The manuscript addresses the problem of finding all solutions of power flow equations or other similar non‐linear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly the direct methods for transient stability analysis and voltage stability assessment. Here, the authors introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. Since finding real solutions is much more challenging, first the authors embed the real form of power flow equation in complex space, and then track the generally unphysical solutions with complex values of real and imaginary parts of the voltages. The solutions converge to physical real form in the end of the homotopy. The so‐called gamma‐trick mathematically rigorously ensures that all the paths are well‐behaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelisable . The authors demonstrate the technique performance by solving several test cases up to the 14 buses. Finally, they discuss possible strategies for scaling the method to large size systems, and propose several applications for security assessments.

We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope region constructed around a given feasible operating point. This certificate is based on the explicit bounds on the nonlinear terms that hold within the self-mapped polytope. The shape of the polytope is adapted to find the largest approximation of the steady-state security region. While the corresponding optimization problem is nonlinear and non-convex, every feasible solution found by local search defines a valid inner approximation. The number of variables scales linearly with the system size, and the general framework can naturally be applied to other nonlinear equations with affine dependence on inputs. Test cases, with the system sizes up to 1354 buses, are used to illustrate the scalability of the approach. The results show that the approximated regions are not unreasonably conservative and that they cover substantial fractions of the true steady-state security regions for most medium-sized test cases.

In this paper, a blockchain-enabled distributed market framework is proposed for the bi-level carbon and energy trading between coal mine integrated energy systems (CMIESs) and a virtual power plant (VPP) with network constraints. To maximize the profits of these two entities and describe their complicated interactions in the market, the bi-level trading problem is formulated as a Stackelberg game considering integrating the energy market and the “cap-and-trade” carbon market mechanism. Meanwhile, in the CMIES, energy recovery units and belt conveyors can be flexibly scheduled and the pumped hydroelectric storage in the VPP is scheduled for energy management. To tackle uncertainties from PV outputs, the joint trading, and the energy management is solved by the distributionally robust optimization (DRO) method. In addition, for participants' privacy, the alternating direction method of multipliers (ADMM) - based DRO algorithm is applied to solve the trading problem in a distributed framework. Further, the Proof-of-Authority (PoA) blockchain is deployed to develop a safe and anonymous market platform. Finally, case studies along with numerous comparison cases are conducted to verify the effectiveness of the proposed method. Simulation results indicate that the proposed method can effectively reduce the system operation cost and regional carbon emission, reduce the conservativeness and protect the privacy of each participant.

Currently, the distribution systems are moving towards decentralized operation due to the high penetration of distributed energy resources (DERs). Peer-to-peer (P2P) energy trading has been an emerging concept that promotes autonomous DER participation in energy markets while preserving their privacy concerns. In this work, a novel P2P energy trading enabled decentralized market framework is proposed for the optimal operation of distribution grids. Nodal agents and P2P agents are established as market participants, and market equilibrium is iteratively achieved via alternating direction method of multipliers based algorithms. The proposed market framework guarantees grid constraint satisfaction, market equilibrium, and global optimality for all market participants without violating their privacy concerns. The agent coordination and local optimization are designed such that fairness of the market clearing mechanism, prosumer autonomy, and prosumer anonymity is preserved without compromising the market efficiency. Further, costs/rewards of ancillary services associated with the P2P energy transactions are considered as trade-offs within the market mechanism, and those are accurately allocated to the respective trading pairs. The case studies illustrate the effectiveness and scalability of the proposed market framework.