa’s own estimate over 80-percent of their customers live on less than 2 dollars per day. An average off-grid Kenyan household spends 75 cents per day on energy. Kerosene is the most common source of energy – used to cook food and light homes. A customer saves about 750 dollars over four years after switching to M-Kopa’s basic solar kit the company claims. Kerosene is not only expensive, it is also very pollutive – its fumes cause nose and throat irritation, respiratory disease, and blacken the walls of homes. Its combustion also releases greenhouse gases. Yet, M-Kopa is a profitable, private firm – the green benefits are almost an accident.

Self-sufficient energy islands

In Haiti, the poorest country in the Western hemisphere and devastated by the 2010 earthquake, more than 75-percent of the population does not have access to electricity. Non-profit EarthSpark International estimates that rural Haitians spend 6.5-percent of their annual income on kerosene and candles for home lighting, whereas the average American family contributes only 0.5-percent. The inhabitants of Les Anglais had no electricity and relied on kerosene until Earthspark brought a self-sufficient, solar microgrid online last year. The pay-as-you-go system has connected hundreds of homes and reduced households’ energy costs.  Earthspark has ambitions to install a further 25 microgrids throughout Haiti.

The grid-connected electricity that does exist in Haiti  is generated  by burning diesel imported from Venezuela. Isolated islands, such as Haiti, suffer disproportionately from upswings in global energy prices, being dependent on fuel imports. In this context, renewables can become highly competitive with imported fuels for electricity generation.  

An abundance of wind and sun also makes islands well-suited to renewable energies.   Electricity storage technologies are needed to ensure reliable supply from the grid though, since islands also lack interconnections to other regions. Akuo Energy, a French renewables company, has used lithium-ion batteries, existing technology, alongside solar power plants in French overseas island territories to provide a reliable source of clean power. A number of facilities are now in operation in Corsica and Ile de la Réunion that have contributed to improving the islands’ energy self-sufficiency.

Islands with tropical climates also have the necessary oceanic conditions to take advantage of an established renewable energy technology called Ocean Thermal Energy Conversion. OTEC relies on a temperature difference between colder deep water and warmer shallow water. The difference is exploited to vaporise a working fluid circulating in a closed circuit, which in turn spins a turbine coupled to a generator. This provides a reliable, steady source of electricity and no pollution. An OTEC demo facility began operations in Hawaii in 2015 and is currently powering around 150 homes. A pilot project in Martinique is being jointly developed by Akuo Energy and DCNS. Construction is expected to get underway this year. Installation will make this facility the largest OTEC project to date.

The COP21 agreement signed in Paris last year specifically mentions small island nations in the text. This recognises their unenviable position as victims of both climate change and energy poverty. The climate change related calamities visited upon islands include rising seas levels, more intense and more frequent droughts and cyclones, as well as a heightened vulnerability to airborne diseases. Clean energy development is imperative for such island nations, as well as other developing countries: to reduce their energy bills, lift communities out of energy poverty and to improve their self-sufficiency. Incidentally, this will also help bring greenhouse gases under control . 


 

Coal condemned

During the last decade, the majority of the OECD countries decoupled their economic growth from energy consumption. Normally these rise in tandem – a trend that persists in developing countries and world’s soon-to-be fastest growing and most populous nation, India.

This decoupling happened as developed nations shifted to providing services and building knowledge economies, which is less energy-intensive than industrial production and manufacturing. China too has started down this path. Policy-makers now talk of “decarbonising” the economy. That is, only producing and consuming energy which does not release greenhouse gases into the atmosphere and contributing to climate change.

Decarbonisation is currently focussed in the electricity sector where it is being helped along by policy incentives. Subsidies, guaranteed prices for electricity and tax-breaks dramatically boosted the growth in renewable electricity generation across Europe in the last few years. The liberalisation of Europe’s electricity markets and new regulation improving competition also played a role. Although, falling prices and technology gains spurred the sector’s expansion more than any government policy, particularly for solar power.

