# Designing an Incentive Contract Menu for Sustaining the Electricity Market

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Description of the Menu of Incentive Contracts

#### 2.1.1. Target GenCo and Contracted Generating Companies

^{0}as set of all possible combinations of target GenCos. For each a

^{0}∈ A

^{0}, ${a}^{0}=({a}_{1}^{0},{a}_{2}^{0},\mathrm{...},{a}_{m}^{0})$, where ${a}_{i}^{0}$ represents whether GenCo i is chosen as a target GenCo or not. Further, we define $I=\left\{k|{a}_{k}^{0}=1,k\in M\right\}$ as a set of target GenCos.

_{i}represents whether GenCo i decides to accept the menu of the incentive contracts or not. If a

_{i}= 0, it is “not”, and a

_{i}≠ 0 is “yes”. If a

_{i}≠ 0, a

_{i}= k, k ∈ I, meaning GenCo k accepts the incentive contract and becomes the target GenCo. The GenCos with a

_{i}≠ 0 are termed as contracted GenCos.

#### 2.1.2. Bidding Curve and Market Clearing Price

_{it}= α

_{it}+ β

_{it}q

_{it}, where α

_{it}and β

_{it}are the bidding coefficients of GenCo i at time t. Here p

_{it}and q

_{it}respectively, represent the bidding price and the bidding power output of GenCo i at time t.

Blocks | Price ($/MWh) | Power output level (MW) |
---|---|---|

Block 0 | 10 | 30 |

Block 1 | 15 | 60 |

Block 2 | 20 | 90 |

_{it}= α

_{it}+ β

_{it}q

_{it}, and the electricity demand at time t is D

_{t}. The MCP at time t, which is denoted as p

_{t}

_{,}can be obtained by solving the following power balance equation:

_{t}= α

_{it}+ β

_{it}q

_{it}, for i = 1,2 …, m

#### 2.1.3. Menu of the Incentive Contracts

_{i}, β

_{i}, π

_{i}), where α

_{i}and β

_{i}represent the thresholds of bidding coefficients respectively, and π

_{i}is the relevant reward for meeting the incentive contract. Though each incentive contract is originally tailored to the rational and incentive-compatibility constraints of a certain target GenCo, it is also expected that these contracts are designed appropriately to motivate non-target GenCos to participate in the incentive program.

_{i0}and β

_{i0}, and the amount of the reward is π

_{i0}by calculating the target GenCo’s individual rationality and incentive compatibility conditions. This contract could be expressed by the triplet (α

_{i0}, β

_{i0}, π

_{i0}) which specifies the reward and the obligation associated with the contracted GenCo: if the dispatched power output of the contracted GenCo during the contract period is always greater than the required level, which is prescribed as $\underset{\_}{q}=({p}_{t}-{\mathsf{\alpha}}_{i0})/{\mathsf{\beta}}_{i0}$ with p

_{t}being the MCP at time t, a reward of π

_{i}

_{0}would be received at the end of the contract period.

_{i}, β

_{i}, π

_{i}) are further incorporated into (AL, B, π), where AL, B, π are the vectors of α

_{i}, β

_{i}, π

_{i}, respectively. Hence the menu of the incentive contracts could be concisely specified in the form of (AL, B, π).

#### 2.2. Scenario-Based Approach

#### 2.3. The Workflow of Multi-Agent System

**Figure 2.**Flow chart of multi-agent system (MAS) scheduling mechanism (note: Y = yes and N = no). ISO: independent system operator.

_{it}= α

_{it}+ β

_{it}q

_{it}. After the ISO receives all the bids from the participants, the relevant information is aggregated and stored in a central repository. Based on the estimated hourly electricity demand of the next day, the ISO decides the unified hourly MCP of the next day, and announces the hourly power output schedule of individual GenCos for the next day.

## 3. Multi-Agent Stackelberg Game Model

#### 3.1. Model Assumption and Description

_{t}denotes a realized scenario in Λ at time t. In addition, Λ

^{B}is used to represent a set of bad scenarios.

