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Unit commitment with frequency‐arresting adequacy constraint: Modelling and pricing aspects

Unit commitment with frequency‐arresting adequacy constraint: Modelling and pricing aspects INTRODUCTIONFrequency response after contingencies is essential for reliable power delivery to consumers. The frequency often experiences relatively high fluctuations after a severe loss of load or trip of a generation unit. Accordingly, practical methods should be devised to avoid the frequency from violating the desirable interval after occurrence of contingencies [1, 2]. The frequency response measures to be taken after a contingency are categorized differently in the literature. A number of the previous works in this area have categorized frequency response actions in three groups of primary, secondary and tertiary frequency response activities [3, 4].The inertial support consists of the inherent and natural response of rotating masses, which are synchronously connected to the grid. The primary frequency response occurs within a few seconds after the contingency and it is mainly driven by governor action [2]. The inertial response problems have captured particular attention due to increasing penetration of intermittent renewable energy resources [5, 6]. The intermittent wind and solar generators are connected to the power system via power electronic converters and they do not provide synchronous inertia [7]. As a result, the system is losing its inherent ability of inertial response through replacing the conventional synchronous generators with converter‐based intermittent wind and solar generators [8–10], becoming more vulnerable to a large loss of generation [4]. The work in this paper is mainly focused on frequency‐arresting (FA) adequacy. The FA adequacy can be measured by Nadir and rate of change of frequency (ROCOF). The Nadir requirement defines a minimum frequency that must not be violated to prevent load shedding after any contingency. The ROCOF is the derivative of frequency with respect to time. It should not violate from interval determined by TSO (Transmission System Operator) [11, 12].In [13], four market designs for ensuring FA adequacy are proposed and analysed. However, none of these proposals considers UC for ensuring FA adequacy. Reference [14] proposes adding a frequency‐responsive reserve constraint (spinning reserve) to the UC formulation to avoid load shedding caused by frequency drop.References [15–17] propose market approaches for FA adequacy. Authors in [15] propose a methodology for optimal allocation of primary reserves for frequency control considering power system events. This work does not provide a detailed formulation considering inertia.Reference [18] has presented a simplified expression for frequency response and analysed the risk of under frequency load shedding (UFLS) wrong action. However, the market aspects of frequency response have not been investigated in detail.Authors of [11] present sufficient conditions for both FA and automatic generation control (AGC). They first study primary and secondary frequency response separately and then propose a formulation for joint primary and secondary frequency response. The work is more focused on control model of frequency response and it has not investigated market aspects.Reference [19] proposes a unit commitment (UC) model considering FA adequacy. The model is implemented on a simple case study and no standard test network is investigated. Furthermore, the work has not investigated market aspects in detail.Here in this paper, non‐convex pricing techniques are employed to calculate the cost of FA adequacy constraint. Among literature in this area, [20] has proposed a modified convex hull pricing technique to deal with the non‐convexities originating from non‐zero minimum output of generators. However, the UC formulation only considers the energy‐balance constraint and no constraints to model ancillary services are considered. Also, the authors of [21] have applied a semi‐Lagrangian method to ensure non‐negative profits of generators in a market with non‐convexities. However, their optimization problem does not include any of ancillary service products. Authors in [22] and [23] have proposed recovery payment mechanisms in a two‐part paper. The same authors have also carried out a review of non‐convex pricing methods and practices in [24]. References [25, 26] have put forward another formulation for market design of frequency response provision in a two‐part paper. The proposed method first calculates the frequency drop and ROCOF through external software. Then, the FA adequacy is ensured through another optimization problem. This method needs iterations between two stages (external‐software simulation and optimization‐problem simulation) which are computationally demanding. Besides, the obtained solution might be suboptimal as compared to the joint‐optimization approach proposed in our paper.Reference [27] has proposed a method for simultaneous scheduling of frequency response and energy. The work has implemented a comprehensive stochastic security constrained UC to achieve its results. However, the impact of considering the frequency response concerns on the pricing schemes is not investigated by the authors of this reference.In [28], the authors focus on upregulating and downregulating reserve allocation for fast ramping capability. This paper is more focused on AGC and secondary frequency response. It proposes a compensation mechanism for the generation units participating in AGC, based on their response performance to AGC signals.The work in [29] is focused on financial compensation of frequency regulation actions. It evaluates market rules in terms of fairness to provide reasonable compensation for the ancillary service provided. However, this reference is focused on up‐ and downregulation. The report is more descriptive and the related computational works are not discussed.In [30], the authors have analysed the impact of integrating renewable energy sources, particularly wind and solar energy, and tried to determine to what extent the side payments are reasonable in energy markets with non‐convexities.Furthermore, chapter 8 in [31] analyses uplift payments in pool‐based electricity markets. They propose a game‐theoretic approach for uplift payment. They investigate their proposed approach using a simple case study. This work does not discuss frequency response control or any ancillary services.In reference [32], an approach based on SCUC (Security Constrained Unit Commitment) has been proposed and implemented to deal with the problem of frequency response and the model has been solved through branch‐and‐bound techniques. The authors have implemented a detailed formulation and also performed a comparison on the maximum frequency drop after a contingency, based on different arrangements of online generators. The formulation provided in this reference is quite thorough and comprehensive.It is important to note that since the total inertia available in the network depends on the online generators, the optimal way to provide energy and ensure FA adequacy is to perform a joint optimization of providing energy and ensuring FA adequacy. The joint optimization of providing energy and ensuring FA avoids suboptimality that may occur in sequential approaches (ensuring FA and then providing energy), such as those discussed in [33–36].There are also some references in the literature which have gone through the details of pricing techniques in electricity markets with non‐convexities. Reference [37] reviews the impact of a number of pricing methods in a comprehensive technical report, including dual pricing, semi‐Lagrangian, integer programming and a number of other methods. Also, the work in [38] has studied a primal–dual approach in detail in an electricity market with non‐convexities.Accordingly, the contributions of the current paper are: First, it models Nadir and ROCOF requirements as a set of non‐linear constraints and differential expressions in standard UC formulation to ensure FA adequacy. The original FA‐constrained UC (FA‐UC) model is an MINLP (Mixed Integer Non‐Linear Programming) with differential equations. Through a series of proposed linearization techniques, the FA‐UC model is transformed to a mixed‐integer linear program (MILP) model which can be solved to global optimality. Second, to price the FA adequacy, non‐convex pricing techniques based on explicit and implicit uplift payment are proposed and discussed. The implicit uplift payment is based on the primal–dual formulation in optimization theory. Furthermore, the proposed MILP models and non‐convex pricing techniques are carefully studied through an illustrative example and also Nordic 44‐node test network. The proposed methodology in this work (1) ensures that the optimal amount of inertia can be provided, considering technical constraints and social‐cost objective along with FA constraints; (2) provides the possibility to consider a conservative assumption on the contingent generator and also load‐damping factor; (3) is capable of providing the optimal solution that ensures FA adequacy in a one‐shot optimization (as compared to possible suboptimal solution of sequential approach); (4) in contrast to some current practices which simply requires every synchronous generator to provide a specific amount of inertial response (which may not necessarily be the optimal amount), ensures that the adequate inertial response is provided in an optimal manner.FA‐CONSTRAINED