Convex Hull Pricing for Unit Commitment: Survey, Insights, and Discussions

Abstract

Energy prices are usually determined by the marginal costs obtained by solving economic dispatch problems without considering commitment costs. Hence, generating units are compensated through uplift payments. However, uplift payments may undermine market transparency as they are not publicly disclosed. Alternatively, energy prices can be obtained from the unit commitment problem which considers commitment costs. But, due to non-convexity, prices may not monotonically increase with demand. To resolve this issue, convex hull pricing has been introduced. It is defined as the slope of the convex envelope of the total cost function over the convex hull of a unit commitment (UC) problem. Although several approaches have been developed, a relevant survey has not been found to aid the understanding of convex hull pricing from the current limited literature. This paper provides a systematic survey of convex hull pricing. It reviews, compares, and links various existing approaches, focusing on the modeling and computation of convex hull prices. Furthermore, this paper explores potential areas of improvement and future challenges due to the ongoing efforts for power system decarbonization.

1. Introduction

In the US, since the deregulation of electricity systems in the 1990s, the focus has been on reducing the cost of electricity, improving price signals, increasing competition, and encouraging innovation and investment in the power sector [1]. It lead to the establishment of independent system operators (ISOs) and regional transmission operators (RTOs) to run the day-ahead and real-time electricity markets by solving the unit commitment (UC) and economic dispatch (ED) problems. Under the UC problem, the objective function consists of the generator’s commitment costs (e.g., start-up and no-load costs) as well as its generation costs. The decision variables include discrete (e.g., commitment status) and continuous variables (e.g., power generation). The objective function is piecewise linear (modeled in a linear way), and the constraints are linear, resulting in a mixed binary linear programming problem [2]. Hence, the UC problem is non-convex, resulting in energy prices that may not monotonically increase with demand [3,4]. Under the ED problem, the commitment decisions from the UC problem are fixed and all the decision variables are continuous, resulting in a linear programming problem. Hence, the ED problem is convex and the resulting energy prices are monotonically non-decreasing. Therefore, ISOs and RTOs set the energy prices from the ED problem, defined as the slope of the total cost function with respect to demand [3]. Since the ED problem does not consider commitment costs, significant uplift payments (Uplift payments are supplementary payments provided to generators to compensate for specific costs that are not accounted for in the market price, such as start-up and no-load costs. Uplift payments include lost opportunity costs and make whole payments) are often needed to compensate the generating units [5]. However, such uplift payments are not uniform for different units even on the same bus, causing market transparency issues since they are not publicly disclosed. Resulting in significant research efforts directed towards the reduction in uplift payments [4,6].

To resolve the issues mentioned above, the convex hull pricing mechanism where convex hull prices are obtained as the slope of the convex envelope of the total cost function over the convex hull of a UC problem, was first introduced in [4]. The convex envelope is the “tightest” convex function that supports the cost function from below, and the convex hull is the smallest convex set that contains all feasible solutions as shown in Figure 1. For illustrative purposes, a general mixed-integer linear programming (MILP) problem is considered in Figure 1. Convex hull pricing can possibly resolve the aforementioned issues since the slope of the convex envelope is non-decreasing with respect to demand, and the convex hull avoids non-convex binary commitment variables. These benefits drew attention from ISOs/RTOs in convex hull pricing, prompting the development of various approaches [7,8,9,10,11,12,13,14,15,16,17,18]. However, a relevant survey from the currently limited literature has not been found to provide a better understanding of convex hull pricing for UC from a modeling standpoint.

This paper provides a systematic survey of convex hull pricing, focusing on the modeling and computation of convex hull prices for UC. For an economic analysis of convex hull pricing readers are encouraged to read [19,20,21]. Beyond UC, convex hull pricing has been used for AC optimal power flow [22], real-time pricing in smart grids [23] and demand response [24,25]. As per the Web of Science database, there are 27 papers published on convex hull pricing. Under convex hull pricing for UC, most of the current approaches start from the UC problem and its convexified version for convex hull pricing. They can be categorized into the following two types:

• The first type is to solve the convex hull pricing problem corresponding to convexified UC which requires the convex envelope of the total cost function over the convex hull of the problem, such a convexified problem can be equivalently formulated by using the convex envelope and the convex hull of each unit [7,8,9,10,11,12,13,14].
• The second type is to solve the Lagrangian dual problem of a UC problem. The resulting optimal multipliers are equal to the convex hull prices calculated by using approaches of the first type mentioned above [15,16,17,18].

The structure of this paper is as follows: The UC problem and its convexified version for convex hull pricing are elaborated in Section 2. Then the aforementioned two categories of convex hull pricing approaches are discussed in Section 3 and Section 4, respectively. The discussions and insights are presented in Section 5, followed by conclusions in Section 6.

2. UC Problem and Its Convexification for Convex Hull Pricing

A UC problem is formulated in Section 2.1, followed by its convexified version for convex hull pricing in Section 2.2.

2.1. UC Problem

Let G and T delineate the sets of units and time slots, respectively. For simplicity, a two-bin formulation without transmission constraints is adopted [2]. For unit g, its decision variables at time slot t consist of binary startup $u_{g,t}$, commitment decisions on/off $x_{g,t}$, and continuous generation level decision $p_{g,t}$. The objective of the UC problem is to minimize the total cost of operation while meeting the system-level and unit-level constraints, as discussed below:

(1)

Objective function

The objective function or the cost function of the UC problem minimizes the total fuel and commitment costs (including start-up and no-load costs) of all the units, i.e.,

$$\underset{\begin{array}{c}\left\{p_{g,t},u_{g,t},x_{g,t}\right\},\\ \forall g\in G,\forall t\in T\end{array}}{min}z=\sum _{g\in G}{z}_g=\sum _{\begin{array}{c}g\in G,\\ t\in T\end{array}}\left(\begin{array}{c}{f}_g\left(p_{g,t},x_{g,t}\right)\\ +c_g^s{u}_{g,t}+c_g^n{x}_{g,t}\end{array}\right),$$

(1)

where $f_g$ is the piecewise linear fuel cost function of unit g and its slope is monotonically non-decreasing within blocks (i.e., segments) [26]; $c_g^s$ and $c_g^n$ are the start-up and no-load costs of unit g, respectively.

(2)

System-level constraints

The system-level constraints capture the operational requirements that couple generating units. For simplicity, only the power balance constraints, where the total power generation at any time slot equals the demand at that time slot are considered in this paper as follows:

$$\sum _{g\in G}{p}_{g,t}=D_t,\forall t\in T,$$

(2)

where $D_t$ is system demand at time slot t.

