On Decision-Dependent Uncertainties in Power Systems with High-Share Renewables

Yunfan Zhang; Yifan Su; Feng Liu

doi:10.1016/j.eng.2025.07.013

Engineering ›› 2025, Vol. 51 ›› Issue (8) :104 -124. DOI: 10.1016/j.eng.2025.07.013

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On Decision-Dependent Uncertainties in Power Systems with High-Share Renewables

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Abstract

The continuously increasing renewable energy sources (RES) and demand response (DR) are becoming crucial sources of system flexibility. Consequently, decision-dependent uncertainties (DDUs), interchangeably referred to as endogenous uncertainties, impose new characteristics on power system dispatch. The DDUs faced by system operators originate from uncertain dispatchable resources such as RES units or DR, while reserve providers encounter DDUs from the uncertain reserve deployment. Thus, a systematic framework was established in this study to address robust dispatch problems with DDUs. The main contributions are drawn as follows. ① The robust characterization of DDUs was unfolded with a dependency decomposition structure. ② A generic DDU coping mechanism was manifested as the bilateral matching between uncertainty and flexibility. ③ The influence of DDU incorporation on the convexity/non-convexity of robust dispatch problems was analyzed. ④ Generic solution algorithms adaptive for DDUs were proposed. Under this framework, the inherent distinctions and correlations between DDUs and decision-independent uncertainties (DIUs) were revealed, laying a fundamental theoretical foundation for the economic and reliable operation of RES-dominated power systems. Illustrative applications in the source and demand sides are provided to show the significance of considering DDUs and demonstrate the proposed theoretical results.

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Keywords

Decision-dependent uncertainty / Endogenous uncertainty / Robust optimization / Renewable energy / Power system dispatch

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Yunfan Zhang, Yifan Su, Feng Liu. On Decision-Dependent Uncertainties in Power Systems with High-Share Renewables. Engineering, 2025, 51(8): 104-124 DOI:10.1016/j.eng.2025.07.013

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1. Introduction

1.1. Background

Driven by the urgent need to reduce greenhouse gas emissions and mitigate the impacts of climate change, recent years have witnessed a worldwide unprecedented proliferation of renewable energy sources (RES) in power systems. Alongside the remarkable benefits of the transition towards a RES-dominated power system, the inherent variability and intermittency of RES, particularly the photovoltaic (PV) and wind generations, exacerbate the varying characteristics of the power system and bring about significant challenges to ensure the reliability and economic efficiency of system operation.

Various uncertainty handling methods, such as stochastic optimization [1], robust optimization (RO) [2], distributionally RO (DRO) [3], and the information gap decision theory (IGDT) [4], have been studied to hedge against the uncertainties in power systems. Attributed to the high dispatch decision robustness and low requirement in uncertainty modeling, RO has been widely adopted in real power systems. PJM Interconnection and Alstom Grid have developed the two-stage RO (TSRO) since 2012 [5]. CAISO also incorporates heterogeneous uncertainties into its optimization framework through RO [6]. There are still some challenges on heavy computational burden and trade-off between robustness and performance, which, however, out of scope for this paper.

These optimization methods distinguish themselves in uncertainty characterization and risk management techniques. Moreover, the considered uncertain factors stemming from the inaccurate prediction of RES generations or load demand, the disruptions of network or generation units, and the variable market clearing prices, are typically exogenous variables independent of the decision-making outcome.

Decision-dependent uncertainties (DDUs), also known as endogenous uncertainties, have received increasing attention recently due to the escalating RES penetration and the emerging novel features of power systems. In real-world decision problems, uncertainties could be affected by decisions in three possible ways [7]:

•Decisions alter the underlying probability distribution of the random variable. For example, the survival probability of a link in a network subject to random failures could be increased if it is strengthened by investment [8].

•Decisions affect whether or when the random variable materializes, namely, retains physical meaning. For example, the uncertain output of a wind farm will materialize only if the decision is made to construct it.

•Decisions influence the resolution of the uncertainty, that is, whether or when the decision maker can observe the uncertain variable. For example, the product demand curve is fixed yet unknown to the retailer, whereas the retailer can exploit and explore the characteristics of customer behavior through different pricing decisions [9].

Contrary to DDUs, uncertainties unaffected by decisions are termed decision-independent uncertainties (DIUs) or exogenous uncertainties interchangeably.

1.2. DDU in power system dispatch

DDUs in the dispatch problems of power systems typically arise from three aspects: ① the uncertain dispatchable resources on the source side, ② the uncertain dispatchable resources on the demand side, and ③ the uncertain reserve deployment from the operators.

(1) DDUs originating from uncertain dispatchable resources: source side. In high-RES-penetration power systems, the RES units, in addition to conventional synchronous generators, can serve as dispatchable resources. By incorporating frequency-related signals within the active power control loop of the wind turbine, the wind turbine can be responsive to frequency variations, contributing to the stabilization of the system deviations. However, the kinetic energy stored in the rotating mass is limited, and the discharge of energy to the grid is only available for a short period. For the sake of participating in primary, secondary, and tertiary frequency control, the wind turbine needs to retain a certain amount of spinning power reserve by deviating from the maximum power point (MPP) through rotor speed or pitch angle control to support system balance in case of frequency contingency. Similarly, PV generations can be curtailed below MPP through voltage control to provide reserve power. Alternatively, the output of a wind farm or a PV plant can be quickly regulated through controlled shutdown or start-up of the units.

It is worth noting that, the regulation effects of RES are jointly determined by the variable weather conditions and the proactive power control strategies or actions of the operators. Therefore, the available power and the reserved power of RES units manifest as DDUs. In this paper, an illustrative example was provided to exemplify the RES-induced DDUs. Specifically, the RES units generally operate in the following three de-loading schemes [[10], [11], [12]] to provide sustainable up-reserve for frequency regulations, where the decision variable

0 \leq λ \leq 1

denotes the de-loading ratio.

•The de-rating scheme: The maximal output of the RES unit is capped at a de-rated point, which equals 1 −

λ

of the rated power.

•The delta scheme: The reserve is set at a fixed amount that equals

λ

of the rated power.

•The percentage scheme: The RES unit reserves a percentage

λ

of the maximum available power.

Fig. 1 reveals that the available and preserved power of a wind turbine is uncertain due to the volatile wind speed. Additionally, the decisions of the system operator, including the de-loading scheme and the reserve ratio

λ

, affected the probability distribution characteristics of these two uncertain parameters.

The DDUs of RES units have attracted increasing concerns from researchers. For instance, a day-ahead strategic wind curtailment schedule was formed to reduce wind variability, and the two-stage robust dispatch model was built to address the corresponding DDU of the real-time wind generation output [13,14]. DDUs arising from the solar feed-in curtailment were handled in a jointly chance-constrained stochastic direct current (DC) optimal power flow model, where affine decision rules were applied to adjust generator output once the uncertainties were revealed [15]. A frequency-constrained stochastic planning model [10] was presented with frequency response from de-loaded wind farms, under the consideration of the support capability uncertainty and its dependency upon the combination of different support schemes. The DDUs from frequency reserve allocation among RES units were tackled without assuming a fixed portion of power reserve used for the synthetic inertial response and the droop response during the process of modeling the post-contingency system frequency dynamics [12].

(2) DDUs originating from uncertain dispatchable resources: demand side. In the context of demand response (DR), customers adjust their electricity consumption patterns to contribute flexibility to system balance by responding to price signals or incentives. The actual DR behaviors of voluntary customers are inevitably uncertain, while closely related to price signals or other DR policies, manifesting as DDUs. The exploration of DDUs in DR is reviewed below, involving both price-elastic models [[16], [17], [18], [19]] and non-price models [[20], [21], [22], [23]].

The price-elastic models: The uncertain price–elastic demand curve was integrated into a robust unit commitment problem [16]. A deviation term was introduced to the demand–price function, as illustrated in Fig. 2 [16]. For a given price, the consumer response varies within a certain range, with this range dependent on the price decision. The uncertainty of the price–elastic demand curve through scenarios in the stochastic unit commitment problem was demonstrated [17], with each scenario corresponding to a curve. It was assumed that the uncertain price elasticity coefficients vary within a specified range, leading to a price-sensitive range within which the actual load fluctuates, as depicted in Fig. 3 [19]. The DR uncertainty model was established considering consumer psychology [18]. Specifically, as the incentive price rises, the willingness of users to participate in DR would increase, while the associated randomness of the participation would decrease. An illustrative example is provided in Fig. 4 [18].

The non-price models: A two-stage robust economic dispatch model was proposed in Ref. [20], wherein the DR setpoint determined by the system operator influences the high-dimensional box-type DDU set (DDU-S) of the load demand. Subsequently, DDUs of deferrable loads were modeled explicitly in Ref. [21] since the cross-time load shifting decisions would reshape the temporal distribution of load and the corresponding uncertainty set. Consequently, a multi-stage robust dispatch model was formulated, accompanied by the design of an enhanced constraint generation algorithm. Additionally, DDUs stemming from the cold load pickup (CLPU) phenomenon after an outage were investigated to develop a two-stage stochastic restoration model [22]. Different load pickup times result in distinguished probability distributions of the CLPU peak amplitude. DDUs emerging as uncertain consumer participation in a DR program can only be resolved upon actual deployment of the DR scheme [23]. To address this issue, a scenario-wise multi-stage stochastic planning model was proposed, incorporating DDUs through the non-anticipativity constraints in the scenario tree.

(3) DDUs originating from uncertain reserve deployment. In the reserve market or energy-reserve market, participants autonomously decide on their reserve capacity provision through strategic bidding [24,25] or price-based self-scheduling [26,27]. The uncertainty of the actual deployment of reserve power during real-time operation is determined by the occurrence of disturbances and regulation signals from the operator. However, the uncertain deployed reserve power must fall within the capacity available for dispatch, manifesting as DDUs for reserve providers. For example, commercial buildings provide frequency reserves to the power grid [28]. Thus, a robust optimal control problem with an adjustable uncertainty set was formulated to reveal the optimal offered reserve capacity. Some researchers managed the robust self-scheduling of a virtual power plant (VPP) participating in the day-ahead energy-reserve market [29], wherein DDUs pertaining to real-time reserve deployment requests were captured through a polyhedral uncertainty set with a decision-dependent right-hand-side vector.

Remark 1 (time scales of DDUs). DDU exists in optimization problems of power systems across different time scales. Day-ahead decisions of wind curtailment can lead to decision-dependent wind variability [13,14]. Deferrable loads were shifted across hours, causing intra-day DDU of power demand [21]. In the field of power system planning, monthly and yearly time-scale DDUs of wind generation were commonly considered [30,31]. Since the proposed theorems and algorithms in this paper can be applied to problems involving various time scales, the following sections of this paper will not emphasize specific time scales.

1.3. Contributions

While the DDUs in power systems have been extensively investigated, more systematic theoretical studies on this topic are needed. In our previous work [32], the solution techniques for DDU-associated RO were reviewed, involving both the static RO models [[33], [34], [35], [36], [37]] and the two-/multi-stage RO models [7,32,[38], [39], [40]]. The previous studies collectively indicate that DDUs would lead to more sophisticated optimization models, posing challenges in developing exact and efficient solution algorithms. However, the underlying reasons for these challenges and common solutions have not been thoroughly explored.

This work develops a systematic framework for two-stage robust dispatch with DDUs. The fundamental differences and relations between DDUs and DIUs are explicitly investigated from the perspectives of uncertainty characterization, coping mechanisms, problem solvability, and solution paradigms.

The main contributions of this paper are as follows.

