Value of power-to-gas as a flexibility option in integrated electricity and hydrogen markets

This paper analyzes the economic potential of Power-to-Gas (PtG) as a source of flexibility in electricity markets with both high shares of renewables and high external demand for hydrogen. The contribution of this paper is that it develops and applies a short-term (hourly) partial equilibrium model of integrated electricity and hydrogen markets, including markets for green certificates, while using a welfare-economic framework to assess the market outcomes. We find that strongly increasing the share of renewable electricity makes electricity prices much more volatile, while the presence of PtG reduces this price volatility. However, a large demand for hydrogen from outside the electricity sector reduces the impact of PtG on the volatility of electricity prices. In a scenario with a high external hydrogen demand, PtG can deliver positive benefits for some groups as it can provide hydrogen at lower costs than Steam Methane Reforming (SMR) during hours when electricity prices are low, but these positive welfare effects are outweighed by the fixed costs of PtG assets plus the costs of replacing a less expensive energy carrier (natural gas) with a more expensive one (hydrogen). Investments in PtG are profitable from a social-welfare perspective when the induced reduction in carbon emissions is valued at 150 – 750 euro/ton. Hence, at lower carbon prices, PtG can only become a valuable provider of flexibility when installation costs are significantly reduced and conversion efficiencies of electrolysers increased.


Introduction
To reduce the carbon emissions resulting from electricity systems, governments are promoting the share of Variable Renewable Energy sources (VREs). This transition in these systems makes grid balancing more challenging, and, as a result, increases both the long-term and short-term demand for flexibility. Flexibility in electricity systems can come from various types of supply-and demand-side options, including flexible conventional power plants, integration with neighboring markets through more cross-border transport capacity, curtailment of renewable production, storage, and more flexible loads. Many governments, including the EU, give a prominent role to hydrogen as a provider of flexibility (EU hydrogen strategy, [1]). After all, Power-to-Gas (PtG) plants, which produce hydrogen via electrolysis, can offer flexibility in three ways [2]. First, hydrogen can be stored for a shorter or longer period which enables producers of hydrogen to adapt the timing of electricity use to the situation in the electricity market, while the produced hydrogen does not need to be immediately supplied to users. This type of flexibility is called the time flexibility. Second, hydrogen can transfer renewable energy to other sectors that need energy in a liquid or gaseous form instead of electricity, which is called the end-use flexibility. This type of flexibility enhances the sector coupling between electricity and gas/hydrogen markets, as not only gas is used to generate electricity, but electricity is also used to produce a gas that can be used in, for instance, heating, transport, or industrial sectors. Third, hydrogen can be used to transfer renewable energy to regions where the electricity grid transmission is less developed, which is the locational flexibility. Hence, technically speaking, PtG has a wide potential to offer flexibility to the power system.
There is a growing body of literature that recognizes the important role of PtG in providing flexibility in energy sectors. The role of PtG as a provider of flexibility has been investigated and compared to other alternatives such as pumped hydro, battery, and compressed air [3]. They find that PtG can store enough energy in reasonably sized facilities compared to alternatives, however, it has a relatively low efficiency and high cost. Because of these high costs, it is generally not seen as an economically feasible storage option [4,5]. Recent studies, however, focus on the role of PtG in coupling electricity and gas sectors [6][7][8][9]. In these studies, PtG consumes electricity to produce hydrogen and subsequently injects the produced hydrogen into the gas network. In this way, PtG can play an important role in electricity markets when the level of VREs penetration is high and, hence, the electricity price is low.
In studies regarding the business feasibility of PtG, authors often assume an energy hub concept where electrolysers use low-priced electricity to produce hydrogen, while the produced hydrogen is stored to be used by fuel cells to generate electricity when the electricity price is high. In this way, PtG offers flexibility to the electricity market [5,10]. In these conversions there is, however, a significant energy loss of about 60% [11]. Besides, storage of hydrogen is economically less attractive since it is difficult to compress [4]. Hence, PtG's potential is not fully exploited if it is only used to provide flexibility to the electricity sector. In fact, most of the large-scale PtG demonstration projects are producing hydrogen for use by other sectors [12,13]. Because of hydrogen's potential in transport, in heating of premises, as well as in industry sectors, it is expected that hydrogen markets will become mature in the coming decades.
From the above follows that to determine the potential role of PtG in flexibility provision, we need to include the potential contribution of PtG as a provider of hydrogen to other sectors as well. Hence, we have to assess the costs and benefits of the provision of flexibility in combination with providing hydrogen to end-users. The costs of investing in PtG are relatively easy to estimate, however, estimating the benefits is more complicated because they depend on how electricity, gas, and hydrogen markets interact. Currently, electricity, gas, and hydrogen sectors are weakly linked by gas-fired power plants as well as by hydrogen plants based on steam reforming of gas. PtG could intensify the sector coupling in two different intensities: one-way and two-way. One-way sector coupling occurs when PtG uses electricity to produce hydrogen while the produced hydrogen is supplied to external users such as the industry, where this hydrogen replaces either natural-gas-based hydrogen or just natural gas. Two-way sector coupling occurs when the hydrogen produced via PtG is also used to generate electricity and, as a result, PtG is able to offer not only demand-side, but also supply-side flexibility to the electricity market.
Hence, to assess the market value of PtG as a flexibility option, we have to study electricity and hydrogen markets in an integrated way, looking more closely at the role of PtG. When this interaction is not taken into account, the estimates of the value of PtG as a flexibility provider may be biased. To overcome this bias, we develop and apply a stylized short-term (hourly) model of an integrated energy system where PtG bridges electricity and hydrogen markets and provides flexibility to the electricity market as well hydrogen to end-users, taking into account preferences of energy users for various types of electricity and hydrogen (e.g. green or grey) through markets for green certificates. As we want to focus on electricity and hydrogen markets, we use exogenous (daily) values for the prices of gas and carbon. We assess the outcomes of various model variants in terms of social welfare. To obtain numerical results, we calibrate this model on the Dutch energy system, but in fact, any other system could have been chosen as well. This integrated welfare-economic analysis of two commodity markets with two markets for certificates to estimate the economic value of flexibility provided by PtG in an actual energy system is the novelty of our study.
We measure the value of flexibility provided by PtG in two ways: the reduction of electricity price volatility measured through the shape of the price-duration curve, and the change in social welfare. The impact on the price-duration curve is used to see how PtG affects the price formation in the electricity market, while the impact on social welfare is used to determine to what extent adding PtG to an energy system has a positive overall effect on social welfare. Using our model and applying it to the Dutch energy system, we find that PtG indeed makes the electricity price-duration curve flatter, but it is not a valuable flexibility option under current market conditions, because its overall welfare effect is negative. PtG can be a valuable flexibility option if its fixed costs reduce significantly and/or carbon price becomes much higher. The break-even price of CO2 to give PtG an overall positive welfare effect is in the range of 150-750 euro/ton depending on the magnitude of the installed PtG capacity and the magnitude of external hydrogen demand. As this carbon break-even price is much higher than the prevailing carbon price in for instance, the European Emissions Trading scheme, one must conclude that PtG is currently not a profitable solution, seen from a social-welfare perspective, to provide flexibility to electricity systems with high shares of renewables.
The paper is organized as follows. Section 2 reviews the related literature. Section 3 provides the economic theory that guides our analysis. Methodology is presented in Section 4. Results are shown in Section 5. We conclude the paper with Section 6.