UNIT COMMITMENTThe FA‐UC is set out in (1).1aMinimizeΩ1OFPP=∑i,kCiPpi,k+CiRri,k+Civvi,k+Ciwwi,k$$\begin{align} &\underset{\Omega _1}{\text{Minimize}}\nobreakspace OF^{PP} = \sum _{i,k} C^P_i p_{i,k} + C^R_i r_{i,k} + \nonumber \\ &\hspace{73.81102pt} C^v_i v_{i,k} +C^w_i w_{i,k} \end{align}$$1bsubjectto∑ipi,k=∑nDn,k∀k:(λk)$$\begin{align} &\text{{s}ubject\nobreakspace to}\nobreakspace \nonumber \\ & \sum _i p_{i,k} = \sum _{n} D_{n,k} \quad \forall k \nobreakspace \nobreakspace :(\lambda _k) \end{align}$$1c∑i≠jri,k≥P¯juj,k∀j,k:(μj,k)$$\begin{align} & \sum _{i\ne j} r_{i,k} \ge \overline{P}_j u_{j,k} \quad \forall j,k \nobreakspace \nobreakspace :(\mu _{j,k}) \end{align}$$1dP̲iui,k≤pi,k≤P¯i−ri,k∀i,k:μ̲i,k,μ¯i,k$$\begin{align} &\underline{P}_{i}u_{i,k} \le p_{i,k} \le \overline{P}_{i}-r_{i,k} \quad \forall i,k \nobreakspace \nobreakspace :\left(\underline{\mu }_{i,k},\overline{\mu }_{i,k}\right) \end{align}$$1eui,k−ui,k−1=vi,k−wi,k−1∀i,k:(σi,k)$$\begin{align} & u_{i,k}-u_{i,k-1}=v_{i,k}-w_{i,k-1} \quad \forall i,k \nobreakspace \nobreakspace :(\sigma _{i,k}) \end{align}$$1f∑t=k−tiUp−1kvi,t≥ui,k∀i,k:(σ¯i,k)$$\begin{align} & \sum ^k_{t=k-t^{Up}_i-1}{v_{i,t}\ge u_{i,k}\ } \quad \forall i,k \nobreakspace \nobreakspace :(\overline{\sigma }_{i,k}) \end{align}$$1g∑t=k−tiDn−1kwi,t≥1−ui,k∀i,k:(σ̲i,k)$$\begin{align} & \sum ^k_{t=k-t^{Dn}_i-1}{w_{i,t}\ge 1-u_{i,k}\ } \quad \forall i,k \nobreakspace \nobreakspace :(\underline{\sigma }_{i,k}) \end{align}$$1hzϱ,k=∑iΓϱ,ipi,k−∑nΨϱ,nDn,k∀ϱ,k:(δϱ,k)$$\begin{align} &\hspace*{-12pt} z_{\varrho ,k} = \nonumber \\ &\sum _{i} \Gamma _{\varrho ,i}p_{i,k} - \sum _{n} \Psi _{\varrho ,n}D_{n,k} \quad \forall \varrho ,k \nobreakspace \nobreakspace :(\delta _{\varrho ,k})\end{align}$$1i−Zϱ≤zϱ,k≤Zϱ∀ϱ,k:δ̲ϱ,k,δ¯ϱ,k$$\begin{align} & -Z_{\varrho } \le z_{\varrho ,k} \le Z_{\varrho } \quad \forall \varrho ,k \nobreakspace \nobreakspace :\left(\underline{\delta }_{\varrho ,k} , \overline{\delta }_{\varrho ,k}\right) \end{align}$$1jmj,k=∑i≠jMiui,k∀k:(λ̲j,k)$$\begin{align} & m_{j,k} = \sum _{i \ne j} M_i u_{i,k} \quad \forall k \nobreakspace \nobreakspace :(\underline{\lambda }_{j,k}) \end{align}$$1kPjri,k−2Yimj,kF0−FNadir−Fdb≤0∀j,k:(πj,k)$$\begin{align} & P_j r_{i,k} - 2Y_{i} m_{j,k} {\left(F_0 - F_{Nadir} - F_{db} \right)} \le 0 \quad \forall j,k \nobreakspace \nobreakspace :(\pi _{j,k}) \end{align}$$1lmj,kdfkdt=−BΔfk+Pm−Pe∀k$$\begin{align} & m_{j,k} \frac{df_k}{dt} =- B {\Delta f_k} + {\left(P_{m} - P_{e} \right)} \quad \forall k \end{align}$$1mF̲≤dfkdt≤F¯$$\begin{align} & \underline{F} \le \frac{df_k}{dt} \le \bar{F} \end{align}$$1npi,k,ri,k,fk∈R+∀i,k$$\begin{align} & p_{i,k}, r_{i,k}, f_k \in \mathbb {R}_+ \quad \forall i,k \end{align}$$1ovi,k,wi,k,ui,k∈{0,1}∀i,k,$$\begin{align} & v_{i,k}, w_{i,k}, u_{i,k} \in \lbrace 0,1\rbrace \quad \forall i,k, \end{align}$$where the Ω1=$\Omega _1 =$ {pi,k$\lbrace p_{i,k}$, ri,k$r_{i,k}$, vi,k$v_{i,k}$, wi,k$w_{i,k}$, ui,k$u_{i,k}$, zϱ,k$z_{\varrho ,k}$, mj,k$m_{j,k}$, yj,k$y_{j,k}$, fk}$f_k\rbrace$ is the set of decision variables. OFPP$OF^{PP}$ stands for the primal objective function. pi,k$p_{i,k}$ and ri,k$r_{i,k}$, respectively, represent generation level and reserve and vi,k$v_{i,k}$, wi,k$w_{i,k}$ and ui,k$u_{i,k}$ indicate start‐up, shut‐down and online status for generator i in hour k. zϱ,k$z_{\varrho ,k}$ shows the flow of line ϱ during hour k. mj,k$m_{j,k}$ shows the total inertia available in hour k in absence of the contingent generator j. The CiP$C^P_i$ and CiR$C^R_i$ are marginal cost of providing energy and reserve while Civ$C^v_i$ and Ciw$C^w_i$ model start‐up and shut‐down cost for generator i. Dn,k$D_{n,k}$ represents the demand connected to node n for hour k. Pj$P_{j}$ shows the production capacity of the contingent generator j. The P̲i$\underline{P}_{i}$ and P¯i$\overline{P}_{i}$ are minimum and maximum generation of generator i. tiUp$t^{Up}_i$ and tiDn$t^{Dn}_i$ show minimum up and down time of each generator. B is the load damping factor of the system. Pm−Pe$P_{m}-P_{e}$ is the difference between the mechanical and electrical power after the contingency (in worst case, it is equal to the capacity of the contingent generator j). Also, Yi$Y_i$ is the fastest ramping capability of generator i. The Γϱ,i$\Gamma _{\varrho ,i}$ and Ψϱ,n$\Psi _{\varrho ,n}$ are generation and demand shift factors. Zϱ$Z_{\varrho }$ shows the line flow limit. FNadir$F_{Nadir}$ is the nadir requirement in the network and Fdb$F_{db}$ is the dead‐band of generator governor reaction. The optimization problem (1) is an MINLP with differential equations. Objective function (1a) is sum of energy, reserve, start‐up and shut‐down costs of the system. The energy balance and reserve constraints are modelled in (1b) and (1c), respectively. Constraint (1d) models the limit on the generation level and reserve capacity. The UC logic is presented in (1e)–(1g). Transmission constraints are represented by (1h)–(1i).Constraint (1j) defines mj,k$m_{j,k}$ as the system inertia provided by all rotating masses except the contingent generator j. Constraint (1k) models Nadir requirement [39]. The ROCOF requirement considering load damping factor B is modelled using differential expression (1l) and constraint (1m).The term dfkdt$\frac{df_k}{dt}$ is non‐linear in FA‐UC model (1). This non‐linearity can be dealt with through a simplified formula that can provide the maximum ROCOF as a function of inertia and the size of contingency along with the load damping factor, without having to calculate the actual dfkdt$\frac{df_k}{dt}$. The linearization method is described in what follows.MILP MODEL FOR INTEGRATION OF FA IN UCFrom expression (1l), we have2dfkdt=−BΔfk+Pm−Pemj,k∀j,k.$$\begin{equation} \frac{df_k}{dt} =\frac{- B {\Delta f_{k}} + {\left(P_m - P_e \right)}}{m_{j,k}} \quad \forall j,k. \end{equation}$$Using (2), constraint (1m) can be written equivalently as:3F̲≤−BΔfk+Pm−Pemj,k≤F¯∀j,k.$$\begin{equation} \underline{F}\le \frac{- B {\Delta f_k} + {\left(P_m - P_e \right)}}{m_{j,k}}\le \bar{F} \quad \forall j,k. \end{equation}$$Since mj,k$m_{j,k}$ can only take positive values, we will have the following expressions for ROCOF limit:4aF̲mj,k≤−BΔfk+Pm−Pe∀j,k:(κ̲j,k)$$\begin{align} & \underline{F}m_{j,k} \le -B {\Delta f_k} + {\left(P_m - P_e \right)} \quad \forall j,k :\big(\underline{\kappa }_{j,k}\big) \end{align}$$4b−BΔfk+Pm−Pe≤F¯mj,k∀j,k:(κ¯j,k).$$\begin{align} & - B {\Delta f_k} + {\left(P_m - P_e \right)}\le \bar{F}m_{j,k} \quad \forall j,k:\big(\bar{\kappa }_{j,k}\big). \end{align}$$Now, FA‐UC problem can be formed as an MILP problem as presented in (5).5aMinimizeΩ1OFs(PP)$$\begin{align} &\underset{\Omega _1}{\rm Minimize}\nobreakspace \nobreakspace OF^{(PP)}_s \end{align}$$5bsubjectto(1b)−(1k),(4a)−(4b)$$\begin{align} &\text{subject\nobreakspace to}\nobreakspace (1b)-(1k), (4a)-(4b)\end{align}$$5cpi,k,ri,k,fk∈R+$$\begin{align} & p_{i,k}, r_{i,k}, f_{k} \in \mathbb {R}_+ \end{align}$$5dvi,k,wi,k,ui,k∈{0,1}.$$\begin{align} & v_{i,k}, w_{i,k}, u_{i,k} \in \lbrace 0,1\rbrace. \end{align}$$For each credible contingency, we assume that the difference between electrical and mechanical power (in its worst case) is equal to Pj$P_{j}$ right after the contingency [40]. This MILP problem in (5) explicitly ensures the FA adequacy in its optimal dispatch instructions. This model can be solved using state of the art solvers such as CPLEX [41]. For the purpose of pricing and cost calculation in this paper, MILP model (5) will be used, due to less computational burden.NON‐CONVEX PRICING OF FA ADEQUACYThe FA‐UC model (5) finds the optimum commitment, energy and reserve levels considering FA adequacy constraints. The FA adequacy constraints impose additional costs on the system operator resulting from committing new generators. The new committed generators ensuring FA adequacy may also affect electricity price. The FA adequacy pricing requires non‐convex pricing techniques because of non‐convex MILP model (5). In this section, the FA adequacy pricing with explicit and implicit uplift payments is proposed and discussed.The FA adequacy pricing with explicit modelling of uplift paymentIn this approach, we first solve the MILP model (5) to global optimality. Then, the integer variables of the MILP model are fixed to the levels obtained from the MILP solution (ui,k∗$u^*_{i,k}$, vi,k∗$v^*_{i,k}$, wi,k∗$w^*_{i,k}$). The LP model of —UC formulation (5) is obtained by replacing constraint (1o) with (6).6aui,k=ui,k∗,vi,k=vi,k∗,wi,k=wi,k∗∀i,k,ι$$\begin{align} & u_{i,k}=u^*_{i,k}, v_{i,k}=v^*_{i,k}, w_{i,k}=w^*_{i,k}\quad \forall i,k,\iota \end{align}$$6bui,k,vi,k,wi,k∈R+∀i,k.$$\begin{align} & u_{i,k}, v_{i,k}, w_{i,k} \in \mathbb {R}_+ \quad \forall i,k. \end{align}$$The proposed LP model of (5) and (6) can be used to obtain nodal prices. However, these nodal prices suffer from the fact that they do not necessarily ensure profit adequacy and market equilibrium conditions [42]. Some generators cannot recover their costs by nodal prices from LP model [42, 43]. Hence, some uplift payments are necessary to ensure profit adequacy [42, 44]. Authors in [44] propose different approaches for calculating uplift payment, among which, two approaches are investigated for FA adequacy pricing along with another approach proposed here in this paper.Approach A1: In this approach, a generator with negative profit receives an amount equal to (1+α1)(CiPpi,k+CiRri,k)+Civvi,k+Ciwwi,k−ζn,kΛi,npi,k$ (1+ \alpha _1) (C^P_i p_{i,k}+C^R_i r_{i,k}) + C^v_i v_{i,k} + C^w_i w_{i,k} -\zeta _{n,k}\Lambda _{i,n}p_{i,k}$ where α1 is a positive factor to increase the profit of generator in approach A1, ζn,k$\zeta _{n,k}$ is the price at node n and Λi,n$\Lambda _{i,n}$ is the generator‐node mapping matrix. ζn,k$\zeta _{n,k}$ is derived in (7) with Ψϱ,n$\Psi _{\varrho ,n}$ as the power transfer distribution matrix.7ζn,k=λk−∑ϱ(δ̲ϱ,k+δ¯ϱ,k)Ψϱ,n∀n,k.