For renewables’ expansion to make any difference to greenhouse gas emissions coal-fired power production has to be tackled. Although it is cheap, burning coal releases significantly more greenhouses gases than other fossil fuels including gas in the electricity sector and oil in transportation. Europe’s aging fleet of coal-fired plants are also extremely inefficient at generating electricity compared to newer gas-fired units. A quarter of electricity in the European Union and almost forty-percent in the United States is still generated by burning coal. It is around two-thirds of the electricity mix in China where the resulting air pollution in its major cities is fuelling a sense of urgency.

Political leaders are aware of this danger and are acting to reduce coal production and consumption in many countries around the world. By 2025 all coal-fired power in the United Kingdom will be shut down according to current plans. New Zealand will close its two remaining large-scale coal-fired power plants in 2018. The provincial government of Alberta in Canada, where the tar sands industry alone produces more emissions than Portugal, has announced plans to phase-out coal power over the next fifteen years. China’s goal is to cap coal consumption in 2025 and accelerate its decline thereafter.

President Obama’s Clean Power Plan intends to restrict emissions from current coal-fired power plans, substitute coal with gas-fired or zero-carbon generation and impose strict emissions standards on new plants. The goal is to cut emissions in the electricity sector by a third relative to 2005 levels. Coal mining states have fiercely contested this “war on coal”, which is bound to be difficult for certain towns and regions whose local economy and workforce are dependent on coal mining, not just in the US. Nevertheless, coal needs to eventually exit the electricity sector if the commitments made by the US and 195 other countries at COP21 in Paris late last year are to materialise.

Yet, none of the above is enough to slow climate change. India is set to contribute the greatest share of growth in global coal demand in the future, mostly from increased domestic production. How it intends to reach its goal to produce forty-percent of its electricity from non-fossil fuel sources by 2020 is unclear. In Germany, coal’s resurgence in the power sector has cast a shadow over its achievements in increased generation from renewable resources. Angela Merkel’s government is working on a plan to phase out coal by mid-century. From the European Unions’s biggest economy this is too long to wait. Decarbonising electricity production by phasing out coal remains a long way off. Coal has been condemned by the world’s leaders but not yet replaced.

The New Zealand test

When machines permitting payment by credit or debit card were first developed New Zealand was one of the first countries within which this EFTPOS technology was deployed. Today one can buy a coffee or even a 50c bag of sweets with their VISA or Mastercard. Most businesses do not have a minimum purchase for which you can use your bank card. Few of us carry cash.

New Zealand’s market is often considered something of a test environment for new technologies. Our small island nation is isolated in the middle of the Pacific Ocean, but we have an advanced economy and large middle class. This, our small population and an open, competitive marketplace makes New Zealand the perfect place to trial new products and innovations. If the product meets a certain need it will rapidly penetrate the market. You will soon know if whether it can be profitable or not – and whether you should launch the product elsewhere in the world.

In May, US company Tesla teamed up with Vector, New Zealand’s biggest electricity distributor, to bring their much lauded lithium-ion batteries to New Zealand homes and businesses.

Like cellphones these batteries do not require heavy investments in supporting infrastructure networks. They permit households and businesses to install PV solar panels whilst managing solar power’s intermittency. The main problem with solar power is that the sun does not shine all of the time. When the skies are cloudy or night falls your photovoltaic rooftop panels stop generating electricity. So households and businesses still need to be connected to the main electricity grid to guarantee their supply, in spite of solar panels installations.

You can resolve this issue by stockpiling electricity during daylight hours to use at night. This seems simple enough. However, batteries boasting the voltage and lifespan needed to supply an average household with enough electricity to keep the lights on have not been brought to market. Basically it is too expensive. Prototypes are also massive in size.

In principle if compact, powerful and affordable batteries hit the market then you would not need to be connected to the electricity distribution network. In fact you or your local community could go off grid.