Literature review
The economic feasibility of PtG is analysed by a number of authors. Most of the studies focus on the economic conditions under which PtG can be profitable, while some others are directed at the potential contribution of PtG as a provider of flexibility to energy systems with high shares of renewables.
The common finding of the various studies which have assessed the business model of PtG is that PtG is not profitable under current market conditions. [14] perform a scenario analysis using the Dutch system as an example to investigate the economic and environmental consequences of the large-scale storage of PtG, pumped hydro, and compressed air energy in an electricity market at different wind penetrations. They find that PtG is the least economically feasible option. [4] investigate the economic feasibility of PtG systems based on a net present value model in which prices for electricity and gas are stochastic. They study three investment cases: a Base Case where the produced gas is directly sold in the market, a Storage and Arbitrage Case where the gas is stored for temporal arbitrage between the electricity and the gas market, and a Storage and Balancing Case where PtG is used for balancing. They find that PtG cannot be economically profitable in the Storage and Balancing Case as well as the Storage and Arbitrage Case, while directly selling the gas is a more favorable option. [15] investigate the potential of an integrated balancing system based on different combinations of gas-turbine power plants and PtG plants. By comparing a wind park with a balancing system consisting of a gas-fired plant and a PtG plant with a baseline case where no balancing system is installed, they find that the current economic conditions are not favorable to the PtG concept. However, if the price of hydrogen goes up and the carbon emission price becomes higher, PtG could be profitable. [16] study the value of a PtG system that produces hydrogen for a hydrogenfueling station while it also provides ancillary services to the grid. Based on a PtG project in Toronto, they first determine the optimal size of hydrogen compression and storage systems, then they derive the optimal operation of the PtG plant. Their analysis shows that the PtG project could be financially successful if the price of hydrogen and the value of carbon-emission reduction both go up. [2] evaluate a PtG plant's willingness to pay for electricity given its cost and revenue estimations. By comparing the PtG plant's willingness to pay for electricity, based on the break-even condition for electrolysis, with historical electricity prices, they find that PtG is unprofitable under current market conditions. PtG may, however, become economically feasible when the investment costs of PtG are reduced in combination with an increase in both the electrolyser efficiency and the price of hydrogen.
A number of studies on PtG pay particular attention to its role in energy storage and as a provider of flexibility. From these studies, one can conclude that PtG is able to provide some valuable flexibility services to electricity systems with high shares of renewables. A comprehensive review of PtG's role of storage can be found in [3]. [6] use a unit-commitment model to determine an optimal PtG capacity in an 85% renewable energy scenario for Germany. In that energy system, PtG is modeled as a storage concept linking power and gas networks by converting power into gas. The produced gas is stored in the natural-gas infrastructure and later used for reconversion to electricity or other purposes such as heating. Power-to-Heat and typical short-term electricity storage systems are competing with PtG to provide flexibility to the electricity sector. PtG's benefits are defined as the reduction of the overall cost of the modeled energy system. They find that the overall reduction of the system costs and CO2 emissions is at its maximum when PtG and Power-to-Heat are both included in the system as balancing options. [8] treat PtG as a seasonal storage option in an integrated gas and electricity transmission network model. They assess the technical feasibility and benefits of PtG over a daily time frame with a direct current optimal power flow model. They find that PtG can absorb more renewable generation and reduce the cost of natural gas use by 4%. [5] investigate the economic potential of an energy hub with both power-tohydrogen and fuel cell as a flexibility option for a wind farm. In their model, the wind farm participates in a day-ahead market based on forecasts and uses the energy hub to correct its forecast errors. Based on data for Germany over the year 2013, their study shows that the energy hub could correct the forecast errors a lot, but operating the hub only for correcting the wind farm's forecast errors is not economically sensible. However, if a fuel cell is used to reduce forecast errors and PtG is used to provide a negative balance in the secondary control reserve market, the operating hub could make sense.
In some other studies, the attention is not only directed at the potential role of PtG as a provider of flexibility, but also at other potential relationships between hydrogen and other energy markets. [7], for instance, develop an operational model of an energy system comprised of gas, electricity, and CO2 markets to analyze the impact of PtG on the demand for flexibility, import profile, seasonal storage in gas sector, the marginal electricity cost in the electricity sector, and the demand for CO2 allowances. In this energy system, the gas-fired power plants are equipped with CO2 capture devices, hence they supply electricity and CO2. PtG consumes electricity and captured CO2 to produce methane that is injected into the gas network, which is how PtG links the gas, electricity, and carbon markets. The authors find that PtG can lower the curtailment in an electricity sector based on 100% renewable electricity and gas-fired power plants as backup. In addition, they conclude that PtG may create a downward pressure on both gas prices and the need for long-term CO2 storage.
Another example of this type of studies on sector coupling is [10] who model an energy hub with both a power-to-hydrogen and a gas-topower facility to assess the impact of PtG on the energy system. With a security-constrained unit-commitment model, they compare the case without the proposed energy hub with the case with the energy hub to show PtG's impact on wind-power curtailments and hydrogen production. They find that PtG reduces wind curtailments more if the produced hydrogen could be directly supplied to other industry sectors. The long-term relationship between electricity and gas markets is analyzed by [9] who develop a stylized model to simulate the long-term equilibrium of an integrated electricity and gas market coupled via PtG. In this system, PtG consumes electricity to produce hydrogen that is blended and injected into the gas network. PtG is modeled as both an electricity consumer and a gas supplier. By maximizing each producer's profit and consumer's utility subject to market clearing constraints, they find that PtG may not be profitable unless its installed capacity is limited such that it does not absorb all spillage and its revenues from arbitrage between low and high electricity price periods are higher than its costs. In addition, the welfare-optimal PtG capacity could lead to a loss for the PtG investor, which implies that a subsidy to PtG may be a welfareimproving policy.
Our paper proceeds on this literature by assessing the economic feasibility of PtG by taking into account both its potential supply of flexibility to the electricity market and its supply of the commodity hydrogen to end-users. With a few exceptions, the previously cited studies on the role of PtG in sector coupling assume that the electricity demand is exogenously given, which underestimates the electricity sector's need for flexibility, hence the value of PtG as a flexibility option. In contrast, we assess the value of PtG as a flexibility option in a model with hourly fluctuating load and generation by renewable sources which both also respond to market prices, leading to a more accurate estimate of the functioning of electricity markets. In addition, in assessing the flexibility option, we explicitly model the influence of external demand for hydrogen as this demand must be seen as a competing use for hydrogen. Another contribution of our paper is that we determine the economic value of PtG from a social-welfare perspective, which means that we estimate the impact of PtG on consumer and producer surplus, both on an aggregated level and for each separate group in the energy system. Finally, we explicitly estimate the minimum CO2 prices needed to make PtG a profitable provider of flexibility.

Micro-economic theory
In markets, demand and supply are cleared through equilibrium prices. Given the price elasticity of supply and demand, if the supply curve or the demand curve shifts upwards, the equilibrium price will rise; and the other way around: if the supply curve or the demand curve shifts downwards, the equilibrium price will fall. Consequently, if demand and supply change over time, we can observe a series of equilibrium prices over time. Ordering the equilibrium prices from the highest to the lowest and displaying them over time, we get the so-called priceduration curve.
The steepness of these curves depends on the presence of flexibility in a market. Flexibility refers to the ability to increase or decrease supply and demand over various time scales (so, it can be expressed as energy per time unit, such as MWh/h). Economically, this flexibility can be measured through the price elasticity of the demand and supply over various periods. More flexibility in a market implies that demand and/or supply are more sensitive to prices, which means, in other words, higher price elasticities of the demand and/or supply. As a result, for the same amount of supply or demand change over a given period, the equilibrium price decreases or increases less than before, which reduces the volatility of the series of equilibrium prices, leading to a flatter priceduration curve. Hence, the steepness of the price-duration curve is a measure of the flexibility within an energy system.
The economic value of a change in the price-duration curve is the change in social welfare caused by adding a unit of flexibility, where the social welfare is measured by the sum of consumer and producer surplus. Hence, the marginal value of flexibility is the change in social welfare as a result of introducing one additional unit of flexibility to the electricity market. Only if this value of flexibility is positive, it is worth extending the capacity of the flexible source from a societal (welfare-economic) point of view. The optimal level of that capacity is found where the marginal change in social welfare is zero. In the application of the model to the Dutch energy system, we will not search for the optimal size of PtG investment, but we will determine the CO2 price which is needed to find a zero social welfare effect of an investment in PtG which is related to the Dutch policy objectives. This price is called the break-even CO2 price of that PtG policy.