(3)

Unit-level constraints of unit g

Unit-level constraints capture the operational requirements enforced on individual units, as illustrated below:

(3.1)

Requirements on generation capacities: When unit g is online, its generation level $p_{g,t}$ must be between its minimum $P_g^{min}$ and maximum $P_g^{max}$, i.e.,

$$x_{g,t}{P}_g^{min}\le p_{g,t}\le x_{g,t}{P}_g^{max},\forall g\in G,\forall t\in T.$$

(3)

(3.2)

Requirements on start-up: The binary start-up variable $u_{g,t}$ is one if unit g is online at time slot t, i.e.,

$$x_{g,t}-x_{g,t-1}\le u_{g,t},u_{g,t}\le x_{g,t},\forall g\in G,\forall t\in T.$$

(4)

Also, unit g cannot start up at time slot t if unit g is online at $t-1$, i.e.,

$$x_{g,t-1}+u_{g,t}\le 1,\forall g\in G,\forall t\in T.$$

(5)

(3.3)

Requirements for minimum up- and down-time: The unit g stays online (resp. offline) for its minimum-up (resp. minimum -down) time slots, i.e.,

$$\sum _{i=t-L_g+1}^t{u}_{g,i}\le x_{g,t},\forall g\in G,\forall t\in T,$$

(6)

$$\sum _{i=t-l_g+1}^t{u}_{g,i}\le 1-x_{g,t-l},\forall g\in G,\forall t\in T,$$

(7)

where $L_g$ and $l_g$ are the minimum up and down time of unit g, respectively.

It is important to note that the initial conditions of the unit g affect the commitment statuses. If unit g has been online for $s_g$ slots, unit g should be online during the next $L-s_g$ slots, i.e.,

$$u_{g,t}=1,\forall g\in G,t\in \left\{1,2,\dots ,L-s_g\right\},$$

(8)

If unit g has been offline for $s_g$ slots, unit g should be offline during the next $l-s_g$ slots, i.e.,

$$u_{g,t}=0,\forall g\in G,t\in \left\{1,2,\dots ,l-s_g\right\}.$$

(9)

(3.4)

Requirements on ramp rates: The change in generation levels between two consecutive time slots must not exceed the unit’s ramp rate, i.e.,

$$p_{g,t}-p_{g,t-1}\le R_g{x}_{g,t-1}+V_g\left(1-x_{g,t-1}\right),\forall g\in G,\forall t\in T,$$

(10)

$$p_{g,t-1}-p_{g,t}\le R_g{x}_{g,t}+V_g\left(1-x_{g,t}\right),\forall g\in G,\forall t\in T,$$

(11)

where $R_g$ is the ramp-up/-down rate; $V_g$ is the start-up/shut-down ramp rate.

For ease of discussion, the UC problem (1)–(11) is compacted as follows:

$$\mathrm{UC}\text{-}\mathrm{Orig}:\underset{\{p,u,x\}}{min}z(p,u,x),$$

(12)

$$\mathrm{s}.\mathrm{t}.Ap=D,$$

(13)

$$(p,u,x)\in X.$$

(14)

The objective function in the UC-Orig problem (12) is the compact representation of the cost function (1), where the vectors p, u and x stack $p_{g,t}$, $u_{g,t}$ and $x_{g,t}$ of each unit g at each time slot t, respectively. The equality constraint (13) in the UC-Orig problem is the compact form of the system-wide constraints (2), where D stacks $D_t$ at each time interval t and A is the constant matrix. The feasible space defined by (14) is the compact form of the unit-level constraints (3)–(11) for all the units across all the time slots. Here, $X={\cup }_{g\in G}{X}_g$ is the feasible region of all units together and $X_g$ is the feasible region of the specific unit g. Note, that $X_g$ is called the unit formulation of unit g hereafter.

2.2. Convexified UC Problem for Convex Hull Pricing

Under the convex hull pricing scheme, the convex hull prices are defined as the slope of the convex envelope of the optimal objective function (12) over the convex hull of the entire unit-level constraint (14), with respect to demand vector D [4]. In the implementation, convex hull prices can be interpreted as the optimal dual multipliers corresponding to the system-wide constraint in the following convexified UC problem:

$$\begin{array}{cc}\hfill \mathrm{UC}\text{-}\mathrm{Convex}:& \underset{\{p,u,x\}}{min}{z}^c(p,u,x),\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left[\lambda \right]:Ap=D,\hfill \\ \hfill & (p,u,x)\in X^c,\hfill \end{array}$$

(15)

where $z^c$ is the convex envelope of the total cost function z in (12); $X^c$ is the convex hull of the unit-level constraint set X in (14); $\lambda$ is the dual multipliers associated with the system-wide constraint (13).

The equivalence between convex hull prices and the optimal dual multipliers ${\lambda }^{*}$ in the UC-Convex-Dual problem is shown in (16). The optimal value $z^{c^{*}}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$ in the UC-Convex problem (15) can be treated as a function of demand D, where $\left(p^{*},u^{*},x^{*}\right)$ is the optimal solution of the UC-Convex problem. Thanks to the strong duality of the UC-Convex problem, the value of $z^{c^{*}}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$ equals the optimal dual value $q^{c^{*}}\left({\lambda }^{*}\left(D\right)\right)$ in the following Lagrangian dual problem of the UC-Convex problem:

$$\begin{array}{c}\mathrm{UC}\text{-}\mathrm{Convex}\text{-}\mathrm{Dual}:{\displaystyle \underset{\lambda }{max}}{q}^c\left(\lambda \right),\mathrm{with}\\ q^c\left(\lambda \right)\equiv {\displaystyle \underset{\{p,u,x\}\in X^c}{min}}\left\{z^c(p,u,x)-{\lambda }^T(Ap-D)\right\}.\end{array}$$

(16)

Consequently, the slope of $z^{c^{*}}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$ with respect to demand D is equal to the slope of $q^{c^{*}}\left({\lambda }^{*}\left(D\right)\right)$ with respect to demand D. Moreover, as observed in the UC-Convex-Dual problem (16), the slope of $q^{c^{*}}\left({\lambda }^{*}\left(D\right)\right)$ with respect to demand D equals the optimal dual multipliers ${\lambda }^{*}$ associated with the system-wide constraint in the UC-Convex problem (15), since $\lambda$ is the coefficient prior to D. This makes convex hull prices exact ${\lambda }^{*}$. Note, that it was recently reported in [27] that the convex hull prices are dependent on the UC formulation. Specifically, the two types of formulation variations considered by ISOs for improving the solution time of the UC problem: the formulation tightening and contingency constraint screening. It is pointed out that although such attempts to reduce computational burden give the same primal optimal UC solutions, the convex hull prices which depend on the dual function of the primal formulation might not be the same. Given such dependence, sufficient conditions under which the convex hull prices are preserved were introduced in [27]. For example, conditions were derived under which multiple representations of nodal balance constraints give the same convex hull prices. In addition, numerical testing shows that convex hull prices can be affected by non-binding security constraints. Hence, it is concluded that any change in the UC formulation must be carefully examined before implementation for convex hull pricing.

