Abstract
Increasing penetration of renewable energy generation poses a challenge to power system inertia adequacy. It is vital to provide long-term incentive signals to induce a generation mix with adequate inertia supply. However, existing literature rarely studies inertia incentive mechanisms or considers inertia constraints when making generation investment decisions. Thus, we propose an inertia market to quantify the value of inertia and to remunerate inertia provision. To examine the impacts of the inertia market on generation mix, we then propose a stochastic bilevel generation investment equilibrium model that depicts a multi-leader and multi-follower Stackelberg game. The lower level of the model considers the proposed inertia market, along with the energy, reserve, and capacity markets. The upper level considers multiple profit-maximizing strategic producers, and each producer is able to build gas-fired generators, wind generators, and energy storages. Numerical experiments demonstrate that a generation mix with adequate inertia supply can be induced with the proposed inertia market whereas there can be inertia shortage without the inertia market. Interestingly, considering carbon taxes, it is more cost-competitive to invest in wind resources with virtual inertia facilities than to substitute wind resources by thermal generators. Correspondingly, the introduction of an inertia market does not significantly reduce wind generation shares but boosts virtual inertia facility penetration. Our findings imply a future power system powered by fully decarbonized power resources with adequate inertia.
A fundamental characteristic of decarbonization in the electricity industry is that rapidly-growing renewable energy resources are replacing traditional thermal power plants. High penetration of renewable generation, inherently featuring uncertainty and low inertia [
In terms of industrial practice, the only inertia market was run in Australia, but it was canceled later [
Here, we propose an auction-based inertia market and derive the corresponding generation mix via a game-theoretical generation investment model. The main contributions of our work are as follows.
1) This paper proposes an inertia market to remunerate inertia provision and to guarantee long-term inertia adequacy. Inspired by existing capacity markets, we present an auction-based inertia market, where all generation assets receive payments based on the marginal inertia price. The proposed inertia market can be incorporated as a new product in existing electricity markets. In this paper, the energy and reserve markets are cleared jointly and hourly whereas the capacity and inertia markets are cleared annually.
2) To examine the impacts of the proposed inertia market on generation mix, this paper proposes a bilevel stochastic generation investment equilibrium model. The upper-level problem yields investment decisions of each strategic producer aiming to maximize its profit. Multiple generation technologies are equally accessible to each strategic producer, including gas-fired generation, wind generation, and grid-level energy storages. The lower-level problem clears the energy, reserve, capacity, and inertia markets.
Reference | Model type | Incentive product |
---|---|---|
[ | Single-level equilibrium model | Capacity market |
[ | Centralized least-cost planning | Scarcity pricing and capacity payments |
[ | Bilevel equilibrium model | Energy, reserve, and capacity markets |
[ | Bilevel equilibrium model | Energy and gas markets |
[ | Bilevel equilibrium model | Energy, reserve, and capacity markets |
[ | Single-level equilibrium model | Energy-only market |
[ | Bilevel equilibrium model | Energy and capacity payments |
[ | Bilevel equilibrium model | Energy market |
[ | Bilevel equilibrium model | Energy market (day-ahead market and long-term contract) |
[ | Single-level equilibrium model | Energy market |
Proposed model | Bilevel equilibrium model | Energy, reserve, capacity, and inertia markets |
The rest of this paper is organized as follows. Section II gives a detailed market design of the proposed inertia market. In Section III, we present the mathematical formulation of the proposed stochastic generation investment equilibrium model. In Section IV, we give the solution techniques. In Section V, we run case studies on a 5-bus system and the IEEE 118-bus system and analyze the impact of the inertia market on the generation mix. We give the conclusions in Section VI.
We propose an auction-based inertia market. The inertia market aims to provide a clear incentive for producers to invest in generators with large inertia or inertia augmentation technologies. On top of this, the market can provide upfront payments to producers and stabilize the volatile incomes of generators. This market mechanism is equally applied to all types of generation technologies. The consumers pay for the inertia provision that keeps the system frequency stable.
In this section, we introduce the inertia market. And then, we give the inertia market clearing algorithm.