Theory applied to PtG
To provide flexibility to an electricity market, PtG plants have to be combined with hydrogen storage and hydrogen-fired power plants. When electricity prices are low, PtG plants use electricity to produce hydrogen which is stored afterwards. When electricity prices are high, the produced and stored hydrogen can be used by hydrogen-fired power plants to generate electricity. To evaluate the value of PtG as a flexibility option, we need to see how the supply and demand curves in the electricity market change after a PtG plant in combination with storage and a hydrogen-fired power plant are introduced. A PtG plant as an electricity consumer directly influences the electricity demand curve through load shifting or shedding, while a hydrogen-fired power plant as an electricity producer directly influences the electricity-supply curve. When the electricity price is lower than the level at which an existing PtG plant breaks even, the PtG plant uses electricity to produce hydrogen and, hence, the total electricity demand is higher than before. When the electricity price is higher than the level at which an existing hydrogen-fired power plant breaks even, the hydrogen-fired plant produces electricity and, hence, the total electricity supply is higher than before. These changes in electricity demand and supply make the total demand and supply for electricity more elastic, which is reflected by Fig. 1. The upward movement of the demand curve and the shift of the supply curve to the right result in, respectively, higher and lower electricity prices, which implies that the price-duration curve becomes flatter.

Setup of the model
We develop a stylized model to simulate the equilibria in an hourly wholesale electricity market, where hydrogen as an energy carrier is able to shift electricity from low-price periods to high-price periods via PtG, hydrogen storage, and hydrogen-to-power(Hydrogen-fired producers). In this case, the hydrogen produced by PtG is stored for later use by hydrogen-fired producers. To analyze the interaction with external hydrogen demand on the role of PtG as a provider of flexibility to the electricity market, we add a hydrogen market to the model. In this market, the supply of hydrogen comes from two sources: SMR and PtG. The demand for hydrogen comes from both external users, reflecting demand from industry or transport, and demand from hydrogen-fired power plants. In this way, PtG and hydrogen-fired power plants connect electricity and hydrogen markets. Hence, the electricity and the hydrogen markets are both modelled, with endogenous hourly equilibrium quantities and prices. The gas and carbon prices are treated as exogenous variables, as both are based on international markets which do not much depend on the circumstances in one market (i.e. our market of analysis). We assume, therefore, a series of daily gas and carbon prices, based on actual data and scenarios regarding the gas market and international climate policy. In addition, we also includes markets for green-electricity certificates and green-hydrogen certificates to model the demand for green electricity and green hydrogen. Table 1 gives an overview of markets included in the model and the participants in each market.
We assume that the electricity spot market is a competitive market where no player is able to raise profits by behaving strategically. Production and consumption respond to prices, but are also affected by volatile exogenous factors such as weather circumstances. Electricity producers utilize a number of generation techniques, each with a constraint on available generation capacity: renewable power producers with a small variable cost; conventional gas-fired plants with variable costs related to the gas price; hydrogen-fired power plants with variable costs related to the hydrogen price. In addition, we assume that the market is connected to neighbouring markets, which results in a potential (net) import supply. This supply is modelled as a net import which is a function of the spread between the (endogenous) domestic hourly electricity price and the (exogenous) foreign hourly electricity price. Renewable power production is a function of weather circumstances with a small marginal cost, which means that its production stops when the market price is very low. Consumers of electricity include a PtG producer and an aggregate demand from other consumers which is a function of the electricity price. In addition, the intercept of the demand function depends on time of the day and exogenous information on weather circumstances. The model is deterministic, but we include a time pattern of wind speed/sun shine based on historical data, which implies that both actual renewable production and load change from hour to hour.
We include climate policy measures by assuming an external carbon price (added as a time-varying exogenous value to the marginal costs of the electricity producers) and an energy/carbon tax on energy use (added as an exogenous constant to the price in the demand function).
The hydrogen market is also assumed to be a competitive spot market where no player is able to raise profits by behaving strategically. Producers are distinguished in two types of producers: PtG producer and SMR producer. There is one aggregated demand for hydrogen.
We also include two certificate markets: one for green electricity and the other for green hydrogen. In the electricity certificates market, the supplier is the renewable producer. In the hydrogen certificates market, the supplier is the PtG producer. In both markets, the demand for certificates is represented by an aggregated demand function. The  In all markets, each producer maximizes its profit by choosing an optimal production (quantity) given the price and its capacity constraint. Each consumer maximizes its utility by choosing an optimal consumption if its demand is price elastic; otherwise, it just consumes what it needs based on the fluctuation in external information (reflected by fluctuations in the intercept) with the constraint being that supply equals demand. See Appendix B how the demand curves have been estimated.
In each market, prices balance the demand and supply. The interaction between different markets is illustrated in Fig. 2.

Methodology
This section describes how the economic mechanisms which are described in the previous section are translated into a mathematical model (Section 4.1), how the market equilibria are determined (Section 4.2.), how this model is calibrated using data on the Dutch energy system (Section 4.3), how this model is used by formulating scenarios and variants (Section 4.4) and, finally, how the costs of PtG installations are determined which are used to determine the long-term welfare effects (Section 4.5).

Mathematical model
Since the power market is generally operated on an hourly basis, we model an energy system that has 8760 h, which is a year in total. Hence, the market clearing prices and quantities are solved at an hourly resolution. As the model is on an hourly basis, it is a short-term model, in which fixed costs of investments do not play any role. We do, however, take these costs into account when we compare the short-term welfare effects resulting from the model with the annualized values of the fixed cost (see Section 4.5). In the following, we denote a day of the year by d ∈ {1, 2, …365} and an hour of each day by h ∈ {1, 2, 3, …24}.

Electricity market
4.1.1.1. Renewable producer. We model the supply of wind power to represent the renewable-energy supply. The maximum possible production is determined by the hourly variations in wind speed, expressed as the hourly capacity factor. As this capacity factor changes over time, the maximum production of renewable energy also changes over time. The producer chooses a production level between zero and this maximum depending on market circumstances. The relevant economic variables for this producer are its marginal costs, which are assumed to be constant, the electricity price and the price of green certificates. Note that the fixed (investments) costs are ignored, since the model is directed at the short-term market equilibria and the fixed costs can be viewed as sunk. The decision variable of the renewable producer is its hourly generation q E,R dh , which depends on the changing weather and market circumstances. Hence, its optimization problem is defined as where p E dh is the electricity price; p GC dh is the green-certificate price per unit of electricity; c R is the variable cost of the renewable producer, assumed to be a constant; K R is the available installed capacity of the renewable, assumed to be a constant; A dh is the capacity factor between 0 and 1, which depends on weather circumstances and follows an exogenous hourly pattern.