How many households do not bother to install a landline phone these days? Could new houses avoid connecting to the main electricity grid in the near future? It is only a matter of time before battery technology hits that sweet spot. You can read about how Tesla plans to achieve economies of scale that surmount the current cost problem here.

To take a residence off-grid you would also need a smart monitoring system that conserves energy and warns you to turn off unnecessary devices when the household is running low on juice. Vector is investing in energy management systems that would provide this kind of service. They’ve also been investing in photovoltaic solar power and micro wind turbines. The company is future-proofing its main business – just in case distribution services are no longer needed in New Zealand.

If a decentralised electricity supply model works in New Zealand it will probably fly elsewhere. I still can’t pay for a coffee by credit card in Europe though.

Future electricity grids: the rise of the prosumer

Unlike other commodities such as gas and oil, electricity cannot  be easily stored. It must be consumed almost as soon as it is produced.

Consumer demand follows a fairly predictable pattern. Electricity prices are higher on weekdays when “peaking” power plants come online to satisfy increased demand. The first demand peak occurs in the morning – when a high proportion of the population is getting ready for work or school.  A second peak occurs in the evening when consumers return home and start cooking dinner, or turn on the television.

Conventional power generators such as coal and gas turbines can respond quickly to variable consumer demand by increasing fuel input and ramping up output during the day. Nighttime is a cool-off period.

Intermittent renewables changed this model. We must now factor in unpredictable supply peaks and increased price volatility. In the UK and Germany consumers bear the brunt of these new costs. You can read more about how renewables are shaking up the traditional power model here. 

There are several ways to manage this new supply intermittency and smooth prices.

One is more interconnections. These reduce bottlenecks and diversify supply sources so that electricity-rich areas can service electricity-poor ones. Nations hoping to boost electricity production from renewable sources will need a well-connected grid, as an oversupplied area can experience shortages as soon as the weather changes. New power lines require public support and investment.

Short-term (spot) electricity trading can optimise electricity flows between areas and facilitate price arbitrage. Spot trading services are offered by EPEX Spot in Central Western Europe or ERCOT in Texas, for example. These services also permit renewable energy producers to rebalance their books if the weather forecast was inaccurate and they produce much more or much less electricity than predicted.

Smart grid technology uses real-time information about supply and demand to automatically adjust electricity flows curtailing price peaks (or negative prices). Again public money is needed to roll-out this infrastructure at the national level.

Another means is electricity storage.

Pumped hydro-storage has existed for a long time. It is the only large-scale storage technology used commercially. It involves pumping water uphill when electricity prices are low, then running water downhill, through turbines, during peak-price hours to generate electricity. Pumped hydro projects are nevertheless hugely dependent on local geography and rainfall, as well as regulations regarding water-use.

The lithium-ion batteries used in electric cars pack a lot of energy density for their size. They cost around US$10 000, even for a small vehicle, and can only run for about 175km before recharging. This could be better. Crucially, lithium-ion batteries do not suffer from “memory” issues. Meaning that don’t need to be drained before being recharged.

Battery manufacturers across Asia and the USA are struggling to cut costs and upscale their technology to plug into the electricity grid. Yet, electricity generation is decentralising. Small-scale industrial and household solar production is on the rise. Rather than selling their excess power back to the grid some could go off-grid.

Most experimental batteries would need to be bigger than a house in order to store enough solar electricity to power one household for a day. And they remain prohibitively expensive. However, Tesla caused a lot of excitement last month when it announced plans to market lithium-ion batteries at prices starting from US$3500. It costs a household a further US$5000 or so to install solar panels. Nevertheless, this much-anticipated battery is priced lower than any other technology on the market. The Tesla battery should be small enough and safe enough to install in your basement. Plus, you don’t need to be a rocket scientist to operate it.