$$\begin{align} & \zeta _{n,k}=\lambda _{k}-\sum _{\varrho } \big(\underline{\delta }_{\varrho ,k}+\overline{\delta }_{\varrho ,k} \big)\Psi _{\varrho ,n} \quad \forall n,k. \end{align}$$Considering the amount of uplift paid, the profit after uplift payment will be α1(CiPpi,k+CiRri,k)$\alpha _1(C^P_i p_{i,k}+C^R_i r_{i,k})$ which is a non‐negative profit for the generator.Approach A2: In the second design, the uplift payment to the generator with negative profit is equal to (1+α2)(CiPpi,k+CiRri,k+Civvi,k+Ciwwi,k−ζn,kΛi,npi,k)$(1+ \alpha _2) (C^P_i p_{i,k}+C^R_i r_{i,k} + C^v_i v_{i,k} + C^w_i w_{i,k} -\zeta _{n,k}\Lambda _{i,n}p_{i,k})$ where 0≤α2≤1$0 \le \alpha _2 \le 1$. Hence, the profit after uplift payment is α2(CiPpi,k+CiRri,k+Civvi,ki+Ciwwi,k−ζn,kΛi,npi,k)$\alpha _2 (C^P_i p_{i,k}+C^R_i r_{i,k} + C^v_i v_{i,k}i + C^w_i w_{i,k} -\zeta _{n,k}\Lambda _{i,n}p_{i,k})$ which is non‐negative, where α2 is a positive factor to increase the profit of generator in approach A2.The FA adequacy pricing with implicit modelling of uplift paymentIn this approach, the MILP model (5) is first converted to an LP model by replacing constraints (1o) with (6b), 0≤ui,k≤1$0 \le u_{i,k} \le 1$, 0≤vi,k≤1$0 \le v_{i,k} \le 1$ and 0≤wi,k≤1$0 \le w_{i,k} \le 1$. The dual objective function of this reformulated LP mode is derived in (8).8OFsDP=∑k(λk∑nDn,k)+∑i,k(μ̲i,kP̲i−μ¯i,kP¯i)+∑i,k(μ¯i,k(u)+μ¯i,k(v)+μ¯i,k(w))+∑ϱ,t(δϱ,k(∑nΨϱ,nDn,k))+∑ϱ,k(δ̲ϱ,kZ̲ϱ+δ¯ϱ,kZ¯ϱ)−∑k(κ̲j,k(F̲mj,k−Pm+Pe))+∑k(κ¯j,k(Pm−Pe−F¯mj,k)).$$\begin{align} OF^{DP}_s = &\,\sum _k {\big (\lambda _{k}\sum _n D_{n,k}\big )} +\sum _{i,k} {\big (\underline{\mu }_{i,k}\underline{P}_{i} -\overline{\mu }_{i,k} \overline{P}_{i}\big )} \nonumber \\ & +\sum _{i,k} {\Big (\bar{\mu }^{(u)}_{i,k}+\bar{\mu }^{(v)}_{i,k}+\bar{\mu }^{(w)}_{i,k}\Big )} +\sum _{\varrho ,t} {\Big(\delta _{\varrho ,k}{\Big( \sum _{n} \Psi _{\varrho ,n}D_{n,k} \Big)}\Big)}\nonumber \\ & +\sum _{\varrho ,k}{\Big (\underline{\delta }_{\varrho ,k}\underline{Z}_{\varrho } +\overline{\delta }_{\varrho ,k}\bar{Z}_{\varrho }\Big )}- \sum _{k}{\Big (\underline{\kappa }_{j,k}{\Big (\underline{F}m_{j,k} - P_m + P_e \Big)} \Big )} \nonumber \\ & +\sum _{k}{\Big (\bar{\kappa }_{j,k}{\big ( P_m - P_e - \bar{F}m_{j,k}\big)}\Big )}. \end{align}$$Selected stationary conditions of reformulated LP model are derived in (9).9a−Pi¯μj,k−Pi¯μ¯i,k−P̲iμ̲i,k+σi,k−σi,k+1+σ̲i,k+σ¯i,k+∑j≠iMjλ̲j,k=0,∀i,k$$\begin{align} & -\bar{P_{i}}\mu _{j,k}-\bar{P_{i}}\bar{\mu }_{i,k}-\underline{P}_{i}\underline{\mu }_{i,k} + \sigma _{i,k}-\sigma _{i,k+1}\nonumber \\ & + \underline{\sigma }_{i,k}+\bar{\sigma }_{i,k} + \sum _{j \ne i} M_j\underline{\lambda }_{j,k} = 0, \quad \forall i,k \end{align}$$9b−Pi¯μj,k−Pi¯μ¯i,k−P̲iμ̲i,k+σi,k+μ¯i,k(u)+μ̲i,k(u)+∑j≠iMjλ̲j,k=0,∀i,k$$\begin{align} & -\bar{P_{i}}\mu _{j,k}-\bar{P_{i}}\bar{\mu }_{i,k}-\underline{P}_{i}\underline{\mu }_{i,k} \nonumber \\ & + \sigma _{i,k} + \bar{\mu }^{(u)}_{i,k}+\underline{\mu }^{(u)}_{i,k} + \sum _{j \ne i} M_j\underline{\lambda }_{j,k} = 0, \quad \forall i,k \end{align}$$9cσi,k+μ¯i,k(v)+μ̲i,k(v)=Civ,∀i,k$$\begin{align} & \sigma _{i,k} + \bar{\mu }^{(v)}_{i,k}+\underline{\mu }^{(v)}_{i,k} = C^{v}_{i}, \quad \forall i,k \end{align}$$9d−σi,k+μ¯i,k(w)+μ̲i,k(w)=Ciw,∀i,k$$\begin{align} & -\sigma _{i,k} + \bar{\mu }^{(w)}_{i,k}+\underline{\mu }^{(w)}_{i,k} = C^{w}_{i}, \quad \forall i,k \end{align}$$9eλk−μ¯i,k−μ̲i,k−∑nΓn,iδn,k=Cip,∀i,k$$\begin{align} & \lambda _{k} - \bar{\mu }_{i,k}-\underline{\mu }_{i,k}-\sum _{n} \Gamma _{n,i}\delta _{n,k} = C^{p}_{i}, \quad \forall i,k \end{align}$$9f∑j≠iμj,k−μ¯i,k≤Cir,∀i,n,k$$\begin{align} & \sum _{j \ne i} \mu _{j,k} - \bar{\mu }_{i,k} \le C^{r}_{i}, \quad \forall i,n,k \end{align}$$9gδn,k+δ¯n,k+δ̲n,k=0,∀n,k$$\begin{align} & \delta _{n,k} + \bar{\delta }_{n,k} + \underline{\delta }_{n,k} = 0, \quad \forall n,k \end{align}$$9hλ̲j,k+F¯κ¯j,k+F̲κ̲j,k−∑i≠j2Yi(f0−FNadir−Fdb)π(j,k)=0$$\begin{align} &\hspace*{-12pt}{\underline{\lambda }}_{j,k}+\bar{F}{\bar{\kappa }}_{j,k}+\underline{F}\underline{\kappa }_{j,k}\nonumber \\ &-\sum _{i \ne j}2Y_i(f_{0}-F_{Nadir}-F_{db})\pi (j,k)=0 \end{align}$$9i−Bκ̲j,k−Bκ¯j,k≤0$$\begin{align} & -B{\underline{\kappa }}_{j,k}-B{\bar{\kappa }}_{j,k}\le 0 \end{align}$$9jκ¯j,k,κ̲j,k,μ¯i,k,μ̲i,k,λ̲j,k∈R+,δ¯n,k,δn,k,σi,k,πj,k,λk,λ¯j,k∈R+.$$\begin{align} & \overline{\kappa }_{j,k}, \underline{\kappa }_{j,k},\overline{\mu }_{i,k}, \underline{\mu }_{i,k}, \underline{\lambda }_{j,k}\in \mathbb {R}_+ , \nonumber \\ & \overline{\delta }_{n,k}, \delta _{n,k} , \sigma _{i,k}, \pi _{j,k}, \lambda _{k}, \overline{\lambda }_{j,k} \in \mathbb {R}_+. \end{align}$$Here, μ¯i,k(w)$\bar{\mu }^{(w)}_{i,k}$ and μ̲i,k(w)$\underline{\mu }^{(w)}_{i,k}$ are the Lagrange multipliers of the constraints 0≤wi,k≤$0\le w_{i,k}\le$ 1 which allow wi,k$w_{i,k}$ to take values between zero and one. μ¯i,k(v)$\bar{\mu }^{(v)}_{i,k}$ and μ̲i,k(v)$\underline{\mu }^{(v)}_{i,k}$ are the Lagrange multipliers of the constraints 0≤vi,k≤1$0\le v_{i,k}\le 1$, and μ¯i,k(u)$\bar{\mu }^{(u)}_{i,k}$ and μ̲i,k(u)$\underline{\mu }^{(u)}_{i,k}$ as the Lagrange multipliers for constraints 0≤ui,k≤1$0\le u_{i,k}\le 1$.Using the nodal prices ζn,k$\zeta _{n,k}$ derived in (7), the non‐negativity profit can be ensured by constraint (10).10∑kζk,nΛn,ipi,k−(CiPpi,k+Civvi,k+CiRri,k)≥0∀i,n.$$\begin{align} & \sum _{k} {\left(\zeta _{k,n}\Lambda _{n,i}p_{i,k} - (C^P_i p_{i,k} + C^v_i v_{i,k}+C^R_i r_{i,k})\right)}\nonumber \\ & \ge 0 \nobreakspace \nobreakspace \quad \forall i,n. \end{align}$$The bilinear term ζk,npi,k$\zeta _{k,n}p_{i,k}$ in (10) are linearized in (11) using discrete approximation technique.11a∑ρP̂ρ,ibi,ρ,k≤pi,k∀i,k$$\begin{align} &\sum _{\rho } \hat{P}_{\rho ,i}b_{i,\rho ,k} \le p_{i,k} \quad \forall i,k \end{align}$$11b∑ρP̂ρ,ibi,ρ,k≥pi,k−Pi¯Υ∀i,k$$\begin{align} &\sum _{\rho } \hat{P}_{\rho ,i}b_{i,\rho ,k} \ge p_{i,k}-\frac{\bar{P_{i}}}{\Upsilon } \quad \forall i,k \end{align}$$11c∑ρbi,ρ,k=1∀i,k$$\begin{align} &\sum _{\rho } b_{i,\rho ,k}=1 \quad \forall i,k \end{align}$$11dei,n,ρ,k≤X2Λn,i∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \le X_2\Lambda _{n,i}\quad \forall i,k,\rho , n \end{align}$$11eei,n,ρ,k≥−X2Λn,i∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \ge -X_2\Lambda _{n,i}\quad \forall i,k,\rho , n \end{align}$$11fei,n,ρ,k≤X2bi,ρ,k∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \le X_2b_{i,\rho ,k} \quad \forall i,k,\rho , n \end{align}$$11gei,n,ρ,k≥−X2bi,ρ,k∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \ge -X_2b_{i,\rho ,k} \quad \forall i,k,\rho , n \end{align}$$11hζk,n−ei,n,ρ,k≥−X2(1−bi,ρ,k)∀i,k,ρ$$\begin{align} &\zeta _{k,n}-e_{i,n,\rho ,k} \ge -X_2(1-b_{i,\rho ,k}) \quad \forall i,k,\rho \end{align}$$11iζk,n−ei,n,ρ,k≤X2(1−bi,ρ,k)∀i,k,ρ$$\begin{align} &\zeta _{k,n}-e_{i,n,\rho ,k} \le X_2(1-b_{i,\rho ,k}) \quad \forall i,k,\rho \end{align}$$11j∑k,ρ,nei,n,ρ,kP̂i,ρ−(CiPpi,k+Civvi,k)≥0∀i,$$\begin{align} &\sum _{k,\rho ,n} {\left(e_{i,n,\rho ,k}\hat{P}_{i,\rho } - (C^P_i p_{i,k} + C^v_i v_{i,k})\right)} \ge 0 \quad \forall i, \end{align}$$where P̂ρ,i$\hat{P}_{\rho ,i}$ is the discrete approximation of generation and bi,ρ,k$b_{i,\rho ,k}$ is associated binary variable. X2 is a suitably large constant. ei,n,ρ,k$e_{i,n,\rho ,k}$ shows the components of the generator‐node incidence matrix. Also, the factor 1Υ$\frac{1}{\Upsilon }$ is for dividing units' capacity to Υ discrete capacities. Obviously, smaller divisions result in more accurate approximation but it requires more binary variables and increases computational time.Now we form the MILP optimization problem (12) where the integrality constraints are enforced.12aMinimizeΩ2OFPP−OFDP$$\begin{align} &\underset{\Omega _2}{\text{Minimize}}\nobreakspace \nobreakspace OF^{PP} - OF^{DP} \end{align}$$12bsubjectto(1b)−(1k),(4a)−(4b),(SCs),(11)$$\begin{align} &\text{subject\nobreakspace to}\nobreakspace (1b)-(1k),(4a)-(4b), \nonumber \\ &(SCs), (11)\end{align}$$12cpi,k,ri,k,fk∈R+∀i,k$$\begin{align} & p_{i,k}, r_{i,k}, f_k \in \mathbb {R}_+ \quad \forall i,k \end{align}$$12dvi,k,wi,k,ui,k∈{0,1}∀i,k$$\begin{align} & v_{i,k}, w_{i,k}, u_{i,k} \in \lbrace 0,1\rbrace \quad \forall i,k \end{align}$$12eκ¯j,k,κ̲j,k,μ¯i,k,μ̲i,k,λ̲j,k∈R+,δ¯n,k,δn,k,σi,k,πj,k,λk,λ¯j,k∈R+,$$\begin{align} & \overline{\kappa }_{j,k}, \underline{\kappa }_{j,k},\overline{\mu }_{i,k}, \underline{\mu }_{i,k}, \underline{\lambda }_{j,k}\in \mathbb {R}_+ , \nonumber \\ & \overline{\delta }_{n,k}, \delta _{n,k} , \sigma _{i,k}, \pi _{j,k}, \lambda _{k}, \overline{\lambda }_{j,k} \in \mathbb {R}_+, \end{align}$$where SCs stands for stationary conditions. Ω2={pi,k$\Omega _2 = \lbrace p_{i,k}$, ri,k$r_{i,k}$, vi,k$v_{i,k}$, wi,k$w_{i,k}$, ui,k$u_{i,k}$, zϱ,k,κ¯j,k,κ̲j,k,μ¯i,k,μ̲i,k,λk,λ¯j,k,λ̲j,k,δ¯n,k,δn,kσi,k,πj,k}$z_{\varrho ,k},\overline{\kappa }_{j,k}, \underline{\kappa }_{j,k},\overline{\mu }_{i,k}, \underline{\mu }_{i,k}, \lambda _{k}, \overline{\lambda }_{j,k}, \underline{\lambda }_{j,k},\overline{\delta }_{n,k}, \delta _{n,k}\, \sigma _{i,k}, \pi _{j,k} \rbrace$ is the set of decision variables.It is also noteworthy to mention that the mentioned methods for price formation are discussed in literature and investigation of the mathematical basis of the methods is out of the scope of this paper. However, further details can be found in previous works such as [37, 38].ILLUSTRATIVE