_{i,t}= 1) or place a normal bid (i.e., b

_{i,t}= 0), where b

_{i,t}is the bidding strategy of GenCo i at time t. The coefficients for different bidding options are defined as follows:

_{i}

^{c}, β

_{i}

^{c}are parameters of the normal bidding curve for GenCo i, and α

_{i}

^{h}, β

_{i}

^{h}are parameters of the corresponding high bidding curve. Obviously, if a GenCo has accepted an incentive contract, we have b

_{it}= 0, α

_{i,t}= α

_{i}

^{c}, β

_{i,t}= β

_{i}

^{c}for $t\in \tilde{t}$.

#### 3.2. Single-Period Decision-Making Model of GenCo

_{t}) represents the probability of λ

_{t}, and ${\mathsf{\pi}}_{k}$ is the reward specified in the incentive contract for target GenCo k.

_{i,t}in a scenario λ

_{t}. c

_{i}

_{1}and c

_{i}

_{2}are cost coefficients of GenCo i. b

_{t}

^{i,c}= {b

_{1,t}, b

_{2,t}, b

_{i-}

_{1,t}, 0, b

_{i+}

_{1,t},…,b

_{m,t}} represent a bidding combination when GenCo i places a normal bid. Note that pos

_{j}(b

_{j,t}) is the probability for GenCo j to take action b

_{j,t}. Here p(λ

_{t}, b

_{t}(a)) is the expected electricity price when the bidding combination of GenCos is b

_{t}(a) in scenario λ

_{t}. Note that p(λ

_{t}, b

_{t}(a)) is p(λ

_{t}, b

_{t}

^{i,C}) when GenCos’ bidding action is b

_{t}

^{i,C}in scenario λ

_{t}, and is p(λ

_{t}, b

_{t}

^{i,}

^{H}) when GenCos’ bidding action is b

_{t}

^{i,H}in scenario λ

_{t}.

_{k}, β

_{k}, π

_{k}) triplet, we have:

#### 3.3. Optimization Problem of Independent System Operator

^{0}) as well as designing the incentive menu for attracting contracted GenCos, so that its objectives could be optimized. Since an incentive contract, which is specified in the triplet form of (α

_{i}, β

_{i}, π

_{i}), is dependent upon a

^{0}, how to target suitable GenCos is the key to designing an optimal incentive menu of contracts. Hence the ISO’s initial decision is to choose optimal a

^{0}, so as to minimize the total cost with certain price stability.

^{0}. A two-level programming model is proposed to facilitate ISO’s decision-making. The sub-problem at the first level enables the ISO to minimize the total cost with price stability by finding an optimal value of a

^{0}. The sub-problem at the second level can be treated as GenCos’ reaction model upon the release of the menu of incentive contracts from the first level decision:

_{i}is obtained by solving follows:

^{0}) is the total power purchasing cost when the combination of the target GenCos is a

^{0}, and $\mathsf{\delta}$ is a balance parameter. EP(a

^{0}) is the expected electricity price when the combination of the target GenCos is a

^{0}, and EP* is the best expected price. BP(a

^{0}) is the variance of mean price versus EP* when the combination of the target GenCos is a

^{0}. Here P(λ

_{t},a) is the expected MCP when the combination of contracted GenCos’ is a in scenario λ

_{t}. The MCP is p(λ

_{t}, b

_{t}(a)) when the bidding behavior of GenCos is b

_{t}(a) in scenario λ

_{t}.

^{0}. Equations (22) and (23) are the mathematical descriptions of the cost and the reward, respectively. Equation (24) calculates the average price in one period under multiple scenarios. Equation (25) calculates the variation of mean price versus EP* in one period in multiple scenarios. Equation (26) ensures that the electricity demand is always satisfied. Equation (27) computes the average price by multiplying the price for certain bid combination in a specified scenario with its occurrence possibility. Equations (28)–(30) provide the mathematical descriptions for $p\left({\mathsf{\lambda}}_{t},{b}_{t}\left(a\right)\right)$, ${{b}_{t}}_{\left(a\right)}$, ${b}_{i,t}\left({a}_{i}\right)$, respectively. Equation (31) represents the GenCos’ objective which is also their incentive-compatibility constraint with a

_{i}being the decision variable for GenCo i. Equation (32) gives the personal rational constraint of the GenCos who is willing to accept an incentive contract. Finally, Equations (33)–(35) defines the constraints of contracted GenCos including power output capacity of contracted GenCos, and the possibilities of contracted GenCos to place high bids or normal bids.