3. Convex Hull Pricing Problem

As stated in Section 2, convex hull prices are derived from the solution to the UC-Convex problem, which can be regarded as a “convex hull pricing problem”. The key to solving the UC-Convex problem requires the explicit formulation of the convex envelope $z^c\left(p,u,x\right)$ over the convex hull $X^c$ by considering all the units. This can be tackled by reformulating $z^c\left(p,u,x\right)$ over $X^c$ as the summation of the convex envelope of the cost function of each unit over the convex hull of its unit formulation, as will be explained in Section 3.1. Consequently, the task reduces to determining the convex hull and convex envelope for each unit. Depending on the tightness (i.e., a unit formulation is tight when integer relaxation delineates its convex hull), the following three cases appear in the current approaches as will be successively examined from Section 3.2 to Section 3.4:

(1)

Case 1. A unit formulation is tight, and integer relaxation leads to its convex hull, as studied in [7,8,9];

(2)

Case 2. When a unit formulation is not tight, its convex hull can be formulated by various approaches, as discussed in [10,11,12,13,14], with a primary focus on convex hull formulation. However, solving the resulting convex hull pricing problems might increase the computational burden, as the complexity of accurately formulating a convex hull increases exponentially with the number of time slots [28];

(3)

Case 3. As a compromise, the employment of an approximate convex hull is also discussed in [8] when a unit formulation is not tight.

3.1. Convex Hull Pricing Problem from Each Unit

Let us consider the following problem for each unit:

$$\begin{array}{cc}\hfill \mathrm{UC}\text{-}\mathrm{CHP}:& \underset{\begin{array}{c}\left\{p_g,u_g,x_g\right\},\\ \forall g\in G\end{array}}{min}\sum _{g\in G}{z}_g^c\left(p_g,u_g,x_g\right),\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left[\lambda \right]:Ap=D,\hfill \\ \hfill & \left(p_g,u_g,x_g\right)\in X_g^c,\forall g\in G,\hfill \end{array}$$

(17)

where $z_g^c$ is the convex envelope of the cost function $z_g$ of unit g in the UC-Orig problem; $X_g^c$ is the convex hull of the unit formulation $X_g$ of unit g in the UC-Orig problem.

The equivalence between the UC-CHP problem (17) and the UC-Convex problem (15) is explained starting from the concepts of a conjugate function; the relevant concepts can be found in [29]. Particularly, the following two properties exist for the conjugate function:

• Property 1: The convex envelope of a function is the double conjugate of the function itself.
• Property 2: Additivity applies to a conjugate function.

The first property allows us to consider the convex envelope $z^c\left(p,u,x\right)$ in the UC-Convex problem as the double conjugate function of the total cost function $z\left(p,u,x\right)$ over the convex hull $X^c$ of the unit-level constraint set X. By the same token, the convex envelope $z_g^c\left(p_g,u_g,x_g\right)$ in the UC-CHP problem as the double conjugate function of the cost function $z_g$ over the convex hull $X_g^c$ of its unit formulation $X_g$. Moreover, Property 2 allows $z^c\left(p,u,x\right)$ to be the summation of $z_g^c\left(p_g,u_g,x_g\right)$. This fact indicates that the equivalence between the UC-CHP problem and the UC-Convex problem in convex hull pricing, i.e., the UC-CHP problem can be utilized for convex hull pricing once the formulations of $z_g^c\left(p_g,u_g,x_g\right)$ and $X_g^c$ are known.

3.2. Convex Hull and Convex Envelope under a Tight Formulation

With the basic unit-level constraints (3)–(9) that do not include ramp rates, the unit formulation $X_g$ is tight, as proved in [8] through polyhedron and [7,9] through network flow models. Therefore, integer relaxation of the unit formulation $X_g$ leads to the convex hull $X_g^c$. As for the convex envelope, when the cost function $z_g\left(p_g,u_g,x_g\right)$ is convex over the convex hull $X_g^c$, the convex envelope $z_g^c$ over the convex hull $X_g^c$ can be obtained by its integer relaxation; see [7,9]. When the cost function $z_g\left(p_g,u_g,x_g\right)$ is not convex over the convex hull $X_g^c$, where the slope of the fuel cost function $f_g(p_{g,t},x_{g,t})$ is non-monotonic, the convex envelope $z_g^c\left(p_g,u_g,x_g\right)$ is derived in [8]. The approaches in [7,8,9] are investigated in detail next. For each approach, the convex hull $X_g^c$ is first studied, followed by its convex envelope $z_g^c\left(p_g,u_g,x_g\right)$.

(1)

Network flow approach

Ref. [7] uses the basic constraints (3)–(9) without ramp rate constraints (10) and (11). The constraints (3)–(9) are regarded as an unimodal network flow model with integral capacities by treating variables as nodes. Such a network flow model can provide integral vertices based on the integrality theorem [30]. Therefore, the constraints (3)–(9) lead to a tight unit formulation, and thus integer relaxation yields the convex hull $X_g^c$. Consider the UC problem without ramp rate constraints in the compact form as shown in (18).

$$\begin{array}{cc}\hfill \mathrm{UC}\text{-}\mathrm{Constraints}(3–9):& \underset{\{p,u,x\}}{min}z(p,u,x)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& Ap=D,\hfill \\ \hfill & g(p,u,x)=H,\hfill \\ \hfill & p_{g,t}\ge 0,\hfill \\ \hfill & x_{g,t}\in 0,1,\hfill \\ \hfill & u_{g,t}\in 0,1.\hfill \end{array}$$

(18)

Here, $g(p,u,x)=H$ represents the compact form of the constraints (3)–(9). Note, that the variables $x_{g,t}$ and $u_{g,t}$ are binary. Under the network flow approach used in [7], it is possible to relax the binary restriction on $x_{g,t}$ and $u_{g,t}$ without affecting the optimal discrete values of $x_{g,t}$ and $u_{g,t}$, as shown in (19). Hence, (18) and (19) are equivalent. As a result, linear programming (LP) techniques (e.g., Simplex method) can be employed to solve the UC in (19).

$$\begin{array}{cc}\hfill \mathrm{UC}\text{-}\mathrm{Network}\mathrm{Flow}:& \underset{\{p,u,x\}}{min}{z}^c(p,u,x)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& Ap=D,\hfill \\ \hfill & g(p,u,x)=H,\hfill \\ \hfill & p_{g,t}\ge 0,\hfill \\ \hfill & x_{g,t}\ge 0,\hfill \\ \hfill & u_{g,t}\ge 0.\hfill \end{array}$$