CASE STUDYTo illustrate the results of proposed —UC model and the non‐convex pricing techniques in Section 4, the five‐node example system in Figure 1 is carefully analysed and discussed.1FIGUREThe single‐line diagram of the five‐node systemThe basic data is presented in Table 1 and the detailed data is provided in [45]. The load damping factor is assumed to be 0.85 [46]. Furthermore, Nadir requirement is set to be 49.4 Hz and maximum ROCOF to be 0.5 Hz/S. The following models are used and discussed:FA‐UC‐A1(‐A2 or ‐B) model which refers to standard UC model with FA adequacy constraints. The non‐convex pricing approach A1 (A2 or B) is used.UC‐A1(‐A2 or ‐B) model which is standard UC model without FA adequacy constraints. The non‐convex pricing approach A1 (A2 or B) is used here as well.All models are coded in GAMS platform and solved using the CPLEX solver [41]. It is also assumed that the lack of coherency in the frequency of different buses is negligible [40, 47]. We have also benchmarked our results with the case where FA adequacy constraints are not modelled (UC‐A1, UC‐A2 and UC‐B models).1TABLEThe generation data of the five‐node systemUnitSRMC ($/MWh)SUC ($)IC (MWSec/Hz)MC (MW)G125000140G210700170170G3301000350260G4401000450200G525800240300Abbreviations: SRMC, short run marginal cost; SUC, start‐up cost, IC, inertia constant; MC, maximum capacity.As can be noted in the Table 1, G1 is providing no inertia and also has a low marginal cost. This generator is modelling a typical renewable energy source (e.g. wind) here. It is to be noted that the price and inertia are the main factors which affect our study here. The optimal dispatch instructions under UC and FA‐UC models are reported in Table 2.2TABLEGeneration levels (MW) in both UC and FA‐UC models for the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6G1140140140140140140G267.5112.9140.1170143.8127G3000000G4000000G50023.035.218.70As can be seen in Table 2, both UC and FA‐UC models have led to same commitment and generations levels. Generators G3 and G4 produce 0 MWh in both cases. This is while G1 is dispatched at 140 MW in all hours because of having lowest marginal cost. Generators G2 and G5 also contribute in providing energy in some hours of the dispatch period. The nodal prices for the 6‐h period using FA‐UC‐A1 and FA‐UC‐A2 models are presented in Table 3. The nodal prices have also been depicted in Figure 3. The height and colour of the bars are showing the prices in different nodes and hours.3TABLENodal prices ($/MWh) from FA‐UC‐A1, FA‐UC‐A2, UC‐A1 and UC‐A2 models in the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6Node 1101010101010Node 2101017.118.717.010Node 3101016.318.416.110Node 4101016.318.416.110Node 51010252525102FIGUREImpact of FA adequacy constraints on payment to generators in five‐node system3FIGURENodal prices ($/MWH) from FA‐UC‐A1, FA‐UC‐A2, UC‐A1 and UC‐A2 models in the 5‐node systemAs shown in Table 3, the nodal prices for UC and FA‐UC models are the same. This is because the generator G3 (which is just committed for ensuring FA adequacy) does not provide energy (see Table 2). This means nodal prices under approaches A1 and A2 in FA‐UC model do not reflect the extra cost of FA adequacy constraints. The profit of generators using these nodal prices for FA‐UC‐A1 model is set out in Table 4.4TABLEProfits ($) before uplift paymentsModelG1G2G3G4G5UC6720−70000−800FA‐UC6720−700−10000−800As in Table 4, generators G2 and G5 experience negative profits in UC, while in FA‐UC, generators G2, G3 and G5 are all experiencing negative profits. An interesting observation is that, in this case, the only generator with positive profit is G1, which does not contribute to FA adequacy.For calculating uplift payments under approaches A1 and A2, the α1 and α2 are, respectively, assumed to be 0.05 and 0.1. The uplift payments for these generators are reported in Table 5.5TABLEUplift payment ($) from FA‐UC‐A1 and FA‐UC‐A2 models in the five‐node systemModelG1G2G3G4G5UC‐A10773.500859.3UC‐A2077000880FA‐UC‐A10753.71032.50850.8FA‐UC‐A2077011000880The FA adequacy constraints have a clear impact on uplift payments (while it did not have any impact on nodal prices). Generator G2 receives $753.7 uplift payment in FA‐UC model and approach A1 (FA‐UC‐A1). This is while its uplift payment without )s (UC‐A1) is $773.5 (a decrease of 753.7−773.5773.5=−2.5%$\frac{753.7-773.5}{773.5}=-2.5 \%$). Also, when the FA adequacy constraints are considered, G3 needs to receive uplift payment of $1032.5, while it was not being paid any uplift in UC‐A1. This is because this generator is only committed to ensure the FA adequacy constraints and does not generate so much energy to cover its start‐up costs. Generator G5 receives slightly less uplift payment when the FA adequacy constraints are modelled ($859.3 as compared to $850.8 in FA‐UC‐A1 and UC‐A1 cases). For approach A2, the FA adequacy constraints do not change the uplift payment for generators G2 and G5. Generator G2 receives $770 and generator G5 receives $880 uplift payment with (FA‐UC‐A2) and without (UC‐A2) FA adequacy constraints. Generator G3 receives $1100 which is slightly higher than method A1. No uplift payment is received by G1 in these cases. Table 6 shows raw profit (profit without any extra uplift payment) and also uplift payment with and without FA adequacy constraints.6TABLERaw profit, uplift payment and profit (= raw profit + uplift payment) of all generators over different hours in five‐node systemPayment approachModelRaw profit ($)Uplift payment ($)Profit ($)A1UC52201632.86852.8FA‐UC422026376857Difference−1000 (−19.2%)1005 (62%)5 (0.1%)A2UC522016506870FA‐UC422027506970Difference−1000 (−19.2%)1100 (67%)100 (1.5%)As we can see from Table 6, raw profit is decreased while uplift payment and profit are increased when the FA adequacy constraints are explicitly modelled (−19.2%$-19.2 \%$ for raw profit, +62%$+62 \%$ for uplift payment and +0.1%$+0.1 \%$ for profit). The same pattern is observed under approach A2. The raw profit, uplift payment and profit are changed by −19.2%$-19.2 \%$, +67%$+67 \%$ and +1.5%$+1.5 \%$, respectively, when the FA adequacy constraints are considered. Approach B is also applied to the illustrative five‐node system. The dispatch levels of generators are provided in Table 7. The generation levels under FA‐UC‐B model are presented in Table 8.7TABLEGeneration levels (MW) in UC‐B model for the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6G1109140140140140140G298.5112.974.0138.5130.5116.9G3000000G4000000G50081.135.713.110.08TABLEGeneration levels (MW) in FAUC‐B model for the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6G179.2105.5140140140140G260.12095.189.983.171G30117.420.322.923.916.1G4000000G568.210.039.753.444.639.9Furthermore, the nodal prices using the UC‐B and FA‐UC‐B models are reported in Tables 9 and 10. In order to illustrate the differences better, the results have been also plotted in Figures 4 and 5. As can be observed from the height and colour of the bars, the prices obtained from FA‐UC are generally higher than the prices from UC. This is because of the fact that in approach B, the differences in payments are reflected in prices. The uplift payments have been implicitly included in these nodal prices such that the non‐negative profitability is ensured. This is in contrast to the explicit approaches (A1 and A2), where there are no higher prices to ensure inertia adequacy but the explicit out‐of‐market uplift payments are added.9TABLENodal prices ($/MWh) using UC‐B model in five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6Node 117.110.010.010.010.010.0Node 213.710.214.518.116.310.0Node 313.310.215.918.417.916.2Node 414.110.016.318.418.416.3Node 513.310.734.925.025.025.010TABLENodal prices ($/MWh) using FA‐UC‐B model in five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6Node 121.81010101010Node 221.116.317.018.619.422.6Node 319.938.630.030.030.030.0Node 419.415.422.323.623.721.6Node 536.925.025.025.025.025.04FIGURENodal prices ($/MWH) using UC‐B model in 5‐node system5FIGURENodal prices ($/MWH) using FA UC‐B model in 5‐node systemComparing Table 9 with Table 10, the impact of FA adequacy constraints on nodal prices can be seen. The nodes and hours which experience different prices because of FA adequacy constraints are highlighted in Tables 9 and 10. The total profit of generators using UC‐B and FA‐UC‐B models are presented in Table 11.11TABLEGenerators' profits ($) over different hours in five‐node systemModelG1G2G3G4G5UC‐B7091.10000FA‐UC‐B6710.90000Difference−380.2 (−5.2%)0000Figure 2 compares raw profit, uplift payment and profit of generators under different investigated models. The difference between bars for different approaches is caused by the FA adequacy constraints.NUMERICAL RESULTSFor further discussions, the Nordic 44‐node system is studied. All simulations are carried out on a computer with 8 GB of RAM and 2.6 GHz CPU. The GAMS platform and the CPLEX solver are used.Regarding the computational time, we should note that there is a trade‐off between computational time and employing detailed formulation to assess FA adequacy. It is clear that a more detailed formulation of FA such as the one in [32] can lead to more precise results but at the cost of higher computational complexity. However, improving the current formulation in our paper using more detailed models such as those in [32] is a good extension of the current paper. Besides the linearization techniques proposed in this paper, the computational complexity of our proposed modified UC