_{i}is obtained by solving follows:

_{max}is the maximum available C(a

^{0}); C

_{min}is the minimum available C(a

^{0}); EPM(a

^{0}) is a balance between price minimization and price variation minimization when the decision variable is a

^{0}. EPM

_{max}is the maximum available EPM(a

^{0}); and EPM

_{min}is the minimum available EPM(a

^{0}). Equations (38)–(52) are the same with Equations (21)–(35).

## 4. Q-Learning for Agents’ Optimal Decision Making

#### 4.1. Periodic and Daily Q-Learning Methods for Generating Companies

#### 4.1.1. State Identification

#### 4.1.2. Action Selection

#### 4.1.3. Reward Calculation

_{s}is the number of the days elapsed in the period.

#### 4.1.4. Q-Value Update

#### 4.2. Q-Learning for the Leader of the Stackelberg Game (Independent System Operator)

#### 4.2.1. State Identification

#### 4.2.2. Action Selection

^{0}is defined as the set of action selection of ISO agent. The ISO takes the action at each step, or at the starting point of each period. ${\left({a}^{0}\right)}_{\tilde{t}}$ denotes the action selection of ISO in period $\tilde{t}$.

#### 4.2.3. Reward Calculation

#### 4.2.4. Q-Value Update

#### 4.3. Solution Methodology for Independent System Operator’s Initial Q Value

- Step 1:
- set initial parameters incorporating bidding coefficients of GenCos and its power capacity.
- Step 2:
- set $\nu =1$.
- Step 3:
- generate a non-zero chaos variable ${\mathsf{\eta}}_{\nu +1}$ using cube mapping method as shown below:$${\mathsf{\eta}}_{\nu +1}=4{\mathsf{\eta}}_{\nu}^{3}-3{\mathsf{\eta}}_{\nu}$$
- Step 4:
- decoding the chaos variables into a binary variable which represents a value for the sets of target GenCos.
- Step 5:
- calculate the tailoring values of (α
_{i}, β_{i}, π_{i}) for all target GenCos using Equation (18). - Step 6:
- for the designed menu of the incentive contracts, check each GenCo’s optimal reaction by solving Equation (17).
- Step 7:
- calculate the corresponding objective value and the state ${\overrightarrow{s}}_{\tilde{t}}$ for ISO’s Q-learning. The latter includes the mean electricity price during the period, and the menu of the incentive contracts. If the obtained objective value is larger than the existing one, substitute the existing one.
- Step 8:
- substitute the chaos variables into Equation (64) to yield new chaos variables:$${x}_{\nu}={c}_{\nu}-\left[{d}_{\nu}{\mathsf{\eta}}_{\nu +1}\right]$$
- Step 9:
- Set $\nu =\nu +1$, k = k + 1.
- Step 10:
- If $\nu >{\nu}_{\mathrm{max}}$, stop searching, else go to Step 4.

## 5. Simulations and Analysis

- Case 1:
- no menu of incentive contracts or Q-learning.
- Case 2:
- Case 3:
- menu of incentive contracts with Q-learning in multiple periods. Note that load demand over the multiple periods varies between 170 MW and 230 MW.

**Table 2.**Parameters of GenCos’ bidding curves. (Unit for α

_{i}

_{,t}

^{c}, α

_{i}

_{,t}

^{h}: $/MW per hour, unit for β

_{i}

_{,t}

^{c}, β

_{i}

_{,t}

^{h}: $/(MW)

^{2}per hour).