(19)

As for the convex envelope, the key of the convex envelope through integer relaxation lies in whether the cost function $z_g$ is convex over the convex hull $X_g^c$. From the objective function in the UC-Orig problem, the cost function $z_g\left(p_g,u_g,x_g\right)$ is the summation of the following three components: the start-up cost $c_g^s{u}_{g,t}$, the no-load cost $c_g^n{x}_{g,t}$ and the fuel cost $f_g(p_{g,t},x_{g,t})$. The start-up and no-load costs are convex over the convex hull $X_g^c$. To this end, integer relaxation leads to the convex envelope $z_g^c\left(p_g,u_g,x_g\right)$ over the convex hull $X_g^c$ if and only if the fuel cost function $f_g(p_{g,t},x_{g,t})$ is convex over the convex hull $X_g^c$. This case is illustrated in Figure 2a through a two-block fuel cost function $f_g(p_{g,t},x_{g,t})$), where the cost function is marked in blue and the feasible operating region is marked in green. Note, that (1) the fuel cost function $f_g(p_{g,t},x_{g,t})$ is originally defined in the feasible operating region $\left\{0\cup \left[P_g^{min},P_g^{max}\right]\right\}$ for $p_{g,t}$ as evident from (3), and (2) $\left\{0\cup \left[P_g^{min},P_g^{max}\right]\right\}$ is convexified as $\left[0,P_g^{max}\right]$ in the convex hull $X_g^c$. As can be observed in Figure 2b, the convexity of the convex envelope $f_g(p_{g,t},x_{g,t})$ (marked in red) over the convex hull $X_g^c$ (marked in green) relies on its convexity in the domain of $p_{g,t}$. For the feasible operating region $\left[P_g^{min},P_g^{max}\right]$, the fuel cost function $f_g(p_{g,t},x_{g,t})$ is inherently convex as required in current markets [26]. Moreover, if the slope of $f_g(p_{g,t},x_{g,t})$ with respect to $p_{g,t}$ from the range $\left[0,P_g^{min}\right]$ is smaller than that in the first block of $f_g(p_{g,t},x_{g,t})$ with respect to $p_{g,t}$, it is not difficult to find the convexity of the fuel cost function $f_g(p_{g,t},x_{g,t})$ over the convex hull $X_g^c$; see Figure 2b.

In summary, integer relaxation of the cost function $z_g\left(p_g,u_g,x_g\right)$ and the unit formulation $X_g$ can lead to convex hull prices from the UC-CHP problem when (1) there are no ramp rates, and (2) the fuel cost function $f_g(p_{g,t},x_{g,t})$ over the convex hull $X_g^c$ is fully convex. However, the absence of ramp-rate constraints may be impractical in real-world power systems. Also, the fuel cost function $f_g(p_{g,t},x_{g,t})$ may not always be convex over the convex hull $X_g^c$; see an example in Figure 3.

(2)

Polyhedron approach

Ref. [8] focuses on the basic constraints (3)–(9) without ramp rate constraints (10) and (11). Similar to [7], the integer relaxation of the unit formulation $X_g$ delineates the convex hull $X_g^c$ as $X_g$ is tight, as shown in (19). This statement is proved from a polyhedron perspective by proving that any point in the convex hull $X_g^c$ can be expressed as a convex combination of the integral vertices of the basic constraints (3)–(9). If the cost function is quadratic, the UC problem is modeled as a second-order cone program and it is solved using interior-point methods. Instead, when the cost function is piecewise linear, the UC problem is solved as an LP problem.

Additionally, Ref. [8] provides a general framework for calculating a convex envelope when the fuel cost function $f_g(p_{g,t},x_{g,t})$ is not convex over the convex hull $X_g^c$. Particularly, the fuel cost function $f_g(p_{g,t},x_{g,t})$ over the convex hull $X_g^c$ can be non-convex when its slope with respect to $p_{g,t}$ in the feasible operating range $\left[0,P_g^{max}\right]$ is larger than that in the first block, as shown in Figure 3. In such a case, the key to formulating the convex envelope lies in convexifying the fuel cost function $f_g(p_{g,t},x_{g,t})$ over the convex hull $X_g^c$. Ref. [8] utilizes $x_{g,t}$ to scale the original cost function $f_g(p_{g,t},x_{g,t})$ for its convexification. Since $x_{g,t}$ is not larger than one, the values of the scaled cost function underestimate or equal those of the original cost function (marked in blue) to finally result in a convex envelope, as shown in Figure 3b.

(3)

State transition approach

Ref. [9] starts from a variation of the unit formulation $X_g$ by enumerating commitment statuses built upon a state transition diagram shown in Figure 4. A binary variable $x_e$ is introduced to indicate the activation of the edge e that represents the transition from one possible status to another possible status. Also, a continuous variable $y_e$ is used to represent the generation level over the minimum generation level $P_g^{min}$ (per unit) of the edge e.

In the variation of the unit formulation $X_g$, the basic constraints (3)–(9) without ramp rate constraints (10) and (11) are considered, similar to [7,8]. The basic constraints (3)–(9) are reformulated in the domain of $\left(x_e,y_e\right)$ based on the state transition diagram. These constraints based on $x_e$ and $y_e$ can be regarded as a network flow model with integral capacities, where vertices of its integer relaxation are integral, similar to [7]. Equations (18) and (19) shown earlier are applicable to this approach as well. Therefore, integer relaxation of the unit formulation leads to its convex hull in the domain of $\left(x_e,y_e\right)$. However, the convex hull is established based on the enumeration of commitment statuses. This will lead to a substantial number of constraints required to formulate the convex hull; thus, a heavy computational burden exists to solve the resulting UC-CHP problem. Therefore, the Bienstock–Zuckerberg algorithm is used as the decomposition scheme for the large-scale LP to reduce the computational burden and achieve higher convergence rates. However, when ramp rates are involved, the unit formulation in the domain of $\left(x_e,y_e\right)$ cannot be expressed as a network flow model with integral capacities. This indicates that integer relaxation only works for the convex hull when there is no ramp rate, similar to [7]. An extension of network flow model-based convex hull pricing with maximum start-up limit is provided in [31].

