Risk-aware Distributed Optimal Power Flow in Coordinated Transmission and Distribution System

Aamir Nawaz; Hongtao Wang

网刊加载中。。。

使用Chrome浏览器效果最佳，继续浏览，你可能不会看到最佳的展示效果，

确定继续浏览么?

复制成功，请在其他浏览器进行阅读

Risk-aware Distributed Optimal Power Flow in Coordinated Transmission and Distribution System PDF

- ORCID：
Aamir Nawaz (Member, IEEE)
- ORCID：
Hongtao Wang (Senior Member, IEEE)

Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education, Shandong University, Jinan 250000, China

Updated：2021-05-19

DOI：10.35833/MPCE.2019.000005

OUTLINE

Abstract

Active distribution grids cause bi-directional power flow between transmission system (TS) and distribution system (DS), which not only affects the optimal cost but also the secure operation of the power system. This paper proposes a hybrid coordination method to solve the risk-aware distributed optimal power flow (RA-DOPF) problem in coordinated TS and DS. For operation risk evaluation, the weather-based contingencies are considered in both TS and DS. A hybrid coordination method is developed that entails analytical target cascading (ATC) and Benders decomposition (BD). Moreover, the risk-aware optimal power flow (RAOPF) in TS and risk-based security-constrained optimal power flow in DS have been performed using the BD method considering basic optimal power flow as a master problem, whereas $N - 1$ and $N - 2$ contingencies are considered as sub-problems. Different case studies are performed using the IEEE 30-bus system with generation reserves as a TS and a 13-bus system as a DS. The results demonstrate the efficacy of the proposed method.

Keywords

Analytical target cascading; Benders decomposition; distribution system; preventive and corrective contingencies; risk awareness; transmission system

NOMENCLATURE
A. Indices, Sets, and Parameters
$α_{T S O}^{t'}$ , $α_{T S O}^{t' + 1}$ , $β_{T S O}^{t'}$ , $β_{T S O}^{t' + 1}$	Quadratic penalty parameters for risk adjustment of TSO at outer loop iterations $t'$ and $t' + 1$
$ε_{1}$ , $ε_{2}$ , $ε_{3}$	Mismatch thresholds
$ρ_{D S O i}^{t'}$ , $ρ_{D S O i}^{t' + 1}$ , $η_{D S O i}^{t'}$ , $η_{D S O i}^{t' + 1}$	Quadratic penalty parameters for generation mismatch of the i^th DSO with TS operator (TSO) at outer loop iterations $t'$ and $t' + 1$
σ₁, σ₂, σ₃	Pre-defined factors
b, j, l, n	Indices of buses, loads, lines, and generators
i, I	Indices and total number of distribution system operators (DSOs)
k, $k'$ , K	Indices of $N - 1$ , common-mode ( $N - 2$ ), and total number of contingencies
m, M	Indices and total number of boundary buses between transmission system (TS) and distribution system (DS)
N_G, N_B, N_L	Sets of generators, buses, and lines
$N_{G}^{D S}$ , $N_{B}^{D S}$ , $N_{L}^{D S}$	Sets of generators, buses, and lines for DS
t, $t'$	Indices of inner and outer loop iterations
x, y	Indices of two internal lines related to common-mode outage
B. Functions and Variables
α	Curtailment load factor
$λ_{l}^{n w}$ , $λ_{l}^{a w}$ , $λ_{l}^{e w}$ , $λ_{l}^{a v g}$	Failure rates in normal, adverse, extreme weather states and average failure rates of different weather states
π^k	Probability of the k^th contingency
ρ_l	Outage probability of the $l^{t h}$ line
ρ_x_\|\|_y	Outage conditional probability of the $x^{t h}$ with respect to the $y^{t h}$ transmission line
$ϕ_{1 l}^{k}$ , $ϕ_{2 l}^{k}$ , $τ_{1}^{k'}$ , $τ_{2 l}^{k'}$ , $τ_{3 l}^{k'}$ , $τ_{4 n}^{k'}$ , $τ_{5 n}^{k'}$	Dual variables
$C_{j}^{b, k'}$ , $C_{j}^{b, k', #}$ , $C D_{j}^{k}$	Load curtailments due to pre-contingency, post-contingency and generalized contingency at the j^th bus
$D S_{c o s t}^{i, t'}$ , $T S_{c o s t}^{t'}$	DS and TS costs
ENS^k	Energy not supplied
F_n, f_n, $f_{n}^{R A}$	Generalized cost, base case cost, and risk aware cost of the n^th generator
$f_{c}^{S C}$ , $f_{c}^{S C, D S O i}$	Security-constrained cost of transmission and the i^th DSO
$f^{k}$ , $f^{k'}$	Objective functions of sub-problems
g_n, h_n	Equality and inequality constraints
$H_{l}^{b w}$ , $H_{l}^{e w}$	Proportions of failures that occur in bad (adverse and extreme), and extreme weather
I_m	Line flow direction at boundary bus
$P_{g, m a x}^{D S O i, n}$ , $P_{g, m i n}^{D S O i, n}$	The maximum and minimum generation limits of the n^th generator in the i^th DSO
$P_{l}^{n w}$ , $P_{l}^{a w}$ , $P_{l}^{b w}$	Steady-state probabilities of the duration of normal, adverse, and extreme weather
$P_{n}^{G}$ , $P_{b}^{L}$ , P_l, $Q_{n}^{G}$ , $Q_{b}^{L}$ , Q_l	Active and reactive power of the n^th generator, the b^th bus and the l^th line
$P_{n}^{G, m i n}$ , $P_{n}^{G, m a x}$ , $Q_{n}^{G, m i n}$ , $Q_{n}^{G, m a x}$	The minimum and maximum active and reactive power of the n^th generator
$P_{n}^{0}$ , $P_{n}^{0, #}$	Pre-contingency and post-contingency generation of the n^th generator
$P G_{T}^{D S O i, t}$ , $P G_{R}^{D S O i, t}$	Active generation target from TSO to the i^th DSO and response from the i^th DSO to TSO
$P_{L}^{D S O 1}$ , $P_{L}^{D S O 2}$ , $. . .$ , $P_{L}^{D S O I}$	Total DSO system loads from 1 to I
$P_{n}^{T S O, m i n}$ , $P_{n}^{T S O, m a x}$	The minimum and maximum generation limits of the n^th generator
$P_{n}^{m i n}$ , $P_{n}^{m a x}$	The minimum and maximum dispatches
$P_{n}^{D S O 1, m a x}$ , $P_{n}^{D S O 2, m a x}$ , $. . .$ , $P_{n}^{D S O I, m a x}$	The maximum generation limits of the n^th generator in DSO system from 1 to I
$\sum_{i} P_{L}^{D S O i}$ , P_d	Total load of the i^th DSO system and total demand of the system scale on all buses of TS
$P_{l}^{m a x}$ , $P B^{A}$	The maximum line flow of the $l^{t h}$ line and the boundary area power from TS to DS
$P_{n}^{0}$ , $P_{n}^{0, #}$ , $P_{n}^{k'}$ , $P_{n}^{k', #}$	Pre-contingency dispatch and post-contingency re-dispatching for base case and the ${k'}^{t h}$ contingency
$P_{j}^{b}$	Total available demand at the b^th bus and $j^{t h}$ load
PT_m, $P T_{m}^{#}$	Pre-contingency and post-contingency line flow between TS and DS
$P T_{m}^{m i n}$ , $P T_{m}^{m a x}$	The minimum and maximum line flows from TS and DS
R, X	Resistance and impedance of transmission lines
$R_{}^{u p}$ , $R_{}^{d n}$ , $R_{}^{u p, #}$ , $R_{}^{d n, #}$	Pre-contingency and post-contingency up and down reserves
$R_{j}^{u p, m a x}$ , $R_{j}^{d n, m a x}$	The maximum up and down reserve dispatches
RISK	System risk calculated based on severity and probability of contingencies
Risk_n	Risk constraints
$R i s k_{T}^{T S O}$ , $R i s k_{R}^{T S O}$	Risk targets and responses for TSO
$R i s k_{i s l}^{D S O i}$	Islanding risk of the i^th DSO
$R i s k^{T S O}$ , $R i s k^{D S O i}$ , $R i s k_{t h r e s h o l d}^{m a x}$	TS risk, the i^th DS risk, and the risk threshold for the system
Sev_k	Severity of contingency
$S F_{b}^{0}$ , $S F_{b}^{k}$ , $S F_{b}^{k'}$	Shift factors of base case and the k^th and ${k'}^{t h}$ contingency
$S y s_{c o s t}^{t'}$ , $S y s_{c o s t}^{t' - 1}$	Overall system costs at the ${t'}^{t h}$ and ( $t' - 1$ )^th hours
$Δ$ T	Short-term operation time frame
V_b, $V_{b}^{m i n}$ , $V_{b}^{m a x}$ , $δ_{b}$ , $δ_{b}^{m i n}$ , $δ_{b}^{m a x}$	Voltage magnitude, angle, and their minimum and maximum values at the b^th bus
$V_{P B^{A}}^{D S O i}$ , $V_{P B^{A}}^{T S O}$	Magnitudes of boundary voltages of the i^th DSO and TSO
VOLL	Value of lost load
$v_{1 l}^{k}$ , $v_{2 l}^{k}$ , $u_{1}^{k^{'}}$ , $u_{2}^{k^{'}}$ , $u_{3 l}^{k^{'}}$ , $u_{4 l}^{k^{'}}$ , $u_{5 n}^{k^{'}}$ , $u_{6 n}^{k^{'}}$	Positive slack variables

I. INTRODUCTION

THE participation of distribution networks in power system balancing and cost optimization has been increased due to the exponential escalation in distributed energy resources (DERs) such as diesel generators (DG), wind farm, and solar farm, etc. Thus, a distribution system (DS) partakes along with the transmission system (TS) in the process of matching supply and demand to achieve the minimum cost. Therefore, previous researchers have investigated the distributed unconstrained economic dispatch (ED) [

1], [2], distributed security-constrained ED [3], distributed optimal power flow (OPF) [4], distributed unconstrained unit commitment (UC) [5], [6], and distributed security-constrained UC [7], [8].