GenCo No. | Case 1, Cases 2.1–2.5, Case 3 | Cases 2.6–2.8 | ||||||
---|---|---|---|---|---|---|---|---|

α_{i}_{,t}^{c} | β_{i}_{,t}^{c} | α_{i}_{,t}^{h} | β_{i}_{,t}^{h} | α_{i}_{,t}^{c} | β_{i}_{,t}^{c} | α_{i}_{,t}^{h} | β_{i}_{,t}^{h} | |

1 | 10 | 0.5 | 10.5 | 0.525 | 10 | 0.5 | 10.5 | 0.525 |

2 | 11 | 0.8 | 12 | 0.84 | 11 | 0.8 | 12 | 0.84 |

3 | 8 | 0.6 | 8.4 | 0.63 | 8 | 0.9 | 8.4 | 0.945 |

4 | 15 | 0.5 | 15.75 | 0.525 | 15 | 0.5 | 15.75 | 0.525 |

5 | 20 | 0.9 | 21 | 0.945 | 20 | 0.6 | 21 | 0.63 |

GenCo No. | Cases 2.1–2.4 and 2.6–2.8 | Case 2.5 | ||
---|---|---|---|---|

c_{i}_{1} | c_{i}_{2} | c_{i1} | c_{i}_{2} | |

1 | 10 | 0.5 | 5 | 0.25 |

2 | 11 | 0.55 | 6 | 0.35 |

3 | 8 | 0.5 | 4 | 0.25 |

4 | 15 | 0.8 | 7 | 0.4 |

5 | 20 | 0.9 | 10 | 0.45 |

Cases | Cost | EPM (0.5 × EP + 0.5 × BP) |
---|---|---|

2.1, 2.5, 2.6 | 1 | 0 |

2.2, 2.7 | 0 | 1 |

2.3, 2.8, 3 | 0.5 | 0.5 |

2.4 | Without menu of incentive contracts |

**Table 5.**Comparative results for Cases 2.1–2.3 and 2.5 (unit: $,). (Note: Y = saying “Yes” to offer of the incentive contract menu and N = saying “No” to the offer).

Items | Case 2.1 | Cases 2.2 and 2.3 | Case 2.5 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Target GenCos | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |

GenCos’ response | Y | Y | Y | N | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y |

Expected reward | 208,750 | 543,530 | 123,070 | ||||||||||||

Expected cost saving (compared with Case 2.4) | 597,830 | 512,120 | 932,580 | ||||||||||||

Expected price (EP) | 37.17 | 36.94 | 36.94 | ||||||||||||

BP (mean price variance in bad scenarios) | 2.78 | 2.53 | 2.53 | ||||||||||||

EPM (0.5 × EP + 0.5 × BP) | 19.97 | 19.74 | 19.74 |

Items | Case 2.6 | Cases 2.7 and 2.8 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Target GenCos | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | |

GenCos’ response | Y | Y | Y | N | Y | Y | Y | Y | Y | Y | |

Expected reward | 215,980 | 581,130 | |||||||||

Expected cost saving (compared with Case 2.9) | 607,780 | 497,270 | |||||||||

EP | 38 | 36.94 | |||||||||

BP (meanprice variance in bad scenarios) | 2.78 | 2.53 | |||||||||

EPM (0.5 × EP + 0.5 × BP) | 20.39 | 20.15 | |||||||||

Threshold for GenCo’s power output | 1 | 6558 | 6477 | ||||||||

2 | 7145 | 7091 | |||||||||

3 | 7145 | 7091 | |||||||||

4 | N/A | 9852 | |||||||||

5 | 6558 | 6477 |

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Notations

## N1. Set Parameters

A^{0} | Set of all possible a ^{0}, which denotes a combination of target GenCos. |