How did they do it? It’s not new technology. Rather, Tesla is building a US$5 billion gigafactory in Nevada’s desert, where it hopes to realise enormous economies of scale. While the market is still waiting for a technological revolution – the step-change that would make batteries as portable and powerful as microchips which are continuously delivering ever cheaper computing power – Tesla intends to reduce manufacturing costs for current battery technologies.

There is a sizable market of homeowners prepared to fit out their homes with solar panels, battery storage and adopt other energy efficient technologies. These early adopters need enough cash to  invest upfront, before they reap the benefits in reduced or zero-cost electricity bills in the months and years that follow.

For most middle class homeowners US$8500 is no small fee. Companies such as SolarCity in the US provide another piece of the jigsaw. Financed by high net worth individuals, as well as Google and Goldman Sachs, the company pays for solar panel installations, aggregates the earnings from energy savings and grid buybacks, then sells bonds based on a predicted revenue stream. Such creative financing will hasten the prosumer revolution and eventually take some of us off-grid.

No one technology will solve all the problems intermittent renewable energies have introduced into electricity markets. A patchwork of different solutions looks likely to emerge – with some consumers taking matters into their own hands.

Low-hanging fruit

Energy efficiency wants more energy for the same amount of fuel. This means both energy bills and pollution from burning fossil fuels fall – to the delight of government and environmental agencies alike.

There are three key sectors within which efficiency gains can have a significant impact in both the developed and developing world: transportation, buildings and electricity.

Simply replacing old cars and trucks with newer versions reduces overall oil usage per vehicle. New vehicles are built to higher fuel efficiency standards as the technology continues to improve, so that you can drive your car further and further using less and less petrol. Manufacturers were busily engineering new models whose improved fuel use and decreased gasoline bills made them attractive to consumers, even before regulation insisted on higher fuel efficiency. Inefficient and dirty, (but cheap) diesel is now highly regulated in the developed world. It is all but obsolete for passenger vehicles. Low-quality fuels for marine transportation and long-distance trucking have yet to be attacked with the same rigour.

It’s also about not wasting energy. Inefficient buildings release huge amounts of unused heat.  Simple measures include nailing shut the last few millimetres between insulation boards – this final step brings the greatest benefits – or using straight, fat water pipes rather than slim, angular ones. These are not universally understood or implemented.

Insulation, heat pumps and newer appliances compliant with current efficiency standards make a huge difference. The invention of light emitting diodes (LED) revolutionised lighting. Previously incandescent light bulbs lost most of their power as heat. Solar and geothermal installations can make buildings energy neutral or turn them into prosumers.

Although this involve additional costs, many energy savings measures pay for themselves within a few years, as heat and electricity bills are cut.

Retrofitting older buildings and replacing appliances is necessary to address standing building stock. Unlike cars, buildings are not replaced every few years. Most of today’s buildings will still be standing in fifty years – but we suffer from an agency problem. Landlords do not pay the energy bills and tenants do not wish to invest in someone else’s property. Yet, even property-owning households and businesses hesitate to retrofit. This is where government incentives can play a role. Heating, cooling and electrifying buildings makes up a third of global energy consumption, so lifting efficiency by just a few percentage points gets purchase and demonstrates the worth of such efforts.

Efficiency was transforming electricity production until renewables shook up the model making even the most flexible and efficient Combined Cycle Gas Turbine (CCGT) plants, which save and reuse heat produced during power production, unprofitable. Nevertheless efficiency can still give thermal power producers an edge on the competition, since decreased fuel use cuts operating costs. Further, governments are imposing tariffs on heavy polluters including inefficient diesel and coal-fired relics. This additional marginal cost crowds some of them out of the marketplace saving energy and reducing pollution.

Demand-side management can address some of the short-fallings of today’s decentralised electricity system. With smart metering industrial and household consumers can react when electricity is scarce (wind or solar production is low). The higher prices signal factories to run less energy-intensive processes or wait for off-peak prices and hours, and household consumers can decide to take a shower or do their washing later on. In fact  smart grids can even automate some of these decisions, at both the local and national level.