Gas-fired producer.
The decision variable of the gas-fired producer is its hourly generation q E,G dh . Its optimization problem is defined as where p G d is the gas price, which results from an international gas market and is treated as an exogenous variable in our model; c G d is the carbon price for each ton of emission in a given day, treated as an exogenous variable; γ C measures how many tons of carbon emission is generated by burning each unit of gas (in MWh), which is a constant; γ G is the conversion efficiency from gas to electricity, assumed to be a constant; K G is the available installed capacity of the producer, assumed to be a constant.

Hydrogen-fired producer.
The decision variable of the hydrogen-fired producer is its hourly generation q E,H dh . Its optimization problem is defined as where p H dh is the hydrogen price; γ H is the conversion efficiency from hydrogen to electricity, assumed to be a constant; K H is the available installed capacity of the producer, assumed to be a constant.

International trader.
We model the international trader as a net importer. The decision variable of the trader is its hourly net import q E,I dh . Its optimization problem is defined as subject to where p F,E dh is the electricity price in the neighbouring country and this foreign price is the variable cost of the importer, which is treated as an exogenous variable implying that we do not model the international market; K I is the available transmission capacity of the importer, assumed to be a constant.

Electricity demand from PtG producer. The decision variable of
the PtG producer is its hourly hydrogen production q H,PtG dh . Its optimization problem is defined as where p H dh is the hydrogen price; p GH dh is the price for green-hydrogen certificates; p GC dh is the price for green-electricity certificates; γ PtG is the conversion efficiency from electricity to hydrogen (from MWh of electricity to MWh of hydrogen), assumed to be a constant; K PtG is the available installed PtG capacity, assumed to be a constant. Given the conversion efficiency from electricity to hydrogen, we know the PtG producer's electricity consumption l E,PtG

Electricity demand from other consumers.
We assume the electricity demand from other consumers is represented by the following linear demand function where l E,O dh is the consumption; t E,O is the electricity tax for consumers.
The intercept α E,O dh and the slope β E,O dh both are positive. The intercept and the slope change over periods and follow an exogenous hourly pattern.

Electricity market-clearing constraint.
Given the derived total supply and total demand for each period, the price p E dh clears the electricity market by meeting the following condition.

Green-certificates market for electricity
A green certificate, also called a renewable-energy certificate, is a tradable asset that proves energy has been produced from a renewable energy source. In a green-certificates market, renewable producers receive certificates for each megawatt-hour (MWh) of produced energy and the certificates can be sold to consumers/retailers, which may result in extra income for the renewable producers depending on the price of the certificates. In our model, the supply of the green-electricity certificates equals the production by the renewable producer. PtG producers demand green-certificates of electricity when it supplies green hydrogen to the hydrogen market and the other demand for green-certificates of electricity is represented by the following linear demand function (13) where the intercept α GC dh and the slope β GC both are positive and the intercept α GC dh is less than the total electricity demand. The certificate price p GC dh clears the green certificate market by meeting the following condition. . Its optimization problem is defined as (15) subject to 0⩽q H,SMR dh ⩽K SMR (16) where γ SMR is the conversion efficiency from gas to hydrogen, assumed to be a constant; as defined before, γ C measures how many tons of carbon are generated by reforming a unit of gas and c G d is the carbon price; c CCS is the cost of carbon capture and storage for each ton of carbon, assumed to be a constant; λ is the fraction of carbon being emitted and 1 − λ is the fraction being captured; K SMR is the available installed SMR capacity, assumed to be a constant.

PtG producer.
The PtG producer's optimization has been discussed in the electricity market. Its hourly hydrogen production is q H,PtG dh , which is a function of the electricity price p E dh and the hydrogen price p H dh . In this way, the electricity sector and the hydrogen sector are coupled via the PtG producer.

4.1.3.3.
Hydrogen-fired producer. The hydrogen-fired producer's optimization has been discussed in the electricity market. Its hourly which is a function of the electricity price p E dh and the hydrogen price p H dh . In this way, the electricity sector and the hydrogen sector are coupled via the hydrogen-fired producer.

Demand from other consumers.
We assume that the hourly hydrogen demand from other consumers is represented by the following linear demand function (17) where l H,O dh is the consumption; the intercept α H,O dh and the slope β H,O both are positive.

4.1.3.5.
Hydrogen storage operator. The storage operator behaves like an arbitrager in the hydrogen market. For simplicity, we assume the operator buys hydrogen when the price is lower than p H and sells hydrogen when the price is higher than p H (> p H ) given its available storage capacity and storage level. The margin between buying and selling is meant to cover the costs of the storage operator. Here, we assume this margin is an exogenous constant. The decision variable of the storage operator is its hourly net storage l H,S dh subject to where K S is the storage capacity and u S dh− 1 is the storage level at the previous period h − 1. We assume the initial storage level u S 10 = 0. The storage level is updated according to the following formula.

Green-certificates market for hydrogen
Similar to the green certificate for renewable electricity, we assume that there is a green-certificates market for green hydrogen. A greenhydrogen certificate is a tradable asset that proves hydrogen has been produced from a renewable energy source. In a green-hydrogen certificate market, green hydrogen producers such as PtG receive certificates for each megawatt-hour (MWh) of produced hydrogen and the certificates can be sold to consumers/retailers, which may result in extra income for the green hydrogen producers depending on the price of the certificates. In our model, the supply of the green-hydrogen certificates equals the production of the PtG, while the demand is represented by the following linear demand function where the intercept α GH dh and the slope β GH both are positive and the intercept α GH dh is less than the total hydrogen demand. The certificate price p GH dh clears the green-hydrogen certificate market by meeting the following condition.

Equilibria
We use the concept of noncooperative market equilibrium to find the equilibrium price in each market. This means that we assume that all markets are competitive and that every market participant has to take the market outcomes (i.e. prices) as exogenous information which cannot be influenced. Regarding the decision criteria, it is assumed that each of the market participants maximizes its profit (in case of firms) or utility (in case of residential consumers) under market clearing constraints (i.e. that aggregated demand should equal aggregated supply) and given the technological constraints regarding supply (such as regarding capacity and conversion efficiencies). This means that every market participant submits its bid (i.e. minimum price required by suppliers and maximum price offered by consumers) to the market operator which solves this by searching for that (equilibrium) price which results in the highest welfare. This is the same way how, for instance, electricity power exchanges function. These market equilibrium prices are often computed by solving an appropriate optimization problem based on maximizing social welfare (consumer surplus plus producer surplus) with constraints. Alternatively, market equilibria can be calculated by solving an appropriate mixed complementarity problem (MCP) based on the Karush-Kuhn-Tucker conditions for all market participants with market-clearing conditions [17]. In this paper, we use the MCP approach to find market equilibrium outcomes. Appendix A explains how the problem is solved.