M | Set of all serial numbers of GenCos. |

I | Set of target GenCos, $I=\left\{k|{a}_{k}^{0}=1,k\in M\right\}$. |

Λ | Set of uncertain scenarios during a bidding period. |

Λ^{B} | Set of all bad scenarios. |

(AL, B, π) | Data set for the menu of incentive contracts. |

## N2. Decision Variables

${a}_{i}^{0}$ | Whether a GenCo is chosen as the target GenCo. ${a}_{i}^{0}=1$ is “yes” and 0 is “not”. |

a^{0} | ${a}^{0}=({a}_{1}^{0},{a}_{2}^{0},\mathrm{...},{a}_{m}^{0})$, a ^{0}∈ A^{0}. |

a_{i} | Whether GenCo i accepts the incentive menu. a _{i} = 0 means “not”, and a_{i} ≠ 0 means “yes”. Moreover, if a_{i} ≠ 0, a_{i} = k, k∈I, meaning GenCo i accepts the incentive contract with GenCo k as the target GenCo. |

a | i ∈M. |

## N3. Model Parameters

C(a^{0}) | Total electricity purchasing cost when the combination of the target GenCos is a ^{0}. |

π (a_{i}) | Award received by GenCo i. |

t | Time interval in one period. |

$\tilde{t}$ | Length of contract period. |

λ_{t} | A certain scenario in Λ at time t. |

ρ(λ_{t}) | Probability of λ _{t}. |

EP(a^{0}) | Expected electricity price when the combination of the target GenCos is a ^{0}. |

EP* | The best expected price. |

$\mathsf{\delta}$ | Balance parameter. |

w | Weight of the objective functions. |

BP(a^{0}) | Variance of mean price in bad scenarios versus EP* when the set of the target GenCos is a ^{0}. |

EPM | A representative symbol of the model objective which combines the expected price (EP) and robustness of the price (BP) with a balance factor. |

P(λ_{t},a) | In scenario λ _{t}, the expected market price when the set of the GenCos’ options for the accepted incentive contract is a. |

m | Number of GenCos. |

K_{i,t} | Power output of GenCo i when the bidding combination of GenCos is b _{t}(a) in scenario λ_{t}. |

α_{it}, β_{it} | Parameters of GenCo i’s bidding curve at time t. |

b_{it} | Bidding strategy of GenCo i at time t. b _{it} = 1 means a high bidding; and b_{it} = 0 implies a normal bidding. |

pos_{i}(b_{it}) | Probability for GenCo i to make a bid b _{it} at time t. |

П_{i}(a_{i}) | Expected profit of GenCo i when its contract decision is a _{i}. |

П_{i}(λ_{t}, a_{i}) | Expected profit for GenCo i in scenario λ _{t}. When a_{i} = 0, GenCo i accepts the menu of incentive contracts; when a_{i} ≠ 0 or a_{i} = k, GenCo i does not accept the incentive contract which is tailored to the target GenCo k (k = a_{i}). |

b_{t}^{i,c} | b _{t}^{i,c} = {b_{1t}, b_{2t}, b_{i-}_{1,t}, 0, b_{i+}_{1,t}, ..., b_{mt}}. |

b_{t}^{i,H} | b _{t}^{i,H} = {b_{1t}, b_{2t}, b_{i-}_{1,t}, 1, b_{i+}_{1,t}, ..., b_{mt}}. |

b_{t}(a) | Combination of GenCos’ bidding strategy with ${b}_{t}\left(a\right)=\left({b}_{1,t}\left({a}_{1}\right),{b}_{2,t}\left({a}_{2}\right),\cdots ,{b}_{m,t}\left({a}_{m}\right)\right)$. |

p(λ_{t}, b_{t}(a)) | The expected electricity price when the bidding combination of GenCos is b _{t}(a). The value of b_{t}(a) could be b_{t}^{i,c} or b_{t}^{i,H}. |

D(λ_{t}) | Electricity demand in scenario λ _{t}. |

c_{i}_{1}, c_{i}_{2} | Cost coefficients of GenCo i. |

(α_{i}, β_{i}, π_{i}) | Parameters of an incentive contract. Note α _{i} and β_{i} represent the bidding coefficients of a target GenCo, and π_{i} denotes per-period reward. |

p_{it} | Bidding price of GenCo i at time t. |

p_{t} | MCP at time t. |

q_{it} | Bidding power output of GenCo i at time t. |

D_{t} | Electricity demand at time t. |

α_{i}^{c}, β_{i}^{c} | Parameters of the normal bidding curve for GenCo i. |

α_{i}^{h}, β_{i}^{h} | Parameters of the high bidding curve for GenCo i. |

## N4. Q-Learning Parameters

${s}_{\tilde{t}}$ | State identification for GenCo’s Q-learning method in a period. |

