## 6 Conclusions  
Bilateral spot EM is a more complicated type of EM in many countries in the world, where every player (GenCO and DisCO) chooses its bidding strategy within a continuous interval of values simultaneously in order to make more profit. Considering renewable resource penetration, the random fluctuation and continuous variation of renewable resource power generation make it more difficult to model the dynamic bidding process and the equilibrium in the bilateral spot EM. Since the GDCAC algorithm has been demonstrated to be an effective method in dealing with continuous state and action variables, in this paper we have proposed the application of a GDCAC algorithm for bilateral hour-ahead EM modeling considering renewable power penetration. The simulation results have verified the feasibility and scientific nature of our proposed approach, and some conclusions can be drawn as follows:  
1. In our proposed GDCAC-based EM modeling approach, every agent needs no common knowledge about other agents' costs or revenue functions, etc. and can make the decision to select an optimal bidding strategy within a continuous interval of values according to many renewable power generations randomly changing within a continuous state space, which can avoid the \"Curse of Dimensionality\". The randomly fluctuating nature of renewable resource output does not affect the proposed EM approach's ability to reach NE after enough iterations;
2. In our proposed GDCAC EM modeling approach, after enough iterations, although with the increase of renewable resource output some agents may have their bidding functions deviate more from their actual marginal cost or revenue functions because of congestions in some transmission lines, the overall SW still increases, which is the same as the conclusions drawn in [2];
3. Our proposed GDCAC EM modeling approach is superior to the fuzzy Q-learning approach (mentioned in [2]) in terms of increasing the profit of a specific agent and the overall SW and lowering the overall LMP level;
4. According to [40], the time complexity of GDCAC is O(n) (the relevant mathematical proof is proposed in [40]). However, when applying the GDCAC algorithm to EM modeling, because in  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenarios** & **GDCAC Approach** & **Fuzzy Q-Learning Approach** & **Q-Learning Approach** \\ \hline Average AVLMP & 37.0352 & 38.2237 & 42.8395 \\ Average SW & 5329.5 & 5242.9 & 4617.1 \\ Computational time & 3.21 m & 3.16 m & 3.12 m \\ \hline \hline \end{tabular}
\end{table}
Table 9: The simulation results of the three approaches.  
every iteration we use the active set (AS) algorithm to solve the congestion management model for ISO, which needs 500 extra iterations, the time complexity of our proposed GDCAC-based EM modeling approach is O(500n). Nevertheless, our simulation with the proposed GDCAC-based EM modeling approach takes only about 3.2 min to reach the final result. That is to say, the time complexity of our GDCAC-based EM modeling approach is acceptable so that we can extend it to the modeling and simulation of more realistic and complex EM systems.  
Moreover, our proposed approach can provide a bidding decision-making tool for participants to get more profit in EM with renewable power penetration. In addition, our proposed approach can provide an economic analysis tool for governments to design proper policies to promote the development of renewable resources.  
This study was supported by the National Natural Science Foundation of China under Grant No. 71373076, the Fundamental Research Funds for the Central Universities under Grant No. JB2016183, and the Major State Research and Development Program of China under Grant Nos. 2016 YFB0900500 and 2016 YFB0900501.  
[PERSON] guided the research; [PERSON] established the model, implemented the simulation, and wrote this article; [PERSON] collected references; [PERSON] and [PERSON] revised the language of this article.  
The authors declare no conflict of interest.