(1) Robust characterization of DDUs. The distinctions and correlations between DDUs and DIUs are elucidated within the framework of set-based uncertainty characterization and TSRO dispatch. A novel concept, the separability of a DDU-S, is introduced to articulate the dependency structure between decision variables and random variables. The existence of separability is further investigated, revealing that any generic DDU-S can be decomposed into a decision-independent support set, pertaining to the robust formulation of uncertainty, and a coupling function that describes the dependency on decisions. This finding bridges DDUs and conventional DIUs, implying that well-developed algorithms for DIUs could be extended to DDUs.

(2) Mechanism of TSRO dispatch with DDUs. The mechanism of TSRO dispatch under DDUs is revealed within the framework of set-based uncertainty characterization and region-based flexibility characterization [41,42]. It is discovered that a TSRO dispatch problem essentially aims to obtain the optimal allocation of system flexibility to hedge against uncertainties and that DDUs enable the bi-directional matching between them, which implies impicit operational flexibility.

(3) Convexity of TSRO dispatch with DDUs. The effects of DDUs on the characteristics and solvability of TSRO dispatch problems are researched. Concerning a TSRO problem that contains only linear constraints and DIUs, its robust feasibility region (RFR) is convex [43]. Our study unveils that, in contrast to DIUs, the RFRs of TSRO dispatch problems with DDUs may be non-convex. The non-convexity, if any, is attributed to the dependency of the uncertainty set on decisions and is irrelevant to the convexity of the uncertainty set itself. With the concept of separability, the convexity of the coupling function is proved to be a sufficient condition for ensuring the convexity of the TSRO problem with DDUs.

(4) Generic solution algorithm for TSRO dispatch with DDUs. Traditional cutting-plane algorithms widely used to solve robust dispatch problems, such as Benders’ decomposition and the column-and-constraint generation (C&CG) algorithm, are not adapted to the nonconvexity arising from DDUs. The ruined robust feasibility and optimality of the solution evidence their limitations in addressing TSRO with DDUs. To fix this issue, an improved solution algorithm is proposed based on the separability of DDU-S. The convexity of the enhanced cuts therein automatically adapts to that of the TSRO problem.

1.4. Organization

The rest of this paper is organized as follows. Section 2 reveals the dependency decomposition structure of DDU-S in robust frameworks. In Section 3, the mechanism of robust dispatch with DDUs is derived. Section 4 presents the discussion on the convexity of robust dispatch problems with DDUs. In Section 5, a general improved solution strategy is designed for DDU-associated RO problems. Section 6 presents applications of robust dispatch with DDUs. Lastly, the conclusions of this study are summarized in Section 7.

2. The robust characterization of DDUs

Consider the TSRO model in Eq. (1a), (1b), (1c), (1d), (1e), which has been widely used as the dispatch model in many real systems.

(1a)

\min_{x} f (x) + S (x)

(1b)

s .t . x \in X \cap X_{R}

(1c)

S (x) : = \max_{u \in U (x)} \min_{y \in Y (x, u)} c^{T} y

(1d)

Y (x, u) : = \{y | A x + B y + C u \leq b, y \geq 0\}

(1e)

X_{R} : = \{x | Y (x, u) \neq \emptyset, \forall u \in U (x)\}

where x denotes the here-and-now decision within constraint set X;

u

represents the DDU variables affected by x;

A

B

, and

C

are constant matrices;

b

is a constant vector. The uncertainty set of

u

, denoted by

U (x) : X ⇉ U

, is a set-valued map parameterized in

x

, and

U

is a set consisting of all possible realizations of

u

. According to the set-based uncertainty representing method in RO, for any given x,

U (x)

characterizes the possible realizations of

u

under a certain level of confidence.

f (x)

is the first-stage cost function.

S (x)

, which is the optimal value of a max–min optimization problem, indicates the worst-case second-stage cost. The second-stage dispatch problem upon the resolution of the uncertainty is assumed to be linear programming parameterized in

x

and

u

with wait-and-see decision variable

y

, as described in Eqs. (1c), (1d). Additionally,

Y (x, u)

designates the feasible region of

y

X_{R}

is the RFR.

The following mild assumptions are made for problem (1).

Assumption 1 For problem (1),

•Set X is a convex set.

•The function

f (x) : X \to R^{1}

is convex on X.

•Set U is a bounded polytope.

•For any

x \in X

and

u \in U

Y (x, u)

is a bounded polytope.

Remark 2 (assumption on polyhedral model). Assumption 1 restricts that sets U and

Y (x, u)

are bounded polytopes, which are common in power system robust dispatch problems. Operational constraints of fuel or wind generators, energy storage, and other devices are primarily linear. Though the power flow model is nonlinear and nonconvex, the linearized models, such as the DC flow model for transmission networks [44] and the linearized distribution flow (LinDistFlow) model for distribution networks [45], have been widely applied. Therefore, Assumption 1 is practical for TSRO dispatch in power systems.

Remark 3 (tri- and multi-stage RO with DDUs). Though this paper focuses on TSRO problems like Eq. (1a), (1b), (1c), (1d), (1e), the proposed theorems and algorithms can be adopted to tri- or multi-stage problems. C&CG and Benders’ decomposition algorithms for TSRO can be nested to solve tri-stage problems. Regarding multi-stage one, affine policies are commonly employed to simplify the models, which thus can be solved by cutting-plane algorithms. Therefore, it is possible to extend the results in TSRO to tri- and multi-stage RO with DDUs.

Next, the dependency structure of DDU-S

U (x)

is investigated by introducing a novel concept of separability.

2.1. The complete separability of DDU-S

2.1.1. Definition

Definition 1 (complete separability of DDU-S). If a DDU-S

U (x)

can be equivalently rewritten into Eq. (2),

(2)

U (x) = \{u = C (ξ, x) | ξ \in Ξ\}, \forall x \in X

Then

U (x)

is called (completely) separable, where,

ξ

denotes the auxiliary random variable with a decision-independent support set

Ξ

;

C (\cdot) : Ξ \times X \to U

represents the coupling function parameterized in

ξ

and

x

. A DDU-S

U (x)

with a separable formulation like Eq. (2) is decomposed into two parts: the dependency function on decision variable

x

and the decision-irrelevant stochasticity implied by the random variable

ξ

and its support set

Ξ

2.1.2. Ubiquity

Next, the ubiquity of the separability of DDU-S is discussed in space

U

as the decision-irrelevant support set.

Without loss of generality, a DDU-S

U (x) : X ⇉ U

can be denoted as the intersection of two sets:

(3)

U (x) = U^{sub} \cap U^{sup} (x), \forall x \in X

where

U^{sub} \subseteq U

represents a decision-independent set in space

U

, and

U^{sup} (x) : X ⇉ U

denotes a set-valued map parameterized in decision

x

. Notably, the intersecting form in Eq. (3) does not impose additional requirements onto the specific formulation of DDU-S

U (x)

because one can always take

U^{sub}

U

and

U^{sup} (x)

U (x)

. The following lemma holds for any DDU-S with the intersecting form of Eq. (3).

Lemma 1 Suppose for a DDU-S

U (x) : X ⇉ U

that can be denoted by the intersection of two sets as in Eq. (3), there exists a coupling function

C (u, x) : U^{sub} \times X \to U

such that:

•for any

x \in X

and

u \in U (x)

C (x, u) = u

;

•for any

x \in X

and

u \in U^{sub}

C (u, x) \in U (x)

Then,

U (x)

is completely separable with:

(4)

U (x) = {u = C (u^{'}, x) | u^{'} \in U^{sub}}

Proof. Define

U^{†} (x) = \{u = C (u^{'}, x) | u^{'} \in U^{sub}, \forall x \in X\}

. Next, we would like to justify the equivalence of

U (x)

and

U^{†} (x)

. First, according to the property of

C (\cdot)

, we have

U^{†} (x) \subseteq U (x), \forall x \in X

. Moreover, for any

u \in U (x)

, there is

u = C (u, x)

. Set

u^{'} = u

, then there exists

u^{'} \in U^{sub}

such that

u = C (u^{'}, x)

implying that

u \in U^{†} (x)

. Therefore,

U (x) \subseteq U^{†} (x), \forall x \in X

. This completes the proof that

U (x) = U^{†} (x), \forall x \in X

Based on Lemma 1, the ubiquity of complete separability of a DDU-S can be justified by the following corollary.

Corollary 2 For any DDU-S

U (x) : X ⇉ U

, it is of complete separability as in Eq. (4) by setting the

U^{sub}

U

and the coupling function

C (\cdot)

as:

(5a)

C (u^{'}, x) = \arg {min}_{u} {∥u^{'} - u∥}_{2}

(5b)

s .t . u \in U (x)

Corollary 2 can be justified by checking whether the coupling function in Eq. (5a), (5b) satisfies the conditions in Lemma 1.

2.1.3. Examples

Example 1 The adjustable uncertainty sets in Ref. [28] are completely separable since they are restricted to having a specific family of geometric forms.

Ball DDU-S:

r \geq 0

denotes the radius,

x

represents the ball center, and

{∥\cdot∥}_{p}

indicates the p-norm of a vector. The ball DDU-S

U (x, r)

parameterized in

x \in R^{n}

and

r \in R^{1}

is completely separable with:

(6)

U (x, r) : = \{u | {∥u - x∥}_{p} \leq r\}

(7)

= \{u = C (u^{'}, x, r) | u^{'} \in U^{0}\}

where

u^{'}

embodies the auxiliary uncertain variable, the support set

U^{0}

is defined as

U^{0} : = \{u^{'} | {∥u^{'}∥}_{\infty} \leq 1\}

, and the coupling function

C (\cdot)

is defined as

u = C (u^{'}, x, r) = x + r I u^{'}

, where

I \in R^{n \times n}

denotes an identity matrix with a proper rank.

Box DDU-S:

x \in R^{n}

denotes the center of the box, and

r \in R^{n}

represents the range of the box. The box DDU-S

U (x, r)

parameterized in

x \in R^{n}

and

r \in R^{n}

is completely separable with:

(8a)

U (x, r) : = \{u | {∥u - x∥}_{\infty} \leq r\}

(8b)

= \{u = C (u^{'}, x, r) | u^{'} \in U^{0}\}

where

u^{'}

designates the auxiliary uncertain variable with support set

U^{0} : = \{u^{'} | {∥u^{'}∥}_{\infty} \leq 1\}

, and the coupling function

C (\cdot)

is defined as

u = C (u^{'}, x, r) = x + diag (r) u^{'}

, where

diag (r) \in R^{n \times n}

represents a diagonal matrix with diagonal elements

r

Ellipsoidal DDU-S:

x \in R^{n}

denotes the center of the ellipsoid, and

Σ \in ℝ^{n \times n}

represents a symmetric and positive definite matrix. The ellipsoidal DDU-S

U (x, Σ)

parameterized in

x

and

Σ

is completely separable with:

(9a)

U (x, r) : = \{{u | (u - x)}^{T} Σ^{- 1} (u - x) \leq 1\}

(9b)

= \{u = C (u^{'}, x, Σ) | u^{'} \in U^{0}\}

where

u^{'}

indicates the auxiliary uncertain variable with support set

U^{0} : = \{u^{'} | {∥u^{'}∥}_{2} \leq 1\}

, and the coupling function

C (\cdot)

is defined as

u = C (u^{'}, x, Σ) = x + Σ^{1 / 2} u^{'}

Polyhedral DDU-S:

U (x^{(1)}, \dots, x^{(m)})

parameterized in

x^{(1)}, \dots, x^{(m)} \in R^{n}

is the convex hall of

m

(

m \geq n

) vectors

x^{(1)}, \dots, x^{(m)}

. Then, polyhedral DDU-S

U (x^{(1)}, \dots, x^{(m)})

is completely separable with:

(10a)

U (x^{(1)}, \dots, x^{(m)}) : = conv (x^{(1)}, \dots, x^{(m)})

(10b)

= \{u = C (u^{'}, x^{(1)}, \dots, x^{(m)}) | u^{'} \in U^{0}\}

where

u^{'}

denotes the auxiliary uncertain variable with support set

U^{0} : = conv (e_{1}, \dots, e_{m})