Previous researches have studied mostly TS risk and DS risk separately, but have not considered the mutual effect of contingencies. TSs mainly face contingencies related to line outages [

9]. These line outages can generate

N - 1

N - 1 - 1

N - 2

, and

N - k

contigencies. Power systems are designed to operate under normal weather conditions. However, unexpected and extreme weather conditions can cause forced outage rates to rise drastically, which may result in multi-concurrent outages and system blackouts such as the 2005 Hurricane Katrina blackouts in the U.S., the 2008 Snow Storm blackouts in China, and the 2012 Typhoon Yolanda blackout in the Philippines. Moreover, [10] depicts that 43.6% of the blackouts in the United States from 1984 to 2006 occurred under extreme weather conditions, e.g., tornadoes, tropical cyclones, wind/rain, and other extreme weather, and ten of the fifteen largest ones were due to severe weather. Thus, these outages have motivated us to focus on weather-based risk assessment in conjunction with the TS and DS. Additionally, line outages in DS can cause islanding, which can lead to load curtailment [10], [11]. Thus, an active distribution network (ADN) can help TS reduce the load curtailment by taking up an expected curtailed load in island vicinity, which can thus reduce the curtailment risk. Hence, the mutual effect of line outages in both TS and DS has been considered in this paper.

Previously, researchers have solved risk assessment and cost assessment separately for coordinated TS and DS [

12], [13], though contingencies in transmission and DSs can mutually affect the coordinated optimized cost [9], [14]. For example, it is reported in California, USA that a high penetration of DGs in DSs has created significant difficulties for TS operations, and has caused outages which are difficult to manage via the current separate management manner [15]. Still, this event has caused approximately 2.7 million customers out of power in a hot day. Besides, a report by the Federal Energy Regulatory Commission (FERC) and the North American Electric Reliability Corporation (NERC) indicates that one of the major reasons for the outage is neglecting the mutual impacts of transmission and distribution contingencies on the overall system while performing coordinated cost optimization and risk assessment [16]. Consequently, some issues that previous researches have left unaddressed are summarized below.

1) In previous studies, the impact of emergency occurrence in the TS on transmission and distribution coordinated system has not been explored. In future power grids, ADN will not only participate in minimizing the overall system cost, but also play a key role in reducing system risk. As shown in Fig. 1(a), TS risk increases with common-mode interruption of buses B1-B2 and B2-B5. If the resources of the TS are insufficient for re-dispatching, load curtailment is inevitable. In such a case, DERs from the DS can help reduce the curtailment by supplying power to local loads and the main grid.

Fig. 1 Physical structure of coordinated TS and DS with contingency locations. (a) Line outage contingency case in TS. (b) Line outage contingency in DS causing islanding.

2) Furthermore, previous researchers have not studied the impact of the DS contingencies such as the impact of isolated islands on the coordinated cost of TS and DS. TS risks not only originate from TS contingencies but also DS contingencies. In Fig. 1(b), line outage in the DS causes the island problem, and the reduction of loads on the island increases the system risk. To solve this problem, the presence of DERs is crucial. DERs can support the survival of some loads or full loads depending on the capacity, thereby reducing the load curtailment and overall system risk.

3) Besides, previous researchers have not investigated the optimization of transmission and distribution coordination systems under simultaneous contingent situations in both systems. System risks can arise from simultaneous contingencies in both TS and DS. Even in this case, the DS can help the TS reduce risks and optimize the synergy costs. Likewise, DERs available in DSs can reduce the island risk.

Further, the above-discussed cases are considered as basic problems in this research. To solve the above-mentioned issues, it is requisite to investigate the minimum cost re-evaluation based on contingent situations of TS and DS. Hence, it paves the way to study on the risk-aware distributed OPF (RA-DOPF) problem, which focuses on $N - 1$ and $N - 2$ contingencies in the TS and $N - 2$ contingency in the DS. Besides, $N - 1$ and $N - 2$ contingencies fall under the category of prevention and correction risk analysis, respectively.

In smart grids, switch operations are automated in DS and can handle the $N - 1$ contingency problem efficiently. However, for $N - 2$ contingency in DS, DERs play a key role and can mitigate the risk. Due to the complexity of the proposed algorithm, only $N - 2$ contingency after $N - 1$ contingency is considered for simplification.

In this paper, an RA-DOPF algorithm with transmission and distribution coordination has been proposed. Additionally, the RA-DOPF solution is utilized to protect the system from the expected emergency state while maintaining the minimum cost. The contributions of this paper are summarized as follows.

1) The RA-DOPF algorithm has been applied to the coordinated TS and DS while considering: ① $N - 1$ and $N - 2$ contingencies in TS only and no contingency in DS; ② $N - 2$ contingency in DS only, which causes islanding and no contingency in TS; ③ $N - 1$ and $N - 2$ contingencies in TS and $N - 2$ contingency in DS simultaneously.

2) A two-level hierarchical model is utilized for RA-DOPF involving TS and DS for evaluating both risk and cost. Furthermore, analytical target cascading (ATC) has been utilized as TS and DS risk-averse coordination algorithm, and the Benders decomposition (BD) as an algorithm for solving the RAOPF and risk-based security-constrained OPF (RBSCOPF) problem.

The rest of the paper is organized as follows. Section II presents a hierarchical coordination framework. Section III explains the problem formulation and solution. Section IV describes different case studies. Section V concludes the paper and gives future directions.

II. Hierarchical Coordination Framework for RA-DOPF

The proposed framework consists of two sections: optimization and coordination. Initially, the problems of TS and DS are optimized, and then coordinated to achieve the optimal state of coordinated operation. Figure 2 shows the hierarchical structure of the proposed RA-DOPF problem.

Fig. 2 TSO-DSO framework for RAOPF- and RBSCOPF-based coordination.

A. Weather-based Contingency Risk Evaluation

At this point, the weather-based contingencies are considered as the main reason for contingent situations in TS and DS [

17].

As noted earlier, system risks include risks from TS and DS. Therefore, both TS and DS require separate risk assessments for line outages. Additionally, the contingencies in the two cases are different. In this paper, $N - 1$ and $N - 2$ contingencies are considered in TS and $N - 2$ contingency in DS.

1) Risk Originated from TS

In most cases, the main causes of transmission line interruption are line overload, weather changes, etc. These unexpected events can cause the system to go from a normal state to an emergency state [

17]. Further, a list of contingencies with credibility includes not only

N - 1

contingency but also

N - 2

contingency.

N - 2

contingency is also known as common-mode contingency, which occurs frequently under extreme weather conditions with large influences on the system security [18]. For example, a single lightning strike can cause trip-outs of two circuits on the common tower, while a tornado or a storm can cause outages of adjacent lines in a common corridor. In addition, one can evaluate

N - k

contingencies but the system solution will be computationally expensive and considerably time-inefficient [19]. Thus,

N - 1

and

N - 2

contingencies are studied in the TS as a prerequisite to avoid the system from going into an emergency state.

The three-state weather model is utilized for evaluation of the impact of weather conditions such as normal, adverse, and extreme conditions. This model is also developed in [

17].

λ_{l}^{n w} = λ_{l}^{a v g} (1 - H_{l}^{b w}) / P_{l}^{n w}

(1)

λ_{l}^{a w} = λ_{l}^{a v g} H_{l}^{b w} (1 - H_{l}^{e w}) / P_{l}^{a w}

(2)

λ_{l}^{e w} = λ_{l}^{a v g} H_{l}^{b w} H_{l}^{e w} / P_{l}^{b w}

(3)

The distribution of forced outage probability of overhead line within a short time can be calculated as:

ρ_{x ‖y} = 1 - e^{- λ_{a v g} Δ T}

(4)

Moreover, the probabilities of each common-mode outages are evaluated using (5), while the severity of each possible contingency on system security is quantified by the load interruption cost, which is the product of VOLL and $E N S^{k}$ as shown in (6). In addition, the operation risk is calculated using the severity and probability of all contingencies [

20] as depicted in (7).

π^{k} = ρ_{x | | y} \prod_{l \neq x, y} (1 - ρ_{l})

(5)

S e v_{k} = V O L L \cdot E N S^{k} = V O L L \cdot \sum_{j} C D_{j}^{k} \cdot Δ T

(6)

R I S K = \sum_{k} π^{k} \cdot S e v_{k}

(7)

Equations (1)-(3) describe the failure rates of the $l^{t h}$ transmission line; (4) describes the outage probability of line l; (5) describes the probability of the $k^{t h}$ contingency; (6) describes the severity index of risk; and (7) describes the formulation of risk using outage probability and severity index of risk.