Model calibration
To be able to make reasonable simulations, we calibrate the model on the Dutch energy system. Appendix B provides our assumptions regarding the technology parameters, the electricity and the hydrogen market, and other costs and tax variables. Using these parameter and variable values, we are able to run the model and calculate the equilibrium prices and quantities. Fig. 3 shows the duration curve for the electricity price resulting from the model in comparison to the curve based on the actual Dutch day-ahead electricity prices in 2019. Comparing both price-duration curves, we can see that our model represents the Dutch electricity day-ahead market fairly well. Only at both ends of the distribution, the actual values are more extreme than the model results (i.e. a few hours with relatively high prices as well as a few hours with relatively low prices).
The set of assumptions which are made to get the model result which more or less resembles the current Dutch electricity system is called the departure scenario. In this scenario, we assume that the installed capacity of renewable electricity is 10,000 MW, resulting in a share of renewable electricity in annual production of about 22%, which was more or less in line with the actual situation in the Netherlands 1 . We call this departure scenario the 'Low Renewable' scenario. The Dutch government, like many other governments, is committed to increase the share of renewable energy strongly. Therefore, we also consider a 'High Renewable' scenario with an installed capacity of renewable electricity of 60,000 MW resulting in a share of renewable electricity in the annual production of 69%. This level is related to the 2050 objectives of the Dutch government (National Climate Agreement of the Netherlands, 2019). In this scenario, we assume that the level of gas-fired capacity is lower than in the departure scenario, but not to the same extent as the increase in renewable production, as investments in flexible gas-fired power plants are profitable when there are more hours with high prices. For further motivation regarding the assumed composition of the electricity mix, see Appendix C. Fig. 4 shows the change in the electricity price-duration curve when we change the electricity mix with way more installed renewable capacity.
This figure clearly shows that an increase in the share of renewable generation results in a higher price volatility, even controlling for a relatively higher share of gas-fired power plants, as the price-duration curve becomes much steeper. There are more hours with higher prices, while there are also more hours with much lower prices. As a result,

Use of model: variants and scenarios
We use the results of the above 'High Renewable' scenario for the analysis of the welfare effects of adding PtG to the electricity system. The value of PtG as a flexibility option depends on how strongly the market needs flexibility. As shown in the above section, when the share of renewable electricity is high, the electricity price becomes more volatile, as a result, more flexibility is needed. Hence, to demonstrate the potential value of PtG as a flexibility option, we focus on the case that the share of renewable electricity is high.
To assess the impact of PtG, we introduce three policy variants regarding the capacity of PtG: no PtG, low PtG and high PtG (see Table 2). The variant no PtG provides a baseline for assessing the impact of PtG on the electricity and hydrogen markets. In the variant with low PtG, the capacity of PtG is 6% of the total available capacity of electricity generation. While high PtG refers to the case that the capacity of PtG is about 30% of the total available capacity of electricity generation. Since PtG can only offer flexibility to the electricity market when it is combined with storage and hydrogen-fired power plants, we also have to make assumptions regarding their size. We assume that the installed capacity of hydrogen-fired power plants is half of the installed capacity of PtG, while regarding the storage capacity we assume that 20 h of full PtG production can be stored.
The welfare effects of PtG will also depend on the demand for hydrogen outside the electricity sector. Therefore, we define three scenarios regarding industrial hydrogen demand. When there is no industrial hydrogen demand, PtG produces hydrogen only to be used by hydrogen-fired power plants to produce electricity. Hence, in this scenario, PtG only offers flexibility to the electricity market. When there is an industrial hydrogen demand, PtG has two options for its supply: both as input for electricity generation and as fuel or feedstock in the industry. Hence, in these scenarios, PtG bridges the electricity and hydrogen markets as a sector coupling channel. Table 3 summarizes the combination of variants and scenarios.
The Baseline case is the reference case for analysing the effects of PtG on prices and social welfare. We are having two situations in which PtG only supplies hydrogen to the electricity market, notably when the industrial hydrogen demand is zero and PtG capacity is low or high. There are also two situations in which hydrogen is only supplied to the industry by SMR, notably when there is no PtG capacity. These two situations are not further analyzed in this paper. In the other four situations, both PtG and SMR produce hydrogen. Comparing the benefits with the cost of PtG, we can see whether PtG is an economically feasible flexibility option in each of these situations in comparison with the Baseline. As said above, this analysis is done for the situation in which the share of renewables is high, but as a sensitivity analysis, we also show the value of PtG in the case that the share of renewable electricity is low (see Appendix F).

Costs of PtG
Although bringing PtG to the market may have benefits for both electricity and hydrogen markets, whether PtG is an economically feasible option to provide flexibility depends on how costly it is. Therefore, we compare the short-term welfare effects resulting from the model, with exogenous information on the costs of investing in PtG assets. Regarding the capital cost of PtG, a wide range of estimates has been reported. Table 4 lists the cost range of PtG, its sources, and our assumptions as well.
For convenience of comparison, we calculate the yearly cost of PtG as follows. We first determine the total investments for each type of installation. Then we calculate the capital recovery factor (CRF) for each type of investment by the following formula where i is a discount rate and n is the expected lifetime of the investment in years. Multiplying the investment with its CRF, we get the yearly capital cost for each type of installation. Summing the yearly capital costs of three types of investments, we get the yearly capital cost of PtG. Table 5 lists the yearly capital cost of PtG with two different discount rates for both Low and High PtG. In the following cost and benefit comparisons, we assume a discount rate 5%.

PtG used only for providing flexibility to electricity market
First, we analyze the contribution of PtG as a provider of flexibility to the electricity market by assuming that hydrogen produced through electrolysis can only be used as a fuel for generating electricity, which is the scenario of zero industrial hydrogen demand. Hence, in this analysis, hydrogen as an energy carrier shifts electricity from low-price periods to high-price periods. Below, we present the effects on the price-duration curve as well as on social welfare.

Price impact of PtG
When PtG is present in a market with a high amount of renewables, it changes the price-duration curve. From Fig. 5, we can see that PtG decreases the electricity prices when they are high and increases the electricity prices when they are low, which makes the price-duration curve of electricity flatter than before. Hence, PtG indeed offers flexibility to the electricity market. The impact of PtG increases in its capacity. Hence, we see a higher impact when more PtG is installed. How the change of gas price impacts the price-duration curve of electricity is discussed in Appendix D.
To understand the impact of PtG on the electricity price in details, we list the average price weighted by hours as well as the average price weighted by domestic consumption in Table 6. Overall, PtG reduces the average price of electricity. The explanation for the decrease of the average electricity price can be found by looking at which type of supplier is the price setter in the market. Fig. 6 shows the percentage that each type of electricity supplier sets the price. PtG decreases the average price of electricity because hydrogen-fired power plants are motivated to increase the supply of electricity when the price is very high. Hence, these generators become more often the marginal     To understand the impact of PtG on the price-duration curve of electricity, we can also look at how the capacity of PtG is used. Table 6 shows the number of hours during which electrolysers and hydrogenfired power plants are producing, as well as their capacity factors that measure the ratio between the used capacity and the installed capacity. We can see that the capacity of PtG is used at a very low rate when PtG only offers flexibility to the electricity market.
The introduction of PtG also changes the price of green-electricity certificates because it induces extra supply of green certificates. Fig. 7 shows the impact of PtG on the price-duration curve of the certificates. We see that PtG decreases the price of green certificates to a very small extent, which is confirmed by Table 6.
In relation to the changes of electricity prices and green-electricity certificate prices, the composition of electricity production also changes. Fig. 8 shows the share of each type of electricity production in different situations. We see that gas-fired electricity generation is to some extent replaced by hydrogen-fired electricity generation because of the presence of PtG.

Welfare impact of PtG
The introduction of PtG affects electricity prices, which leads to changes in consumer surplus, producer surplus, and government tax. Fig. 9 shows the welfare change due to PtG, paying attention to the effects for different groups of participants.
Renewable-electricity producers benefit from PtG because they benefit from the lower number of hours with low electricity prices, while hydrogen-fired power producers benefit from the production during hours with high prices. Gas-fired electricity producers and electricity importers suffer, however, because they can less benefit from high prices. The benefit of PtG for renewable electricity producers comes from the fact that only renewable power plants are producing when the electricity prices are low and that just these prices are raised when PtG is introduced. Hence, the extra electricity demand from PtG in these hours increases the profits of renewable electricity producers. In contrast, hydrogen-fired power plants only produce when electricity prices are very high. For these periods, it is often the case that gas-fired power plants are producing. The extra electricity supply from hydrogen-fired power plants may decrease the electricity price, which reduces the profit of gas-fired power producers. Since the introduction of PtG reduces the volatility of the electricity price, electricity importers have a smaller room for price arbitrage between markets, which decreases the profits of importers.
For hydrogen producers, it comes as no surprise that PtG producers and hydrogen storage operators benefit from the introduction of PtG.
For consumers, green-certificate consumers benefit from PtG because PtG induces renewable electricity producers to produce more electricity raising the supply of certificates, which reduces the certificate price. PtG increases consumers' surplus because overall PtG brings electricity prices down.
Although the benefits for various groups seem to exceed the costs for others, the overall welfare effects are negative due to the (annual) fixed costs related to the investments in the PtG assets. Hence, PtG appears to have a negative impact on social welfare when it is only used for providing flexibility to the electricity market.