, where

e_{i} \in R^{n}

indicates the unit vector with the ith element being 1. The coupling function

C (\cdot)

is defined as

u = C (u^{'}, x^{(1)}, \dots, x^{(m)}) = [x^{(1)}, \dots, x^{(m)}] u^{'}

Example 2

P^{mppt}

denotes the maximal available power of the wind turbine under the MPP tracking mode,

P^{avail}

represents the available power,

P^{rate}

is the rated power,

R

indicates the preserved power, and

v

embodies the uncertain wind speed. With given decisions on the loading scheme

I_{mode} \in {de-rating, delta, percentage}

and the de-loading ratio

λ \in [0, 1]

, the relations between

P^{avail}

R

, and

P^{mppt}

can be captured as:

(11a)

De - rating: \{\begin{matrix} R = \max \{P^{mppt} - (1 - λ) P^{rate}, 0\} \\ P^{avail} = \min \{P^{mppt}, (1 - λ) P^{rate}\} \end{matrix}

(11b)

Delta: \{\begin{matrix} R = \min \{P^{mppt}, λ P^{rate}\} \\ P^{avail} = \max \{P^{mppt} - λ P^{rate}, 0\} \end{matrix}

(11c)

Percentage: \{\begin{matrix} R = λ P^{mppt} \\ P^{avail} = (1 - λ) P^{mppt} \end{matrix}

In the above formulas,

P^{mppt}

designates a DIU variable solely determined by the uncertain wind speed

v

(12)

P^{mppt} = \{\begin{array}{l} 0, & v < v_{in} \\ μ_{0} v^{3}, & v_{in} \leq v \leq v_{rate} \\ P^{rate}, & v_{rate} \leq v \leq v_{out} \\ 0, & v > v_{out} \end{array}

where

v_{in}

v_{rate}

, and

v_{out}

are the cut-in, rated, and cut-out wind speed, respectively;

μ_{0}

is the generation parameter.

Eqs. (11a), (11b), (11c), (12) can be viewed as the function of

P^{avail}

and

R

with arguments

I

λ

, and

v

. The compact form of this function is expressed as:

(13)

[\begin{matrix} P^{avail} \\ R \end{matrix}] = C (I, λ, v)

Among the arguments of coupling function

C (\cdot)

I

and

λ

are decision variables of the operator, and the wind speed

v

is an exogenously stochastic variate. Then, the DDU-S of DDU variables

P^{avail}

and

R

is completely separable with the coupling function

C (\cdot)

in Eq. (13), and the uncertainty set of

v

is denoted by

V

(14)

U (λ, I) = \{[\begin{matrix} P^{avail} \\ R \end{matrix}] = C (I, λ, v) | v \in V\}

Example 3 The DDU-S

U (x)

in Fig. 4 is completely separable with:

(15)

U (x) = \{u = g^{lb} (x) + u^{'} (g^{ub} (x) - g^{lb} (x)) | u^{'} \in [0, 1]\}

where

u^{'}

denotes the auxiliary random variable falling into the range [0, 1], which is independent of the decision variable

x

The DDU-S of the uncertain demand

d

in Fig. 2 is also completely separable with:

(16)

U (p_{d}) = \{d = A_{d} {(p_{d} + ∊)}^{κ} | - \hat{∊} \leq ∊ \leq \hat{∊}\}

The auxiliary random variable

∊

represents the deviation from the deterministic price–elastic demand curve and varies within the range

[- \hat{∊}, \hat{∊}]

, which is irrelevant to the decision

p_{d}

In Fig. 3, the uncertain price–elastic coefficient

∊

is regarded as the decision-independent auxiliary random variable, and then the DDU-S of demand

d

is completely separable with:

(17)

U (p_{d}) = \{u = u_{0} (1 + ∊ \frac{p_{d} - p_{0}}{p_{0}}) | ∊ \in [∊^{min}, ∊^{max}]\}

2.2. The partial separability of a DDU-S

2.2.1. Definition

The partial separability of a DDU-S is defined as follows.

Definition 2 (partial separability of DDU-S). If the following two conditions hold for a DDU-S

U (x)

, then

U (x)

is called partially separable.

(1) There exists a set-valued map

U^{sub} (x) : X ⇉ U

such that the graph of

U^{sub} (x)

is the subset of the graph of

U (x)

, that is,

graph (U^{sub} (\cdot)) \subseteq graph (U (\cdot)) .

(2) The set-valued map

U^{sub} (x)

in condition (1) is separable:

U^{sub} (x) = \{u = C (ξ, x) | ξ \in Ξ\}, \forall x \in X .

2.2.2. Ubiquity

Considering the DDU-associated TSRO problem (1), any given DDU-S

U (x)

is partially separable and can be equivalently substituted by the set-valued map with complete separability.

Concerning the TSRO problem (1) and any given DDU-S

U (x)

, define the separable DDU-S with regard to robust feasibility and robust optimality as follows:

The DDU-S

U_{fea}^{sep} (x) : X ⇉ U

is formulated considering the robust feasibility of the TSRO problem (1):

(18a)

U_{fea}^{sep} (x) : = \{u = C_{fea} (ξ, x) | ξ \in Ξ\}, \forall x \in X

where

Ξ

indicates the decision-independent support set of the auxiliary random variable

ξ

(18b)

Ξ : = \{ξ | B^{T} ξ \leq 0, - 1 \leq ξ \leq 0\}

and

C_{fea} (\cdot)

denotes the coupling function defined as:

(18c)

C_{fea} (ξ, x) : = \arg \{\begin{matrix} \max_{u} & - ξ^{T} C u \\ s .t . & u \in U (x) \end{matrix}\}

The DDU-S

U_{opt}^{sep} (x) : X ⇉ U

is formulated considering the robust optimality of the TSRO problem (1):

(19a)

U_{opt}^{sep} (x) : = \{u = C_{opt} (π, x) | π \in Π\}, \forall x \in X

where

Π

represents the decision-independent support set of the auxiliary random variable

π

(19b)

Π : = \{π | B^{T} π \leq c, π \leq 0\}

and

C_{opt} (\cdot)

stands for the coupling function formulated by:

(19c)

C_{opt} (π, x) : = \arg \{\begin{matrix} \max_{u} & - π^{T} C u \\ s .t . & u \in U (x) \end{matrix}\}

Since the graph of

U_{fea}^{sep} (x)

and

U_{opt}^{sep} (x)

are the subsets of the graph of

U (x)

, condition (1) in Definition 2 is satisfied. Combining with the fact that

U_{fea}^{sep} (x)

and

U_{opt}^{sep} (x)

are completely separable, it is revealed that the DDU-S

U (x)

is partially separable regardless of the specific formulation of

U (x)

. This finding sets a key cornerstone for solving complex RO problems with DDUs.

Next, we justify that the DDU-S

U (x)

in the TSRO problem (1) can be equivalently substituted by the two separable DDU-Ss (i.e.,

U_{fea}^{sep} (x)

and

U_{opt}^{sep} (x)

Theorem 3 The TSRO problem (1) with an arbitrary DDU-S

U (x)

has a surrogate model as follows.

The RFR

X_{R}

is expressed as:

(20a)

X_{R} : = \{x \in X | Y (x, u) \neq \emptyset, \forall u \in U (x)\}

(20b)

= \{x \in X | Y (x, u) \neq \emptyset, \forall u \in U_{fea}^{sep} (x)\}

where Eq. (20a) follows the definition of

X_{R}

in Eq. (1e), (20b) implies the surrogate formulation of

X_{R}

The robust optimality sub-function

S (x)

is described as

(21a)

S (x) : = \max_{u \in U (x)} \min_{y \in Y (x, u)} c^{T} y, \forall x \in X

(21b)

= \max_{u \in U_{opt}^{sep} (x)} \min_{y \in Y (x, u)} c^{T} y, \forall x \in X

where Eq. (21a) follows the definition of

S (x)

in Eq. (1c), (21b) and Eq. (1c), (21b) implies the surrogate formulation of function

S (x)

Proof. Assertion (1): First, denote by

R (x, u)

the optimal value of the optimization problem and its dual formulation as follows:

(22a)

R (x, u) : = \{\begin{matrix} \min_{y, s} 1^{T} s \\ s .t . A x + B y + C u \leq b + s \\ s \geq 0, y \geq 0 \end{matrix}\}

(22b)

= \{\begin{matrix} \max_{ξ} ξ^{T} (b - A x - C u) \\ s .t . ξ \in Ξ \end{matrix}\}

where Eq. (22a) is to define value function

R (x, u)

and Eq. (22b) is from the dual transformation of the minimization problem in Eq. (22a). It is easy to verify that for any

x \in X

and

u \in U

Y (x, u) \neq \emptyset \Leftrightarrow R (x, u) \leq 0

Denote by

U_{fea}^{*} (x)

the set-valued map as follows:

(23)

U_{fea}^{*} (x) = \arg_{u} \{\begin{matrix} \max_{u, ξ} ξ^{T} (b - A x - C u) \\ s .t . u \in U (x), ξ \in Ξ \end{matrix}\}

For any

x \in X

(24a)

Y (x, u) \neq \emptyset, \forall u \in U (x)

(24b)

\Leftrightarrow R (x, u) \leq 0, \forall u \in U (x)

(24c)

\Leftrightarrow \{\begin{matrix} \max_{u, ξ} ξ^{T} (b - A x - C u) \\ s .t . u \in U (x), ξ \in Ξ \end{matrix}\} \leq 0

(24d)

\Leftrightarrow \{\begin{matrix} \max_{u, ξ} ξ^{T} (b - A x - C u) \\ s .t . u \in U_{fea}^{*} (x), ξ \in Ξ \end{matrix}\} \leq 0

(24e)

\Leftrightarrow R (x, u) \leq 0, \forall u \in U_{fea}^{*} (x)

(24f)

\Leftrightarrow Y (x, u) \neq \emptyset, \forall u \in U_{fea}^{*} (x)

where Eqs. (24a), (24f) are according to the equivalent formulation of

Y (x, u) \neq \emptyset

; Eqs. (24b), (24e) are according to the definition of

R (x, u)

; Eq. (24d) follows the definition of

U_{fea}^{*} (x)

According to the definition of

U_{fea}^{*} (x)

in Eq. (23) and that of

U_{fea}^{sep}

in Eq. (1e), (20b), the following relationship holds:

(25)

U_{fea}^{*} (x) \subseteq U_{fea}^{sep} (x) \subseteq U (x), \forall x \in X

Then, robust feasibility set

X_{R}

is with equivalent formulations as follows:

(26a)

X_{R} = \{x \in X | Y (x, u) \neq \emptyset, \forall u \in U (x)\}

(26b)

= \{x \in X | Y (x, u) \neq \emptyset, \forall u \in U_{fea}^{*} (x)\}

(26c)

= \{x \in X | Y (x, u) \neq \emptyset, \forall u \in U_{fea}^{sep} (x)\}

where Eq. (26b) is from the relationship in Eq. (24a), (24b), (24c), (24d), (24e), (24f), (26c) and Eq. (24a), (24b), (24c), (24d), (24e), (24f), (26c) is from Eq. (25).