2) Risk Originated from DS

The outage event in DS divides DS into various islands. This causes the loads within the islands to disconnect from the main bus, and curtailment may occur. However, if DGs are available in those islands, the loads can survive during a contingency. Consequently, curtailment risk needs to be re-evaluated considering loads that survive with the help of DGs. Thus, DGs are necessary for the survival of loads.

B. Hierarchical RA-DOPF Problem

The problem at hand is complex considering the risk awareness. Further, OPF in TS is referred to direct current OPF (DCOPF) because the resistance is negligible in TS. Therefore, the voltage magnitude at every transmission and distribution node is assumed to be 1.0 p.u., and only active power is considered. Conversely, OPF in DS is referred to as alternating current OPF (ACOPF) because the ratio of line resistance to line reactance is large. Hence, the voltage at distribution terminals can vary between 0.95 p.u. and 1.05 p.u.. Additionally, the risk assessment based on load curtailment is considered, so the DS shares active power information and risk information, while reactive power sharing information is not included. To model the reactive power mismatch at the border between TS and DS, the voltage at TS boundary terminal is considered to be 1.0 p.u., while the boundary terminal voltage of DS can vary between 0.95 p.u. and 1.05 p.u., though the reactive power mismatch is reduced by bringing the boundary terminal voltage of DS to approximately 1 p.u.. Further, the reactive power at the boundary of DS is considered to be 50% of the assigned boundary active power (load or generator), which is sent as a shared variable by TSO to the relevant DSO. In this way, the DSOs will only need to take up the assigned active power and reactive power (half of assigned active power shared by TSO). Here, the assigned active power means the shared variable from TSO. Since the shared variable does not involve the reactive power variable, DSOs will take up 50% of active power (load or generator) shared by TSO, while the remaining reactive power is taken up by TSO. It should be noted that the scope of this paper will not involve the evaluation of voltage stability.

The proposed problem has a hierarchical structure where the TS operator (TSO) acts as an upper-level problem, while the DS operator (DSO) acts as a lower-level problem as shown in Fig. 2.

1) Upper level: TS risk-aware based cost assessment

TSO will serve as the upper level in the hierarchy to solve RA-DOPF. TSO will evaluate its RAOPF at TS boundaries considering weather-based line outages as shown in Fig. 2. Here, $N - 1$ and $N - 2$ contingencies are assessed in the risk evaluation of TS.

2) Lower level: DS risk mitigation support and islanding risk-aware based cost assessment

DS acts as a lower-level section in the hierarchical structure of the problem. DSOs will evaluate the RBSCOPF and receive the targeted power generation from DERs. $N - 2$ contingency after $N - 1$ contingency is considered in DS. But it will be dealt with as $N - 1$ contingency (RBSCOPF) in the formulation because the $N - 1$ is already handled by the automated operation of switches. DS can help TS reduce both the total system cost and total system risk.

III. Problem Formulation and Solution

The RAOPF and RBSCOPF are formulated for TS and DS, respectively, and solved by using BD. However, the coordination problem between TS and DS are handled using ATC. The following sub-sections explain the problem formulation and solution.

A. Formulation of RAOPF and RBSCOPF

RAOPF and RBSCOPF are different from basic OPF in terms of pre- and post-contingency constraints. The base-case constraints are taken as pre-contingency constraints, while $N - 1$ and $N - 2$ contingency constraints are taken as post-contingency constraints. Initially, this paper has solved RAOPF for TS, which is a convex non-linear problem (NLP) as shown below:

m i n \sum_{n \in N_{G}} F_{n} (P_{n}^{G})

(8)

s.t.

g_{n} (P_{n}^{G}) = 0

(9)

h_{n} (P_{n}^{G}) \leq 0

(10)

R i s k_{n} (P_{n}^{G}) \leq R i s k_{t h r e s h o l d}^{m a x}

(11)

Equation (8) describes a generalized objective of DCOPF; (9) and (10) describe its equality and inequality constraints; respectively; and (11) is a risk evaluation constraint in this problem. Unlike RAOPF, the RBSCOPF problem involves $N - 1$ contingency for risk evaluation, which involves $N - 1$ and $N - 2$ contingencies. Consequently, (8)-(11) are also applicable to the RBSCOPF problem, but with the involvement of reactive power. It should be noted that no corrective measures have been considered for voltage limit violations, but only voltage limits are followed as problem constraints as described earlier. Hence, the risk is dependent on the active power curtailment, and so, (11) is the same for the RBSCOPF problem.

1) Base OPF

The objective and basic constraints of OPF for TS are defined as:

m i n \sum_{n \in N_{G}} f_{n} (P_{n}^{G})

(12)

s.t.

0 = \sum_{n = 1}^{N_{G}} P_{n}^{G} - \sum_{b = 1}^{N_{B}} P_{b}^{L} - \sum_{l = 1}^{N_{L}} P_{l} + P B^{A} \forall n \in N_{G}, \forall b \in N_{B}, \forall l \in N_{L}

(13)

P_{n}^{G, m i n} \leq P_{n}^{G} \leq P_{n}^{G, m a x} \forall n \in N_{G}

(14)

δ_{b}^{m i n} \leq δ_{b} \leq δ_{b}^{m a x} \forall b \in N_{B}

(15)

Here, $P B^{A} = P G_{T}^{D S O i, t}$ for TS. Since DS has utilized ACOPF, (12)-(15) can be rewritten as:

m i n \sum_{n \in N_{G}} f_{n} (P_{n}^{G}, Q_{n}^{G})

(16)

s.t.

\begin{array}{l} 0 = \sum_{n = 1}^{N_{G}} P_{n}^{G} - \sum_{b = 1}^{N_{B}} P_{b}^{L} - \sum_{l = 1}^{N_{L}} P_{l} + P B^{A} \\ \forall n \in N_{G}^{D S}, \forall b \in N_{B}^{D S}, \forall l \in N_{L}^{D S} \end{array}

(17)

0 = \sum_{n = 1}^{N_{G}} Q_{n}^{G} - \sum_{b = 1}^{N_{B}} Q_{b}^{L} - \sum_{l = 1}^{N_{L}} Q_{l} \forall n \in N_{G}^{D S}, \forall b \in N_{B}^{D S}, \forall l \in N_{L}^{D S}

(18)

P_{n}^{G, m i n} \leq P_{n}^{G} \leq P_{n}^{G, m a x} \forall n \in N_{G}^{D S}

(19)

Q_{n}^{G, m i n} \leq Q_{n}^{G} \leq Q_{n}^{G, m a x} \forall n \in N_{G}^{D S}

(20)

V_{b}^{m i n} \leq V_{b} \leq V_{b}^{m a x} \forall b \in N_{B}^{D S}

(21)

δ_{b}^{m i n} \leq δ_{b} \leq δ_{b}^{m a x} \forall b \in N_{B}^{D S}

(22)

In RBSCOPF, the boundary power $P B^{A} = - P G_{R}^{D S O i, t}$ for DS. To reduce reactive power mismatch at the boundary, the voltage angles on the transmission side are considered to be 1 p.u., while on the distribution side, they are considered within the range of 0.95-1.05 p.u..

2) Base-case Optimization Constraints

The operation constraints given below are obligatory for the pre-contingency state of the system:

\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} = \sum_{j} P_{j}^{b}

(23)

\sum_{b} S F_{b}^{0} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b}) \leq P_{l}^{m a x}

(24)

- \sum_{b} S F_{b}^{0} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b}) \leq P_{l}^{m a x}

(25)

P_{n}^{m i n} - R_{j}^{d n} \leq P_{n}^{0} \leq P_{n}^{m a x} - R_{j}^{u p}

(26)

\{\begin{matrix} 0 \leq R_{j}^{u p} \leq R_{j}^{u p, m a x} \\ 0 \leq R_{j}^{d n} \leq R_{j}^{d n, m a x} \end{matrix}

(27)

P T_{m}^{m i n} \leq P T_{m} \leq P T_{m}^{m i n}

(28)

Equations (23)-(28) represent the power balance, power flow limits of internal transmission lines, the lower and upper bounds of the power output of each online generator, the upward/downward generation reserves of each generating unit, and the coupling flows for the base case, respectively.

Besides, the transmission loss is not negligible in DS. Therefore, the power loss in each line should be added. Equations (23)-(25) are then converted to (29)-(31) by adding $P_{l}$ .

\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} = \sum_{j} P_{b}^{j} + \sum_{l} P_{l}

(29)

\sum_{b} S F_{b}^{0} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b} - P_{l}) \leq P_{l}^{m a x}

(30)

- \sum_{b} S F_{b}^{0} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b} - P_{l}) \leq P_{l}^{m a x}

(31)

3) Preventive $N - 1$ Contingency Constraints

In the case of $N - 1$ contingency, the preventive dispatching of generation units controls the branch flow violations to meet the $N - 1$ criteria.

\sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b}) \leq P_{l}^{m a x}

(32)

- \sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b}) \leq P_{l}^{m a x}

(33)

Here, (32) and (33) are used as the constraints for satisfying branch flow limits in $N - 1$ contingent situations. The RAOPF and RBSCOPF both involve preventive constraints.

In RBSCOPF, (32) and (33) are rewritten as (34) and (35), which involve transmission losses in the evaluation of OPF.

\sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b} - P_{l}) \leq P_{l}^{m a x}

(34)

- \sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} P_{j}^{b} - P_{l}) \leq P_{l}^{m a x}

(35)

4) Corrective $N - 2$ Contingency Constraints

In the case of $N - 2$ contingencies caused by common-mode outages, the corrective measures such as generation re-dispatching and load curtailment eliminate post-fault branch overflows.