PtG supplies both flexibility to electricity sector and hydrogen to hydrogen market
In the previous section, it was assumed that PtG could only supply flexibility to the electricity sector, but is likely that PtG will also be used to provide hydrogen for users in industry or transport. Hence, in such a situation, the demand for hydrogen may compete with the demand for flexibility in the electricity market, which may affect the value of the latter. Below, we will first present the effects of PtG on the priceduration curves of electricity and hydrogen as well as on social welfare for a number of scenarios and variants. Then, we will relate these short-term welfare effects to the fixed costs to determine the overall welfare effects.

Price impact of PtG
As we have seen in the previous section, the introduction of PtG (i.e., electrolysers, hydrogen storage, plus hydrogen-fired power plants) is able to make the price-duration curve of electricity flatter. As shown in Fig. 10, in the scenario of low industrial hydrogen demand the impact of PtG on the price-duration curve is similar to the previous situation without any sector coupling. When the hydrogen demand is high, however, PtG mainly drives the low electricity prices up, but it does not bring high prices down too much. This is due to the fact that there is less hydrogen available to be used during hours of high electricity prices. When the industrial hydrogen demand is high, hydrogen prices are more likely to be high as well, which incentivizes PtG plants to produce and  supply more hydrogen and, as a result, hydrogen-fired power plants will produce less electricity. Consequently, the high electricity prices are not reduced by extra electricity production based on hydrogen. This also results in a higher on average electricity price, both weighted by hours and weighted by domestic consumption (see Table 7).
To understand the impact of PtG on the price-duration curve of electricity, we can also look at how intensive the capacity of PtG is used. Table 7 shows the number of hours during which electrolysers and hydrogen-fired power plants are producing, as well as their capacity factors that measure the ratio between the used capacity and the installed capacity. We can see that the capacity factor of electrolysers increases with hydrogen demand while the capacity factor of hydrogenfired power plants slightly decreases with hydrogen demand, which pushes the electricity price up with the increase of hydrogen demand.
To further understand the mechanism behind the impact of PtG on the electricity price, we have to look at the price-setting power plants. Fig. 11 shows the percentage that each type of supplier sets the market price of electricity. In the scenario of low industrial hydrogen demand, hydrogen-fired power plants are more often the marginal, price-setting power plant than in the scenario with a high industrial demand for hydrogen.
When hydrogen produced by PtG can be used in industry or transport, there may also be a higher demand for green hydrogen. To be able to supply green hydrogen, electrolysers need to buy green-electricity certificates. As a result, the price of these certificates will go up. Fig. 12 shows how the price of green-electricity certificates changes due to the introduction of PtG. From Table 7 we can see that the annual average price of green-electricity certificates increases when there is a higher demand for hydrogen from the industry.
In combination with the change in the electricity prices, the mix of electricity production also changes. Fig. 13 shows the share of each type of electricity production in different situations. The share of gas-fired electricity generation decreases with the introduction of PtG, which is due to the fact that PtG replaces gas to some extent as supplier of electricity during hours of high prices. Moreover, the share of renewable electricity generation increases with a high capacity of PtG as this results in a higher demand for electricity, reducing the number of hours with too low prices.
The supply of hydrogen by PtG also affects the price of hydrogen. As shown in Fig. 14, this impact varies from scenario to scenario. PtG decreases hydrogen prices much more when the industrial hydrogen demand is low than when it is high. When the industrial hydrogen demand is low, PtG can supply hydrogen to meet the whole market demand. Due to its low marginal cost, PtG drives high-cost SMR out of the market, which brings hydrogen prices down. When the industrial hydrogen demand is high, however, PtG cannot supply the whole market demand. As a result, the high-cost SMR producer remains the price-setter, which makes that hydrogen prices are close to the Baseline level most time. When we increase the capacity of PtG from low to high, however, hydrogen prices can decrease strongly as now more often PtG becomes the price-setting supplier. From Table 7, we can see that PtG reduces the average annual hydrogen price significantly.
When PtG offers hydrogen to the industry, a supply of green hydrogen emerges. When the electrolysers buy green-electricity certificates in combination with their electricity, they are able to supply both hydrogen and green-hydrogen certificates. Hence, in the scenario of no PtG, the market for green certificates of hydrogen does not exist, but in the scenario with PtG, the market for green-hydrogen certificates is created. When there is no supply of green-hydrogen certificates, the price of green certificate of hydrogen is 20 Euro/MWh, which is the  Fig. 11. Share of various types of marginal (price-setting) electricity supplier with sector coupling.
(assumed) maximum willingness to pay for it. When there is oversupply, the price drops to 0. The price of green-hydrogen certificates decreases with the supply (see Fig. 15). As a result, PtG reduces the average certificates prices (see Table 7). In combination with the change prices, the mix of hydrogen production also changes. Fig. 16 shows that SMR hydrogen generation is replaced by PtG and this replacement effect is stronger in the scenario of low industrial hydrogen demand. Hence, when the industrial demand is relatively small, while the capacity of PtG is large, the supply of hydrogen by SMR can almost be fully replaced by hydrogen produced through electrolysis. How the change of gas price impacts the market share of types of hydrogen supply in case of sector coupling with high      industrial hydrogen demand is discussed in Appendix E.

Welfare impact of PtG
As the introduction of PtG affects both electricity and hydrogen markets, it also has an impact on welfare. The short-term welfare effects consist of changes in consumer surplus and producer surplus. The introduction of PtG has a positive short-term welfare effect, but this positive effect is outweighed by the annualized costs of the investments in PtG assets. Overall, the welfare effect is negative (see Fig. 17).
This welfare effect results, however, from different effects for the various groups of participants. In the model, we have defined the following groups: electricity producers, hydrogen producers, electricity consumers, and hydrogen consumers.
Among the group of electricity producers, those using renewable resources or hydrogen benefit, while those who use natural gas suffer. Importers suffer from PtG. The mechanism is the same as described in Section 5.1.2.
Among the group of hydrogen producers, PtG producers and hydrogen storage operators benefit from PtG, which is the same as in Section 5.1.2. The welfare effects for SMR producers result from a mixture of factors. In the baseline without PtG, SMR producer is the price-setter in the market, where the hydrogen price equals to the marginal cost of SMR. This leads to a zero profit for SMR producers. When PtG is introduced, generally, SMR produces less hydrogen due to the competition from PtG, and, consequently, the hydrogen price decreases. However, when electricity prices are very high, hydrogen-fired power producers could drive hydrogen prices up. When the hydrogen prices exceed the marginal costs of SMR, these producers realize a profit.
Among the group of consumers, hydrogen consumers benefit from PtG because PtG decreases the hydrogen price on average. In the scenario of low industrial hydrogen demand, electricity consumers benefit from PtG because PtG flattens the price-duration curve. In this case, they consume more electricity, and, as a result, the tax revenue from electricity also increases. In the scenario of high industrial hydrogen demand, electricity consumers suffer because PtG mainly drives the electricity price up. In this case, they consume less electricity, and, as a result, the tax revenue from electricity decreases. The consumers of green-hydrogen certificates benefit from PtG because only through PtG these certificates are offered. In contrast, the consumers of greenelectricity electricity suffer because PtG raises the demand for the green-electricity certificates resulting in higher prices.
After taking the annualized costs of the PtG assets into consideration, we find that PtG only has a tiny positive welfare effect in case of a low installed PtG capacity combined with a high hydrogen demand. In all other situations, the overall welfare effect is negative.