We first briefly introduce the concept of inertia and the inertia response period. Inertia is a kind of energy to hinder any changes from current states. As a response to a sudden power imbalance, it reduces the rate-of-change-of-frequency (RoCoF) to prevent frequency from dropping to an out-of-control level before the primary frequency response is activated. In a grid with all generators at a nominal frequency, inertia is usually quantified by normalized inertia constant H (s) due to ease of calculation [
Inertia response is the period up to a few seconds in response to a sudden power imbalance before primary frequency response, as shown in

Fig. 1 System frequency response after a contingency.
Inertia, as a product to be auctioned, is quantified by the inertia constant H. This product has the following features.
1) For each generator, the level of inertia depends on its technical parameter: the inertia constant H. Once a generator has been built, its inertia constant is fixed and is irrelevant to its dispatched output. Also, once it is connected to the grid, its inertia can be absorbed by the power system.
2) For a power system, the overall system inertia is a sum of the inertia of all generators, weighted by rated power of each machine [
3) The overall inertia demand is inelastic and is related to the RoCoF requirement of the grid. Usually, large-scale grids can withstand a relatively high RoCoF due to their adequate flexibility to better conduct primary frequency regulation.
In terms of the inertia contributions from the demand side, this paper ignores them, leading to a conservative generation mix with adequate inertia provision. This is because the inertia provisions from the demand side have various types due to the complexity of load types. It is hard to model them, especially in a planning-scale horizon [
This paper ignores PV installations as sources of inertia. Although both wind generation and PV equipped with energy storages can theoretically provide virtual inertia via advanced converter control, we usually only consider wind turbines to provide inertia. This is because wind turbines are rotational masses, resembling rotators of thermal generators, with kinetic energy to be released during frequency deviation. Compared with static masses such as PV, wind generation has more stored energy to contribute to frequency response. Also, besides the provision of virtual inertia, converters of PV are more suitable to provide many other functions to help stabilize the grid, such as voltage control through reactive power. Therefore, providing inertia might not be the best application of converters for PV [
As sellers, different types of generators equally participate in the inertia market. Not only thermal generators with conventional rotating inertia, but also renewable generators and energy storage systems with virtual inertia facilities can participate.
As buyers, the independent system operator (ISO) or load-serving entities act on behalf of electricity consumers to pay for the inertia provision. This payment compensates for the installed capacity of thermal generators and the technical improvement of renewable generators for virtual inertia provision.
The regulator defines the total required system inertia based on a desired RoCoF target. The RoCoF target varies across power systems according to stability criteria and adequacy of flexible resources [
The procedure is based on a pool-based market where inertia provisions are auctioned. In this way, the inertia market determines the clearing price and allocation of inertia products among different generators. The mechanism is implemented as follows.
1) Initially, the regulator collects some basic parameters for the auction: ① the inertia constant and installed capacity of each generator; ② the minimum RoCoF and corresponding inertia requirement of the grid; and ③ the frequency of the auction, typically on an annual basis.
2) The generators submit a price-quantity pair per bid to the inertia auction. The quantity in the bid is the maximum inertia contribution that the generator can commit. The price represents the minimum fee that the generator is willing to accept for inertia provision.
3) The price of the last accepted bid determines the clearing price in the inertia market, as shown in

Fig. 2 Illustrative auction procedure and surplus of sellers and buyers.
4) A generator with an accepted bid (s) accepts payment ($). All of this procedure applies once in each auction and the payment covers the following whole year.
In this section, we form a bilevel model for each strategic producer. The upper-level problem maximizes the profit of a producer and determines the optimal investment decisions while anticipating the revenues from multiple markets. The lower-level problems represent the market clearing processes, taking the upper-level investment decisions as given parameters. Each producer has its bilevel model to maximize its profits while sharing the same lower-level constraints.
The objective function (1) aims to minimize the investment cost and maximize the profits from the energy, reserve, capacity, and inertia markets for a producer. We choose several representative scenarios to mimic the variability of renewable generation and demand. The energy and reserve markets are cleared hourly, resulting in an hourly price. The capacity market and inertia market are cleared annually and each of them results in a single annual price.