model can be handled through decomposition algorithms such as Benders decomposition or Lagrangian Relaxation algorithms (which are common technique for dealing with computational complexity of UC models [48] and [49]).The GAMS platform and the CPLEX solver are used. In order to consider the worst‐case scenario for FA adequacy, we assume load damping factor to be zero. We also assume no transmission congestion for our case studies. The intertemporal system prices for a period of 12 h from FA‐UC‐A1 and FA‐UC‐A2 models are reported in Table 12. The benchmark prices from UC‐A1 and UC‐A2 are also reported. For hours 1–8, the system prices with and without FA adequacy constraints are the same. However, for hours 9–12, the FA adequacy constraints result in an increase in intertemporal system prices.12TABLESystem prices ($/MWh) for the Nordic 44‐node system from FA‐UC and UC modelsHourUCFA‐UCHourUCFA‐UC1–715415410154205815415411154205917620512154176The prices reported in Table 12 result in negative profit for several generators. The explicit and implicit approaches for non‐convex pricing are applied and the results for total profits are reported in Table 13.13TABLERaw profit, uplift payment and profit (= raw profit + uplift payment) under explicit and implicit non‐convex pricingPayment approachModelRaw profit ($)Uplift ($)Total profits ($)A1UC371,01067,950438,960FA‐UC369,430123,700493,130Difference−158055,75054,170 (13%)A2UC371,01090,189462,450FA‐UC369,430146,435515,865Difference−158056,24653,415 (11.6%)BUC‐‐567,420FA‐UC‐‐1,316,021Difference‐‐748,601(132%)Under approach A1, the FA adequacy constraints decrease the total raw profit of all generators by $1580. The uplift payment is increased by $55, 750 (=123,700−67,950$=123,700-67,950$) and this results in 13% raise in the profit. Similar situation is observed in case A2 with the increase of 11.6% in profit of generators in case of FA‐UC‐A2 model as compared to UC‐A2 model. Under implicit non‐convex pricing approach B, the total profit is changed from $567, 420 in UC‐B model to $1316, 021 in FA‐UC‐B model (an increase of 132%). This significant raise in profit mainly comes from the fact that the generators with low marginal cost of energy, are being paid by system prices which are adjusted to cover the start‐up cost of FA providers.We have also reported the intertemporal system prices from FA‐UC‐B model. In hours 1, 3 and 8–12, the system prices with FA adequacy constraints are higher than those without FA adequacy constraints. In hour 2, the price with FA adequacy constraint is lower than the one without this adequacy constraint. In the rest of hours, the prices are the same with and without FA adequacy constraints.As in Table 14, we see a considerable difference in system prices in hours 1 and 9. This is due to the need to compensate the start‐up cost of extra units committed to ensure FA adequacy. Figure 6 compares the profits for the Nordic 44‐node system under explicit and implicit non‐convex pricing approaches. Figure 6 shows the difference in total payments. It can be seen that the two explicit uplift payment methods (A1 and A2) result in lower amount for total payments to generators, and as a result, lower cost for the operator to provide the energy, especially when considering FA adequacy concerns. However, the advantage of implicit method B over A1 and A2 is that it does not require out‐of‐market payments and this means the entire cost is reflected in prices.14TABLEIntertemporal system prices ($/MWh) from UC‐B and FA‐UC‐B modelsHourUCFA‐UCHourUCFA‐UC1169.3317.071541542163.6155.28154.3155.73154156.39154343.5415415410154.4206.6515415411154.5205.0615415412155.3205.06FIGUREImpact of FA adequacy constraints on payments to generators in Nordic 44‐node systemFURTHER DISCUSSIONOne of the points that can be a subject of discussion is that, what is the reason and motivation behind building up a complex formulation, as the one discussed here, to ensure FA adequacy, while one way to ensure FA adequacy currently is to consider the worst‐case scenario and commit plenty of synchronous units to provide adequate inertial response. The answer is, with a high rate of renewable integration, there might not be necessarily plenty inertia providers available among the committed units [50]. This important limitation has overlooked as inertia and ramping capability has been adequate up to now. There are also reports such as [51] and [52] which state that in a number of power systems such as Nordic system, the lack of frequency response contribution will be more critical in future. Accordingly, in a situation where there is a considerable share of renewable non‐synchronous generation, in order to calculate the FA cost properly and economically, inertia and ramping constraints need to be integrated in UC model and the cost of providing FA should be minimized along with the cost of providing energy. This modelling needs the dynamic equations of FA as we proposed in our paper. In other words, the proper modelling of FA which considers both inertia and ramping constraints adds the level of complexity proposed in our study to the standard UC formulation.In such formulation, due to the need for including non‐linear algebraic and differential equations as constraints, there will be added computational complexity which is unavoidable. Also, a number of other studies that consider FA constraints have a similar level of complexity, as previously reviewed in this paper [25, 26, 32, 40].However, we have tried to reduce the computational complexity as much as possible by linearization of the formulation (improving the computational tractability of our proposed UC formulation by proposing an MILP model of its original MINLP model). Further reduction of computational complexity is out of the scope of this study but it can be a potential for future work and has been added in the conclusion and future work in Section 8.Also, on the importance and motivation behind building up a formulation that integrates dynamic equations inside UC optimization problem, we should state that since the total inertia available in the network depends on the online units, the optimal way to ensure FA adequacy is to integrate FA constraints in the energy optimization (UC) problem to make sure that we can satisfy FA requirements along with other technical constraints by commitment decisions. This approach avoids the suboptimality that may occur in sequential approach as discussed in references [25] and [26]. Our proposed joint optimization needs proper modelling of FA provision cost which in turn involves the dynamic equations (as we proposed in our paper) in the standard UC formulation.One can determine by a Digsilent/PSSE/PowerWorld simulation a set of generators to maintain adequacy, but that set of generators, in general, is a feasible but not necessarily optimal solution to the UC model. In other words, if we solve the FA adequacy problem based on worst‐case scenario approach and UC problem separately, we obtain suboptimum solution as compared to the case where we do joint optimization. More advantages of joint energy and ancillary service markets over the separate designs are discussed in [33–36].CONCLUSIONSThis paper proposes an MILP model for ensuring the FA adequacy constraints. First, Nadir and ROCOF measures are modelled through a set of bilinear constraints and differential equations and are included in standard UC formulation. The resulting model is a mixed‐integer bilinear program with differential equations. Through a series of linearization techniques, an MILP model is proposed. Due to the non‐convexity of proposed MILP model, some generators providing energy or inertia service might experience negative profit. To tackle the negative profit, the explicit and implicit non‐convex pricing techniques are proposed and implemented using our proposed MILP model. In the explicit pricing approaches, the generators with negative profit receive uplift payment. Under the implicit non‐convex pricing approach, all payments are embedded in the calculated nodal prices. The developed MILP model of FA‐UC and all approaches for explicit and implicit non‐convex pricing are applied to an illustrative five‐node system and Nordic 44‐node system and the results are carefully studied.Devising allocation mechanisms for distributing the FA adequacy costs among generators can be a good extension of this work. Furthermore, our proposed model can be improved by more detailed modelling of FA adequacy constraint and changing the assumption of frequency coherency used in our paper.Also, reduction of computational complexity of the proposed modified UC model can be addressed through decomposition algorithms such as Benders decomposition or Lagrangian decomposition (which are common techniques for dealing with computational complexity of UC models [49] and [48]).In addition, a sensitivity analysis on the effect of load damping factor on the overall system frequency response can be a subject for further study.AUTHOR CONTRIBUTIONSEhsan Davari Nejad: Conceptualization, data curation, investigation, methodology, visualization, writing ‐ original draft, writing ‐ review and editing. Mohammad Hesamzadeh: Conceptualization, methodology, resources, software, supervision, writing ‐ original draft, writing ‐ review and editing. 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Unit commitment with frequency‐arresting adequacy constraint: Modelling and pricing aspects

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© 2022 The Institution of Engineering and Technology.