Similar to the reformulation of the basic constraints (3)–(9) above, the cost function $z_g\left(p_g,u_g,x_g\right)$ can be also reformulated in the domain of $\left(x_e,y_e\right)$ using the state transition diagram. Particularly, the cost function $z_g\left(p_g,u_g,x_g\right)$ can be reformulated as a linear combination of $\left(x_e,y_e\right)$ across all the edges. This provides the opportunity where the convex envelope of the cost function of each unit can be easily achieved by integer relaxation. Note, that such a cost function of each unit built from the state transition diagram can bring advantages in formulating the convex envelope under several scenarios:

(1)

No matter whether the fuel cost function of each unit is either convex or non-convex over the convex hull of each unit, the convex envelope of each unit can be obtained by integer relaxation because the cost function based on the state transition diagram is linear with $x_e$ and $y_e$. This arises from the fact that $y_e$ changes the generation capacity from $\left\{0\cup \left[P_g^{min},P_g^{max}\right]\right\}$ to $\left[P_g^{min},P_g^{max}\right]$ (per unit). Therefore, the slope from zero to $P_g^{min}$ does not affect the convexity of the cost function.

(2)

No matter whether the coefficients of commitment costs are either constant or time-dependent, the convex envelope of each unit can be obtained by integer relaxation. This is because time-dependent coefficients of commitment costs are handled by $x_e$ that enumerates transitions of all the commitment statuses.

3.3. Convex Hull and Convex Envelope under a Unit Formulation Which Is Not Tight

With additional ramp rate constraints (10) and (11) added to the basic constraints (3)–(9), the unit formulation $X_g$ is not tight anymore, as verified in [32]. Consequently, the integer relaxation of the unit formulation $X_g$ does not lead to its convex hull $X_g^c$. Various approaches [10,11,12,13,14] have been developed to formulate the convex hull $X_g^c$ together with its convex envelope $z_g^c\left(p_g,u_g,x_g\right)$. In [10,11], the convex hull $X_g^c$ is formulated by a convex combination of the constraints under all the commitment statuses based on different enumeration methods. As for the convex envelope $z_g^c\left(p_g,u_g,x_g\right)$ in [10,11], integer relaxation can work by adding more continuous variables to reflect costs. As an alternative, Refs. [12,13] formulate the convex hull $X_g^c$ and the convex envelope $z_g^c\left(p_g,u_g,x_g\right)$ starting from a novel single-unit commitment problem by which integer relaxation can lead to the convex hull $X_g$ and the convex envelope $z_g^c\left(p_g,u_g,x_g\right)$. Finally, Ref. [14] uses a column generation approach to iteratively construct the convex hull using extreme points generated by the sub-problems. The approaches in [10,11,12,13,14] are reviewed in detail next. For each approach, the convex hull $X_g^c$ is first studied, followed by its convex envelope $z_g^c\left(p_g,u_g,x_g\right)$.

(1)

Disjunctive programming approach

The convex hull $X_g^c$ in [10] is expressed as the convex combination of the constraints under all the commitment statuses. Particularly, commitment statuses are enumerated by disjunctive programming to obtain the unit formulation $X_g$, i.e.,

$$X_g=\underset{j\in J_g}{\cup }{X}_g^j=\underset{j\in J_g}{\cup }\left(A_g^j{p}_g+B_g^j{c}_g\le C_g^j\right),$$

(20)

where $J_g$ is the set of commitment statuses of unit g; $X_g^j$ is the feasible operating region under status j of unit g; $A_g^j$, $B_g^j$, and $C_g^j$ are constants to formulate $X_g^j$; $p_g$ and $c_g$ are generation level variables and cost variables of unit g, respectively.

From [33], the convex hull $X_g^c$ can be expressed as the convex combination of the disjunctive programming-based unit formulation $X_g$ in (20), i.e.,

$$X_g^c=\left\{\begin{array}{c}{p}_g=\sum _{j\in J_g}{\xi }_{pg}^j,c_g=\sum _{j\in J_g}{\xi }_{cg}^j,\hfill \\ A_g^j{\xi }_{pg}^j+B_g^j{\xi }_{cg}^j\le {\xi }_0^j{C}_g^j,\forall j\in J_g,\hfill \\ \sum _{j\in J_g}{\xi }_0^j=1,{\xi }_0^j\ge 0,{\xi }_{pg}^j\ge 0,{\xi }_{cg}^j\ge 0,\forall j\in J_g.\hfill \end{array}\right\},$$

(21)

where ${\xi }_{pg}^j$ and ${\xi }_{cg}^j$ are continuous variables which are introduced for $p_g$ and $c_g$ under status j, respectively; ${\xi }_0^j$ is a continuous variable that represents the selection of commitment statuses. Once the convex hull is established using this approach, it can be solved using LP methods. Although Ref. [10] provides a general framework to obtain the convex hull from any linear constraint (which can contain any time-dependent requirements), such a general formulation of the convex hull requires a large number of constraints under all the commitment statuses. This highlights the significant computational burden associated with this approach.

The cost function of each unit in [10] is directly represented by the continuous cost variable $c_g$ in the constraint based on disjunctive programming (20). It is shown in [10] that $c_g$ is convex over the convex hull $X_g^c$, i.e., an integer relaxation of the cost function of each unit leads to its convex envelope. This is verified by showing that strong duality exists between $c_g$ of this approach and the Lagrangian dual of the UC problem ([10], Equation (7)).

(2)

Interval concept approach

Similar to [10], Ref. [11] (1) formulates the convex hull of each unit starting from the enumeration of commitment statuses, and (2) formulates the convex envelope of each unit from requiring a cost variable $c_g$ that represents the cost function, as detailed below.

In [11], a variation of the unit formulation $X_g$ is used. It is built on the time interval concept by introducing three types of new variables associated with each time interval $[a,b]$: one indicator variable $y_g^{[a,b]}$ that represents whether unit g is online in $[a,b]$ (i.e., $y_g^{[a,b]}=1$ when unit g is online in $[a,b]$; otherwise, $y_g^{[a,b]}=0$), a generation level variable $p_g^{[a,b]}$ that corresponds to the original generation level variable $p_g$ in $[a,b]$, and a cost variable $c_g^{[a,b]}$ that corresponds to the original cost in $[a,b]$. Using the time interval concept, the constraints across all intervals that define the unit formulation $X_g$ can be similarly structured in the form presented in (21). Such constraints in [11] directly delineate the convex hull of each unit because they have been the convex combination of the constraints under all the intervals. Although this approach also provides a general method to formulate a convex hull of each unit, a significant number of constraints are required to formulate the convex hull of each unit. This indicates that a heavy computational burden of solving the UC-CHP problem may exist, the same as [10]. Therefore, the authors employ Benders decomposition approach to improve the computational performance of their convex hull pricing model.

As for the convex envelope of each unit, it can be obtained by integer relaxation of the cost function of each unit because the cost function has been represented by the cost variable $c_g$, the same as [10]. Since the commitment statuses are enumerated, time-dependent costs can be considered in the cost function by introducing one additional indicator variable that represents whether a unit is offline in a certain time interval.