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
Constraint (7) represents that the capacity investments for thermal units, wind resources, and storages are non-negative. Constraint (8) sets the energy-to-power ratio of storages. Constraints (9) and (10) represent that the offers of the candidate and existing units in all markets are non-negative, respectively. Equations (
This paper also considers tie-breaking constraints for energy, reserve, capacity, and inertia markets when different producers offer the same price. For simplification, we incorporate tie-breaking constraints in the upper-level constraints to guarantee the linearity of the lower-level problems, as shown in [
The variable sets of the upper-level problem consist of the upper-level variable set and lower-level variable sets , , and . The upper-level variables are those in the set ={, , , , , , , , , , , , , , , , , , , , , , , }. The lower-level variables related to the energy and reserve markets are those in the set ={, , , , , , , , , , , , , , , , , }. The lower-level variables related to the capacity market are those in the set ={, , , , , , , }. The lower-level variables related to the inertia market are those in the set ={, , , , , , }.
The detailed formulations of energy and reserve markets are given in Appendix A. Note that this paper regards thermal generators, wind generators, and storages as price-makers.
The detailed formulations of the capacity market are given in Appendix A as well. Note that we evaluate the capacity provision of different generation technologies by their firm capacity.
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
(23) |
(24) |
(25) |
In this section, we give solution techniques for the proposed generation investment equilibrium model. Each producer acts given other producers’ investment decisions. An equilibrium among multiple bilevel problems is achieved if no producer alters its decision [
First, we transform each bilevel model into a single-level problem by replacing the lower-level problems with their primal-dual optimality conditions [
Notably, it is possible that there is no equilibrium to the problem and that the proposed algorithm cannot converge. Facing multiple markets, producers can make profits from the energy market, reserve market, capacity market, and inertia market. This complexity of income structure increases the nonlinearity of the problem, which can lead to nonexistence of equilibrium and nonconvergence. With numerical observation, the solution might oscillate in some cases. Although this does not sound elegant in theory, it somehow reflects the reality. In many electricity markets in the world, the generation mix and overall generation capacity usually oscillate around the ideal point. In some years, generation capacities are too adequate, and suppliers decide to curtail generation capacity, leading to underinvestment. Consequently, in the next years, generation capacities are inadequate, and suppliers decide to increase generation capacity, leading to overinvestment in future. If the solution oscillates, the stopping criterion can be terminated at a certain number of iterations [
Here, we introduce the method for choosing an initial value of the diagonalization approach. Different initial values will lead to different solutions. To obtain an appropriate and reasonable solution, we first find the least-cost capacity mix from the aspect of central planning. Then, taking this solution as an initial value, we use the proposed model to obtain the equilibrium.
The problem may also have multiple equilibria. Among different equilibria, we can set different objective functions to pick one equilibrium, such as one that maximizes social welfares or maximizes profits of all producers [
In this section, we run numerical experiments of six scenarios: ① equilibrium model with inertia market but without virtual inertia; ② equilibrium model with inertia market and with virtual inertia; ③ equilibrium model without inertia market; ④ centralized model with inertia requirements but without virtual inertia; ⑤ centralized model with inertia requirements and with virtual inertia; and ⑥ centralized model without inertia requirements. We use an illustrative example based on a 5-bus system and a large system example with the IEEE 118-bus system. All examples have been implemented and solved using GAMS with default GAMS options. We use dedicated MPEC solvers embedded in GAMS, NLPEC, which can give locally optimal solutions to MPEC problems. The simulations are carried out on an Intel Core i5 CPU with 3.20 GHz and 8 GB of RAM.
The model time horizon is one year, and the investment costs are annualized to match the time horizon. We choose four representative days (, ) to account for the variability of renewable generation and demand. Each representative day consists of 24 hours and represents one season. To highlight the impacts of the inertia market, the numerical experiment omits transmission constraints and demand response. And we only consider spinning reserves and ignore other reserve types.
The topology of the 5-bus system is shown in

Fig. 3 Topology of 5-bus system.