\sum_{n} P_{n}^{k^{'}} + \sum_{m} I_{m} \cdot P T_{m} = \sum_{j} (P_{j}^{b} - C_{j}^{b, k^{'}})

(36)

\sum_{b} S F_{b}^{k^{'}} (\sum_{n} P_{n}^{k^{'}} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} (P_{j}^{b} - C_{j}^{b, k^{'}})) \leq P_{l}^{m a x}

(37)

- \sum_{b} S F_{b}^{k^{'}} (\sum_{n} P_{n}^{k^{'}} + \sum_{m} I_{m} \cdot P T_{m} - \sum_{j} (P_{j}^{b} - C_{j}^{b, k^{'}})) \leq P_{l}^{m a x}

(38)

P_{n}^{0} - R_{j}^{d n} \leq P_{n}^{k^{'}} \leq P_{n}^{0} - R_{j}^{u p}

(39)

0 \leq C_{j}^{b, k^{'}} \leq α P_{j}^{b}

(40)

The corrective measures must maintain the balance in the system (36) by generation re-dispatching and load curtailment, and bring the branch flows back within their nominal limits (37)-(38). Moreover, (39) allows the re-dispatching of generation units within the imposed limits by the scheduled reserves in the base case. Constraint (40) imposes the maximum limit of lost load j at bus b due to common-mode failure. It should be noted that RAOPF involves corrective constraints, while RBSCOPF does not evaluate corrective constraints. Accordingly, RAOPF involves the objective (12) with constraints (11), (13)-(15), (23)-(28), (32), (33), and (36)-(40). Conversely, RBSCOPF involves the objective (16) with constraints (11), (17)-(22), (26)-(31), (34), and (35).

5) System Risk Evaluation Formulation

For restricting the risk exposure of TS to its tolerable limit, it is enforced to follow the probabilistic constraint to avoid unnecessary load shedding, and balance the anticipated unsupplied demand that occurs due to common-mode outage failures:

\begin{array}{l} R I S K = \sum_{k} π^{k} \cdot S e v_{k} = \sum_{k} π^{k} \cdot V O L L \cdot E N S^{k} = \\ \sum_{k} π^{k} \cdot V O L L \cdot \sum_{j} C D_{j}^{k} \cdot Δ T \leq R i s k_{t h r e s h o l d}^{m a x} \end{array}

(41)

The constraint (41) can manage and mitigate system risk. Here, the system risk is divided into TS and DS risks. Here, VOLL is in $/MWh while ENS^k is in MWh. Hence, RISK is measured as an increase of extra cost in $. Different case studies are carried out to account for the effect of each system on the total system risk. In addition, the selection of a threshold for the system depends upon the appetite and tolerance of TSO considering different levels of severity and probability. In China, provincial grid incidents are categorized into four levels: level I (less than 10% load loss), level II (10%-13% load loss), level III (13%-30% load loss) and level IV (over 30% load loss) [

21]. In Fig. 3, a risk matrix is shown based on three levels of severity and different weather-based contingency probabilities. Therefore, the severities of levels III and IV are considered as the severity of single-level III.

Fig. 3 Risk matrix under normal, adverse and extreme weather-based contingency conditions.

To avoid critical risk in the power system, system operators should select and follow $R i s k_{t h r e s h o l d}^{m a x}$ for the optimized system operation. One should select the risk threshold keeping in view of the above-explained severity and probability levels in Fig. 3. The online risk-based security assessment should be performed before applying the RAOPF and RBSCOPF [

22]. In this way, one can adjust

R i s k_{t h r e s h o l d}^{m a x}

dynamically while considering: ① time-dependent failure rates of transmission lines and distribution lines; ② varying system operation conditions such as network topology, unit commitment, and variable load.

By choosing risk-averse behavior and risk threshold properly, the proposed algorithm can achieve an optimized risk-based cost assessment.

B. RAOPF and RBSCOPF Solutions Using BD

This paper utilizes BD for solving the problem. For TS, the RAOPF problem is divided into master OPF problem and $N - 1$ and $N - 2$ sub-problems, while for DS, the RBSCOPF problem is divided into master OPF problem and $N - 1$ sub-problem.

1) Master OPF Problem

The master OPF problem consists of basic OPF objective function (12) and constraints (13)-(15). The constraints from (23)-(28) are added as base-case constraints.

Furthermore, the proposed algorithm adds more constraints in each iteration, which act as feasibility and optimality cuts from $N - 1$ and $N - 2$ sub-problems, as shown in (32)-(35) and (36)-(40), respectively. The BD-based RAOPF solution is shown in Fig. 4. Further, the RBSCOPF solution process is the same as shown in Fig. 4, but only $N - 2$ contingency is not considered in solving this problem.

Fig. 4 BD framework with master problem and sub-problems for evaluation of RAOPF and RBSCOPF.

2) $N - 1$ Based Preventive Security Sub-problem

For solving $N - 1$ contingency violations, the preventive security of sub-problem is assessed to satisfy the violation of power flow limits:

m i n f^{k} = \sum_{l} (v_{1 l}^{k} + v_{2 l}^{k})

(42)

\sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0, #} + \sum_{m} I_{m} \cdot P T_{m}^{#} - \sum_{j} P_{j}^{b}) - v_{1 l}^{k} \leq P_{l}^{m a x} ϕ_{1 l}^{k}

(43)

- \sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0, #} + \sum_{m} I_{m} \cdot P T_{m}^{#} - \sum_{j} P_{j}^{b}) - v_{2 l}^{k} \leq P_{l}^{m a x} ϕ_{2 l}^{k}

(44)

Here, (42) shows the objective of preventive security check, (43) and (44) shows the constraints for preventive measures, (45) shows the $N - 1$ feasibility cuts that the proposed algorithm generates for preventive measures and adds to the master problem.

\begin{array}{l} f^{k} + \sum_{n} (\sum_{l} (- ϕ_{1 l}^{k} + ϕ_{2 l}^{k}) \sum_{b} S F_{b}^{k}) (P_{n}^{0} - P_{n}^{0, #}) + \\ \sum_{m} (\sum_{l} (- ϕ_{1 l}^{k} + ϕ_{2 l}^{k}) \sum_{b} I_{m} \cdot S F_{b}^{k}) (P T_{m} - P T_{m}^{#}) \leq 0 \end{array}

(45)

For the $N - 1$ sub-problem evaluation in DS, $P_{l}$ variable is added to (43) and (44) to obtain (46) and (47) with the objective (45) remaining the same.

\sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0, #} + \sum_{m} I_{m} \cdot P T_{m}^{#} - \sum_{j} P_{j}^{b} - P_{l}) - v_{1 l}^{k} \leq P_{l}^{m a x} ϕ_{1 l}^{k}

(46)

- \sum_{b} S F_{b}^{k} (\sum_{n} P_{n}^{0, #} + \sum_{m} I_{m} \cdot P T_{m}^{#} - \sum_{j} P_{j}^{b} - P_{l}) - v_{2 l}^{k} \leq P_{l}^{m a x} ϕ_{2 l}^{k}

(47)

3) $N - 2$ Corrective Security Sub-problem

To solve the $N - 2$ common-mode contingencies that lead to power flow violations and unsupplied demand, the proposed algorithm evaluates corrective security sub-problem to calculate the required generation re-dispatching and load curtailment for system power balance.

m i n f^{k^{'}} = u_{1}^{k^{'}} + u_{2}^{k^{'}} + \sum_{l} (u_{3 l}^{k^{'}} + u_{4 l}^{k^{'}}) + \sum_{n} (u_{5 n}^{k^{'}} + u_{6 n}^{k^{'}})

(48)

\sum_{n} P_{n}^{k^{'}} + \sum_{m} I_{m} \cdot P T_{m}^{#} + u_{1}^{k^{'}} - u_{2}^{k^{'}} = \sum_{j} (P_{j}^{b} - C_{j}^{b, k^{'}, #}) τ_{1}^{k^{'}}

(49)

\sum_{b} S F_{b}^{k^{'}} (\sum_{n} P_{n}^{k^{'}} + \sum_{m} I_{m} \cdot P T_{m}^{#} - \sum_{j} (P_{j}^{b} - C_{j}^{b, k^{'}, #})) - u_{3 l}^{k^{'}} \leq P_{l}^{m a x} τ_{2 l}^{k^{'}}

(50)

- \sum_{b} S F_{b}^{k^{'}} (\sum_{n} P_{n}^{k^{'}} + \sum_{m} I_{m} \cdot P T_{m}^{#} - \sum_{j} (P_{j}^{b} - C_{j}^{b, k^{'}, #})) - u_{4 l}^{k^{'}} \leq P_{l}^{m a x} τ_{3 l}^{k^{'}}

(51)

P_{n}^{k^{'}} - P_{n}^{0, #} - u_{5 n}^{k^{'}} \leq R_{n}^{u p, #} τ_{4 n}^{k^{'}}

(52)

P_{n}^{k^{'}} - P_{n}^{0, #} - u_{6 n}^{k^{'}} \leq R_{n}^{d n, #} τ_{5 n}^{k^{'}}

(53)

\begin{array}{l} f^{k^{'}} + \sum_{l} (τ_{4 n}^{k^{'}} - τ_{5 n}^{k^{'}}) (P_{n}^{0} - P_{n}^{0, #}) + \\ \sum_{n} τ_{4 n}^{k^{'}} (R_{n}^{u p} - R_{n}^{u p, #}) + \sum_{n} τ_{5 n}^{k^{'}} (R_{n}^{d n} - R_{n}^{d n, #}) + \\ \sum_{m} I_{m} (- τ_{1}^{k^{'}} + \sum_{l} (- τ_{2 l}^{k^{'}} + τ_{3 l}^{k^{'}}) \sum_{b} S F^{k^{'}}) (P T_{m} - P T_{m}^{#}) + \\ \sum_{j} I_{m} (- τ_{1}^{k^{'}} + \sum_{l} (- τ_{2 l}^{k^{'}} + τ_{3 l}^{k^{'}}) \sum_{b} S F^{k^{'}}) (C_{j}^{b} - C_{j}^{b, #}) \leq 0 \end{array}

(54)

Equation (48) shows the objective of corrective security check, while (49)-(53) show the constraints that must be satisfied for the required corrective measures. In addition, (54) shows the $N - 2$ corrective feasibility and optimality cut. The proposed algorithm then adds this constraint to the master problem.