${s}_{t}$ | State identification for GenCo’s Q-learning method in each day. |

${a}_{i,\tilde{t}}$ | Periodical action selection of GenCo i at the starting point of a period for GenCo’s Q-learning method. |

${a}_{i,t}$ | Daily action selection of GenCo i in each day for GenCo’s Q-learning method. |

$r({s}_{i,\tilde{t}},{a}_{i,\tilde{t}})$ | Periodical reward function for Q-learning method. |

$R({a}_{i,\tilde{t}})$ | Reward obtained by GenCo i over period $\tilde{t}$. |

$r({s}_{i,t},{a}_{i,t})$ | Daily reward function for Q-learning method. |

$R({a}_{i,t})$ | Reward obtained by GenCo i over period at a day. |

$\mathsf{\phi}$ | Discount factor. |

T | Number of days in the contract period. |

T_{s} | Number of days elapsed over a contract period $\tilde{t}$. |

${Q}_{\tilde{t}+1}({s}_{\tilde{t}},{a}_{i,\tilde{t}})$ | Periodical Q-value function defined for GenCo i. |

${Q}_{t+1}({s}_{t},{a}_{i,t})$ | Daily Q-value function defined for GenCo i. |

${\ell}_{\tilde{t}}$ | Positive learning rate for periodical Q-learning function. |

${\ell}_{t}$ | Positive learning rate for daily Q-learning function. |

${\gamma}_{\tilde{t}}$ | Discount parameter for periodical Q-learning function. |

${\gamma}_{t}$ | Discount parameter for daily Q-learning function. |

${\overrightarrow{s}}_{\tilde{t}}=\left\{\left({s}_{\tilde{t}},{a}^{0}\right)\right\}$ | State identification for ISO’s Q-learning function. |

${\left({a}^{0}\right)}_{\tilde{t}}$ | Periodic action selection of ISO. |

$r\left({s}_{\tilde{t}},{\left({a}^{0}\right)}_{\tilde{t}}\right)$ | Periodic reward calculation for ISO’s Q-learning function. |

${{Q}^{0}}_{\tilde{t}+1}\left({s}_{\tilde{t}},{\left({a}^{0}\right)}_{\tilde{t}}\right)$ | Periodic Q-value function defined for ISO. |