(3)

Single-unit commitment approach

As an alternative, the convex hull $X_g^c$ and the convex envelope $z_g^c$ in [12,13] are formulated based on a single-unit commitment problem. Convex hull is established using dynamic programming (DP) equations and then the DP equations are converted to an LP problem whose dual problem is further derived for the convex hull $X_g^c$ and the convex envelope $z_g^c$, as reviewed below.

The constraints of this dual problem directly delineate the convex hull $X_g^c$ in the dual-variable domain, due to the mapping from the optimal solution of DP equations to the optimal solution in the dual problem, as shown in Figure 5. The optimal solution of the DP equations is one vertex of the convex hull of the original single-unit commitment problem. The optimal solution to the dual problem is also one vertex of its convex hull. It is proved in [12,13] that the variables in this dual problem represent commitment statuses, generation levels, and costs. Hence, each vertex of the convex hull of the original single-unit commitment problem corresponds to one vertex of the convex hull in this dual problem. However, DP equations lead to a large number of constraints in formulating a convex hull due to the combination properties. Finally, the original MILP problem is solved as an LP problem.

The cost function of this dual problem is equivalent to that in the single-unit commitment problem based on the physical meanings of dual variables shown in Figure 5. Since the cost function of this dual problem is fully convex, the convex envelope over the convex hull can be obtained by integer relaxation. Note, that because of the DP equations, time-dependent costs and constraints can be easily involved in [12,13].

(4)

Dantzig-Wolfe (DW) decomposition approach

In [14] a new approach is developed to find the convex hull prices, where the convex hull of the overall UC problem is constructed iteratively using the DW decomposition. This approach relaxes the integrality requirement in UC and then solves the resulting large-scale LP problem using the DW column generation approach. The process also involves solving the Lagrangian dual of the integer relaxed UC problem to calculate the uplift payments in every iteration, assuming strong duality. Initially, a subset of feasible unit schedules is considered under a restricted problem. New columns (i.e., new feasible schedules) are added to the restricted problem in each iteration provided they have a negative reduced cost (which is also the profit maximization problem). Note, that each iteration provides at least one extreme point. Since the restricted problem characterizes any feasible point as a convex combination of extreme points, the DW approach iteratively finds the extreme points of the convex hull and converges finitely. The resulting prices give the exact convex hull prices as they are obtained from the dual problem. It is important to note that this approach does not necessitate a reformulation of the cost function.

3.4. Approaches Using Approximate Convex Hulls

The above approaches focus on the exact convex hull prices by accurately delineating the convex hull and convex envelope of each unit. However, the number of constraints to delineate the exact convex hull grows exponentially as the number of time slots increases [28]. This can result in a significant computational burden when solving the UC-CHP problem for convex hull pricing. As a trade-off, using an approximate convex hull can reduce the computational load, although it means sacrificing the precision of the exact convex hull prices. In [8] where a unit formulation is not tight due to ramp rate constraints (10) and (11), the entire convex hull of each unit is approximated by using the convex hulls for two/three consecutive slots [34]. The idea is to derive explicit convex hulls for $T=2$ and $T=3$, and use the resulting valid inequalities to obtain approximate convex hulls for $T>3$.

More generally, when a unit formulation is not tight, caused by ramp rate constraints or other time-dependent constraints, the convex hull across all time slots can be approximated by the systematic approach in [32]. Ref. [32] provides a general way to formulate the convex hull for two/three consecutive slots and then facilitates the approximation of the convex hull in all the slots. This idea is illustrated in Figure 6, where the constraint is shown in blue and the convex hull in red. Consider a problem with the constraint $x_1+x_2\ge 0.5$, where $x_1$ and $x_2$ are binary, shown on the left. Since obtaining the convex hull is challenging, the following procedure is adopted: (1) relax the integrality requirement and convert the constraints into vertices, as shown on the right. (2) drop fractional vertices (open blue dots) as $x_1$ and $x_2$ are binary (solid blue dots). (3) convert vertices to constraints, the resulting constraints are tight, i.e., the convex hull is obtained. (4) the final step is to parameterize (express numerical coefficients as combinations of unit parameters) the tight constraints for re-use. Since it is challenging to generate tight constraints for all time slots, the above systematic formulation tightening approach is typically applied for two or three intervals. The resulting tightened constraints can be applied to any number of time slots to obtain approximate convex hulls, as shown in [35].

4. Convex Hull Pricing by the Lagrangian Dual Problem of a UC Problem

In addition to solving the UC-CHP problem for convex hull prices, convex hull prices can be also taken as the optimal multipliers in the Lagrangian dual problem of the UC-Orig problem, as will be shown in Section 4.1. Although solving the Lagrangian dual problem does not require the convex hull and convex envelope of each unit, the non-smooth nature of the Lagrangian dual problem can pose computational issues. The convergence issues have been studied and parties addressed in [15,16,17,18] as discussed Section 4.2. Ultimately, the Surrogate Lagrangian Relaxation (SLR) method [36], which fundamentally resolves convergence issues, will be presented in Section 4.3 for convex hull pricing.

4.1. Convex Hull Pricing from the Lagrangian Dual Problem of the UC-Orig Problem

The Lagrangian dual problem of the UC-Orig problem can be expressed as

$$\begin{array}{c}\mathrm{UC}\text{-}\mathrm{Orig}\text{-}\mathrm{Lagrangian}\text{-}\mathrm{Dual}:{\displaystyle \underset{\lambda }{max}}q\left(\lambda \right),\mathrm{with}\\ q\left(\lambda \right)\equiv {\displaystyle \underset{\{p,u,x\}\in X}{min}}\left\{z(p,u,x)-{\lambda }^T(Ap-D)\right\},\end{array}$$

(22)

where $\lambda$ is the Lagrangian dual multipliers associated with the relaxed system-wide constraints $Ap=D$.

At optimum, the optimal Lagrangian dual value $q^{*}\left({\lambda }^{*}\left(D\right)\right)$ can be regarded as the double conjugate of $z^{*}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$, as well as the value of the convex envelope $z^{c*}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$; see [29] for the relationships among double conjugate functions, dual functions and convex envelopes. Therefore, by regarding $q^{*}\left({\lambda }^{*}\left(D\right)\right)$ and $z^{*}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$ as a function of demand D, the slope of $z^{c*}\left(p^{*}\left(D\right),u^{*}\left(D\right),x^{*}\left(D\right)\right)$ with respect to demand D is equal to the slope of $q^{*}\left({\lambda }^{*}\left(D\right)\right)$ with respect to D. As observed in the UC-Orig-Lagrangian-Dual problem, the slope of $q^{*}\left({\lambda }^{*}\left(D\right)\right)$ with respect to D equals ${\lambda }^{*}$ which is the coefficient prior to D. In other words, the optimal Lagrangian dual multipliers ${\lambda }^{*}$ corresponding to the relaxed system-wide constraints give the convex hull prices.