The master problem deals with these cuts as constraints, and optimizes iteratively to achieve the minimum cost for the required generation re-dispatching and load curtailment.

C. Formulation of Hierarchical Structure of RA-DOPF

The hierarchical structure of RA-DOPF involves the objective functions of TSO and DSO. The coordination method is shown in Fig. 5. Further explanation is provided in this sub-section about the coordinated formulation of TSO and DSOs.

Fig. 5 Coordination of TSO and DSOs.

1) Formulation of Coordinated TSO RAOPF

TSO is responsible for solving its RAOPF problem with common-mode contingencies due to the weather-based transmission outages. TSO also generate the targets, which are sent to DSOs. The following formulation (55) is utilized for the evaluation of coordinated TSO RAOPF, subject to (13)-(15), (23)-(28), (45), (54), and (56)-(59).

m i n \{\sum_{n} f_{n}^{R A} (P_{n}^{G}, R i s k_{T}^{T S O, t}) + \sum_{i = 1}^{I} [ρ_{D S O i}^{t^{'}} (P G_{R}^{D S O i, t} - P G_{T}^{D S O i, t}) \overset{}{\underset{}{+}} {‖η_{D S O i}^{t^{'}} \circ (P G_{R}^{D S O i, t} - P G_{T}^{D S O i, t})‖}^{2}] + α_{T S O}^{t^{'} + 1} (R i s k_{R}^{T S O, t} - R i s k_{T}^{T S O, t}) + {‖β_{T S O}^{t^{'}} \circ (R i s k_{R}^{T S O, t} - R i s k_{T}^{T S O, t})‖}^{2}\}

(55)

P G_{T}^{D S O i, t} = P_{L}^{D S O i, t}

(56)

\sum_{i = 1}^{I} P G_{T}^{D S O i, t} \leq \sum_{i = 1}^{I} \sum_{n = 1}^{N_{G}} P_{g, m a x}^{D S O i, n}

(57)

\sum_{i = 1}^{I} P G_{T}^{D S O i, t} \geq \sum_{i = 1}^{I} \sum_{n = 1}^{N_{G}} P_{g, m i n}^{D S O i, n}

(58)

R i s k_{T}^{T S O, t} \leq R i s k_{t h r e s h o l d}^{m a x}

(59)

The symbol $\circ$ is the Hadamard product for element-wise multiplication. If the contingency risk is located within TS, $R i s k_{R}^{T S O}$ is considered as the risk threshold $R i s k_{t h r e s h o l d}^{m a x}$ . On the other hand, $R i s k_{T}^{T S O} = R i s k^{T S O} + \sum_{i = 1}^{I} R i s k^{D S O i}$ is considered as overall system risk. When $R i s k_{T}^{T S O}$ achieves less value than $R i s k_{t h r e s h o l d}^{m a x}$ with less cost, the value of $R i s k_{t h r e s h o l d}^{m a x}$ is updated to $R i s k_{t h r e s h o l d}^{m a x} = R i s k_{T}^{T S O}$ . In this way, the cost can be reduced further beyond the risk threshold. Although, an operator can decide whether to allow the algorithm to search for further least cost with less risk, or stop the algorithm at the initial risk threshold $R i s k_{t h r e s h o l d}^{m a x}$ .

2) Formulation of Coordinated DSO OPF

In modern power systems, ADNs play a vital role in the system cost optimization. DSO needs to optimize its cost objective function given in (60), subject to (11), (17)-(22), (26)-(31), and (61)-(63).

m i n \{\sum_{n} f_{n} (P_{n}^{G}, Q_{n}^{G}) + [ρ_{D S O i}^{t^{'}} (P G_{T}^{D S O i, t} - P G_{R}^{D S O i, t}) \overset{}{\underset{}{+}} {‖η_{D S O i}^{t^{'}} \circ (P G_{T}^{D S O i, t} - P G_{R}^{D S O i, t})‖}^{2}]\}

(60)

V_{P B^{A}}^{D S O i} = V_{P B^{A}}^{T S O}

(61)

P G_{R}^{D S O i, t} \leq \sum_{n = 1}^{N_{G}} P_{g, m a x}^{D S O i, n}

(62)

P G_{R}^{D S O i, t} \geq \sum_{n = 1}^{N_{G}} P_{g, m i n}^{D S O i, n}

(63)

The TSO utilizes the DCOPF, thus, $V_{P B^{A}}^{T S O} = 1$ , and (61) can take care of the reactive power mismatch at the boundary of TS and DSs. If there is no contingency risk in the DS, TS risk $R i s k_{T}^{T S O}$ will be considered as the overall system risk.

3) Formulation of Coordinated DSO RBSCOPF for Islanding

Due to the contingencies in DS, we need to investigate the islanding risk for the secure operation of the power system. For this reason, (64) has been utilized for evaluating the islanding-risk-based security-constrained OPF.

\{\begin{array}{l} \begin{array}{l} m i n \{\sum_{n} f_{n}^{S C} (P_{n}^{G}, Q_{n}^{G}, R i s k_{i s l}^{D S O i}) + [ρ_{D S O i}^{t^{'}} (P G_{T}^{D S O i, t} - P G_{R}^{D S O i, t}) \overset{}{\underset{}{+}} \\ {‖η_{D S O i}^{t^{'}} \circ (P G_{T}^{D S O i, t} - P G_{R}^{D S O i, t})‖}^{2}]\} \end{array} \\ s . t . (11), (17) - (22), (26) - (31), (45), (61) - (63) \end{array}

(64)

DSO needs to evaluate separately the islanded section, which is disconnected from the main transformer bus. It is worth mentioning that $R i s k_{i s l}^{i}$ will be added with $R i s k_{T}^{T S O}$ to calculate the system risk for the simultaneous occurrence of islanding problem in DS and contingent problem in TS. This will increase the overall system risk, and a better optimized operation state can be achieved by reducing the risk.

D. Hierarchical RA-DOPF Solution Using ATC

Figure 6 shows the flowchart of the RA-DOPF algorithm with the inner and outer loops of ATC. The following algorithm flow is proposed for solving the RA-DOPF problem for TS and DS.

Fig. 6 Flowchart of coordinated risk-based OPF of TSO and DSO.

Step 1: at the upper level, TSO evaluates the RAOPF problem and targets $P G_{T}^{D S O i, t}$ . Then, TSO communicates them to all DSOs. The risk calculated in the RAOPF problem is set to be a risk response for the system.

Step 2: at lower levels, each DSO receives targets from TSO and solves its RBSCOPF problem along with curtailment load calculation with and without islanding, respectively. DSOs send back the responses $P G_{R}^{D S O i, t}$ to TSO. Additionally, each DSO calculates the curtailed load for the cases with and without the support of DERs in DS, respectively. In the proposed algorithm, DSOs share this information with TSO for the evaluation of risks, i.e., expected system risk and reduced system risk. When the coordinated OPF of coupled TS and DSs is evaluated, the total evaluated risk using (7) is called the expected system risk. RAOPF and RBSCOPF are risk corrective methods. When these OPF methods are applied to coupled TS and DSs, the total evaluated risk is called reduced system risk.

Step 3: after receiving data from all DSOs, TSO solves its RAOPF problem to generate new targets $P G_{T}^{D S O i, t}$ for the next iteration while evaluating the minimum risk $R i s k_{T}^{T S O, t} = R i s k^{T S O} + \sum_{i = 1}^{I} R i s k^{D S O i}$ that is achievable.

Step 4: in the IL, TSO checks whether the following stopping rules are satisfied:

|P G_{R}^{D S O i, t} - P G_{R}^{D S O i, t - 1}| \leq ε_{1}

(65)

|P G_{T}^{D S O i, t} - P G_{T}^{D S O i, t - 1}| \leq ε_{1}

(66)

|R i s k_{T}^{T S O, t} - R i s k_{T}^{T S O, t - 1}| \leq ε_{2}

(67)

Step 5: in the OL, the following sufficient conditions are checked for convergence:

|\frac{S y s_{c o s t}^{t^{'}} - S y s_{c o s t}^{t^{'} - 1}}{S y s_{c o s t}^{t^{'}}}| \leq ε_{3}

(68)

S y s_{c o s t}^{t^{'}} = T S_{c o s t}^{t^{'}} + \sum_{i = 1}^{i} D S_{c o s t}^{i, t^{'}}

(69)

If the above conditions are satisfied, the iteration is ended. Otherwise, the multipliers are updated by using the following equations:

ρ_{D S O i}^{t^{'} + 1} = ρ_{D S O i}^{t^{'}} - 2 {(η_{D S O i}^{t^{'}})}^{2} (P G_{R}^{D S O i, t^{'}} - P G_{T}^{D S O i, t})

(70)

η_{D S O i}^{t^{'} + 1} = σ_{1} η_{D S O i}^{t^{'}}

(71)

α_{T S O}^{t^{'} + 1} = α_{T S O}^{t^{'}} - 2 {(β_{T S O}^{t^{'}})}^{2} (R i s k_{R}^{T S O, t^{'}} - R i s k_{T}^{T S O, t^{'}})

(72)

β_{T S O}^{t^{'} + 1} = σ_{2} β_{T S O}^{t^{'}}

(73)

Step 6: the algorithm repeats Steps 2-4 until the stopping criteria are met.