Costs of demand substitution to hydrogen
In the above analysis, we implicitly assumed that the extra demand for hydrogen from the industry occurred without any costs. In reality, however, this is not the case. As long as natural gas is less expensive than hydrogen, industries will continue to use gas. Since gas is more competitive in cost than hydrogen under current market conditions, the shift of energy consumption from gas to hydrogen leads to a welfare loss. Hence, one should account for the fact that the extra demand for hydrogen can only be created with policies boosting hydrogen demand such as a tax on gas consumption or a subsidy on hydrogen consumption. After all, the hydrogen demand is not just an extra demand, but it replaces the demand for gas. The costs of this substitution depend on the spread between the costs of consuming natural gas (based on the price of gas and the price of carbon), and the price of hydrogen, as well as on the amount of the gas consumption that is replaced by hydrogen. Using the above model results, we find that the welfare loss is 49 million Euro per year in the scenario of low industrial hydrogen demand and 493 million Euro per year in the scenario of high industrial hydrogen demand. When we take this welfare loss into account, PtG has a negative welfare effect in all circumstances.

Break-even CO2 price
The above welfare effects are based on exogenous values for CO2 emissions. These values are based on the actual daily carbon prices in 2019. The higher the carbon price, the lower the costs of the substitution from gas to hydrogen consumption. The costs of this substitution may even become negative when reducing carbon emissions is highly valued, which may also have as a result that the overall welfare effect of PtG turns positive. To fully assess the welfare effects of PtG, we calculate the break-even CO2 price, which is the price resulting in zero overall welfare effect. Since PtG has the largest welfare effect in the case of high industrial hydrogen demand, we focus on the case with high industrial hydrogen demand to find the break-even price of CO2 for PtG. Fig. 18 reports the net welfare of PtG with different carbon prices. We can see that a CO2 price between 650 and 700 euro/ton will make a high PtG capacity break even in a situation with high amount of renewables and high hydrogen demand.
The break-even carbon price is related to the assumptions regarding in particular the investment costs and the conversion efficiencies. The break-even CO2 price will be lower if the costs of PtG decrease or the efficiency of electrolysers increases. Table 8 lists the break-even CO2 price for different levels of cost reduction and efficiency enhancement. When the annualized costs of PtG reduce with 10 percent, the breakeven price reduces with about 150 euro/ton. When also the efficiency of electrolysers is increased with 10 percent points, then the break-even price reduce further with about 100 euro/ton. The break-even price is reduced to about 100 euro/ton when the annualized costs of PtG is reduced by 50 percent.

Conclusion
To reduce carbon emissions, governments are strongly promoting renewable energy. The increasing shares of renewable electricity generation will result in higher volatility in electricity supply due to the weather dependency of the generation by in particular wind turbines and solar PV. This higher volatile supply of electricity calls for more flexible options to manage the balance of the electricity grid. This need for more flexibility can be met by a variety of technologies, including Power to Gas (PtG). Therefore, it is expected that PtG will become more important in future electricity markets which are characterised by higher shares of renewable generation. In such markets, PtG can use electricity during hours of low prices to produce hydrogen which is stored for generating electricity when electricity prices are much higher. At the same time, hydrogen produced through PtG is increasingly seen as key energy carrier to reduce carbon emissions outside the electricity sector, such as in industry and transport. The question now is how this external demand for hydrogen affects the economic value of flexibility in the electricity market provided by PtG.
In this paper, we have analysed the economic potential of PtG by developing and applying a short-term partial equilibrium model of integrated electricity and hydrogen markets. Our approach differs from most other studies on the feasibility of PtG, as we model welfare effects by simulating the interaction between market participants in these markets. Our model has been calibrated to reflect the main characteristics of the Dutch electricity system, just to have reference to a real market. We used this model to analyse the welfare effects of PtG as both provider of flexibility to the electricity market and provider of hydrogen to other industries outside the electricity market.
We find that strongly increasing the share of renewable electricity from the current 10 GW to 60 GW installed capacity (which is consistent with the Dutch policy target for 2050), makes the electricity prices indeed much more volatile, even if we control for an increase in the share of flexible CCGT power plants. When we add PtG to this energy system, the price volatility is reduced, indicated by a less steep priceduration curve. In such a situation, there are less hours with very high power prices and also less hours with very low prices. These effects depend, as can be expected, on the magnitude of the installed PtG capacity. However, even when the PtG capacity is set at about 8 GW (which is way above the current Dutch ambition), the impact on prices is fairly modest. Hence, although PtG is technically a potential supplier of flexibility, the business case will not be sufficient which is due to its too small impact on the price-duration curve. Because of this relatively small impact, even when the size of PtG is fairly large, the overall welfare effects are negative.
As usual, there are winners and losers. To the group of winners belong the producers of renewable electricity (as they have less hours with low electricity prices), the consumers of electricity (as they have less hours of very high prices) as well as those consumers who consume renewable electricity (as they have lower prices for the green certificates which is due to the higher supply of renewable electricity). The group of losers are formed by the producers of gas-fired power plants (because they enjoy less hours of high prices) as well as the international traders (because there are less international price differences). Hence, while the overall welfare effect of PtG as a flexibility provider are negative, for some groups it may be beneficial giving them an incentive to promote this.
The above conclusions hold when we ignore any relationship with an external demand for hydrogen. As known, governments are also promoting the use of hydrogen in industrial process (as feedstock and for heating) and transport. We have also analysed how this external demand affects the economic value of PtG as a provider of flexibility in electricity markets.
We find that when there is a large demand for hydrogen from outside the electricity sector, the impact of PtG on the price-duration curve is reduced. When an industrial sector is demanding large amounts of hydrogen, as is expected to happen in many countries, then the PtG producers are more likely to supply hydrogen to these consumers instead of storing the hydrogen for electricity generation in hours of high prices, because they can realize higher prices. In such a scenario, the hydrogen price is determined by the more expensive supply from SMR producers, and as a result, the hydrogen-power producers cannot compete with gasfired power plants (as the costs of hydrogen are then always higher than the costs of gas). As a result, when there is a high industrial hydrogen demand, hydrogen will not be used to generate electricity. However, in such a scenario there will be more demand for electricity when prices are low, which means that there will be less hours with low electricity prices, resulting in a bit less steep price-duration curve.
Nevertheless, a high hydrogen demand by an industrial or transport sector has some positive welfare effects. This is due to the fact that PtG is able to provide less expensive hydrogen than SMR during hours when electricity prices are low. This is generally the argument used by governments to pursue this policy (EU hydrogen strategy, 2020). Also in this situation, there are winners and losers. The winners are, again, the producers of renewable electricity (because they experience less hours of low prices), the producers of PtG producers (because they can realize higher volumes and higher prices) as well as hydrogen consumers (because they have to pay lower prices compared to situation without PtG). One of the major losers consists of the (other) electricity consumers (because they have less hours of low prices). This also implies that in such a scenario, industrial users benefit at the expense of other electricity users.
The overall welfare effect, however, is negative when we take the annualized costs of the PtG assets into account. This negative effect becomes even larger when we control for the costs of creating the hydrogen demand. When industrial energy users are stimulated, for instance, through fossil-energy taxes, to switch from natural gas to hydrogen, then one should also take into account the costs and benefits of this substitution. After all, the industries are replacing a less expensive by a more expensive energy carrier. When we include these costs, then the conclusion is that the overall welfare effects of introducing PtG are negative. The required carbon price to neutralize this negative effect for a low level of PtG is about 150 euro/ton carbon and for a high level of PtG about 700 euro/ton, which is much higher than many other options for carbon reduction. As this carbon break-even price is much higher than the prevailing carbon price in for instance, the European Emissions Trading scheme, one must conclude that PtG is currently not a profitable solution, seen from a social-welfare perspective, to provide flexibility to electricity systems with high shares of renewables. These break-even values of the carbon price will, however, become lower when the required investment costs of PtG are significantly reduced or when the conversion efficiencies of electrolysers become much higher.
From this analysis, we conclude that although PtG installations are technically able to provide flexibility which is required by electricity systems characterised by large amounts of renewables, its economic value appears to be negative. This value is negatively affected by the presence of hydrogen demand from outside the electricity sector, which makes that hydrogen is less used to generate electricity during hours of scarcity. This is due to the fact that hydrogen demand from outside the electricity sector reduces the value of hydrogen as a provider of flexibility in the electricity market. The economic value of PtG can be turned positive by significantly reducing the fixed costs of the PtG installations, raising the conversion efficiencies and/or assuming much higher monetary values for the reduction in carbon emissions.  The problem is a generalized Nash equilibrium in the sense that each player maximizes its profit or utility given the relevant prices, while the equilibrium prices are determined by the clearing of all markets. In the equilibrium, no player has an incentive to change its decision because each one chooses the optimal decision given the market prices, while the decisions of all players satisfy the market clearing conditions in each market. If one player changes its decision, it will influence other players' decision via the linking constraints which connect different markets. This problem can be solved since that there are 22 variables and each of those variables is exactly associated with one condition. The above conditions form a Mixed Complementarity Problem (MCP), which can be solved by the General Algebraic Modeling System (GAMS) software.