## N5. Algorithms Parameters

$\nu $ | An iteration number. |

${\mathsf{\eta}}_{\nu}$ | Non-zero chaos variable. |

${c}_{\nu}$, ${d}_{\nu}$ | Constant vectors. |

${\nu}_{\mathrm{max}}$ | Max iteration times. |

## References

- Zhang, D.; Wang, Y.; Luh, P.B. Optimization based bidding strategies in the deregulated market. IEEE Trans. Power Syst.
**2000**, 15, 981–986. [Google Scholar] [CrossRef] - Kian, A.R.; Cruz, J.B. Bidding strategies in dynamic electricity markets. Decis. Support Syst.
**2005**, 40, 543–551. [Google Scholar] [CrossRef] - Swider, D.J.; Weber, C. Bidding under price uncertainty in multi-unit pay-as-bid procurement auctions for power systems reserve. Eur. J. Oper. Res.
**2007**, 181, 1297–1308. [Google Scholar] [CrossRef] - Centeno, E.; Renese, J.; Barquin, J. Strategic analysis of electricity markets under uncertainty: A conjectured-price-response approach. IEEE Trans. Power Syst.
**2007**, 22, 423–432. [Google Scholar] [CrossRef] - Sahraei-Ardakani, M.; Rahimi-Kian, A. A dynamic replicator model of the players’ bid in an oligopolistic electricity market. Electr. Power Syst. Res.
**2009**, 79, 781–788. [Google Scholar] [CrossRef] - Li, G.; Shi, J. Agent-based modeling for trading wind power with uncertainty in the day-ahead wholesale electricity markets of single-sided auctions. Appl. Energy
**2012**, 99, 13–22. [Google Scholar] [CrossRef] - Nojavan, S.; Zare, K. Risk-based optimal bidding strategy of generation company in day-head electricity market using information gap decision theory. Int. J. Electr. Power Energy Syst.
**2013**, 48, 83–92. [Google Scholar] [CrossRef] - Qiu, Z.; Gui, N.; Deconick, G. Analysis of equilibrium-oriented bidding strategies with inaccurate electricity market models. Int. J. Electr. Power Energy Syst.
**2013**, 46, 306–314. [Google Scholar] [CrossRef] - Kardakos, E.G.; Simoglou, C.K.; Bakirtzis, A.G. Optimal bidding strategy in transmission-constrained electricity markets. Electr. Power Syst. Res.
**2014**, 109, 141–149. [Google Scholar] [CrossRef] - Anderson, E.J.; Cau, T.D.H. Implicit collusion and individual market power in electricity markets. Eur. J. Oper. Res.
**2011**, 211, 403–414. [Google Scholar] [CrossRef] - Nam, Y.W.; Yoon, Y.T.; Hur, D.; Park, J.; Kim, S. Effects of long-term contracts on firms exercising market power in transmission constrained electricity markets. Electr. Power Syst. Res.
**2006**, 76, 435–444. [Google Scholar] [CrossRef] - David, A.K.; Wem, F.S. Market power in electricity supply. IEEE Trans. Energy Convers.
**2001**, 16, 352–360. [Google Scholar] [CrossRef] - Oh, S.; Hildreth, A.J. Decisions on energy demand response option contracts in smart grids based on activity-based costing and stochastic programming. Energies
**2013**, 6, 425–443. [Google Scholar] [CrossRef] - Faria, P.; Vale, Z.; Baptista, J. Demand response programs design and use considering intensive penetration of distributed generation. Energies
**2015**, 9, 6230–6246. [Google Scholar] [CrossRef] - Ghazvini, M.A.F.; Soares, J.; Horta, N.; Neves, R.; Castro, R.; Vale, Z. A multi-objective model for scheduling of short-term incentive-based demand response programs offered by electricity retailers. Appl. Energy
**2015**, 151, 102–118. [Google Scholar] [CrossRef] - Ghazvini, M.A.F.; Faria, P.; Ramos, S.; Morais, H.; Vale, Z. Incentive-based demand response programs designed by asset-light electricity providers for the day-ahead market. Energy
**2015**, 82, 786–799. [Google Scholar] [CrossRef] - Zhong, H.; Xie, L.; Xia, Q. Coupon incentive-based demand response: Theory and case study. IEEE Trans. Power Syst.
**2013**, 28, 1266–1276. [Google Scholar] [CrossRef] - Fakhrazari, A.; Vakilzadian, H.; Choobineh, F.F. Optimal energy scheduling for a smart entity. IEEE Trans. Smart Grid
**2014**, 5, 2919–2928. [Google Scholar] [CrossRef] - Christopher, O.A.; Wang, L. Smart charging and appliance scheduling approaches to demand side management. Int. J. Electr. Power Energy Syst.