E. Convergence and Optimality of Proposed Algorithm

In this paper, we propose an RA-DOPF model for solving the risk problem with the minimum cost for coupled TS and DSs. Further, ATC is utilized as a coordination algorithm whose convergence and optimality has been proven in [

23] for convex problems even with discrete variables. In addition, three conditions should be satisfied to hold these proofs valid:

1) There will be communication between the upper- and lower-level sub-systems.

2) The communication between the same-level sub-systems is not allowed.

3) At the t^th iteration, a sequential process is followed where target values are required for every sub-system, i.e., DSOs, to evaluate their response values, which are required by the upper-level problem, i.e., TSO, to generate new targets.

To show the superiority of ATC, it is necessary to discuss the capability of the ATC algorithm to solve non-convex problems such as DCOPF with losses and ACOPF. The augmented Lagrange method is used as a penalty method, which involves a linear term and a quadratic term. As discussed in [

21], [24], the quadratic term acts as a convex factor for the non-convex objectives. Consequently, the quadratic penalty function in the objective function of the proposed method can convexify the non-convex problem. Hence, this proves the convergence of the proposed method even for the non-convex problems. The convergence and optimality of the proposed method have been validated using different case studies.

IV. Case Study

In this paper, a modified test system is used for the TS and DS. The IEEE 30-bus system with reserves acts as TS while the IEEE 13-bus radial DS with five DERs acts as DS. It should be noted that the total demand P_d of the system is scaled proportionally on all buses of TS. Likewise, for a particular DS, the load at relevant boundary bus of TS system connected to the DS is scaled proportionally on all buses of the DS. This means that a total load of the DS will be equal to the total load at the relevant bus of the TS system. Further, the impedance of the line, which connects the boundary bus of every DS with TS, is set to be $0.015 + j 0.2$ p.u.. Also, the simulation has been performed on Core i7 2.7 GHz with 8 GB RAM, with MATLAB R2018a. Matpower has been used for OPF analysis and for obtaining the IEEE system datasets [

25].

Figure 7 shows the system model used for the evaluation of results. Tables I and II show the generator and bus data and branch data for IEEE 13-bus radial DS. It is depicted that ten ADNs are considered on different buses of the IEEE 30-bus system. In addition, each ADN is the same as ADN 1 (a 13-bus radial DS) as shown in Fig. 7.

Fig. 7 System model of IEEE 30-bus system as TS and ten IEEE 13-bus radial DSs (ADNs) on shown buses of TS.

TABLE I Generator and Bus Data for IEEE 13-bus Radial DS

Unit	Bus	P_max (MW)	b (MBtu/ MW)	c (MBtu/ MW²)
DG 1	1	3	0.9	0.075
DG 2	4	5	0.7	0.061
DG 3	7	2	0.2	0.038
DG 4	9	3	0.3	0.041
DG 5	11	2	1.0	0.011

TABLE II Branch Data for IEEE 13-bus Radial DS

From-to buses	R (Ω)	X (Ω)	From-to buses	R (Ω)	X (Ω)
1-2	0.0922	0.0470	7-8	0.7114	0.2351
2-3	0.4930	0.2511	8-9	1.0300	0.7400
3-4	0.3660	0.1864	7-10	1.0440	0.7400
2-5	0.3811	0.1941	10-11	0.1966	0.0650
5-5	0.8190	0.7070	7-12	0.3744	0.1238
2-7	0.1872	0.6188	8-13	1.4680	1.1550

This paper involves $N - 2$ contingency after $N - 1$ contingency in the DS considering the automated switch operation. Thus, $N - 2$ contingency in the DS utilizes the formulation of RBSCOPF, as $N - 1$ contingency is already tackled by automated switch operation. To reduce the complexity of the problem, only DGs are considered as DERs. Henceforth, the uncertainty due to wind farms or solar power is not involved in this paper.

Here, the evaluated results are represented as system cost and system risk. The system risk is also the same as reduced system risk when a comparison between expected system risk and reduced system risk is provided. Further, expected system risk means the evaluation of risk without involving risk awareness measures given in the proposed algorithm.

A. Case 1: RA-DOPF in Coordinated TS and DS with Contingent Situations Only in TS

Usually, DSOs are needed to share necessary data for defining the observability area of TSO, but any such request by TSO is subject to Article 4.2 of the system operation (SO) regulation [

26]. Such requests require four principles that should be followed: ① the application of the principles of proportionality; ② non-discrimination; ③ transparency; ④ the principle of optimization between the highest overall efficiency and lowest total costs for all parties involved. Therefore, in some cases, DSOs are unable to share the risk information when it does not come under the SO regulations. This is the reason why case 1 is considered with contingencies only in the TS section, but not in DS sections.

In this case, RAOPF has been performed on TS with reserves while DCOPF is performed on DS. The coordinated risk-aware cost has been evaluated. The line outage in TS can cause the generation resources or load to be cut off from the main system. In this way, TS reserves and DERs in DS are used in a coordinated manner to reduce the total risk and cost. Also, the proposed algorithm allows TS and DS to adjust their generation source outputs to mitigate the projected risk of system and attain a state with low risk and optimal cost.

However, Fig. 8 clearly shows that the overall system cost is increasing consecutively to reduce the post-contingency risk. It depicts that the system cost increases to reduce the system risk. However, DS will help reduce the system risk and bring it within the risk threshold limits. Then, the proposed method has given more priority to risk reduction than cost minimization to make the system operation more secure. This case study will help system operators to prevent the after-effects of contingency in the TS.

Fig. 8 Comparison of reduced system cost and system risk in case 1.

Furthermore, Fig. 9 shows a comparison of expected system risk and reduced system risk. It can be observed that without utilizing the proposed algorithm, the risk is high enough, and the system operation will be insecure.

Fig. 9 Comparison of reduced system risk and expected system risk in case 1.

B. Case 2: RA-DOPF in Coordinated TS and DS with Contingent Situations Only in DS

Some TSOs need a larger observability area including different interconnected systems due to structure and operation conditions of the power system [

26]. However, it is not an efficient way to impose such requirements on other TSOs (with smaller observability arear) to invest more resources. Therefore, it is a non-beneficial application of SO guideline (GL) Article 4.2c to impose the same threshold on these TSOs than that needed for the previous ones. Subsequently, the risk evaluation in some TSOs due to small observability areas is not an efficient way for utilizing resources. This is why we consider case 2 with contingencies only in DSs.

This case study considers the DS islanding impact on TS risk evaluation. Besides, DCOPF is performed in TS while RBSCOPF is applied in DS to evaluate the risk due to islanding. When a line outage occurs in DS, it can create islands within DS. Thus, the islands connected to the main transformer bus will remain intact, and no load loss is expected. While other islands disconnected from the transformer bus will face the problem of curtailment. In this case, DERs in these islanded sections can be handy and help reduce the load curtailment. Besides, large-scale DERs are considered. Subsequently, the load curtailment is the minimum if the proposed algorithm utilizes DERs efficiently. In other words, DERs in DS can help reduce the total system risk by taking up the expected curtailed load. The proposed method will evaluate the risk due to islanding in DS and adjust risk within the threshold limits. Likewise, the proposed method also optimizes the cost with reasonable risk.

Further, Fig. 10 shows the system risk, and the system cost is reduced in consecutive iterations. It is worth noticing that the minimum cost achieved in this case study is yet more than the cost achieved in a non-risk constrained system. Table III shows the results of the cost and risk for this case study. Also, the proposed algorithm is yet trying to reduce the cost beyond the risk threshold in consecutive iterations by utilizing cheap generation resources from TS and DSs. Figure 11 shows that the DERs in expected islands are partially mitigating the load curtailment problem. Thus, the proposed method can reduce the system risk, which is much less than the expected system risk.

Fig. 10 Comparison of reduced system cost and system risk in case 2.

TABLE III Comparison of Three Case Studies on Basis of Different Area Costs and Risks

Case	System cost ($)	TS cost ($)	DS cost ($)	Expected system risk ($)	Reduced system risk ($)	System risk threshold ($)	Simulation time (s)
1	691.7121	602.5646	89.1475	2577.9234	1620.1128	1800	285.0912
2	674.1914	622.9427	51.2487	452.7320	206.8100	250	66.8337
3	720.0192	640.3176	79.7016	3447.4405	1902.9093	2000	354.5172

Fig. 11 Comparison of reduced system risk and expected system risk in case 2.