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10.1049/gtd2.12641
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Abstract

INTRODUCTIONFrequency response after contingencies is essential for reliable power delivery to consumers. The frequency often experiences relatively high fluctuations after a severe loss of load or trip of a generation unit. Accordingly, practical methods should be devised to avoid the frequency from violating the desirable interval after occurrence of contingencies [1, 2]. The frequency response measures to be taken after a contingency are categorized differently in the literature. A number of the previous works in this area have categorized frequency response actions in three groups of primary, secondary and tertiary frequency response activities [3, 4].The inertial support consists of the inherent and natural response of rotating masses, which are synchronously connected to the grid. The primary frequency response occurs within a few seconds after the contingency and it is mainly driven by governor action [2]. The inertial response problems have captured particular attention due to increasing penetration of intermittent renewable energy resources [5, 6]. The intermittent wind and solar generators are connected to the power system via power electronic converters and they do not provide synchronous inertia [7]. As a result, the system is losing its inherent ability of inertial response through replacing the conventional synchronous generators with converter‐based intermittent wind and solar generators [8–10], becoming more vulnerable to a large loss of generation [4]. The work in this paper is mainly focused on frequency‐arresting (FA) adequacy. The FA adequacy can be measured by Nadir and rate of change of frequency (ROCOF). The Nadir requirement defines a minimum frequency that must not be violated to prevent load shedding after any contingency. The ROCOF is the derivative of frequency with respect to time. It should not violate from interval determined by TSO (Transmission System Operator) [11, 12].In [13], four market designs for ensuring FA adequacy are proposed and analysed. However, none of these proposals considers UC for ensuring FA adequacy. Reference [14] proposes adding a frequency‐responsive reserve constraint (spinning reserve) to the UC formulation to avoid load shedding caused by frequency drop.References [15–17] propose market approaches for FA adequacy. Authors in [15] propose a methodology for optimal allocation of primary reserves for frequency control considering power system events. This work does not provide a detailed formulation considering inertia.Reference [18] has presented a simplified expression for frequency response and analysed the risk of under frequency load shedding (UFLS) wrong action. However, the market aspects of frequency response have not been investigated in detail.Authors of [11] present sufficient conditions for both FA and automatic generation control (AGC). They first study primary and secondary frequency response separately and then propose a formulation for joint primary and secondary frequency response. The work is more focused on control model of frequency response and it has not investigated market aspects.Reference [19] proposes a unit commitment (UC) model considering FA adequacy. The model is implemented on a simple case study and no standard test network is investigated. Furthermore, the work has not investigated market aspects in detail.Here in this paper, non‐convex pricing techniques are employed to calculate the cost of FA adequacy constraint. Among literature in this area, [20] has proposed a modified convex hull pricing technique to deal with the non‐convexities originating from non‐zero minimum output of generators. However, the UC formulation only considers the energy‐balance constraint and no constraints to model ancillary services are considered. Also, the authors of [21] have applied a semi‐Lagrangian method to ensure non‐negative profits of generators in a market with non‐convexities. However, their optimization problem does not include any of ancillary service products. Authors in [22] and [23] have proposed recovery payment mechanisms in a two‐part paper. The same authors have also carried out a review of non‐convex pricing methods and practices in [24]. References [25, 26] have put forward another formulation for market design of frequency response provision in a two‐part paper. The proposed method first calculates the frequency drop and ROCOF through external software. Then, the FA adequacy is ensured through another optimization problem. This method needs iterations between two stages (external‐software simulation and optimization‐problem simulation) which are computationally demanding. Besides, the obtained solution might be suboptimal as compared to the joint‐optimization approach proposed in our paper.Reference [27] has proposed a method for simultaneous scheduling of frequency response and energy. The work has implemented a comprehensive stochastic security constrained UC to achieve its results. However, the impact of considering the frequency response concerns on the pricing schemes is not investigated by the authors of this reference.In [28], the authors focus on upregulating and downregulating reserve allocation for fast ramping capability. This paper is more focused on AGC and secondary frequency response. It proposes a compensation mechanism for the generation units participating in AGC, based on their response performance to AGC signals.The work in [29] is focused on financial compensation of frequency regulation actions. It evaluates market rules in terms of fairness to provide reasonable compensation for the ancillary service provided. However, this reference is focused on up‐ and downregulation. The report is more descriptive and the related computational works are not discussed.In [30], the authors have analysed the impact of integrating renewable energy sources, particularly wind and solar energy, and tried to determine to what extent the side payments are reasonable in energy markets with non‐convexities.Furthermore, chapter 8 in [31] analyses uplift payments in pool‐based electricity markets. They propose a game‐theoretic approach for uplift payment. They investigate their proposed approach using a simple case study. This work does not discuss frequency response control or any ancillary services.In reference [32], an approach based on SCUC (Security Constrained Unit Commitment) has been proposed and implemented to deal with the problem of frequency response and the model has been solved through branch‐and‐bound techniques. The authors have implemented a detailed formulation and also performed a comparison on the maximum frequency drop after a contingency, based on different arrangements of online generators. The formulation provided in this reference is quite thorough and comprehensive.It is important to note that since the total inertia available in the network depends on the online generators, the optimal way to provide energy and ensure FA adequacy is to perform a joint optimization of providing energy and ensuring FA adequacy. The joint optimization of providing energy and ensuring FA avoids suboptimality that may occur in sequential approaches (ensuring FA and then providing energy), such as those discussed in [33–36].There are also some references in the literature which have gone through the details of pricing techniques in electricity markets with non‐convexities. Reference [37] reviews the impact of a number of pricing methods in a comprehensive technical report, including dual pricing, semi‐Lagrangian, integer programming and a number of other methods. Also, the work in [38] has studied a primal–dual approach in detail in an electricity market with non‐convexities.Accordingly, the contributions of the current paper are: First, it models Nadir and ROCOF requirements as a set of non‐linear constraints and differential expressions in