Assertion (2): First, denote by

U_{opt}^{*} (x)

the optimal solution set of

u

for optimization problem

S (x)

(27a)

U_{opt}^{*} (x) : = \arg_{u} S (x), \forall x \in X

(27b)

= \arg_{u} \{\begin{matrix} \max_{u, π} π^{T} (b - A x - C u) \\ s .t . u \in U (x), π \in Π \end{matrix}\}, \forall x \in X

where Eq. (27a) is the definition of

U_{opt}^{*} (x)

and Eq. (27b) is according to the dual transformation on the inner minimization problem in

S (x)

. According to the definition of

U_{opt}^{*} (x)

in Eq. (27a), (27b) and that of

U_{opt}^{sep} (x)

in **Eq. (19a), (19b), (19c), the following relationships are derived:

(28)

U_{opt}^{*} (x) \subseteq U_{opt}^{sep} (x) \subseteq U (x), \forall x \in X

Then, assertion 2 is justified as follows

(29a)

S (x) : = \max_{u \in U (x)} \min_{y \in Y (x, u)} c^{T} y, \forall x \in X

(29b)

= \max_{u \in U_{opt}^{*} (x)} \min_{y \in Y (x, u)} c^{T} y, \forall x \in X

(29c)

= \max_{u \in U_{opt}^{*} (x)} \min_{y \in Y (x, u)} c^{T} y, \forall x \in X

where Eq. (29a) is from the definition of

S (x)

in Eq. (29c), Eq. (29b) is from the definition of

U_{opt}^{*} (x)

in Eq. (27a), (27b), and Eq. (29c) is according to the relationship in Eq. (28).

According to Theorem 3, for any DDU-S

U (x)

, the TSRO problem (1) is equivalent to Eq. (30a), (30b), (30c):

(30a)

\min_{x} \{f (x) + \max_{u \in U_{opt}^{sep} (x)} \min_{y \in Y (x, u)} c^{T} y\}

(30b)

s .t . x \in X

(30c)

Y (x, u) \neq \emptyset, \forall u \in U_{fea}^{sep} (x)

where

U_{opt}^{sep} (x)

and

U_{fea}^{sep} (x)

are two separable DDU-Ss.

2.2.3. Example

Next, an illustrative example is provided to demonstrate how the partial separability of DDU-S is applied to TSRO problems.

Example 4. Consider a polyhedral DDU-S given by:

(31)

U (x) = \{u \in R^{2} | \begin{array}{l} u_{1} \leq 7 x_{1} + 8 x_{2} \\ u_{2} \leq 13 x_{2} \\ - u_{1} + 2 u_{2} \leq 15 x_{2} + 8 \\ u_{1} + u_{2} \leq 7 x_{1} + 2 x_{2} + 13 \\ 4 u_{1} - 7 u_{2} \leq 21 x_{1} + 11 x_{2} - 25 \\ - 8 u_{1} - 3 u_{2} \leq - 40 \end{array}\}

where the here-and-now decision variable

x = {(x_{1}, x_{2})}^{T} \in R^{2}

determines the right-hand-side vector of the polytope uncertainty set.

As observed in Fig. 5, the shape, size, and location of the DDU-S

U (x)

in Eq. (31) change with the different values of the decision variable

x

For a TSRO problem with

Y (x, u)

in Eq. (32),

(32)

Y (x, u) = \{y \in R^{2} | \begin{array}{l} - 1 \leq y_{1} \leq 1 (ξ_{1}, ξ_{2}) \\ - 1 \leq y_{2} \leq 1 (ξ_{3}, ξ_{4}) \\ u_{1} \leq y_{1} + y_{2} \leq u_{2} (ξ_{5}, ξ_{6}) \end{array}\}

an auxiliary DDU-S

U_{fea}^{sep} (x)

concerning robust feasibility is formulated as:

(33a)

U_{fea}^{sep} (x) : = \{u = C_{fea} (ξ, x) | ξ \in Ξ\}, \forall x \in X

(33b)

Ξ : = \{ξ \in R^{6} | \begin{array}{l} ξ_{5} - ξ_{6} = ξ_{2} - ξ_{1} \\ ξ_{6} - ξ_{5} = ξ_{3} - ξ_{4} \\ - 1 \leq ξ_{i} \leq 0, \forall i = 1, \dots, 6 \end{array}\}

(33c)

C_{fea} (ξ, x) : = \arg \{\begin{matrix} \max_{u} & - ξ_{5} u_{1} + ξ_{6} u_{2} \\ s .t . & u \in U (x) \end{matrix}\}

Both

U (x)

and

U_{fea}^{sep} (x)

vary with the decision variable

x

and satisfy the conditions in Definition 2. Therefore,

U (x)

can be partially separated, while

U_{fea}^{sep} (x)

is completely separable.

As illustrated in Fig. 5, for any given

x \in R^{2}

U_{fea}^{sep} (x)

indicates the subset of the original DDU-S

U (x)

. According to Theorem 3, the RFR

X_{R}

regarding

x

has two surrogate formulations, one based on

U (x)

in Eq. (20a) and the other based on

U_{fea}^{sep} (x)

in Eq. (20b).

3. Mechanism of robust dispatch with DDUs

In this section, the surrogate model of problem (1) is first derived based on the concept of the dispatchable region [42,46] and its variants. Then, the mechanism of TSRO dispatch with DDUs is revealed.

Definition 3 (dispatchable region). The dispatchable region of problem (1), denoted by

D (x) : X ⇉ U

, is defined as:

(34a)

\forall u \in D (x), Y (x, u) \neq \emptyset

(34b)

\forall u \notin D (x), Y (x, u) = \emptyset

The extended dispatchable region of problem (1), denoted by

D^{ext} (x, α) : X \times R^{1} ⇉ U

with arguments

x \in X

and

α \in R^{1}

, is defined as:

(35a)

\forall u \in D^{ext} (x, α), Y (x, u) \cap \{y | c^{T} y \leq α\} \neq \emptyset

(35b)

\forall u \notin D^{ext} (x, α), Y (x, u) \cap \{y | c^{T} y \leq α\} = \emptyset

For any

x \in X

and

α \in R^{1}

D^{ext} (x, α) \subseteq D (x)

Following the definition of the extended dispatchable region, the surrogate model of Eq. (1a), (1b), (1c), (1d), (1e) is established as:

(36a)

\min_{x \in X, α \in R^{1}} f (x) + α

(36b)

s .t . U (x) \subseteq D^{ext} (x, α)

where α is an auxiliary decision variable indicating the value of function

S (x)

. Eq. (36b) combines the robust feasibility requirement

x \in X_{R}

and the robust optimality function

S (x)

. Notably, the equivalence between Eqs. (1a), (1b), (1c), (1d), (1e), (36a), (36b) has no relation with the specific formulation of DDU-S

U (x)

. Therefore, this equivalence also applies to DIU sets because DIU is a special case of DDU.

Based on problem (36), the mechanism of TSRO hedging against uncertainties can be concluded as the optimal match between the uncertainty set

U (x)

and the extended dispatchable region

D^{ext} (x, α)

. The objective (36a) aims to minimize the matching cost, which is

f (x) + α

The difference in the mechanism of robust dispatch under DIUs and DDUs is illustrated in Fig. 6. In the case of DIU, the decision maker needs to find a feasible

(x, α)

, allowing the extended dispatchable region

D^{ext} (x, α)

that varies with

(x, α)

to cover the fixed uncertainty set

U^{0}

. This process can be concluded as unilateral matching, where the dispatchable region is adapted to suit the uncertainty set. In the case of DDUs, both the extended dispatchable region

D^{ext} (x, α)

and the DDU-S

U (x)

vary with decision variables

(x, α)

, and the mechanism of robust dispatch under DDUs can be summarized as the bilateral matching between the two.

Remark 4 (bilateral matching). The bilateral matching mechanism points out the crucial difference between DDU and DIU in power system robust dispatch. DDU is an active, flexible source rather than a burden. With rational decisions, the DDU-S can be reshaped so that uncertainty can be more easily admitted. In other words, the bilateral matching mechanism explores the potential flexibility hidden in DDUs, facilitating dispatch and operation in RES-dominated power systems.

According to 2.1 The complete separability of DDU-S, 2.2 The partial separability of a DDU-S, both the complete and partial separable formulations for the DDU-S

U (x)

exist in the problem (36). Suppose that

U (x)

is equivalent to or can be equivalently substituted by a separable DDU-S

U^{sep} (x) = {u = C (ξ, x) | ξ \in Ξ}

. Then, constraint (36b) can be transformed into:

(37)

Ξ \subseteq D (x, α)

where

D (x, α)

indicates the extended dispatchable region of the auxiliary random variable

ξ

(38)

D (x, α) : = \{ξ | C (x, ξ) \in D^{ext} (x, α)\}

Therefore, the bilateral matching between the extended dispatchable region

D^{ext} (x, α)

and the DDU-S

U (x)

can be reduced to the unilateral matching from

D (x, α)

to the fixed

Ξ

. An example is provided as follows.

Example 5 On the basis of Example 4, leaving the robust optimality aside, the focus shifts to how the dispatchable region and the uncertainty set match with each other.

For the DDU variable

u

with the DDU-S

U (x)

in Eq. (31), the dispatchable region of

u

is given by:

(39)

D (x) = \{u \in R^{2} | u_{1} \leq 2, u_{2} \geq - 2, u_{1} \leq u_{2}\}

According to Example 4,

U (x)

can be equivalently substituted by the separable DDU-S

U_{fea}^{sep} (x)

in Eq. (33a), (33b), (33c), which reveals an auxiliary random variable

ξ

with decision-independent support

Ξ

in Eq. (33b). The dispatchable region of

ξ

, denoted by

D (x)

, can be obtained according to Eq. (38). Next, we show that

(40a)

x \in X_{R} \Leftrightarrow U (x) \subseteq D (x)

(40b)

\Leftrightarrow U_{fea}^{sep} (x) \subseteq D (x)

(40c)

\Leftrightarrow Ξ \subseteq D (x)

When

x = {(1, 1)}^{T}

, the corresponding

D (x)

U (x)

, and

U_{fea}^{sep} (x)

are plotted in Fig. 7(a); the corresponding

D (x)

and

Ξ

are plotted in Fig. 7(b). As observed from the figures,

D (x)

fails to cover neither

U (x)

nor

U_{fea}^{sep} (x)

, and

D (x)

does not cover

Ξ

. Thus,

x = {(1, 1)}^{T}

is not robustly feasible.

With other conditions unchanged, the DDU-S

U (x)

is modified into:

(41)

U (x) = \{u \in R^{2} | \begin{array}{l} u_{1} \leq x_{1} + x_{2} \\ u_{2} \leq 13 x_{2} \\ - u_{1} + 2 u_{2} \leq 15 x_{2} + 8 \\ u_{1} + u_{2} \leq 7 x_{1} + 2 x_{2} + 13 \\ 4 u_{1} - 7 u_{2} \leq 21 x_{1} + 11 x_{2} - 25 \\ - 8 u_{1} - 3 u_{2} \leq - 40 \end{array}\}

The dispatchable region of

u

and the uncertainty set of

ξ

remain unchanged, whereas

U_{fea}^{sep} (x)

and

D (x)

vary with

U (x)

. When

x = {(1, 1)}^{T}

, these regions are plotted in Fig. 8. In the space of u,

D (x)

covers both

U (x)

and

U_{fea}^{sep} (x)

. In the space of

ξ

D (x)

covers

Ξ

. In other words,

x = {(1, 1)}^{T}

is robustly feasible when the DDU-S is modified into Eq. (41).

4. Convexity of robust dispatch problem with DDUs

As explained previously, DDUs can facilitate bilateral matching between the dispatchable region and the uncertainty set. However, compared with DIUs, the challenge of achieving optimal matching is heightened. This section reveals that the underlying difficulty stems from the fact that DDUs may introduce non-convexity to TSRO problems.

4.1. Case of DIUs

For a TSRO problem that contains only linear constraints and DIUs, its RFR is convex [43]. This conclusion can be extended to the convexity of the entire TSRO problem, see Lemma 4 as follows.

Lemma 4 Let Assumption 1 hold for problem (1). If

U (x) = U^{0}, \forall x \in X

and

U^{0}

is a polyhedron, then the problem (36) is convex.