Technology | Investment cost | Operation cost ($/MWh) | Ramp rate (p.u.) | Inertia constant (s) |
---|---|---|---|---|
Large thermal | 34267 | 4+CT | 1/12 | 5.0 |
Small thermal | 18933 | 7+CT | 2/15 | 4.3 |
Wind | 35034 | 4.9 | 0 | |
Storage | 18730 (power), 2000 (energy) | 1 | 2/15 | 0 |
Existing thermal 1 | 4.8+CT | 1/12 | 4.3 | |
Existing thermal 2 | 5+CT | 1/12 | 4.5 | |
Existing thermal 3 | 5.1+CT | 1/12 | 4.8 |
Three existing thermal generators are located in bus 1, with installed capacities at 80 MW, 80 MW, and 120 MW, respectively. The planning margin reserve for the capacity of the system is 13.75%. The inertia requirement of the system is no less than 3.125 s. The price cap for the energy market is 400 $/MWh and the bid of capacity demand is 8600 $/MWh [
The sum of these two prices is 9000 $/MWh, representing the value of lost load (VOLL) [
The rationale behind the assumed capacity offer costs is explained as follows.
1) Candidate small thermal generators are those peak units. Their operation costs decide the marginal price of the energy market during non-scarcity hours. During scarcity hours, the electricity price will be the price cap, i.e., 400 $/MWh, which is higher than their operation cost, leading to profits. However, in a whole year, the total scarcity hours are few, usually less than 20 hours, due to the reliability requirement of the power system. Therefore, candidate small thermal generators can hardly have profits in the energy market. They rely on the capacity market to recover nearly all their investment cost. As a result, their offers in the capacity market are their investment costs.
2) Candidate large thermal generators can recover part of their investment costs by profits from the energy market. The proportion of cost recovery from the energy market is assumed to be 80% [
3) Existing thermal generators have been working for many years. We assume that they have recovered all their investment costs. Therefore, they offer zero in the capacity market.
4) Wind generators offer zero in the capacity market. Wind generators can recover nearly all their investment costs in the energy market due to their cheap operation costs. Moreover, due to the variability and intermittence of wind generation, ISO usually does not regard them as capacity providers or take their firm capacity to be the capacity provision.
5) Storages offer zero in the capacity market. The generation capacity of storage depends on their stored energy, i.e., state of charge (SOC). Analogous to wind generators, ISO usually does not regard them as capacity providers or take their firm capacity to be the capacity provision.
Per

Fig. 4 Generation mix for 5-bus system of equilibrium model and centralized planning model. (a) Equilibrium model with inertia market but without virtual inertia. (b) Equilibrium model with inertia market and virtual inertia. (c) Equilibrium model without inertia market. (d) Centralized planning model with inertia requirements but without virtual inertia. (e) Centralized planning model with inertia requirements and virtual inertia. (f) Centralized planning model without inertia requirements.
Per
We compare the total cost, including the investment cost and operation cost, across six cases. We use those cases without inertia constraints as benchmarks. The total costs increase when considering inertia constraints with virtual inertia provision. The reason is that inertia provision, similar to reserve provision, is an additional requirement for the power system with high penetration of renewable generation. The total costs increase further when considering inertia constraints but without virtual inertia. The reason is that, with carbon taxes, it is more cost-competitive to invest wind resources with virtual inertia facilities than to substitute wind generators with thermal generators.
Compared with the centralized planning model, the market-based model also yields higher shares of thermal generation. This is because the reserve, capacity, and inertia markets favor thermal generators due to their firm ramping ability, reliable capacity, and high inertia constants. Some researchers criticize that these markets discriminate against renewable resources and may hinder decarbonization. However, we argue that reserve service, reliable firm capacity, and adequate inertia are crucial to the long-term reliability of the electricity supply. To achieve the low-carbon targets, renewable generations are encouraged to participate in these markets such as by equipping virtual inertia devices.
It is also noted that storages only show up in
The overall system inertia of the six cases is shown in

Fig. 5 Overall system inertia of six cases.
The energy and reserve prices for cases 1, 2, and 3 are shown in

Fig. 6 Energy prices and reserve prices for cases 1-3. (a) Energy prices. (b) Reserve prices.