Once upon a time, rising energy consumption was an accurate indicator of how fast an economy was growing. No longer. In the OECD, efficient technologies and smarter policies have decoupled energy consumption and development, proving that environmental concerns need not frustrate economic ones.


Sources:

Invisible Fuel, The Economist

Energy Efficiency topic, International Energy Agency, OECD

Renewables menace traditional power model

Lots of things are shaking up the traditional power model. A decade ago gas and coal power plants were very profitable. Retail companies, which distribute to industrial and household consumers, bought wholesale electricity at a price that always covered operating costs and got a healthy boost during peak demand hours. Even fairly inefficient power plants could expect to have enough profitable operating hours to keep in the money.

Electricity generated from renewable energy sources has altered this dynamic, most noticeably in Germany where Energiewende policies encourage renewable energy development. The upfront costs of new renewable energy projects are subsidised. Once operational wind or solar parks are given priority access to the distribution grid – they can always market the electricity they produce. Furthermore, the government pays out a “feed-in” tariff. That is, guarantees a certain price for every megawatt hour of electricity generated by a wind or solar farm.

These policies have discouraged private investment that might have brought more competitive renewable energy technologies, ones that do not require government subsidies, to market sooner. Nevertheless, Germany’s goal to get 60% of its electricity from renewable sources by 2050 is on track. The eventual success or failure of these policies is the experiment the entire world is watching.

However, the rapid expansion of renewables has upset the incumbents – traditional thermal power generators that use coal and gas as fuel. Renewables harm their profitability for a number of reasons.

First of all, the average wholesale electricity price is lower. Once a wind turbine or solar panel is installed operating costs are near zero because the wind and sun are both free fuel sources. The price of electricity depends on where inflexible consumer demand matches producers’ supply. The producers with the lowest operating costs are always called on first. Then the price of electricity creeps up the supply curve until consumer demand is satisfied. Every day, every hour, producers receive a price for the electricity they produce based on the last generator called up in the so-called “merit-order.” The graph below illustrates this.

meritorder

The last generator is always less efficient. This means that its operating costs are higher and it will only generate electricity when the price covers these operational costs. Now that renewables are part of the merit order, we don’t climb as high up the curve as before. On average, prices have decreased, implying the recurrent “last generator” is more efficient than a few years ago.

Second, thermal power plants’ operating hours are down. A lot of electricity is being generated from renewable sources replacing supply previously provided by gas and coal power plants. This point is obvious – money can only be earned when your power plant is online and generating electricity. This adds to traditional power plants’ woes. Prices are weakened, but their sale volumes are also harmed as renewable energy production grows.

Third, renewables are very variable. Already gas power plants have shut down and new projects have been cancelled because they could not survive the renewables’ economic shake-up. However, some days the sun does not shine, or there is no wind, and traditional generators are still needed. This can vary hour-by-hour, minute-by-minute. Only very modern gas facilities are capable of ramping up and down to balance unpredictable renewable production. Although, this is simply not profitable in a weak price climate where operating hours are down. So, these rapid-response power plants are no longer being built. This is called the “missing money” problem.

Fourth: the rise of the prosumer. Households and businesses have been installing solar panels with the hope of decreasing their electricity bills. In some countries, excess electricity that is generated can be injected into the grid earning you cash back from the local electricity retailer. This is how the word prosumer came about. Households connected to the distribution grid were traditionally pure consumers. Having installed solar panels the consumer is now a producer as well. They may even be electricity self-sufficient on sunny days or exceed their own electricity needs, affording them the opportunity to sell back to the grid.

Alone, one solar powered household cannot produce enough electricity to perturb the traditional power model. Yet, the arrival of hundreds and thousands of prosumers on the grid has the potential to be very destabilising as seen with commercial solar generation.