Proof. Define the set

Γ

as follows:

(42)

Γ : = \cap_{u \in U^{0}} \{(x, α, y) | Y (x, u) \cap {y | c^{T} y \leq α} \neq \emptyset\}

Since

Y (\cdot)

and

U^{0}

are both polyhedrons,

Γ

is equivalent to:

(43)

Γ = \cap_{u \in vert (U^{0})} \{(x, α, y) | Y (x, u) \cap {y | c^{T} y \leq α} \neq \emptyset\}

According to Eq. (43),

Γ

is the intersection of a finite number of polyhedrons. Therefore,

Γ

is a polytope in the space of

(x, α, γ)

By projecting

Γ

onto the subspace of

(x, α)

, Eq. (36b) is derived. Since the projection of a polyhedron onto its subspace is still a polyhedron, Eq. (36b) is proved to be a polyhedron. This completes the proof.

Note that Lemma 4 only guarantees the convexity of the TSRO problem (36) when the uncertainty set is both decision-independent and polyhedral. Next, Corollary 6 suggests that the convexity of Eq. (36a), (36b) actually relies on the decision-independence of the uncertainty set rather than its polyhedrality.

4.2. Case of DDUs

In this subsection, we would like to prove that when the uncertainty set is decision-dependent, the convexity of the TSRO problem (36) is no longer guaranteed.

DDU-S with complete separability: without loss of generality, it is assumed that the DDU-S

U (x)

is separable as

(44)

U (x) = \{u = C (ξ, x) | ξ \in Ξ\}

Then, Theorem 5 gives a sufficient condition for the convexity of the TSRO problem (36).

Theorem 5 Let Assumption 1 hold for problem (1). If

U (x)

is separable, as expressed in Eq. (44), and the composite function

T (ξ, x) : conv (ξ) \times R^{n_{x}} \to R^{m}

given by Eq. (45) is convex, then the TSRO problem (36) is convex.

(45)

T (ξ, x) : = C \cdot C (ξ, x)

where

C

denotes the matrix constant in Eq. (1d).

Proof. In this theorem, Assumption 1 holds and

T

is convex. First, formulate the following optimization problem:

(46a)

\min_{y, s} 1^{T} s_{1} + 1^{T} s_{2} + 1^{T} s_{3}

(46b)

s .t . A x + B y + T (ξ, x) \leq b + s_{1}

(46c)

y + s_{2} \geq 0

(46d)

c^{T} y \leq α + s_{3}

(46e)

s_{1}, s_{2}, s_{3} \geq 0

where the objective function is linear, and the feasible region consists of a polyhedron.

Next, let

h (y, s)

FR (x, α, ξ)

, and

V (x, α, ξ)

denote the objective function, the feasible region, and the optimal value, respectively. We show that

V (x, α, ξ)

is convex with respect to

x

α

, and

ξ

. For any

ρ \in [0, 1]

(x^{1}, α^{1}, ξ^{1})

, and

(x^{2}, α^{2}, ξ^{2})

, the optimal solution of

V (x^{1}, α^{1}, ξ^{1})

is denoted by

(y^{1}, s^{1})

, and the optimal solution of

V (x^{2}, α^{2}, ξ^{2})

is denoted by

(y^{2}, s^{2})

. Therefore,

(47)

(y^{1}, s^{1}) \in FR (x^{1}, α^{1}, ξ^{1})

(48)

(y^{2}, s^{2}) \in FR (x^{2}, α^{2}, ξ^{2})

Since

T

is a convex function and

FR (x, α, ξ)

is a convex polyhedron, we have:

(49)

(\begin{matrix} ρ y^{1} + (1 - ρ) y^{2} \\ ρ s^{1} + (1 - ρ) s^{2} \end{matrix}) \in FR (\begin{matrix} ρ x^{1} + (1 - ρ) x^{2} \\ ρ α^{1} + (1 - ρ) α^{2} \\ ρ ξ^{1} + (1 - ρ) ξ^{2} \end{matrix})

which indicates that

(ρ y^{1} + (1 - ρ) y^{2}, ρ s^{1} + (1 - ρ) s^{2})

is a feasible solution given the parameters

ρ x^{1} + (1 - ρ) x^{2}

ρ α^{1} + (1 - ρ) α^{2}

, and

ρ ξ^{1} + (1 - ρ) ξ^{2}

Therefore, the following relations hold.

(50a)

ρ V (x^{1}, α^{1}, ξ^{1}) + (1 - ρ) V (x^{2}, α^{2}, ξ^{2})

(50b)

= ρ h (y^{1}, s^{1}) + (1 - ρ) h (y^{2}, s^{2})

(50c)

= h (ρ y^{1} + (1 - ρ) y^{2}, ρ s^{1} + (1 - ρ) s^{2})

(50d)

\geq V (ρ x^{1} + (1 - ρ) x^{2}, ρ α^{1} + (1 - ρ) α^{2}, ρ ξ^{1} + (1 - ρ) ξ^{2})

where Eq. (50b) holds because

(y^{1}, s^{1})

and

(y^{2}, s^{2})

are optimal solutions to the problem given the parameters

(x^{1}, α^{1}, ξ^{1})

and

(x^{2}, α^{2}, ξ^{2})

, respectively; Eq. (50c) yields because function

h (\cdot)

is linear; Eq. (50d) holds because

(ρ y^{1} + (1 - ρ) y^{2}, ρ s^{1} + (1 - ρ) s^{2})

is a feasible solution given the parameters

ρ x^{1} + (1 - ρ) x^{2}

ρ α^{1} + (1 - ρ) α^{2}

, and

ρ ξ^{1} + (1 - ρ) ξ^{2}

The above derivation justifies that

V (x, α, ξ)

is a convex function with respect to

x

α

, and

ξ

The definition of

D^{ext} (x, α)

in Eq. (35a), (35b) and the definition of

V (x, α, ξ)

yields:

(51a)

\{(x, α) | U (x) \subseteq D^{ext} (x, α)\}

(51b)

\Leftrightarrow Y (x, C (ξ, x)) \cap {y | c^{T} y \leq α} \neq \emptyset, \forall ξ \in Ξ

(51c)

\Leftrightarrow \{\begin{array}{l} A x + B y + T (ξ, x) \leq b \\ y \geq 0 \\ c^{T} y \leq α \end{array}\} \neq \emptyset, \forall ξ \in Ξ

(51d)

\Leftrightarrow V (x, α, ξ) \leq 0, \forall ξ \in Ξ

Therefore, constraint (36b) equals to

(52)

\{(x, α) | V (x, α, ξ) \leq 0, \forall ξ \in Ξ\}

Next, Eq. (36b) is proved to be convex by contradiction. Suppose there exist

(x^{1}, α^{1})

and

(x^{2}, α^{2})

satisfying Eq. (36b) and

ρ \in [0, 1]

such that

(ρ x^{1} + (1 - ρ) x^{2}, ρ α^{1} + (1 - ρ) α^{2})

not satisfying Eq. (36b). This implies there exists

ξ^{*} \in Ξ

such that

(53)

V (ρ x^{1} + (1 - ρ) x^{2}, ρ α^{1} + (1 - ρ) α^{2}, ξ^{*}) > 0

Since

(x^{1}, α^{1})

and

(x^{2}, α^{2})

satisfy Eq. (36b), it can be obtained that

V (x^{1}, α^{1}, ξ^{*}) \leq 0

and

V (x^{2}, α^{2}, ξ^{*}) \leq 0

. Since

V (\cdot)

is convex, it is derived that

(54a)

V (ρ x^{1} + (1 - ρ) x^{2}, ρ α^{1} + (1 - ρ) α^{2}, ξ^{*})

(54b)

\leq ρ V (x^{1}, α^{1}, ξ^{*}) + (1 - ρ) V (x^{2}, α^{2}, ξ^{*})

(54c)

\leq 0

which cause a contradiction. This completes the proof that Eq. (36b) is convex.

Theorem 5 implies that, when the DDU-S

U (x)

is separable, the convexity of the TSRO problem (36) can be determined by the property of the coupling function

C (\cdot)

and has no other requirement on the decision-independent auxiliary uncertainty set

Ξ

, as long as

Ξ

is independent of decision x.

A corollary of Theorem 5 is provided as follows.

Corollary 6 Let Assumption 1 hold for problem (1). If

U (x) = U^{0}, \forall x \in X

, then the TSRO problem (36) is convex.

Corollary 6 implies that when the uncertainty set is independent of the decision variable x, the convexity of the TSRO Eqs. (1a), (1b), (1c), (1d), (1e), (36a), (36b) hold, and this has no relation with the formulation of the DIU set

U^{0}

Next, an example to illustrate Theorem 5 is provided.

Example 6. On the basis of Example 4, the DDU-S

U (x)

is expressed as:

(55a)

U (x) = \{u = ξ + x | ξ \in Ξ\}

(55b)

Ξ = \{0 \leq ξ \leq 1\} \cup {- 1 \leq ξ \leq 0}

where

x, u, ξ \in R^{2}

Since the

U (x)

in Eq. (55a), (55b) is separable and the coupling function is convex, according to Theorem 5, the corresponding RFR

X_{R}

is convex, even though the DIU set

Ξ

is nonconvex. To verify this, by searching all over the

R^{2}

space, the RFR

X_{R}

that contains all robustly feasible x is plotted in Fig. 9. The result complies with Theorem 5.

Moreover, denote by

x^{1} = {(0, 0)}^{T}

and

x^{2} = {(0, 1)}^{T}

. It is observed in Fig. 9(a) that

x_{1} \notin X_{R}

is not robustly feasible, whereas

x^{2} \in X_{R}

is. To verify this from another perspective, the dispatchable region

D (x)

and the DDU-S

U (x)

with different values of x are depicted in Fig. 10.

It is revealed in Fig. 10(a) that when

x = x^{1}

, the dispatchable region

D (x^{1})

is not able to cover the DDU-S

U (x^{1})

, complying with

x^{1} \notin X_{R}

. In Fig. 10(b), when

x = x^{2}

, the DDU-S

U (x^{2})

is covered by the dispatchable region

D (x^{2})

, which is compatible with

x^{2} \in X_{R}

(1) DDU-S with partial separability. The TSRO problem (36) with a decision-independent polyhedral uncertainty set is guaranteed to be convex, as claimed in Lemma 4 and Theorem 5. However, the TSRO problem (36) can become nonconvex when the polytope varies with the here-and-now decision x. This is further explained as follows.

Example 7. On the basis of Example 4, the DDU-S

U (x)

is expressed in Eq. (56), where the here-and-now decision

x \in R^{1}

varies within [0.8, 2.2].

(56)

U (x) = \{u \in R^{2} | \begin{array}{l} u_{1} \leq 6 - 2 x \\ u_{1} \leq 2 x \\ u_{2} \leq 13 x \\ - u_{1} + 2 u_{2} \leq 15 x + 8 \\ u_{1} + u_{2} \leq 31 - 9 x \\ 4 u_{1} - 7 u_{2} \leq 32 x - 25 \\ - 8 u_{1} - 3 u_{2} \leq - 20 \\ 0 \leq u_{1} \leq 3 \\ 8 \leq u_{2} \leq 13 \end{array}\}

As the

Y (x, u)

is defined in Eq. (32), the dispatchable region

D (x)

remains Eq. (39). To obtain the RFR

X_{R}

, the graph of

U (x)

and

D (x)

are plotted in Fig. 11. Moreover,

U (x)

and

D (x)

with different values of x are shown in Fig. 12.

When

1 < x < 2

, the dispatchable region

D (x)

fails to cover the uncertainty set

U (x)

. By recalling that the RFR

X_{R}

contains all the x that makes

U (x) \subseteq D (x)

hold, the

X_{R}

is obtained as

(57)

X_{R} = [0.8, 1] \cup [2, 2.2]

which is obviously nonconvex.