Appendix C. Gas-fired generation capacity in various scenarios
In the scenario of low renewable electricity, the available electricity capacity is about 15,800 MW, which resembles the current Dutch centralized generation capacity. If we want to keep the available electricity capacity (after controlling for capacity factors related to weather circumstances) unchanged when we increase the capacity of renewable electricity, we must reduce the capacity of gas-fired electricity producers. Such a symmetric change in the composition of the electricity generation has, however, strong effects on the electricity price. To illustrate how the change of capacity composition affects the electricity price, we show a price-duration curve in which the installed capacity of renewable electricity producers is increased from 10,000 MW to 35,000 MW and the installed capacity of gas-fired electricity producers is decreased from 10,000 MW to 3,000 MW. Fig. C.1 shows how this change in the generation mix affects the price-duration curve.
We can see that in the new situation, the electricity price is much higher than before most of the time which is due to capacity scarcity. After all, when the renewable capacity is not able to generate because lack of wind, much smaller amounts of gas-fired plants and import become responsible for the full supply. The scarcity in such situations results in so-called scarcity prices, which gives gas-fired producers incentives to expand their capacity. Hence, from a dynamic perspective, it is not reasonable that the gas-fired generation capacity will be reduced to the same extent as the renewable capacity will increase. Economically, the electricity firms have an incentive to invest relatively more in gas-fired power plants. So, when we increase the capacity of renewable electricity dramatically, the capacity of gas-fired electricity may still remain at a high level. For this reason, in the scenario of high renewable electricity, we choose 60,000 MW as the capacity of renewable electricity (which is related to the Dutch policy target for 2050) combined with 6,000 MW as the capacity of gas-fired electricity. This results in a price-duration curve with much less hours with high prices.

Appendix D. Impact of gas price on the price duration curve of electricity
In the scenarios and variants used in this paper, we have assumed similar exogenous values for the gas price, as the focus of the paper is on the impact of PtG and hydrogen demand on electricity prices. To show the impact of the gas price assumptions, we have conducted a sensitivity analysis where we show the price duration curve with high PtG and high renewables without external hydrogen demand (based on Fig. 5) for three gas-price scenarios: a) as assumed in the baseline, b) 10% higher prices every day and c) 10% lower prices every day. The results in Fig. D.1 show that compared to the baseline, higher gas prices lead to higher electricity prices, vice versa, which is of course due to the fact that the marginal costs of gas-fired power plants are related to the gas price. We also see that when the electricity price is very high, the gas price hardly has any influence, which is due to the fact that in situations of high demand (or low supply) (i.e. in case of a tight electricity market), the consumers' willingness-to-pay determines the electricity price. The gas price is neither relevant when the electricity price is very low, because in hours of high supply (or low demand), the marginal cost of renewable producers determines the electricity price. From this, we conclude that the assumptions regarding the daily gas prices hardly affect the electricity price levels which are relevant for the business case of PtG.  Price-duration curve of electricity with low renewable (baseline) and more renewable plus a similar decrease in gas-fired plant capacity (i.e. equal total amount of generation capacity).

Appendix E. Impact of gas price on the market share of different types of hydrogen supply
In the market for hydrogen, PtG has to compete with SMR-CCS hydrogen. The competitive positions of both technologies depend on their respective marginal costs, the installed capacity, and the demand for hydrogen. Fig. 16 shows the market shares of both technologies in various scenarios. It shows that the SMR-CCS remains the price-setting supplier of hydrogen when the external (industrial) demand for hydrogen is high. In a scenario, however, when this demand is low (compared to the amount of installed PtG capacity), hydrogen prices go down and SMR-CCS is less competitive. These relationships are of course subject to the exogenous values of the gas price. To show this effect, we have conducted a sensitivity analysis where we show the impact of gas price on the market shares of SMR-CCS and electrolysis in the supply of hydrogen. We have three gas-price scenarios: a) as assumed in the baseline, b) 10% higher prices every day, and c) 10% lower prices every day. The results in Fig. E.1 show that compared to the baseline, higher gas prices lead to smaller market share of SMR-CCS, vice versa, which is of course due to the fact that the marginal costs of SMR-CCS are related to the gas price. As we see from the figure, a 10% change of the gas price only changes the market share of SMR-CSS by 1%. From this, we conclude that the assumptions regarding the daily gas prices hardly affect the competitive positions of SMR-CCS and PtG. The explanation for this minor effect of the gas price on the market shares is that both the lowest and the highest electricity prices are hardly related to the marginal costs of gas-fired power plants, and, hence, they do hardly affect the market position of electrolysis hydrogen.

Appendix F. Scenario of low renewable energy
In the current Dutch energy market, the production of renewable energy and in particular the production and use of hydrogen are fairly low [25]. Hence, our scenario of low renewable energy is more or less in line with the current situation. As shown in this paper, PtG brings more welfare when it is used with sector coupling than without, because users outside the electricity sector benefit from lower-priced hydrogen. In the paper, it is also shown that the scenario with high level of renewables is more beneficial for PtG than a scenario with low levels of renewables. Just to illustrate this, Figs. F.1, F.2, and F.3 show the impact of PtG on the prices of electricity and hydrogen as well as the welfare effects respectively in a scenario with low renewables and low hydrogen demand. It appears that the price-duration curves of both electricity and hydrogen are hardly affected by the introduction of PtG, which implies that the overall welfare effects are strongly negative in such a scenario. Hence, without strongly volatile electricity prices or high hydrogen demand from outside the electricity sector, investments in PtG hardly generate benefits.  X. Li and M. Mulder