These four issues are part of a bigger problem: electricity infrastructure and markets are inflexible. They were not designed to manage decentralised and unpredictable electricity production. Nevertheless, this is the model we will have to manage in the future. Distribution lines also have ramping limits constraining how quickly power flows can be increased or decreased. Volatile prosumers and commercial wind and solar farms compromise the grid’s technical stability. And we still need back-up for the days and hours when renewable electricity production is low. Managing variable electricity production demands a model where this responsibility is shared by the market players.


Graph was found at www.powermarket.eu

Electricity prices & the solar eclipse

Electricity cannot be stored. When the sun hits a solar panel, or coal is burnt to turn a turbine and generate an electrical current, this energy is delivered to the distribution grid straight away.

Spot markets are where wholesale electricity producers and consumers go to balance their planned against their actual electricity needs. Those needs become clearer the closer we are to delivery, which is why electricity is often traded the day-ahead, or on the same day as delivery to the grid (intraday). This is particularly true for solar and wind power generators since the weather forecast becomes increasingly accurate from 24 hours out.

Solar eclipse

If you are a solar power farm what do you do if your energy source – the sun – goes offline? This is what happened last Friday morning, March 20th, in North-Western Europe. A solar eclipse, lasting around 75 minutes, during which the moon at least partially blocked the sun, had a big effect on solar electricity production.

Germany was particularly affected. Today it gets approximately seven-percent of its electricity from solar energy.

The celestial event affected French-German intraday spot prices between 9:00 and 11:00am. If you didn’t know better you might’ve thought the traders had pressed the wrong buttons on their keyboards! Bids as low as -975.00 euros and as high as 950.00 euros were tendered. To give you an idea prices are normally closer to 20.00 or 40.00 euros on the intraday market at the moment.

The final prices did eventually settle at 40.79 euros for 9:00-10:00am, and 66.37 euros for 10:00-11:00am, but varied a lot within the hour. Some 15min intervals settled at a negative price. This is not so unusual and has been seen before.[i] Nevertheless, the spot market demonstrated strong resilience to price volatility during the eclipse.

Negative electricity prices

When wind and solar generators have really good days electricity prices can drop below zero.[ii] The negative price means the market is oversupplied.  Everyone produced more electricity than expected and they don’t know what to do with it.

A negative price indicates you would actually pay someone else to use the excess electricity you produced. Why? It might be too late to decrease your production. Gas, coal and nuclear power plants need several hours to warm up (or down). Such facilities do not have simply on/off switches.

Avoiding blackouts

Those in charge of maintaining electrical grid stability, Grid Operators, can impose large fines if you exceed what you committed to delivering to the grid. Or if you do not produce as much as promised. Paying someone else to consume your excess electricity is probably a lesser loss than the fines imposed by Grid Operators.

The Grid Operators impose these rules because electrical currents need to be gently “ramped up” and “ramped down.”[iii] They have to plan ahead to ensure electricity flows safely and avoid blackouts.

What’s more consumers are fickle. You wouldn’t have been happy if your computer crashed, or you couldn’t make a cup of tea during your morning break because there wasn’t enough electricity – solar eclipse or not.

No one knew exactly how the solar eclipse would affect production, which explains traders’ erratic behavior. Somewhere else in Europe a more flexible electricity generator – probably a gas-fired power plant – had to quickly ramp up production to replace the eclipsed solar generation and meet consumer demand. Only the most modern and efficient power plants can react this quickly.

A test for Germany

The sudden drop in solar electricity production was an important test of grid stability. If Germany achieves its 2050 goal to produce 60% of its electricity from renewables, then cloudy days will have an affect on the grid as significant as last week’s solar eclipse.

The European Union also plans to increase the share of renewables in electricity production across the continent. In the future enormous swings in solar production could become commonplace.


[i] Take a look at all the prices here.

[ii] Sunny days tend to be windier then average, so solar and wind production peaks can coincide.

[iii] Imagine pulling your hairdryer out of the wall when it’s on full blast. Sparks fly! Multiply that effect by thousands and you can imagine the challenge for Grid Operators.