The nonconvexity induced by polyhedral DDU-S can be explained from two perspectives:

(1) Consider the TSRO problem (1) with polyhedral DDU-S as follows

(58)

U (x) = \{u | G (x) u \leq g (x)\}

where

G (x), g (x)

are functions with respect to x. According to Section 2.2, the

U (x)

is proved to be partially separable by considering

U_{fea}^{sep} (x)

in Eq. (1e), (20b) and

U_{opt}^{sep} (x)

in Eq. (19a), (19b), (19c). Moreover, according to Theorem 3, the DDU-S

U (x)

in TSRO problem (1) can be equivalently substituted by

U_{fea}^{sep} (x)

and

U_{opt}^{sep} (x)

that are both completely separable. However, the coupling function of

U_{fea}^{sep} (x)

, which is

C_{fea} (\cdot)

in Eq. (18c), and the coupling function of

U_{opt}^{sep} (x)

, which is

C_{opt} (\cdot)

in Eq. (19c), are both nonconvex functions. Therefore, the convexity of the TSRO problem cannot be ensured by applying Theorem 5.

(2) Consider the polytope in the space of

(x, α, u, y)

as follows:

(59)

\{(x, α, u, y) | Y (x, u) \cap \{y | c^{T} y \leq α\}\}

The projection of Eq. (59) onto the space of

(x, α, u)

is:

(60)

\{(x, α, u) | H x + J α + K u \leq M\}

where

H, J

, and

K

are constant matrices and

M

is a constant vector. According to Definition 3, Eq. (60) is the graph of

D^{ext} (x, α)

. Then, constraint (36b) has equivalent formulations as in Eq. (61a), (61b), (61c), (61d), (61e).

(61a)

\{(x, α) | U (x) \subseteq D^{ext} (x, α)\} = \{(x, α) | H x + J α + K u \leq M, \forall u \in U (x)\}

(61b)

= \{(x, α) | \begin{matrix} h_{j}^{T} x + J α + \max_{u \in U (x)} k_{j}^{T} u \leq m_{j}, \\ \forall j = 1, \dots, len (M) \end{matrix}\}

(61c)

= \{(x, α) | \begin{matrix} h_{j}^{T} x + J α + \{\begin{matrix} \min_{λ_{j}} λ_{j}^{T} g (x) \\ s .t . G {(x)}^{T} λ_{j} = k_{j} \\ λ_{j} \geq 0 \end{matrix}\} \\ \leq m_{j}, \forall j = 1, \dots, len (M) \end{matrix}\}

(61d)

= \{(x, α) | \begin{matrix} \{\begin{matrix} h_{j}^{T} x + J α + λ_{j}^{T} g (x) \leq m_{j} \\ G {(x)}^{T} λ_{j} = k_{j} \\ λ_{j} \geq 0 \end{matrix}\} \neq \emptyset, \\ \forall j = 1, \dots, len (M) \end{matrix}\}

(61e)

= \{(x, α) | \underset{(*)}{\underset{⏟}{\{\begin{matrix} H x + J α + Λ g (x) \leq M \\ Λ G (x) = K \\ Λ \geq 0 \end{matrix}\}}} \neq \emptyset\}

where in Eq. (61b),

h_{j}, k_{j}, and m_{j}

are the jth row of

H, K, and M

, respectively, and

len (M)

denotes the number of rows in M; Eq. (61c) is derived by recalling Eq. (58) and applying the dual transformation to the inner maximization problem in Eq. (61b), where

λ_{j}

is the dual multiplier; Eq. (61d) is obtained by equivalently removing the minimization operator in Eq. (61c). In Eq. (61e),

Λ : = {(λ_{1}, \dots, λ_{len (M)})}^{T}

According to the transformation in Eq. (61a), (61b), (61c), (61d), (61e), constraint (36b) is equivalent to the projection of (∗) onto the space of

(x, α)

where (∗) is defined in Eq. (61e). By noting the bilinear term

Λ g (x)

and

Λ G (x)

in (∗), region (∗) may be nonconvex in the space of

(x, α, Λ)

. Therefore, constraint (36b), which is the projection of (∗), can also be nonconvex. When

G (x)

and

g (x)

are decision-independent constant matrices, region (∗) and its projection are guaranteed to be polyhedrons, justifying that constraint (36b) is convex in the case of DIUs.

5. Algorithm for robust dispatch with DDUs

Some algorithms have been proposed to solve specific robust dispatch problems with DDUs. However, the mechanism of DDU’s influence on algorithms is still unclear. Compared to these works, this paper unfolds the separable property of DDU and then designs a generic solution algorithm for DDUs based on the coupling function. The proposed algorithm is not only practical but also inspiring by revealing the essential change when DDUs are considered. It is possible to improve existing algorithms by applying the theoretical results in this section.

This section begins by reviewing two mainstream cutting-plane algorithms developed for TSRO problems and revealing their limitations in handling DDUs. Then, an improved solution strategy is proposed to solve the DDU-associated TSRO problem.

5.1. The cutting-plane algorithms for TSRO problems

Existing solution algorithms for TSRO problems, comprising the renowned Benders’ decomposition and C&CG algorithm, follow the principle of restriction and relaxation (R&R) [47,48], as illustrated in Fig. 13. Restriction indicates fixing the values of some decision variables in an optimization problem, Relaxation means dropping some of the constraints.

The basic idea of R&R is explained as follows. The TSRO problem is solved as the original problem. Without loss of generality, the original problem is assumed to be a minimization problem. The so-called equivalent master problem (EMP) is derived by applying an equivalent transformation to the original problem. A subproblem (SP) is obtained by applying restriction to the original problem, and a relaxed master problem (RMP) is derived by applying relaxation to the EMP. The RMP and EMP share the same optimization objective.

The solution algorithms based on the principle of R&R are designed to solve the SP and the RMP iteratively. The solution to SP and RMP provides an upper bound and lower bound of the optimum of original problem, respectively. Moreover, the values of the fixed decision variables in the RMP are updated by solving the RMP. By solving the SP, additional cutting-plane constraints are appended to the RMP. The iteration terminates when the optimum of SP and RMP are close enough, implying that the optimum of original problem is obtained. Under the same solution framework of R&R, the Benders’ decomposition and the C&CG algorithm mainly differ in the formulations of cutting-planes, EMP, and RMP.

(1) The Benders’ decomposition. Recalling the sets

Ξ

in Eq. (18b) and

Π

in Eq. (19b), the cutting planes in Benders’ decomposition are designed as:

(62)

{CP}_{fea}^{bd} (ξ, u) = \{x | ξ^{T} (b - A x - C u) \leq 0\}

(63)

{CP}_{opt}^{bd} (π, u) = \{(x, α) | π^{T} (b - A x - C u) \leq α\}

The

{CP}_{fea}^{bd} (ξ, u)

parameterized in

ξ \in Ξ

and

u \in U

is called the feasibility cut and is a half-plane in

R^{n_{x}}

. The

{CP}_{opt}^{bd} (π, u)

parameterized in

π \in Π

and

u \in U

is the optimality cut which is a half-plane in

R^{n_{x} + 1}

According to Theorem 3, the EMP in Benders’ decomposition, which is surrogate to the original TSRO problem (1), can be formulated as:

(64a)

\min_{x \in X, α \in R^{1}} f (x) + α

(64b)

x \in {CP}_{fea}^{bd} (ξ, u), \forall ξ \in Ξ, \forall u \in U (x)

(64c)

(x, α) \in {CP}_{opt}^{bd} (π, u), \forall π \in Π, \forall u \in U (x)

By applying partial enumeration to

ξ

Ξ

π

Π

, and

u

U (x)

, the Benders decomposition is intended to relax constraints (64b), (64c) into linear cutting-planes. To be specific, if a tentative solution

x^{j}

does not satisfy constraint (64b), there exists

ξ^{j} \in Ξ

and

u^{j} \in U (x^{j})

such that

x^{j} \notin {CP}_{fea}^{bd} (ξ^{j}, u^{j})

. Then, in the RMP, a cutting-plane

x \in {CP}_{fea}^{bd} (ξ^{j}, u^{j})

is formulated to approximate constraint (64b). Similarly, if a tentative solution

(x^{j}, α^{j})

does not satisfy constraint (64c), there exists

π^{j} \in Π

and

u^{j} \in U (x^{j})

such that

x^{j} \notin {CP}_{opt}^{bd} (π^{j}, u^{j})

. Then, in the RMP, a cutting-plane

(x, α) \in {CP}_{opt}^{bd} (π^{j}, u^{j})

is formulated to approximate constraint (64c). For details of Benders decomposition for TSRO, please refer to Refs. [[49], [50], [51]].

(2) The C&CG algorithm. Denote by

{CP}^{ccg} (u)

the cutting-plane in the C&CG algorithm:

(65)

{CP}^{ccg} (u) = \{(x, α) | Y (x, u) \cap \{y | c^{T} y \leq α\} \neq \emptyset\}

It is a half-plane in

R^{n_{x} + 1}

parameterized in

u \in U

. When the C&CG algorithm is applied to the TSRO problem (1), the EMP is designed as:

(66a)

\min_{x \in X, α \in R^{1}} f (x) + α

(66b)

(x, α) \in {CP}^{ccg} (u), \forall u \in U (x)

The equivalence between the original TSRO problem Eq. (1a), (1b), (1c), (1d), (1e) and the EMP (66) is justified by recalling problem (36). By partially enumerating the

u \in U (x)

in constraint (66b), the C&CG algorithm is intended to relax constraint (66b) into linear cutting-planes. To be specific, if a tentative solution

(x^{j}, α^{j})

does not satisfy constraint (66b), there exists

u^{j} \in U (x^{j})

such that

(x^{j}, α^{j}) \notin {CP}^{ccg} (u^{j})

. Then, in the RMP, a cutting-plane

(x, α) \in {CP}^{ccg} (u^{j})

is formulated to approximate constraint (66b). For details of the C&CG algorithm, please refer to Ref. [52].

5.2. Limitations of cutting-plane algorithms

The inapplicability of the aforementioned cutting-plane algorithms to DDU-associated TSRO problems is discussed from the following two perspectives.

(1) Theoretical perspective. As discussed in Section 4, DDUs may introduce nonconvexity to the original problem (1). The EMPs could also be nonconvex optimization problems because the original problem (1), the EMP (64) in Benders’ decomposition, and the EMP (66) in C&CG algorithm are equivalent. The nonconvexity of EMPs, if any, must reside in the constraints of EMP. As stated in Section 5.1, the two cutting-plane algorithms are intended to relax and approximate the feasible region of EMP by a set of cutting planes. When the feasible region of EMP becomes nonconvex due to DDUs, the RMPs formulated by cutting-plane constraints are not guaranteed to be a relaxation of the EMP, leading to degraded quality of the computation result.

(2) Physical perspective. Recalling the physical meaning of RO, the solution to the TSRO problem hedges against the worst-case scenario within the uncertainty set. As discussed in Section 5.1, additional cutting planes parameterized in

u^{j}

are generated to handle the identified worst-case scenario when a tentative solution

x^{j}

is not content with the constraints in EMPs under scenario

u^{j} \in U (x^{j})

. However,

u^{j}

may reside outside the uncertainty set when

x

takes other values, as the uncertainty set varies with decision

x

. Therefore, the cutting planes parameterized in

u^{j}

may bring over-conservatism to the RMP, preventing the RMP from approximating the EMP.

Next, the potential consequences of applying conventional cutting-plane algorithms to solving DDUs-integrated TSRO problems are revealed and justified by illustrative examples.

Failure in robust feasibility. The RMP, consisting of cutting planes, is not guaranteed to be a relaxation to the EMP. The over-conservative cutting-plane may be an empty set, making the RMP infeasible and interrupting the iterative algorithm. Once the decision maker observes an infeasible RMP, the original problem would be misdiagnosed as infeasible. An example of the failure in robust feasibility is detailed as follows.