standard UC formulation to ensure FA adequacy. The original FA‐constrained UC (FA‐UC) model is an MINLP (Mixed Integer Non‐Linear Programming) with differential equations. Through a series of proposed linearization techniques, the FA‐UC model is transformed to a mixed‐integer linear program (MILP) model which can be solved to global optimality. Second, to price the FA adequacy, non‐convex pricing techniques based on explicit and implicit uplift payment are proposed and discussed. The implicit uplift payment is based on the primal–dual formulation in optimization theory. Furthermore, the proposed MILP models and non‐convex pricing techniques are carefully studied through an illustrative example and also Nordic 44‐node test network. The proposed methodology in this work (1) ensures that the optimal amount of inertia can be provided, considering technical constraints and social‐cost objective along with FA constraints; (2) provides the possibility to consider a conservative assumption on the contingent generator and also load‐damping factor; (3) is capable of providing the optimal solution that ensures FA adequacy in a one‐shot optimization (as compared to possible suboptimal solution of sequential approach); (4) in contrast to some current practices which simply requires every synchronous generator to provide a specific amount of inertial response (which may not necessarily be the optimal amount), ensures that the adequate inertial response is provided in an optimal manner.FA‐CONSTRAINED UNIT COMMITMENTThe FA‐UC is set out in (1).1aMinimizeΩ1OFPP=∑i,kCiPpi,k+CiRri,k+Civvi,k+Ciwwi,k$$\begin{align} &\underset{\Omega _1}{\text{Minimize}}\nobreakspace OF^{PP} = \sum _{i,k} C^P_i p_{i,k} + C^R_i r_{i,k} + \nonumber \\ &\hspace{73.81102pt} C^v_i v_{i,k} +C^w_i w_{i,k} \end{align}$$1bsubjectto∑ipi,k=∑nDn,k∀k:(λk)$$\begin{align} &\text{{s}ubject\nobreakspace to}\nobreakspace \nonumber \\ & \sum _i p_{i,k} = \sum _{n} D_{n,k} \quad \forall k \nobreakspace \nobreakspace :(\lambda _k) \end{align}$$1c∑i≠jri,k≥P¯juj,k∀j,k:(μj,k)$$\begin{align} & \sum _{i\ne j} r_{i,k} \ge \overline{P}_j u_{j,k} \quad \forall j,k \nobreakspace \nobreakspace :(\mu _{j,k}) \end{align}$$1dP̲iui,k≤pi,k≤P¯i−ri,k∀i,k:μ̲i,k,μ¯i,k$$\begin{align} &\underline{P}_{i}u_{i,k} \le p_{i,k} \le \overline{P}_{i}-r_{i,k} \quad \forall i,k \nobreakspace \nobreakspace :\left(\underline{\mu }_{i,k},\overline{\mu }_{i,k}\right) \end{align}$$1eui,k−ui,k−1=vi,k−wi,k−1∀i,k:(σi,k)$$\begin{align} & u_{i,k}-u_{i,k-1}=v_{i,k}-w_{i,k-1} \quad \forall i,k \nobreakspace \nobreakspace :(\sigma _{i,k}) \end{align}$$1f∑t=k−tiUp−1kvi,t≥ui,k∀i,k:(σ¯i,k)$$\begin{align} & \sum ^k_{t=k-t^{Up}_i-1}{v_{i,t}\ge u_{i,k}\ } \quad \forall i,k \nobreakspace \nobreakspace :(\overline{\sigma }_{i,k}) \end{align}$$1g∑t=k−tiDn−1kwi,t≥1−ui,k∀i,k:(σ̲i,k)$$\begin{align} & \sum ^k_{t=k-t^{Dn}_i-1}{w_{i,t}\ge 1-u_{i,k}\ } \quad \forall i,k \nobreakspace \nobreakspace :(\underline{\sigma }_{i,k}) \end{align}$$1hzϱ,k=∑iΓϱ,ipi,k−∑nΨϱ,nDn,k∀ϱ,k:(δϱ,k)$$\begin{align} &\hspace*{-12pt} z_{\varrho ,k} = \nonumber \\ &\sum _{i} \Gamma _{\varrho ,i}p_{i,k} - \sum _{n} \Psi _{\varrho ,n}D_{n,k} \quad \forall \varrho ,k \nobreakspace \nobreakspace :(\delta _{\varrho ,k})\end{align}$$1i−Zϱ≤zϱ,k≤Zϱ∀ϱ,k:δ̲ϱ,k,δ¯ϱ,k$$\begin{align} & -Z_{\varrho } \le z_{\varrho ,k} \le Z_{\varrho } \quad \forall \varrho ,k \nobreakspace \nobreakspace :\left(\underline{\delta }_{\varrho ,k} , \overline{\delta }_{\varrho ,k}\right) \end{align}$$1jmj,k=∑i≠jMiui,k∀k:(λ̲j,k)$$\begin{align} & m_{j,k} = \sum _{i \ne j} M_i u_{i,k} \quad \forall k \nobreakspace \nobreakspace :(\underline{\lambda }_{j,k}) \end{align}$$1kPjri,k−2Yimj,kF0−FNadir−Fdb≤0∀j,k:(πj,k)$$\begin{align} & P_j r_{i,k} - 2Y_{i} m_{j,k} {\left(F_0 - F_{Nadir} - F_{db} \right)} \le 0 \quad \forall j,k \nobreakspace \nobreakspace :(\pi _{j,k}) \end{align}$$1lmj,kdfkdt=−BΔfk+Pm−Pe∀k$$\begin{align} & m_{j,k} \frac{df_k}{dt} =- B {\Delta f_k} + {\left(P_{m} - P_{e} \right)} \quad \forall k \end{align}$$1mF̲≤dfkdt≤F¯$$\begin{align} & \underline{F} \le \frac{df_k}{dt} \le \bar{F} \end{align}$$1npi,k,ri,k,fk∈R+∀i,k$$\begin{align} & p_{i,k}, r_{i,k}, f_k \in \mathbb {R}_+ \quad \forall i,k \end{align}$$1ovi,k,wi,k,ui,k∈{0,1}∀i,k,$$\begin{align} & v_{i,k}, w_{i,k}, u_{i,k} \in \lbrace 0,1\rbrace \quad \forall i,k, \end{align}$$where the Ω1=$\Omega _1 =$ {pi,k$\lbrace p_{i,k}$, ri,k$r_{i,k}$, vi,k$v_{i,k}$, wi,k$w_{i,k}$, ui,k$u_{i,k}$, zϱ,k$z_{\varrho ,k}$, mj,k$m_{j,k}$, yj,k$y_{j,k}$, fk}$f_k\rbrace$ is the set of decision variables. OFPP$OF^{PP}$ stands for the primal objective function. pi,k$p_{i,k}$ and ri,k$r_{i,k}$, respectively, represent generation level and reserve and vi,k$v_{i,k}$, wi,k$w_{i,k}$ and ui,k$u_{i,k}$ indicate start‐up, shut‐down and online status for generator i in hour k. zϱ,k$z_{\varrho ,k}$ shows the flow of line ϱ during hour k. mj,k$m_{j,k}$ shows the total inertia available in hour k in absence of the contingent generator j. The CiP$C^P_i$ and CiR$C^R_i$ are marginal cost of providing energy and reserve while Civ$C^v_i$ and Ciw$C^w_i$ model start‐up and shut‐down cost for generator i. Dn,k$D_{n,k}$ represents the demand connected to node n for hour k. Pj$P_{j}$ shows the production capacity of the contingent generator j. The P̲i$\underline{P}_{i}$ and P¯i$\overline{P}_{i}$ are minimum and maximum generation of generator i. tiUp$t^{Up}_i$ and tiDn$t^{Dn}_i$ show minimum up and down time of each generator. B is the load damping factor of the system. Pm−Pe$P_{m}-P_{e}$ is the difference between the mechanical and electrical power after the contingency (in worst case, it is equal to the capacity of the contingent generator j). Also, Yi$Y_i$ is the fastest ramping capability of generator i. The Γϱ,i$\Gamma _{\varrho ,i}$ and Ψϱ,n$\Psi _{\varrho ,n}$ are generation and demand shift factors. Zϱ$Z_{\varrho }$ shows the line flow limit. FNadir$F_{Nadir}$ is the nadir requirement in the network and Fdb$F_{db}$ is the dead‐band of generator governor reaction. The optimization problem (1) is an MINLP with differential equations. Objective function (1a) is sum of energy, reserve, start‐up and shut‐down costs of the system. The energy balance and reserve constraints are modelled in (1b) and (1c), respectively. Constraint (1d) models the limit on the generation level and reserve capacity. The UC logic is presented in (1e)–(1g). Transmission constraints are represented by (1h)–(1i).Constraint (1j) defines mj,k$m_{j,k}$ as the system inertia provided by all rotating masses except the contingent generator j. Constraint (1k) models Nadir requirement [39]. The ROCOF requirement considering load damping factor B is modelled using differential expression (1l) and constraint (1m).The term dfkdt$\frac{df_k}{dt}$ is non‐linear in FA‐UC model (1). This non‐linearity can be dealt with through a simplified formula that can provide the maximum ROCOF as a function of inertia and the size of contingency along with the load damping factor, without having to calculate the actual dfkdt$\frac{df_k}{dt}$. The linearization method is described in what follows.MILP MODEL FOR INTEGRATION OF FA IN UCFrom expression (1l), we have2dfkdt=−BΔfk+Pm−Pemj,k∀j,k.$$\begin{equation} \frac{df_k}{dt} =\frac{- B {\Delta f_{k}} + {\left(P_m - P_e \right)}}{m_{j,k}} \quad \forall j,k. \end{equation}$$Using (2), constraint (1m) can be written equivalently as:3F̲≤−BΔfk+Pm−Pemj,k≤F¯∀j,k.$$\begin{equation} \underline{F}\le \frac{- B {\Delta f_k} + {\left(P_m - P_e \right)}}{m_{j,k}}\le \bar{F} \quad \forall j,k. \end{equation}$$Since mj,k$m_{j,k}$ can only take positive values, we will have the following expressions for ROCOF limit:4aF̲mj,k≤−BΔfk+Pm−Pe∀j,k:(κ̲j,k)$$\begin{align} & \underline{F}m_{j,k} \le -B {\Delta f_k} + {\left(P_m - P_e \right)} \quad \forall j,k :\big(\underline{\kappa }_{j,k}\big) \end{align}$$4b−BΔfk+Pm−Pe≤F¯mj,k∀j,k:(κ¯j,k).