**2014**, 57, 232–240. [Google Scholar] - Yousefi, S.; Moghaddam, M.P.; Majd, V.J. Optimal real time pricing in an agent-based retail market using a comprehensive demand response model. Energy
**2011**, 36, 5716–5727. [Google Scholar] [CrossRef] - Shariatazadeh, F.; Mandal, P.; Srivastava, A.K. Demand response for sustainable energy systems: A review, application and implementation strategy. Renew. Sustain. Energy Rev.
**2015**, 45, 343–350. [Google Scholar] [CrossRef] - Gu, W.; Yu, H.; Liu, W.; Zhu, J.; Xu, X. Demand response and economic dispatch of power systems considering large-scale plug-in hybrid electric vehicles/electric vehicles (PHEVs/EVs): A review. Energies
**2013**, 6, 4394–4417. [Google Scholar] [CrossRef] - Bradley, P.; Leach, M.; Torriti, J. A review of the costs and benefits of demand response for electricity in the UK. Energy Policy
**2013**, 52, 312–327. [Google Scholar] [CrossRef] - Silva, C.; Wollenberg, B.F.; Zheng, C.Z. Application of mechanism design to electric power markets. IEEE Trans. Power Syst.
**2001**, 16, 1–8. [Google Scholar] [CrossRef] - Liu, Z.; Zhang, X.; Lieu, J.; Li, X.; He, J. Research on incentive bidding mechanism to coordinate the electric power and emission-reduction of the generator. Int. J. Electr. Power Energy Syst.
**2010**, 32, 946–955. [Google Scholar] [CrossRef] - Cai, X.; Li, C.; Lu, Y. Price cap mechanism for electricity market based on constraints of incentive compatibility and balance accounts. Power Syst. Technol.
**2011**, 35, 143–148. [Google Scholar] - Heine, K. Inside the black box: Incentive regulation and incentive channeling on energy markets. J. Manag. Gov.
**2013**, 17, 157–186. [Google Scholar] [CrossRef] - Weber, J.D.; Overbye, T.J. A two-level optimization problem for analysis of market bidding strategies. In Proceedings of the IEEE Power Engineering Society Summer Meeting, Edmonton, AB, Canada, 18–22 July 1999; Volume 2, pp. 682–687.
- Lei, W.; Shahidehpour, M.; Zuyi, L. Comparison of scenario-based and interval optimization approaches to stochastic SCUC. IEEE Trans. Power Syst.
**2012**, 27, 913–921. [Google Scholar] - Wang, B.; Yang, X.; Li, Q. Bad-scenario set risk-resisting robust scheduling model. Acta Autom. Sin.
**2012**, 38, 270–278. [Google Scholar] [CrossRef] - North, M.J.; Collier, N.T.; Vos, J.R. Experiences creating three implementations of the repast agent modeling toolkit. ACM Trans. Model. Comput. Simul.
**2006**, 16, 1–25. [Google Scholar] [CrossRef] - Rahimiyan, M.; Mashhadi, H.R. An adaptive Q-learning algorithm developed for agent-based computational modeling of electricity market. IEEE Trans. Syst. Man Cybern. C Appl. Rev.
**2010**, 40, 547–556. [Google Scholar] [CrossRef] - Naghibi-Sistani, M.B.; Akbarzadeh-Tootoonchi, M.R.; Bayaz, M.H.J.D.; Rajabi-Mashhadi, H. Application of Q-learning with temperature variation for bidding strategies in market based power systems. Energy Convers. Manag.
**2006**, 47, 1529–1538. [Google Scholar] [CrossRef] - Haddad, M.; Altmann, Z.; Elayoubi, S.E.; Altaman, E. A Nash-Stackelberg fuzzy Q-learning decision approach in heterogeneous cognitive networks. In Proceedings of the IEEE Global Telecommunications Conference, Miami, FL, USA, 6–10 December 2010.
- Zuo, X.Q.; Fan, Y.S. A chaos search immune algorithm with its application to neuro-fuzzy controller design. Chaos Solitons Fractals
**2006**, 30, 94–109. [Google Scholar] [CrossRef]

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## Share and Cite

**MDPI and ACS Style**

Yu, Y.; Jin, T.; Zhong, C.
Designing an Incentive Contract Menu for Sustaining the Electricity Market. *Energies* **2015**, *8*, 14197-14218.
https://doi.org/10.3390/en81212419

**AMA Style**

Yu Y, Jin T, Zhong C.
Designing an Incentive Contract Menu for Sustaining the Electricity Market. *Energies*. 2015; 8(12):14197-14218.
https://doi.org/10.3390/en81212419

**Chicago/Turabian Style**

Yu, Ying, Tongdan Jin, and Chunjie Zhong.
2015. "Designing an Incentive Contract Menu for Sustaining the Electricity Market" *Energies* 8, no. 12: 14197-14218.
https://doi.org/10.3390/en81212419