As observed from
As observed from
The annual payment and the average payment are listed in
Case | Annual payment ($) | Average payment ($/MWh) | ||||||||
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Energy | Reserve | Capacity | Inertia | Total | Energy | Reserve | Capacity | Inertia | Total | |
1 |
1.466×1 |
4.193×1 |
4.992×1 |
3.125×1 |
1.520×1 | 33.668 | 0.096 | 1.147 | 0.007 | 34.918 |
2 |
1.303×1 |
4.604×1 |
4.992×1 |
3.125×1 |
1.358×1 | 29.936 | 0.106 | 1.147 | 0.007 | 31.196 |
3 |
1.476×1 |
1.961×1 |
4.992×1 |
1.546×1 | 33.910 | 0.450 | 1.147 | 35.507 |
Consistent with the trends of prices for different products, case 3 has the highest energy payment while case 2 has the lowest one. For reserve payment, case 3 has the highest payment while case 1 has the lowest one.
For the composition of average payment, energy fee is the majority, accounting for around 96% of the total payment. Capacity fee accounts for around 3.4% of the total payment. The sum of reserve and inertia fees hold slight proportions, with around 0.6%.
We conduct numerical experiments on the IEEE 118-bus system to verify the scalability of the proposed model. The initial parameters are from MATPOWER 7.1. Other settings of parameters keep the same as those in the 5-bus system. All parameters can be found in [
The generation mix of different cases for the IEEE 118-bus system is shown in

Fig. 7 Generation mix for IEEE 118-bus system of equilibrium model and centralized planning model. (a) Equilibrium model with inertia market but without virtual inertia. (b) Equilibrium model with inertia market and virtual inertia. (c) Equilibrium model without inertia market. (d) Centralized planning model with inertia requirements but without virtual inertia. (e) Centralized planning model with inertia requirements and virtual inertia. (f) Centralized planning model without inertia requirements.
In this paper, we propose an auction-based inertia market mechanism and present a generation investment equilibrium model to examine the impacts of the inertia incentives on the generation mix. Our findings are as follows. First, it is necessary to incorporate inertia constraints and design an inertia remuneration mechanism to ensure inertia adequacy with high shares of renewable generation. Second, with inertia requirements but without virtual inertia provision, a sizable share of wind generators has to be replaced by thermal generators to ensure inertia adequacy. Third, with inertia requirements and virtual inertia provision, thermal generators only replace 0%-2% shares of wind generation. Fourth, considering carbon taxes, it is more cost-competitive to invest wind resources with virtual inertia facilities than to substitute wind generators with thermal generators. The findings imply that, by increasing virtual inertia facility penetration, a power system with adequate inertia is possible and cost-effective with high shares of renewable generation.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Sets |
—— | Set of buses adjacent to bus n | |
, , | —— | Sets of candidate thermal units, wind units, and storages |
, , | —— | Sets of existing thermal units, wind units, and storages |
, , | —— | Sets of candidate thermal units, wind units, and storages located at bus n |
, , | —— | Sets of existing thermal units, wind units, and storages located at bus n |
—— | Set of buses | |
—— | Set of upper-level variables | |
, , | —— | Sets of lower-level variables, including those in energy and reserve markets, capacity market, and inertia market |
—— | Set of generators of producer y | |
—— | Set of capacity demand curve segments | |
—— | Set of nodes | |
—— | Set of stochastic scenarios | |
—— | Set of time intervals | |
B. | —— | Parameters |
, | —— | Ratios of firm power to rated power of candidate wind unit w and storage b in capacity market |
, | —— | Ratios of firm power to rated power of existing wind unit w and storage b in capacity market |
—— | Time duration of each interval (hour) | |
, | —— | Charging and discharging efficiencies of candidate storage b |
, | —— | Charging and discharging efficiencies of existing storage b |
—— | Probability of scenario s | |
—— | Power-to-energy ratio of storage (hour) | |
—— | Number of days in a year represented by time t | |
—— | Susceptance of transmission line (n,m) (p.u.) | |