Example 8. On the basis of Example 7, the TSRO problem is expressed as:

(67a)

\min_{x \in [0.8, 2.2]} | x - 1.5 |

(67b)

s .t . x \in X_{R}

According to Example 7, the explicit formulation of

X_{R}

[0.8, 1] \cup [2, 2.2]

. Therefore, the optimal solution to Eq. (67a), (67b) is

x^{*} = 1

When applying Benders’ decomposition to solving Eq. (67a), (67b), the tentative solution in the first iteration round is

x^{1} = 1.5

, which corresponds to

u^{1} = {(3, 8)}^{T} \in U (x^{1})

and

ξ^{1} = {(0, - 1, 0, - 1, - 1, 0)}^{T} \in Ξ

such that

x^{1} \notin {CP}_{fea}^{bd} (ξ^{1}, u^{1})

. Then, a cutting-plane constraint

x \in {CP}_{fea}^{bd} (ξ^{1}, u^{1})

with explicit formulation as

(68)

x \in R^{1} : ξ_{1}^{1} + ξ_{2}^{1} + ξ_{3}^{1} + ξ_{4}^{1} - u_{1}^{1} ξ_{5}^{1} + u_{2}^{1} ξ_{6}^{1} \leq 0

which is appended to the EMP. However, it can be justified that constraint (68) is an empty set that makes the RMP infeasible. As observed from an infeasible RMP, the decision maker would be under the delusion that the original problem (67) is infeasible.

When the C&CG algorithm is applied to solving Eq. (67a), (67b), there exists

u^{1} = {(3, 8)}^{T} \in U (x^{1})

in the first iteration round such that

x^{1} \notin {CP}^{ccg} (u^{1})

. Then, a cutting-plane constraint

x \in {CP}^{ccg} (u^{1})

is explicitly formulated as

(69)

x \in R^{1} : \{\begin{array}{l} - 1 \leq y_{1} \leq 1 \\ - 1 \leq y_{2} \leq 1 \\ 3 \leq y_{1} + y_{2} \leq 8 \end{array}\} \neq \emptyset

which is appended to the RMP. However, constraint (69) is an empty set that makes the RMP infeasible. Similarly, the feasibility of Eq. (67a), (67b) would be misconceived by the decision maker.

Failure in robust optimality. The over-conservative cutting planes in RMPs may shrink the feasibility region of the EMP, resulting in a suboptimal solution to the original problem, as suggested by the following example.

Example 9 On the basis of Example 8, the constraints of the second-stage problem are expressed as:

(70)

Y (x, u) = \{y \in R^{2} | \begin{array}{l} - 1 \leq y_{1} \leq 1 (ξ_{1}, ξ_{2}) \\ - 1 \leq y_{2} \leq 1 (ξ_{3}, ξ_{4}) \\ u_{1} - 0.5 x \leq y_{1} + y_{2} (ξ_{5}) \\ y_{1} + y_{2} \leq u_{2} + 0.5 x (ξ_{6}) \end{array}\}

The dispatchable region corresponds to

Y (x, u)

in Eq. (70) is

(71)

D (x) = \{u \in R^{2} | \begin{array}{l} u_{1} \leq 2 + 0.5 x \\ - u_{2} \leq 2 + 0.5 x \\ u_{1} - u_{2} \leq x \end{array}\}

The DDU-S

U (x)

in Eq. (56), the dispatchable region

D (x)

, and their graphs are illustrated in Fig. 14 to derive the RFR.

Fig. 14 reflects that the explicit formulation of the RFR in this example is

X_{R} = [0.8, 4 / 3] \cup [1.6, 2.2]

. Therefore, the optimal solution in this example is

x^{*} = 1.6

, and the optimum is 0.1.

When the Benders’ decomposition is applied, the tentative solution in the first iteration round is

x^{1} = 1.5

, and there exists

u^{1} = {(3, 8)}^{T} \in U (x^{1})

and

ξ^{1} = {(0, - 1, 0, - 1, - 1, 0)}^{T} \in Ξ

such that

x^{1} \notin {CP}_{fea}^{bd} (ξ^{1}, u^{1})

. Then, a feasibility cutting-plane

x \in {CP}_{fea}^{bd} (ξ^{1}, u^{1})

is explicitly formulated as

(72a)

x \in R^{1} : ξ_{1}^{1} + ξ_{2}^{1} + ξ_{3}^{1} + ξ_{4}^{1} + (0.5 x - u_{1}^{1}) ξ_{5}^{1} + (0.5 x + u_{2}^{1}) ξ_{6}^{1} \leq 0

(72b)

x \in R^{1} : 2 \leq x

which is appended to the RMP. In the second iteration round, the RMP generates a tentative solution

x^{2} = 2

, which is proved to be within the RFR

X_{R}

. Then, the iteration algorithm terminates and outturns a suboptimal solution

x^{2} = 2

with an objective value of 0.5.

When the C&CG algorithm is applied, the tentative solution in the first iteration round is also

x^{1} = 1.5

, and there exists

u^{1} = {(3, 8)}^{T} \in U (x^{1})

such that

x^{1} \notin {CP}^{ccg} (u^{1})

. Then, a cutting-plane

x \in \notin {CP}^{ccg} (u^{1})

is explicitly formulated as

(73a)

x \in R^{1} : \{\begin{array}{l} - 1 \leq y_{1} \leq 1 \\ - 1 \leq y_{2} \leq 1 \\ 3 - 0.5 x \leq y_{1} + y_{2} \leq 8 + 0.5 x \end{array}\} \neq \emptyset

(73b)

x \in R^{1} : x \geq 2

which is appended to the RMP. Similarly, the RMP with constraint (73) would outturn a suboptimal solution

x^{2} = 2

with an objective value of 0.5.

5.3. Improved solution algorithm for DDUs

In this subsection, an improved solution algorithm is proposed to solve the DDU-associated TSRO problems based on the separability of DDU-S.

(1) DDU-S with complete separability. Without loss of generality, the DDU-S

U (x)

is assumed to be separable as in Eq. (2). Denote by

{EC}^{ccg} (ξ)

the enhanced C&CG cuts parameterized in

ξ \in Ξ

. Then, it can be formulated as:

(74)

{EC}^{ccg} (ξ) : = \{(x, α, u) |\begin{array}{l} (x, α) \in {CP}^{ccg} (u) \\ u^{j} = C (ξ, x) \end{array}\}

Then, the EMP (64) can be reformulated as

(75a)

\min_{x \in X, α \in R^{1}, u} f (x) + α

(75b)

(x, α, u) \in {EC}^{ccg} (ξ), \forall ξ \in Ξ

By applying partial enumeration to

ξ \in Ξ

in Eq. (75b), the RMP is formulated as:

(76a)

\min_{x \in X, α \in R^{1}, u} f (x) + α

(76b)

over: x \in X, α \in R^{1}, u^{j} \in U

(76c)

(x, α, u^{j}) \in {EC}^{ccg} (ξ^{j}), \forall j

The RMP (76) provides a valid relaxation to the EMP (75). As the constraints in constraint (76c) accumulate, the RMP (76) is approaching the EMP (75).

(2) DDU-S with partial separability. The enhanced feasibility Benders’ cut parameterized in

ξ \in Ξ

is formulated as:

(77)

{EC}_{fea}^{bd} (ξ) : = \{(x, u) |\begin{matrix} x \in {CP}_{fea}^{bd} (ξ, u) \\ u = C_{fea} (ξ, x) \end{matrix}\}

The

{EC}_{fea}^{bd} (ξ)

is a constraint in

R^{n_{x} + n_{u}}

. The enhanced optimality Benders’ cut is:

(78)

{EC}_{opt}^{bd} (π) : = \{(x, α, u) |\begin{array}{l} (x, α) \in {CP}_{opt}^{bd} (π, u) \\ u = C_{opt} (π, x) \end{array}\}

which is a constraint in

R^{n_{x} + n_{u}}

Then the EMP can be formulated as

(79a)

\min_{x} f (x) + α

(79b)

over: x \in X, α \in R^{1}, u_{fea}, u_{opt} \in U

(79c)

(x, u_{fea}) \in {EC}_{fea}^{bd} (ξ), \forall ξ \in Ξ

(79d)

(x, α, u_{opt}) \in {EC}_{opt}^{bd} (π), \forall π \in Π

By applying partial enumeration to

ξ \in Ξ

in constraint (79c) and

π \in Π

in constraint (79d), the RMP is formulated into:

(80a)

\min_{x} f (x) + α

(80b)

over: x \in X, α \in R^{1}, u_{fea}^{j}, u_{opt}^{k} \in U

(80c)

(x, u_{fea}^{j}) \in {EC}_{fea}^{bd} (ξ^{j}), \forall j

(80d)

(x, α, u_{opt}^{k}) \in {EC}_{opt}^{bd} (π^{k}), \forall k

6. Illustrative applications of robust dispatch with DDUs

This section presents three applications of robust dispatch with DDUs: frequency-constrained reserve allocation with wind generators’ DDUs [12], robust dispatch considering DR’s DDUs [20], and robust scheduling of VPP under both DDUs and DIUs [29]. These works have been published previously, and we briefly introduce them here only to showcase the theoretical results proposed in this paper. For detailed information, one can refer to the above references.

6.1. Source side: Frequency-constrained reserve allocation of wind farms

Through synthetic inertia and droop control, RES units such as wind generators can provide fast responses to frequency regulation by reserve allocation. In other words, wind generators can be considered a kind of frequency regulation resource. Meanwhile, the reserve allocation problem of wind generators should consider the uncertainty of wind generation power. The decision to preserve power will influence the realization of the real available wind power, resulting in DDUs.

The frequency-constrained reserve allocation problem of wind generators can be formulated as a TSRO problem. At stage one, the here-and-now decision

x

is composed of the unit commitment decisions of synchronous generators and the energy dispatch and reserve allocation decisions of synchronous and wind generators. The reserve allocation affects the uncertainty of wind power. After the real wind power is observed, the second-stage decision

y

, including the re-dispatch of synchronous and wind generators, is made for frequency regulation.

Example 2 considers DDUs of wind generators in different modes. The Delta mode is adopted in Ref. [12] to provide frequency reserve with a reliability guarantee, expressed as:

(81)

Delta: \{\begin{matrix} R_{t} = \min \{P_{t}^{mppt}, λ_{t}^{r} P^{rate}\} \\ P_{t}^{avail} = \max \{P_{t}^{mppt} - λ_{t}^{r} P^{rate}, 0\} \end{matrix}

where

R_{t}, P_{t}^{avail}, λ_{t}^{r}

, and

P_{t}^{mppt}

represent the preserved power, the available power, the de-loading ratio, and the maximal available power of the wind generator at time slot

t

, respectively;

P_{t}^{mppt}

indicates a DIU variable solely determined by the uncertain wind speed; the preserved power

R_{t}

and the available power

P_{t}^{avail}

embody DDU variables depending on the de-loading ratio decision

λ_{t}^{r}

By imposing a robustness guarantee on

P_{t}^{mppt} \geq {\bar{R}}_{t} : = λ_{t}^{r} P^{rate}

, the DDU model is further simplified into

(82a)

{\bar{P}}_{t}^{avail} = P_{t}^{mppt} - {\bar{R}}_{t}, \forall t

(82b)

{\underset{̲}{P}}_{t}^{mppt} \leq P_{t}^{mppt} \leq {\bar{P}}_{t}^{mppt}, \forall t

(82c)

\sum_{t} | P_{t}^{mppt} - P_{t}^{av} | / P_{t}^{h} \leq Γ^{T}

where

P_{t}^{av}

and

P_{t}^{h}

denote the expected value and fluctuation level of

P_{t}^{mppt}

, respectively;

{\bar{P}}_{t}^{mppt}

and

{\underset{̲}{P}}_{t}^{mppt}

denote the upper- and lower-bounds of maximal available power;

Γ^{T}

represents the temporal robustness budget for the wind generator.