C. Case 3: RA-DOPF in Coordinated TS and DS with Contingent Situations in Both TS and DS

For a system with no such constraints on TS and DS described in earlier case studies, both TS and DS can share data for risk evaluation. Consequently, in case 3, the contingencies in both TS and DS are evaluated. This case study deals with a specific case of simultaneous line outage in TS and DS, which causes common-mode outages in TS and islanding in DS. Such risk is less probable, but it is necessary to study and evaluate such a problem because of its criticality. In this case, RAOPF is performed in TS while RBSCOPF in DS. The islanding in DS can divide it into different sections: one is connected to the main transformer bus while other sections act as an island. TS risk can be reduced using DER generation, which is available in the DS section connected to the main transformer bus. While DERs available in the islanded section can take up the load in that section, and, henceforth, reduce the load curtailment. It should be noticed that the overall system risk is the sum of TS risk and DS risk. Therefore, the overall system risk is greater than that evaluated in cases discussed above.

Figure 12 shows that the risk reduction in this case study is more than that in other case studies. This shows that the simultaneous occurrence of contingencies in TS and DS can lead to high risk. Meanwhile, the proposed method has successfully reduced the risk below the threshold of this case study. Figure 13 shows the risk comparison of reduced system risk and expected system risk. Here, the reduced system risk is evaluated using the proposed method. Likewise, Table III shows a brief comparison of results in cases 1-3.

Fig. 12 Comparison of reduced system cost and system risk in case 3.

Fig. 13 Comparison of reduced system risk and expected system risk in case 3.

D. Comparison of RA-DOPF with Centralized Case and Previous Literature

Table IV shows the comparison of centralized and decentralized coordination of TS and DS. The centralized cost is evaluated considering the IEEE 30-bus system and IEEE 13-bus radial DS as a single system with different voltage levels at different buses. In that single system, $N - 1$ and $N - 2$ contingencies are evaluated in the section of the IEEE 30-bus system, while $N - 1$ contingency is considered in the IEEE 13-bus radial DS. Consequently, the combination of contingencies is prepared in the following manner.

TABLE IV Comparison of Centralized Method with Decentralized Method

Case	Centralized		Decentralized (RA-DOPF)
Case	System cost ($)	System risk ($)	System cost ($)	System risk ($)
1	663.4372	1607.9103	691.7121	1620.1128
2	669.2115	205.3085	674.1914	206.8100
3	698.0785	1897.6250	720.0192	1902.9093

1) Case 1: $N - 1$ and $N - 2$ contingencies in the section of the IEEE 30-bus system while no contingency in the section of IEEE 13-bus radial DS.

2) Case 2: no contingency in the section of the IEEE 30-bus system while $N - 1$ contingency in the section of the IEEE 13-bus radial DS.

3) Case 3: $N - 1$ and $N - 2$ contingencies in the section of the IEEE 30-bus system and $N - 1$ contingency in the section of the IEEE 13-bus radial DS.

The truncated diagonal quadratic approximated (TDQA) algorithm [

27] is utilized for the evaluation of DOPF without risk awareness in coordinated TS and DS. Furthermore, ATC-based DOPF algorithm [28] is utilized for the evaluation of security-constrained DOPF (SCDOPF) in coordinated TS and DS. It should be noted that SCDOPF involves only

N - 1

contingency evaluation in transmission and DSs. In Table V, a comparison is provided between the proposed and the above-mentioned algorithms. The results show that the system cost would increase in the proposed algorithm while reducing the risk to its threshold. Additionally, under adverse or extreme weather conditions, the system security has a higher priority than the system cost for the system operator. Further, this algorithm can help system operators under normal weather conditions at the expense of a small increase in cost, which is discussed in the next section. Then, this algorithm can handle real-time risk-based cost assessment problems, while previous algorithms can only solve simple or security-constrained cost assessment problems.

TABLE V Comparison of Previous Literature with Proposed Algorithm

Case	System cost ($)			System risk ($)
Case	RA-DOPF	DOPF [27]	SCDOPF [28]	RA-DOPF	DOPF [27]	SCDOPF [28]
1	691.7121	601.9101	647.0931	1620.1128	3275.1639	2715.7430
2	674.1914	601.9101	647.0931	206.8100	3275.1639	2715.7430
3	720.0192	601.9101	647.0931	1902.9093	3275.1639	2715.7430

E. Voltage Optimization Assessment of Proposed Algorithm

Since ACOPF is utilized in DSOs for the evaluation of risk-based cost, the bus voltage optimization should be presented for the proposed algorithm. Case 3 involves higher risk than cases 1 and 2. Therefore, more voltage violations are expected in case 3. Hence, in Table VI, the bus voltages of 10 DSOs (p.u.) are shown only for case 3. It is illustrated that the proposed algorithm has kept the bus voltages within a given range of 0.95-1.05 p.u., but also near to 1.0 p.u..

TABLE VI Voltage Magnitudes (in p.u.) at Buses of 10 DSOs in Case 3

Bus No.	DSO 1	DSO 2	DSO 3	DSO 4	DSO 5	DSO 6	DSO 7	DSO 8	DSO 9	DSO 10
1	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000
2	0.999982	1.000030	1.000002	0.999985	1.000012	0.999982	0.999985	0.999998	0.999997	1.000028
3	0.999944	1.000019	0.999961	0.999929	0.999980	0.999941	0.999938	0.999962	0.999953	1.000009
4	0.999949	1.000111	1.000063	0.999938	0.999984	0.999946	0.999945	1.000012	0.999959	1.000041
5	0.999897	0.999772	0.999662	0.999858	0.999941	0.999889	0.999878	0.999800	0.999898	0.999908
6	0.999780	0.999422	0.999201	0.999686	0.999846	0.999763	0.999732	0.999531	0.999765	0.999745
7	1.000023	1.000281	1.000265	1.000066	1.000097	1.000026	1.000050	1.000151	1.000078	1.000177
8	1.000021	1.000463	1.000385	1.000116	1.000184	1.000028	1.000082	1.000290	1.000140	1.000353
9	1.000143	1.001105	1.001052	1.000373	1.000411	1.000165	1.000284	1.000779	1.000374	1.000783
10	0.999984	1.000472	1.000516	1.000010	1.000065	0.999985	1.000003	1.000062	1.000033	1.000123
11	0.999985	1.000531	1.000596	1.000011	1.000066	0.999985	1.000004	1.000063	1.000034	1.000124
12	1.000007	1.000233	1.000203	1.000042	1.000085	1.000009	1.000030	1.000115	1.000060	1.000155
13	0.999935	1.000207	1.000048	0.999990	1.000114	0.999936	0.999975	1.000094	1.000042	1.000234

Furthermore, the boundary voltages in TS and DSs have a quite negligible mismatch as shown in Table VII. This results in a non-significant mismatch in reactive power at boundary buses.

TABLE VII Voltage Magnitudes at Boundary Buses of 10 DSOs and TSO in Case 3

DSO No.	Voltage magnitude at DSO boundary bus (p.u.)	Voltage magnitude at TSO boundary bus (p.u.)	Voltage mismatch error (%)
1	1.002604	1.000000	0.260
2	0.997927	1.000000	0.207
3	0.996311	1.000000	0.379
4	1.001965	1.000000	0.197
5	0.997615	1.000000	0.239
6	1.002729	1.000000	0.273
7	1.001987	1.000000	0.199
8	0.999131	1.000000	0.087
9	1.000026	1.000000	0.003
10	0.997519	1.000000	0.248

F. Tradeoff Between Risk and Cost

Generally, the cost is increased for the power system whenever the risk-averse measures are taken. The same phenomenon has been depicted in Fig. 14 (a)-(c) for cases 1-3. In Fig. 14(a)-(c), the thresholds are set to be 1800, 250, and 2000 for cases 1-3, respectively. When analyzing these for the tradeoff between risk and cost from high risk to low risk, it is observed that the risk reduction rate is higher before the threshold is reached. The risk reduction rate is decreased after passing the threshold point. This shows that the threshold set for cases 1-3 is near the minimum achievable risk for the minimum possible cost. For power system operators, such tradeoff is obligatory for the sake of secure operation of the power system. In the risk mitigation process, pre-contingency assessment is a better way to deal with contingencies. Accordingly, the operator can make decision by utilizing this method, especially when there are high probabilities of adverse and extreme weather conditions.

Fig. 14 Trade-off relationship between risk and system cost. (a) Case 1 with risk threshold of $1800. (b) Case 2 with risk threshold of $250. (c) Case 3 with risk threshold of $2000.

G. Effect of Weather-based Contingencies on Operation Cost

Since the probability of emergencies under extreme weather conditions is much more than normal and adverse weather, the operation cost of the system increases so as to control such a high-risk threshold. Accordingly, in the proposed method, the preventive generation and reserve allocation are adjusted dynamically to limit the contingency risk.

Figure 15 shows the dynamic adjustment of preventive generation dispatching and reserve allocation of generators in TS to limit the contingency risk. The same process is followed for DS for the adjustment of dynamic dispatching. This provides another significant advantage to the proposed method over the DSCED method, as the proposed method considers the variable risk due to varying weather conditions. Additionally, the minimum cost and risk thresholds under different weather conditions are given in Table VIII. This depicts that operation cost increases with an increase of the severity of the weather conditions.

Fig. 15 Dynamic adjustment of generation dispatching of TS generators under different weather conditions.