—— | Bid of capacity demand for segment i ($/()) | |
, | —— | Marginal operation costs of candidate thermal unit g and wind unit w ($/MWh) |
, | —— | Marginal operation costs of existing thermal unit g and wind unit w ($/MWh) |
, | —— | Marginal discharging and charging costs of candidate storage b ($/MWh) |
, | —— | Marginal discharging and charging costs of existing storage b ($/MWh) |
, | —— | Offers of candidate thermal unit g and storage b for reserve market ($/MWh) |
, | —— | Offers of existing thermal unit g and storage b for reserve market ($/MWh) |
—— | Load demand of bus n at time t for scenario s (MW) | |
—— | The maximum capacity demand for segment i (MW) | |
—— | Rated capacity of existing storage b (MWh) | |
—— | Capacity of transmission line (n,m) (MW) | |
, , | —— | Inertia constants of candidate thermal unit g, wind unit w, and storage b (s) |
, , | —— | Inertia constants of existing thermal unit g, wind unit w, and storage b (s) |
—— | Inertia requirement (s) | |
, , | —— | Annual investment costs of thermal unit g, wind unit w, and storage b () |
—— | Annual investment cost of storage b ( | |
, , | —— | Rated power of existing thermal unit g, wind unit w, and storage b (MW) |
—— | Penalty price for load shedding ($/MWh) | |
—— | Ratio of scheduled power to rated power of candidate wind unit w at time t for scenario s | |
—— | Ratio of scheduled power to rated power of existing wind unit w at time t for scenario s | |
, | —— | 10-min ramp rates of candidate thermal unit g and storage b (p.u.) |
, | —— | 10-min ramp rates of existing thermal unit g and storage b (p.u.) |
—— | Reserve requirement at time t (MW) | |
C. | —— | Variables |
—— | Voltage angle of bus n at time t for scenario s (MW) | |
, , | —— | Electricity prices of node n with thermal unit g, wind unit w, and storage b at time t for scenario s ($/MWh) |
—— | Reserve price at time t for scenario s ($/MWh) | |
, | —— | Capacity and inertia prices (, )) |
—— | Investment cost of all candidate units ($) | |
, , | —— | Scheduled capacities of candidate thermal unit g, wind unit w, and storage b (MW) |
, , | —— | Scheduled capacities of existing thermal unit g, wind unit w, and storage b (MW) |
—— | Capacity demand for segment i (MW) | |
—— | Newly-built capacity of storage b (MWh) | |
, | —— | States of charges of candidate and existing storage b at time t for scenario s (MWh) |
, | —— | Profits from energy and reserve markets for scenario s ($) |
, | —— | Profits from capacity and inertia markets ($) |
, , | —— | Scheduled inertia of candidate thermal unit g, wind unit w, and storage b (s) |
, , | —— | Scheduled inertia of existing thermal unit g, wind unit w, and storage b (s) |
, , | —— | Inertia provisions of candidate thermal unit g, wind unit w, and storage b (s) |
, , | —— | Inertia provisions of existing thermal unit g, wind unit w, and storage b (s) |
—— | Load shedding of node n at time t for scenario s (MW) | |
, | —— | Electricity offers of candidate and existing thermal unit g at time t for scenario s ($/MWh) |
, , | —— | Capacity offers of candidate thermal unit g, wind unit w, and storage b ( |
, , | —— | Inertia offers of candidate thermal unit g, wind unit w, and storage b |
, , | —— | Capacity offers of existing thermal unit g, wind unit w, and storage b ( |
, , | —— | Inertia offers of existing thermal unit g, wind unit w, and storage b |
, , , | —— | Scheduled power of candidate thermal unit g, wind unit w, and storage b (discharging and charging) at time t for scenario s (MW) |
, , , | —— | Scheduled power of existing thermal unit g, wind unit w, and storage b (discharging and charging) at time t for scenario s (MW) |
, | —— | Scheduled reserves of candidate thermal unit g and storage b at time t for scenario s (MW) |
, | —— | Scheduled reserves of existing thermal unit g and storage b at time t for scenario s (MW) |
, , | —— | Newly-built power of thermal unit g, wind unit w, and storage b (MW) |
Appendix
In the appendix, we give the detailed formulations of problems described in Section III-B, including energy and reserve markets and the capacity market.
(A1) |
(A2) |
(A3) |
(A4) |
(A5) |
(A6) |
(A7) |
(A8) |
(A9) |
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(A36) |
(A37) |
(A38) |
(A39) |
(A40) |
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