been verified in 22 and 23 Each level is formulated as a stochastic two stage

# Been verified in 22 and 23 each level is formulated

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been verified in [ 22 ] and [ 23 ]. Each level is formulated as a stochastic two-stage problem with the first stage to optimize the base generation and power exchanges of all entities based on the forecasted outputs of RES-based DGs and the second stage to adjust generations according to the variations of real- ized RES-based DG outputs. The uncertain power outputs of wind turbines and PVs are described by scenarios generated from Monte Carlo simulations (MCs). The simultaneous back- ward scenario reduction method [ 24 ] is applied to increase the calculation speed while maintaining the accuracy of the solution. The major contributions of this paper are summarized as follows. 1) Optimal coordinated control of networked MGs with distinct economic and operational objectives in a dis- tribution system is a new topic with limited existing works. 2) Uncertainty and variability of RES-based DG outputs are fully considered. 3) Stochastic bi-level formulation of the control framework with each level modeled as a two-stage problem. The remainder of this paper is organized as follows. Section II presents the local optimization problems of the DNO and MGs. Section III introduces the coordinated con- trol scheme of multiple MGs and transforms the coordinated control problem into a stochastic MPCC formulation and pro- poses the solution methodology. In Section IV, the numerical results are provided. Section V concludes the paper with the major findings. II. M ATHEMATICAL M ODELING OF I NDIVIDUAL S YSTEMS This section introduces a widely used electrical network model and provides the local optimization formulation for individual systems, DNO and MGs. A. Distribution System Model Consider an electrical network as shown in Fig. 1 , there are n buses indexed by i = 0 , 1 , . . . , n . DistFlow [ 25 ] equations can be used to describe the complex power flows at each node i P i + 1 = P i r i ( P 2 i + Q 2 i ) V 2 i p i + 1 (1) Q i + 1 = Q i x i ( P 2 i + Q 2 i ) V 2 i q i + 1 (2) V 2 i + 1 = V 2 i 2 ( r i P i + x i Q i ) + ( r 2 i + x 2 i )( P 2 i + Q 2 i ) V 2 i (3) p i = p D i p g i , q i = q D i q g i . (4) In the above equations, we assume p g i is generated by both RES-based DG units which are subject to uncertainties and controllable DG units, q g i is generated by controllable DG units [ 26 ]. The DistFlow equations can be simplified using linearization. The linearized power flow equations have been extensively used and justified in both traditional distribution systems and MGs [ 7 ], [ 27 ], [ 28 ] P i + 1 = P i p i + 1 (5) Q i + 1 = Q i q i + 1 (6) V i + 1 = V i ( r i P i + x i Q i ) V 2 1 (7) p i = p D i p g i , q i = q D i q g i . (8) B. Optimization Problem for DNO It is assumed that the DNO also owns both dispatchable DGs (MTs in this paper) and RES-based DGs (WTs in this paper) [ 29 ]. The optimization problem of a DNO can be formulated as follows (denote the formulation as M ): min i D c G p G i + c B , DNO η 1 + m c S , MG θ m 1 c S , DNO θ 1 m c B , MG η m 1 + s γ s i D c G p G i , s + C rd i , s + s γ s c B , DNO η 1 , s + m c S , MG θ m 1 , s c S , DNO θ 1 , s m c B , MG η m 1 , s (9) s . t . P i + 1 = P i p D i + 1 + g

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