Due to the non-differentiability of the dual function, subgradient directions are typically non-ascending, rendering the usual necessary and sufficient conditions for extrema inapplicable. Additionally, when the number of units increases, calculating subgradient directions demands significant computational resources. Convergence may also be slow, as multipliers tend to zigzag along the ridges of the Lagrangian dual function, further complicating the process [36], which can lead to many iterations being required for convergence. In addition, traditional subgradient methods require steps that depend on $q^{*}$. The unknown $q^{*}$ is traditionally adjusted by the “subgradient-level” methods [37] or incremental subgradient methods [38], whose heuristic adjustments lead to slow convergence as demonstrated in [36].

4.2. Approaches Using the UC-Orig-Lagrangian-Dual Problem

The current approaches [15,16,17,18] focus on studying the convergence difficulties of solving the UC-Orig-Lagrangian-Dual problem to improve the computational efficiency of convex hull pricing, as reviewed below:

(1)

Subgradient simplex cutting plane approach

A subgradient simplex cutting plane approach is introduced in [15]. In each iteration, subgradients and the current Lagrangian dual variables are used to generate cuts from query points. By iteratively eliminating non-optimal solutions, this approach helps reduce the overall computational effort. Then, the next query points are calculated by using an adaptive three-level scheme. However, the subgradient directions lead to slow convergence in [15] because of the multiplier zigzagging across the ridges of the Lagrangian dual function. Note, that this paper computes Extended Locational Marginal Prices (ELMPs), which are an approximation of the convex hull prices. The subgradient approach used in this paper lays the foundation for the subdifferential approach in [16,17].

(2)

Extreme-point subdifferential approach

As an alternative, an extreme-point subdifferential method is described in [16,17]. The steepest ascent direction is determined by minimizing the squared error between demand and generation levels while maximizing revenues in single-unit commitment problems given the estimated convex hull prices. Figure 7 compares the trajectory of iterates of the extreme-point subdifferential method with a standard subgradient method like the one presented in [15]. Although the steepest ascent direction can alleviate the multiplier zigzagging caused by subgradient directions and does not require $q^{*}$, the computational effort is high to obtain the direction; computations of ascending directions come at a price because all possible generation levels that maximize revenues in single-unit commitment problems should be explored in each iteration. In addition, for a large-scale UC problem, the Lagrangian dual function has many facets, and ridges are very short. Therefore, multipliers may not make significant progress along the ascending direction.

(3)

Level method

Ref. [18] develops a level method based on the Kelley algorithm to compute the convex hull prices. It iteratively constructs the upper bound of the Lagrangian dual function using its supergradients. The lower bound is established by utilizing the maximum value of the Lagrangian dual function. The relative difference between the upper and lower bound is used as the stopping criterion. The level method differs from Kelley’s algorithm in the way it stabilizes hyperplanes in each iteration. The level method updates prices more smoothly by not choosing the optimal price in each iteration, instead using a projection of the price iterated on the level set, as shown below in (23) and Figure 8.

$$q\left(\lambda \right)\ge \alpha U{B}_k+(1-a)L{B}_k,$$

(23)

where $\alpha \in [0,1]$ is the projection parameter, $\alpha =0$ corresponds to the Kelley’s algorithm and means the iterate does not change. Ref. [18] reports that $\alpha =0.2$ is chosen for all experiments presented in the paper. Building on this foundational idea, Ref. [18] proposes a multi-cut level method where a cut is added for each generator subproblem. Although generating multiple cuts gives a more accurate solution, it adds a considerably more computational burden.

4.3. Discussions on Convex Hull Pricing by Solving the Lagrangian Dual Problem through the SLR Method

The SLR method [36] is a decomposition and coordination algorithm that overcomes the major challenges associated with Lagrangian relaxation and performs better than Augmented Lagrangian Relaxation (ALR) and Alternating Direction Method of Multipliers (ADMM) approaches [39,40]. Under SLR, the multipliers are updated based on surrogate subgradients $\tilde{g}\left(p^k,r^k\right)$, as shown in (24):

$${\lambda }_t^{k+1}={\lambda }_t^k+s_{SLR}^k·\tilde{g}\left(p^k,r^k\right),$$

(24)

and a step sizing rule, which is obtained based on a contraction mapping principle, as:

$$s_{SLR}^k=\left(1-{\displaystyle \frac{1}{M·k^{1-\frac{1}{k^{\rho }}}}}\right){\displaystyle \frac{s_{SLR}^{k-1}·\parallel \tilde{g}\left(p^{k-1},r^{k-1}\right)\parallel }{∥\tilde{g}\left(p^k,r^k\right)∥}},$$

(25)

where M and $\rho$ are the parameters that satisfy $M>1$ and $0<\rho <1$. Here, “tilde” indicates that the full minimization of the Lagrangian is not required as long as the surrogate optimality condition (26) ([36], p. 178, Equation (12)) is met as:

$$\tilde{L}\left({\lambda }^k,x^k,u^k,p^k,r^k\right)<\tilde{L}\left({\lambda }^k,x^{k-1},u^{k-1},p^{k-1},r^{k-1}\right),$$

(26)

where $\tilde{L}\left({\lambda }^k,x^k,u^k,p^k,r^k\right)$ is the “surrogate dual value” at iteration k.

The SLR method [36] enjoys three key benefits over previous approaches to solving the Lagrangian dual problem:

• All the subproblems are not required to be solved optimally. Only the surrogate optimality condition must be satisfied, as demonstrated in (26). Consequently, computational efforts are reduced.
• The guesstimate of $q^{*}$ is not needed in the implementation. This avoids the slow convergence caused by inefficient adjustments of $q^{*}$.
• The surrogate directions given by (25) are smoother, and zigzagging is eliminated.

Therefore, the SLR method provides the opportunity to efficiently obtain the optimal multipliers as the exact convex hull prices. Additionally, The SLR method can also be employed to offer a high-quality measure of convex hull prices by generating a new upper bound for the optimal dual value [41], as shown in Figure 9. The novel quality measure is defined as the difference between the upper and lower bounds of the optimal dual value. The lower bound can be established using the best available Lagrangian dual value and a novel procedure is used for calculating the upper bound is derived by: (1) using a notion similar to multiplier oscillations first proposed in the “subgradient-level” method [37], but in a decision-based manner rather than a heuristic one; (2) inferring the “level”, which is a sought-for upper bound. Testing this novel quality measure on the IEEE 118-bus system shows its advantages over the standard duality gap in terms of accuracy and computational efforts needed.

5. Discussion and Insights

Based on the above survey, this section will first present our viewpoints on the remaining limitations and the possible potential improvements of current approaches in Section 5.1. In addition, new challenges for convex hull pricing arising from power system decarbonization will be discussed in Section 5.2.