In the DDU model, the DDU variable is

(83)

u : = {\bar{P}}_{t}^{avail}, \forall t

The variable affecting the DDU in the here-and-now decision

x

{\bar{R}}_{t}, \forall t

Based on the separability of DDU-S, the auxiliary random variable is

(84)

ξ : = P_{t}^{mppt}, \forall t

The DIU support set Ξ is defined by constraints (82b)–(82d), and the coupling function

C (ξ, x)

is defined by Eq. (82a). It is clear that the corresponding DDU-S is completely separable. Hence, the robust reserve allocation problem can be directly solved by the improved C&CG algorithm presented in Section 7.

Numerical results of a modified PJM 5-bus system are provided to verify the significance of considering DDUs in robust dispatch. Fig. 15, Fig. 16 [12] exhibit the frequency trajectories at each time considering the DDU and DIU models, respectively. By considering the DDU model of wind power, the frequency deviation is still maintained in the expected region. On the contrary, when a DIU set is used to model wind power uncertainty, it cannot provide anticipated frequency support, leading to violations of the frequency nadir. The numerical comparison demonstrates the superiority of modeling wind generation output as DDU rather than DIU.

Numerical tests on a real regional power grid of China with 57 conventional generators and six renewable energy power plants [53] are carried out to verify the scalability of the proposed solution algorithm. Two scenarios are considered to test the effect of RESs (including wind farms and PV stations) in the TSRO model with DDUs.

(1) Compared (COMP.) Model: only synchronous generators provide frequency regulation service.

(2) Proposed (PPSD.) Model: synchronous generators and RESs all provide frequency regulation service.

The numerical results are depicted in Fig. 17. In Fig. 17(a), during 11:00–16:00, the proportion of RES generation power in the COMP. model is reduced to about 70% compared to the PPSD. Model. In Fig. 17(b), the penetration of RES decreases as the value of the contingency size increases from 0 to 3%. The curve of PPSD. Model drops much slower than that of COMP. Model. Besides, the problem of the COMP. Model becomes infeasible when the contingency size increases to 2.9%. The post-contingency frequency dynamics at different time slots are plotted in Fig. 17(c), verifying the robust feasibility. Moreover, the computational time of the PPSD. Model is 1914.22 s, which is compatible with the day-ahead dispatch problem in real-world large power systems.

6.2. Demand side: Robust dispatch with DR

DR is a promising solution to reducing operation costs and alleviating system risks by encouraging end-users to participate in system regulation. However, the demand itself is commonly stochastic. Since the decision of DR may change probability distributions or bounds of demand, the uncertainty of DR belongs to DDU.

The robust dispatch problem with DR, belonging to TSRO, has been investigated in our previous work [20]. At stage one, the here-and-now decision

x

is composed of the nominal generations and reserve capacities of conventional units and the set-point of DR

d^{0}

. This problem concentrates on the DDU of DR depending on the set-point

d^{0}

. After the real loads are observed, the stage-two decision

y

, regulated generation outputs of conventional units, is made to maintain power balance and power flow constraints.

The loads themselves own the following uncertainty

(85)

D^{E} = \{d | - δ_{d}^{-} \leq d - d^{E} \leq δ_{d}^{+}\}

where

δ_{d}^{-},

δ_{d}^{+}

, and

d^{E}

denote the upper fluctuation, lower fluctuation, and the forecast value of the load.

Through DR, the forecast loads are changed to the set-point

d^{0}

. Assuming that the fluctuations of loads are proportionally amplified with the forecast load, the new uncertainty set is defined as

(86)

D (d^{0}) = \{d |- \frac{d^{0}}{d^{E}} δ_{d}^{-} \leq d - d^{0} \leq \frac{d^{0}}{d^{E}} δ_{d}^{+}\}

where

d^{0} / d^{E}

represents the amplification ratio. Hence,

D (d^{0})

indicates a DDU-S depending on the here-and-now decision

d^{0}

The DDU-S of DR belongs to box DDUs in Example 1, which is completely separable. Hence, the robust dispatch problem with DR is solved by using the improved C&CG algorithm proposed in Section 7.

Our previous work [20] reveals the influence of DDUs in DR. It validates the effectiveness of the enhanced C&CG algorithm in the IEEE 5-bus system with two loads participating in the DR program. Fig. 18 illustrates the iteration procedure of the improved C&CG algorithm, where Disp-Reg reflects the dispatchable region in Definition 3. Fig. 18 exhibits the bilateral matching between the DDU-S and the dispatchable region. If the uncertainty dependents on decisions, the dispatch result may fail to fulfill the robust feasibility or/and optimality. Fig. 19 demonstrates the iteration result of the standard C&CG algorithm with the DIU model. The dispatchable region covers worst-case scenarios outside the uncertainty set, implying the loss of robust optimality.

6.3. Reserve deployment: Robust scheduling of VPP

VPP can provide power systems with flexibility by aggregating renewable generation units, conventional power plants, energy storage, and flexible demands. The scenario in which a VPP participates in the day-ahead reserve market is adopted in our previous work [29]. The VPP, which does not know the real reserve deployment request in the day-ahead time scale, will provide a flexibility range. Then, the bulk grid will transport a regulating signal varying within the flexibility range. Particularly, the regulating signal is uncertain for the VPP. Meanwhile, the uncertainty depends on the flexibility range it decides. Therefore, the regulating signal from the day-ahead reserve market for the VPP belongs to DDU.

The robust scheduling problem of VPP can be formulated by a TSRO problem. At stage one, the VPP will make the here-and-now decision x composed of the traded power, the upward/downward reserve capacity in the market, and the unit commitment of conventional power plants. The VPP should consider the DDU of regulating signals and the DIU of energy market price and wind generation power. At stage two, RESs, including conventional power plants, energy storage units, and flexible demands, are employed to tackle the uncertainties.

The VPP focuses on the uncertain power exchange between it and the bulk grid, defined as

(87)

P_{t}^{EXCH} = P_{t}^{E} + P_{t}^{R+} - P_{t}^{R -}, \forall t

where

P_{t}^{E}

represents the decision of power traded in the energy market at time slot

t

;

P_{t}^{R+}

and

P_{t}^{R -}

denote the uncertain upward and downward regulating signals to the VPP at time slot

t

, respectively.

The uncertainties of upward and downward regulating signals are described as:

(88a)

0 \leq P_{t}^{R+} \leq P_{t}^{RC+}, \forall t

(88b)

0 \leq P_{t}^{R -} \leq P_{t}^{RC -}, \forall t

(88c)

\sum_{t} P_{t}^{R+} \leq \sum_{t} {sig}_{t}^{+} P_{t}^{RC+}

(88d)

\sum_{t} P_{t}^{R -} \leq \sum_{t} {sig}_{t}^{-} P_{t}^{RC -}

where

{sig}_{t}^{+}

and

{sig}_{t}^{-}

are the average normalized upward/downward reserve energy called on to provide at time slot

t

;

P_{t}^{RC+}

and

P_{t}^{RC -}

indicate the decisions of upward and downward reserve capacities, respectively.

The uncertainty set of

P_{t}^{EXCH}

is formulated by Eqs. (87), (88a), (88b), (88c), (88d). Since the set depends on the stage-one decisions

P_{t}^{E}, P_{t}^{RC+}

, and

P_{t}^{RC -}

, it belongs to DDU-S. According to Section 2, this complex DDU-S is partially separable. Section 4 demonstrates that TSRO problems with partially separable DDU-S could be nonconvex. The improved C&CG algorithm in Section 5 may fail to guarantee the robust feasibility or/and optimality. Section 5 also presents an improved Benders’ decomposition algorithm to deal with partially separable DDU-S.

The effectiveness of the improved Benders’ decomposition algorithm in solving the robust VPP scheduling problem has been verified in our previous research [29]. The standard C&CG algorithm is utilized to solve the same VPP scheduling problem. Table 1 [29] lists the outcomes of both algorithms. The net profit by the standard C&CG algorithm is much smaller than that by the improved Benders’ decomposition algorithm. This is because the feasibility cut is directly generated by the worst-case uncertainty realization, in the C&CG algorithm, ignoring the fact that the uncertainty set varies with decisions. Moreover, the improved Benders’ decomposition algorithm requires fewer iterations and less solution time, demonstrating the efficiency of the Benders-dual cutting-plane.

In this section, several applications are presented to showcase different types of DDUs in power system dispatch and how to handle them. It should be noted that the proposed theorems and algorithms are also applicable in other RESs, such as PV, energy storage units, and market participants.

7. Conclusions

This paper provides a systematic framework for two-stage robust dispatch with DDUs, highlighting the fundamental relations and distinctions between DDUs and DIUs. The structural properties of DDUs are first explored through the concept of separability, with a rigorous definition, proved existence, and illustrative examples. The results reveal that a DDU-S with an arbitrary set-valued mapping formulation can be decomposed into a coupling function parameterized in decisions and a decision-irrelevant uncertainty set. Subsequently, robust dispatch with DDUs is equivalently formulated as the bilateral matching between the DDU-S and the dispatchable region or its variant in combination with the region-based flexibility characterization. Compared with the unilateral matching under DIUs, the bi-directional matching facilitated by DDUs endows more system flexibility, despite the possibility of introducing non-convexity into the robust dispatch problem. It has been proved that non-convexity, if any, is attributed to the non-convex coupling function. Afterward, an improved algorithm is proposed to precisely and efficiently solve the generic DDU-associated TSRO dispatch, as a variant of the iterative algorithm within the framework of restriction and relaxation. Unlike traditional primal or dual cutting planes, the enhanced cuts are designed based on the separability of DDU-S and can autonomously adapt to the convexity or non-convexity of the problem. Finally, applications to robust dispatch problems considering the DDUs of wind generators, DR, and VPP are introduced. These applications emphasize the importance of considering DDUs and verifying the effectiveness of proposed algorithms in dealing with DDUs.

It should be noted that though this paper focuses on the TSRO, its key idea of DDU could still work in other optimization problems, such as multi-stage RO and two/multi-stage distributionally RO. Hopefully, this study will establish a better understanding of the characterizations of DDUs and the algorithms for robust two-stage dispatch with DDUs. We hope it could enlighten more theoretical and practical works concerning DDUs.

CRediT authorship contribution statement

Yunfan Zhang: Conceptualization, Formal analysis, Writing – original draft, Visualization, Validation, Methodology, Data curation. Yifan Su: Visualization, Validation, Methodology, Formal analysis, Data curation, Writing – original draft. Feng Liu: Project administration, Methodology, Formal analysis, Writing – review & editing, Supervision, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the Joint Research Fund in Smart Grid (U1966601) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and State Grid Corporation of China.

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1. Introduction

1.1. Background

1.2. DDU in power system dispatch

1.3. Contributions

1.4. Organization

2. The robust characterization of DDUs

2.1. The complete separability of DDU-S

2.1.1. Definition

2.1.2. Ubiquity

2.1.3. Examples

2.2. The partial separability of a DDU-S

2.2.1. Definition

2.2.2. Ubiquity

2.2.3. Example

3. Mechanism of robust dispatch with DDUs

4. Convexity of robust dispatch problem with DDUs

4.1. Case of DIUs

4.2. Case of DDUs

5. Algorithm for robust dispatch with DDUs

5.1. The cutting-plane algorithms for TSRO problems

5.2. Limitations of cutting-plane algorithms

5.3. Improved solution algorithm for DDUs

6. Illustrative applications of robust dispatch with DDUs

6.1. Source side: Frequency-constrained reserve allocation of wind farms

6.2. Demand side: Robust dispatch with DR

6.3. Reserve deployment: Robust scheduling of VPP

7. Conclusions

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

References

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