TABLE VIII The Minimum Cost and Risk Thresholds Comparison Under Different Weather Conditions

Case	Normal weather		Adverse weather		Extreme weather
Case	System cost ($)	System risk ($)	System cost ($)	System risk ($)	System cost ($)	System risk ($)
1	605.1319	6.1	620.1437	410	691.7121	1800
2	607.1436	2.3	634.5158	40	674.1914	250
3	608.0641	8.5	659.1588	570	720.0192	2000

V. Conclusion and Future Directions

This paper has presented an RA-DOPF model for coordinated TS and DS. The proposed method has emphasized that the RAOPF in coupled TS and DS is inevitable. In previous research, the coordinated cost optimization in these systems mostly leads to a high-risk problem that this paper has addressed efficiently. On one hand, TS and DS both contribute to risk origination. On the other hand, the proposed method efficiently utilizes TS sources and DERs in DS to mitigate the risk coordinately.

Different case studies have described different categories of possible contingencies within both TS and DS. In the case of TS, the preventive and corrective measures are taken into account for mitigating $N - 1$ and $N - 2$ contingencies. Also, $N - 2$ contingencies are considered in DS and its preventive measures are evaluated. Different case studies give a brief view of contingencies in different system boundaries and their effect on the total risk. Besides, the proposed method is evaluated for all cases and its efficacy is demonstrated. The RA-DOPF helps both systems to reduce their risk within their limits along with the optimized cost and less load curtailment. In the future, this research can be enhanced using the probabilistic coordination methods for the sources with uncertainty such as wind farms and photovoltaic. Moreover, the coordination of voltage stability between TS and DS could be considered in future studies.

REFERENCES

I. Kouveliotis-Lysikatos and N. Hatziargyriou, “Fully distributed economic dispatch of distributed generators in active distribution networks considering losses,” IET Generation, Transmission & Distribution, vol. 11, no. 3, pp. 627-636, Feb. 2017. [百度学术]

C. Lin, W. Wu, Z. Li et al., “Decentralized economic dispatch for transmission and distribution networks via modified generalized benders decomposition,” in Proceedings of 2017 IEEE PES General Meeting, Chicago, USA, Jul. 2017, pp. 1-5. [百度学术]

Y. Wen, C. Y. Chung, and X. Liu, “Hierarchical interactive risk hedging of multi-TSO power systems,” IEEE Transactions on Power Systems, vol. 33, no. 3, pp. 2962-2974, Aug. 2018. [百度学术]

Z. Li, Q. Guo, H. Sun et al., “Coordinated transmission and distribution AC optimal power flow,” IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 1228-1240, Jun. 2016. [百度学术]

A. Kargarian, M. Mehrtash, and B. Falahati, “Decentralized implementation of unit commitment with analytical target cascading: a parallel approach,” IEEE Transactions on Power Systems, vol. 33, no. 4, pp. 3981-3993, Dec. 2018. [百度学术]

M. Bragin, Y. Dvorkin, and A. Darvishi, “Toward coordinated transmission and distribution operations,” in Proceedings of 2018 IEEE PES General Meeting, Portland, USA, Aug. 2018, pp. 1-5. [百度学术]

A. Kargarian and Y. Fu, “System of systems based security-constrained unit commitment incorporating active distribution grids,” IEEE Transactions on Power Systems, vol. 29, no. 5, pp. 2489-2498, Mar. 2014. [百度学术]

A. Nawaz, H. Wang, Q. Wu et al., “TSO and DSO with large-scale distributed energy resources: a security constrained unit commitment coordinated solution,” International Transactions on Electrical Energy Systems, vol. 30, no. 3, pp. 1-26, Mar. 2020. [百度学术]

ENTSO-E. (2015, Nov.). General guidelines for reinforcing the cooperation between TSOs and DSOs. [Online]. Available: https://docstore.entsoe.eu/Documents/Publications/Position%20papers%20and%20reports/entsoe_pp_TSO-DSO_web.pdf#search=tso%2Ddso [百度学术]

M.-S. Kim, R. Haider, G.-J. Cho et al., “Comprehensive review of islanding detection methods for distributed generation systems,” Energies, vol. 12, pp. 1-21, Mar. 2019. [百度学术]

P. Mahat, C. Zhe, and B. Bak-Jensen, “Review of islanding detection methods for distributed generation,” in Proceedings of 2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Nanjing, China, Apr. 2008, pp. 2743-2748. [百度学术]

A. Nawaz and H. Wang, “Stochastically coordinated transmission and distribution system operation with large-scale wind farms,” CSEE Journal of Power and Energy Systems, doi: 10.17775/CSEEJPES.2020.02150. [百度学术]

Z. Li, Q. Guo, H. Sun et al., “A new LMP-sensitivity-based heterogeneous decomposition for transmission and distribution coordinated economic dispatch,” IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 931-941, Jul. 2017. [百度学术]

ENTSO-E. (2015, Mar.). Towards smarter grids: developing TSO and DSO roles and interactions for the benefit of consumers. [Online]. Available: https://docstore.entsoe.eu/Documents/Publications/Position%20papers%20and%20reports/150303_ENTSO-E_Position_Paper_TSO-DSO_interaction.pdf [百度学术]

A. Olson, A. Mahone, E. Hart et al., “Halfway there: can California achieve a 50% renewable grid?” IEEE Power and Energy Magazine, vol. 13, pp. 41-52, Jun. 2015. [百度学术]

I. Olivine. (2014, Jan.). Distributed energy resources integration summarizing the challenges and barriers. [Online]. Available: http://www.caiso.com/Documents/OlivineReport_DistributedEnergyResourceChallenges_Barriers.pdf [百度学术]

R. Billinton, C. Wu, and G. Singh, “Extreme adverse weather modelling in transmission and distribution system reliability evaluation,” in Proceedings of 14th Power System Calculation Conference, Seville, Spain, Jun. 2002, pp. 1-7. [百度学术]

W. Li and R. Billinton, “Common cause outage models in power system reliability evaluation,” IEEE Transactions on Power Systems, vol. 18, no. 2, pp. 966-968, May 2003. [百度学术]

M. Papic, S. Agarwal, R. Allan et al., “Research on common-mode and dependent (CMD) outage events in power systems: a review,” IEEE Transactions on Power Systems, vol. 32, no. 2, pp. 1528-1536, Jul. 2016. [百度学术]

E. Ciapessoni, D. Cirio, G. Kjølle et al., “Probabilistic risk-based security assessment of power systems considering incumbent threats and uncertainties,” IEEE Transactions on Smart Grid, vol. 7, no. 6, pp. 2890-2903, Mar. 2016. [百度学术]

H. M. Kim, W. Chen, and M. M. Wiecek, “Lagrangian coordination for enhancing the convergence of analytical target cascading,” AIAA Journal, vol. 44, pp. 2197-2207, Sept. 2006. [百度学术]

N. Ming, J. D. McCalley, V. Vittal et al., “Online risk-based security assessment,” IEEE Transactions on Power Systems, vol. 18, no. 1, pp. 258-265, Feb. 2003. [百度学术]

N. Michelena, H. Park, and P. Y. Papalambros, “Convergence properties of analytical target cascading,” AIAA Journal, vol. 41, pp. 897-905, May 2003. [百度学术]

S. Tosserams, L. Etman, P. Papalambros et al., “An augmented Lagrangian relaxation for analytical target cascading using the alternating direction method of multipliers,” Structural and Multidisciplinary Optimization, vol. 31, pp. 176-189, Feb. 2006. [百度学术]

R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education,” IEEE Transactions on Power Systems, vol. 26, no. 2, pp. 12-19, Jun. 2011. [百度学术]

2 August 2017 Establishing a Guideline on Electricity Transmission System Operation, Commission Regulation (EU) 2017/1485, Aug. 2017. [百度学术]

A. Mohammadi, M. Mehrtash, and A. Kargarian, “Diagonal quadratic approximation for decentralized collaborative TSO + DSO optimal power flow,” IEEE Transactions on Smart Grid, vol. 10, no. 3, pp. 2358-2370, Jan. 2018. [百度学术]

A. Kargarian, J. Mohammadi, J. Guo et al., “Toward distributed/decentralized DC optimal power flow implementation in future electric power systems,” IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 2574-2594, Oct. 2016. [百度学术]

Address:No.19 Chengxin Avenue, Jiangning District, Nanjing 211106, China

E-mail: mpce@alljournals.cn

Tel:86-25-81093060

Fax:86-25-81093040

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Risk-aware Distributed Optimal Power Flow in Coordinated Transmission and Distribution System PDF

Abstract

Keywords

I. INTRODUCTION

II. Hierarchical Coordination Framework for RA-DOPF

A. Weather-based Contingency Risk Evaluation

B. Hierarchical RA-DOPF Problem

III. Problem Formulation and Solution

A. Formulation of RAOPF and RBSCOPF

B. RAOPF and RBSCOPF Solutions Using BD

C. Formulation of Hierarchical Structure of RA-DOPF

D. Hierarchical RA-DOPF Solution Using ATC

E. Convergence and Optimality of Proposed Algorithm

IV. Case Study

A. Case 1: RA-DOPF in Coordinated TS and DS with Contingent Situations Only in TS

B. Case 2: RA-DOPF in Coordinated TS and DS with Contingent Situations Only in DS

C. Case 3: RA-DOPF in Coordinated TS and DS with Contingent Situations in Both TS and DS

D. Comparison of RA-DOPF with Centralized Case and Previous Literature

E. Voltage Optimization Assessment of Proposed Algorithm

F. Tradeoff Between Risk and Cost

G. Effect of Weather-based Contingencies on Operation Cost

V. Conclusion and Future Directions

REFERENCES