5.1. Remaining Limitations and Potential Improvements of Current Approaches

As surveyed in Section 3 and Section 4, current approaches can be categorized into two types: the primal for the UC-CHP problem, and the dual for the UC-Orig-Lagrangian-Dual problem. For the primal category, the key lies in formulating the convex envelope and convex hull of each unit. For the convex envelope, Ref. [8] has provided a general framework to convexify a cost function for its convex envelope. However, the exact convex hull is usually difficult to delineate, since the facial description of the associated high-dimension polytope requires a large number of inequalities/vertices. Although various approaches have been presented in [7,8,9,10,11,12,13,14], certain limitations still exist, such as the impractical assumption where ramp rates are omitted in [7,8,9], heavy computational burden by enumerating all the possible commitment statuses in [10,11,12,13], and a large number of vertices required in [14].

For the dual category, the key is to solve the UC-Orig-Lagrangian-Dual problem without requiring the exact formulations of the convex envelope and convex hull. However, difficulties arise from the non-smooth nature of the associated dual problems that lead to the zigzagging of dual solutions when using standard subgradient methods. While the SLR method is guaranteed to converge and leads to a faster CPU-time-wise convergence by employing surrogate subgradients that require much less effort and lead to the concomitant alleviation of zigzagging as compared to subgradient methods, the “non-summable” nature of stepsizes can only guarantee the linear convergence outside of the neighborhood of ${\lambda }^{*}$ thereby impeding fast iteration-wise convergence. This may restrict the applications of the SLR method in convex hull pricing for large-scale problems.

Given the above observations, the following two potential improvements for convex hull pricing are suggested:

• Exact convex hull pricing approaches: For the primal category, the computational efficiency can be enhanced by bypassing the need for the exact convex hulls. This is inspired by the fact that the solution to a MILP problem is at one vertex of the convex hull, hence, only facets adjacent to the optimal solution needed to be delineated. For the dual category, the utilization of the linear convergence potential, as discussed in [42,43], can accelerate iteration-wise convergence, thereby enabling more efficient computations of convex hull pricing for large-scale problems.
• Approximated convex hull pricing approaches: When calculating exact convex hull prices is not efficient for large-scale problems, approximated convex hull prices offer a viable alternative. In existing studies, the approximation accuracy is usually evaluated by the total uplift payments, i.e., the lower the value, the higher the accuracy. Beyond the existing system-level evaluation based on total uplift payments, a more detailed evaluation can be conducted at the unit level. This involves comparing the uplift payments for each unit across various approaches. For such unit-level analysis, metrics such as the root mean square errors, maximum errors, and minimum errors of each unit’s uplift payments could be used as standards.

5.2. New Challenges from Power System Decarbonization

With the ambitious energy transition toward decarbonization through renewable energy [44], two emerging challenges related to convex hull pricing may arise, as discussed below:

• New binary variables: Looking to the future, at least two sources of new binary variables will be integrated into UC. The first source emerges from new components, such as energy storage. To mitigate the intermittency of variable renewable energy sources such as solar and wind, and to further decarbonize, energy storage is essential for maintaining a reliable and resilient grid operation. In modeling energy storage, binary variables are required to avoid simultaneous charging and discharging of the storage unit [45]. Recent studies developed a convex hull pricing model for pumped hydro storage units [46] and carbon-capture-utilization-and-storage (CCUS) systems [47]. However, further research on convex hull models for emerging chemical battery storage is desired. The second source of binary variables is associated with new, broader operational requirements at the system level. For example, stability constraints, have been encouraged in UC [48], often involving complex formulations. These stability constraints are currently interpreted using machine learning techniques, such as deep neural networks, which necessitate binary variables for their explicit representation [49]. Another source of binary variables is for reserve commitments where additional binary variables are used to ensure that a generator is committed to provide regulation reserves [50]. In our view, incorporating these binary variables necessitates addressing three key questions: (1) How do these new binary variables influence energy prices? (2) How can these effects be incorporated into the convex hull pricing framework? (3) What are the strategies for efficiently implementing convex hull pricing in the presence of new binary variables?
• Variable renewable generation uncertainties: One key feature of variable renewable energy is the uncertain nature of its fluctuations. However, current convex hull pricing majorly focuses on deterministic UC. A recent study [51] employed convex hull pricing as a way of pricing the uncertainty associated with wind production instead of incorporating several wind power generation scenarios into the UC problem. Convex hull prices were determined for different renewable energy penetration levels and added to the cost function. Thereby, transforming a stochastic UC problem into a deterministic problem. While this study uses convex hull pricing for risk mitigation, the impact of variable renewable generation uncertainties on convex hull pricing remains unclear. In addition, renewable energy sources can lead to transmission congestion as they are often located far from load centers. For example, in the state of Texas, most of the wind power is located in the Western region of the state but the population centers are in the East. The resulting network congestion affects convex hull prices throughout the Texas grid. It is thus critical to incorporate renewable uncertainties in convex hull pricing. From our understanding, convex hull pricing under renewable uncertainties may involve addressing four critical questions: (1) What types of renewable uncertainties (e.g., probabilistic distribution functions and uncertainty set) should be considered in electricity markets? (2) What modeling changes will be needed to incorporate renewable uncertainties in electricity markets? (3) What are the impacts of these modeling changes on convex hull pricing (e.g., impacts on the formulations of convex hulls and convex envelopes)? (4) How to address the impact of transmission congestion on convex hull prices?

6. Conclusions

This paper provides a systematical survey on convex hull pricing for unit commitment (UC). There are two major types of approaches for convex hull pricing. One is to solve the convex hull pricing problem with the convex envelope of the total cost function and the convex hull of the entire UC problem, such a pricing problem can be equivalently formulated by using the convex envelope and the convex hull of each unit. The other one is to solve the Lagrangian dual problem of a UC problem. The remaining difficulties of the first type lie in formulating the exact convex hull of each unit under practical requirements (e.g., ramp rates), while the second type falls into unsatisfactory convergence rates in large-scale UC problems. From our understanding, the first type could be improved by finding the solution of a convex hull pricing problem without delineating the entire convex hull of each unit, and the second type could be improved by the linear convergence potential in surrogate Lagrangian relaxation methods pointed out in [42,43]. When convex hull pricing cannot be exactly implemented, a comprehensive evaluation of the approximate computations should be studied, both at the system-wide level and for individual components. Moreover, within the framework of ongoing power system decarbonization, convex hull pricing may face new challenges through (1) new sources of binary variables and (2) variable renewable uncertainty modeling. Further research can refine the convex hull pricing framework, transforming it into a practical tool for minimizing uplift payments.