$$\begin{align} & - B {\Delta f_k} + {\left(P_m - P_e \right)}\le \bar{F}m_{j,k} \quad \forall j,k:\big(\bar{\kappa }_{j,k}\big). \end{align}$$Now, FA‐UC problem can be formed as an MILP problem as presented in (5).5aMinimizeΩ1OFs(PP)$$\begin{align} &\underset{\Omega _1}{\rm Minimize}\nobreakspace \nobreakspace OF^{(PP)}_s \end{align}$$5bsubjectto(1b)−(1k),(4a)−(4b)$$\begin{align} &\text{subject\nobreakspace to}\nobreakspace (1b)-(1k), (4a)-(4b)\end{align}$$5cpi,k,ri,k,fk∈R+$$\begin{align} & p_{i,k}, r_{i,k}, f_{k} \in \mathbb {R}_+ \end{align}$$5dvi,k,wi,k,ui,k∈{0,1}.$$\begin{align} & v_{i,k}, w_{i,k}, u_{i,k} \in \lbrace 0,1\rbrace. \end{align}$$For each credible contingency, we assume that the difference between electrical and mechanical power (in its worst case) is equal to Pj$P_{j}$ right after the contingency [40]. This MILP problem in (5) explicitly ensures the FA adequacy in its optimal dispatch instructions. This model can be solved using state of the art solvers such as CPLEX [41]. For the purpose of pricing and cost calculation in this paper, MILP model (5) will be used, due to less computational burden.NON‐CONVEX PRICING OF FA ADEQUACYThe FA‐UC model (5) finds the optimum commitment, energy and reserve levels considering FA adequacy constraints. The FA adequacy constraints impose additional costs on the system operator resulting from committing new generators. The new committed generators ensuring FA adequacy may also affect electricity price. The FA adequacy pricing requires non‐convex pricing techniques because of non‐convex MILP model (5). In this section, the FA adequacy pricing with explicit and implicit uplift payments is proposed and discussed.The FA adequacy pricing with explicit modelling of uplift paymentIn this approach, we first solve the MILP model (5) to global optimality. Then, the integer variables of the MILP model are fixed to the levels obtained from the MILP solution (ui,k∗$u^*_{i,k}$, vi,k∗$v^*_{i,k}$, wi,k∗$w^*_{i,k}$). The LP model of —UC formulation (5) is obtained by replacing constraint (1o) with (6).6aui,k=ui,k∗,vi,k=vi,k∗,wi,k=wi,k∗∀i,k,ι$$\begin{align} & u_{i,k}=u^*_{i,k}, v_{i,k}=v^*_{i,k}, w_{i,k}=w^*_{i,k}\quad \forall i,k,\iota \end{align}$$6bui,k,vi,k,wi,k∈R+∀i,k.$$\begin{align} & u_{i,k}, v_{i,k}, w_{i,k} \in \mathbb {R}_+ \quad \forall i,k. \end{align}$$The proposed LP model of (5) and (6) can be used to obtain nodal prices. However, these nodal prices suffer from the fact that they do not necessarily ensure profit adequacy and market equilibrium conditions [42]. Some generators cannot recover their costs by nodal prices from LP model [42, 43]. Hence, some uplift payments are necessary to ensure profit adequacy [42, 44]. Authors in [44] propose different approaches for calculating uplift payment, among which, two approaches are investigated for FA adequacy pricing along with another approach proposed here in this paper.Approach A1: In this approach, a generator with negative profit receives an amount equal to (1+α1)(CiPpi,k+CiRri,k)+Civvi,k+Ciwwi,k−ζn,kΛi,npi,k$ (1+ \alpha _1) (C^P_i p_{i,k}+C^R_i r_{i,k}) + C^v_i v_{i,k} + C^w_i w_{i,k} -\zeta _{n,k}\Lambda _{i,n}p_{i,k}$ where α1 is a positive factor to increase the profit of generator in approach A1, ζn,k$\zeta _{n,k}$ is the price at node n and Λi,n$\Lambda _{i,n}$ is the generator‐node mapping matrix. ζn,k$\zeta _{n,k}$ is derived in (7) with Ψϱ,n$\Psi _{\varrho ,n}$ as the power transfer distribution matrix.7ζn,k=λk−∑ϱ(δ̲ϱ,k+δ¯ϱ,k)Ψϱ,n∀n,k.$$\begin{align} & \zeta _{n,k}=\lambda _{k}-\sum _{\varrho } \big(\underline{\delta }_{\varrho ,k}+\overline{\delta }_{\varrho ,k} \big)\Psi _{\varrho ,n} \quad \forall n,k. \end{align}$$Considering the amount of uplift paid, the profit after uplift payment will be α1(CiPpi,k+CiRri,k)$\alpha _1(C^P_i p_{i,k}+C^R_i r_{i,k})$ which is a non‐negative profit for the generator.Approach A2: In the second design, the uplift payment to the generator with negative profit is equal to (1+α2)(CiPpi,k+CiRri,k+Civvi,k+Ciwwi,k−ζn,kΛi,npi,k)$(1+ \alpha _2) (C^P_i p_{i,k}+C^R_i r_{i,k} + C^v_i v_{i,k} + C^w_i w_{i,k} -\zeta _{n,k}\Lambda _{i,n}p_{i,k})$ where 0≤α2≤1$0 \le \alpha _2 \le 1$. Hence, the profit after uplift payment is α2(CiPpi,k+CiRri,k+Civvi,ki+Ciwwi,k−ζn,kΛi,npi,k)$\alpha _2 (C^P_i p_{i,k}+C^R_i r_{i,k} + C^v_i v_{i,k}i + C^w_i w_{i,k} -\zeta _{n,k}\Lambda _{i,n}p_{i,k})$ which is non‐negative, where α2 is a positive factor to increase the profit of generator in approach A2.The FA adequacy pricing with implicit modelling of uplift paymentIn this approach, the MILP model (5) is first converted to an LP model by replacing constraints (1o) with (6b), 0≤ui,k≤1$0 \le u_{i,k} \le 1$, 0≤vi,k≤1$0 \le v_{i,k} \le 1$ and 0≤wi,k≤1$0 \le w_{i,k} \le 1$. The dual objective function of this reformulated LP mode is derived in (8).8OFsDP=∑k(λk∑nDn,k)+∑i,k(μ̲i,kP̲i−μ¯i,kP¯i)+∑i,k(μ¯i,k(u)+μ¯i,k(v)+μ¯i,k(w))+∑ϱ,t(δϱ,k(∑nΨϱ,nDn,k))+∑ϱ,k(δ̲ϱ,kZ̲ϱ+δ¯ϱ,kZ¯ϱ)−∑k(κ̲j,k(F̲mj,k−Pm+Pe))+∑k(κ¯j,k(Pm−Pe−F¯mj,k)).$$\begin{align} OF^{DP}_s = &\,\sum _k {\big (\lambda _{k}\sum _n D_{n,k}\big )} +\sum _{i,k} {\big (\underline{\mu }_{i,k}\underline{P}_{i} -\overline{\mu }_{i,k} \overline{P}_{i}\big )} \nonumber \\ & +\sum _{i,k} {\Big (\bar{\mu }^{(u)}_{i,k}+\bar{\mu }^{(v)}_{i,k}+\bar{\mu }^{(w)}_{i,k}\Big )} +\sum _{\varrho ,t} {\Big(\delta _{\varrho ,k}{\Big( \sum _{n} \Psi _{\varrho ,n}D_{n,k} \Big)}\Big)}\nonumber \\ & +\sum _{\varrho ,k}{\Big (\underline{\delta }_{\varrho ,k}\underline{Z}_{\varrho } +\overline{\delta }_{\varrho ,k}\bar{Z}_{\varrho }\Big )}- \sum _{k}{\Big (\underline{\kappa }_{j,k}{\Big (\underline{F}m_{j,k} - P_m + P_e \Big)} \Big )} \nonumber \\ & +\sum _{k}{\Big (\bar{\kappa }_{j,k}{\big ( P_m - P_e - \bar{F}m_{j,k}\big)}\Big )}. \end{align}$$Selected stationary conditions of reformulated LP model are derived in (9).9a−Pi¯μj,k−Pi¯μ¯i,k−P̲iμ̲i,k+σi,k−σi,k+1+σ̲i,k+σ¯i,k+∑j≠iMjλ̲j,k=0,∀i,k$$\begin{align} & -\bar{P_{i}}\mu _{j,k}-\bar{P_{i}}\bar{\mu }_{i,k}-\underline{P}_{i}\underline{\mu }_{i,k} + \sigma _{i,k}-\sigma _{i,k+1}\nonumber \\ & + \underline{\sigma }_{i,k}+\bar{\sigma }_{i,k} + \sum _{j \ne i} M_j\underline{\lambda }_{j,k} = 0, \quad \forall i,k \end{align}$$9b−Pi¯μj,k−Pi¯μ¯i,k−P̲iμ̲i,k+σi,k+μ¯i,k(u)+μ̲i,k(u)+∑j≠iMjλ̲j,k=0,∀i,k$$\begin{align} & -\bar{P_{i}}\mu _{j,k}-\bar{P_{i}}\bar{\mu }_{i,k}-\underline{P}_{i}\underline{\mu }_{i,k} \nonumber \\ & + \sigma _{i,k} + \bar{\mu }^{(u)}_{i,k}+\underline{\mu }^{(u)}_{i,k} + \sum _{j \ne i} M_j\underline{\lambda }_{j,k} = 0, \quad \forall i,k \end{align}$$9cσi,k+μ¯i,k(v)+μ̲i,k(v)=Civ,∀i,k$$\begin{align} & \sigma _{i,k} + \bar{\mu }^{(v)}_{i,k}+\underline{\mu }^{(v)}_{i,k} = C^{v}_{i}, \quad \forall i,k \end{align}$$9d−σi,k+μ¯i,k(w)+μ̲i,k(w)=Ciw,∀i,k$$\begin{align} & -\sigma _{i,k} + \bar{\mu }^{(w)}_{i,k}+\underline{\mu }^{(w)}_{i,k} = C^{w}_{i}, \quad \forall i,k \end{align}$$9eλk−μ¯i,k−μ̲i,k−∑nΓn,iδn,k=Cip,∀i,k$$\begin{align} & \lambda _{k} - \bar{\mu }_{i,k}-\underline{\mu }_{i,k}-\sum _{n} \Gamma _{n,i}\delta _{n,k} = C^{p}_{i}, \quad \forall i,k \end{align}$$9f∑j≠iμj,k−μ¯i,k≤Cir,∀i,n,k$$\begin{align} & \sum _{j \ne i} \mu _{j,k} - \bar{\mu }_{i,k} \le C^{r}_{i}, \quad \forall i,n,k \end{align}$$9gδn,k+δ¯n,k+δ̲n,k=0,∀n,k$$\begin{align} & \delta _{n,k} + \bar{\delta }_{n,k} + \underline{\delta }_{n,k} = 0, \quad \forall n,k \end{align}$$9hλ̲j,k+F¯κ¯j,k+F̲κ̲j,k−∑i≠j2Yi(f0−FNadir−Fdb)π(j,k)=0$$\begin{align} &\hspace*{-12pt}{\underline{\lambda }}_{j,k}+\bar{F}{\bar{\kappa }}_{j,k}+\underline{F}\underline{\kappa }_{j,k}\nonumber \\ &-\sum _{i \ne j}2Y_i(f_{0}-F_{Nadir}-F_{db})\pi (j,k)=0 \end{align}$$9i−Bκ̲j,k−Bκ¯j,k≤0$$\begin{align} & -B{\underline{\kappa }}_{j,k}-B{\bar{\kappa }}_{j,k}\le 0 \end{align}$$9jκ¯j,k,κ̲j,k,μ¯i,k,μ̲i,k,λ̲j,k∈R+,δ¯n,k,δn,k,σi,k,πj,k,λk,λ¯j,k∈R+.$$\begin{align} & \overline{\kappa }_{j,k}, \underline{\kappa }_{j,k},\overline{\mu }_{i,k}, \underline{\mu }_{i,k}, \underline{\lambda }_{j,k}\in \mathbb {R}_+ , \nonumber \\ & \overline{\delta }_{n,k}, \delta _{n,k} , \sigma _{i,k}, \pi _{j,k}, \lambda _{k}, \overline{\lambda }_{j,k} \in \mathbb {R}_+. \end{align}$$Here, μ¯i,k(w)$\bar{\mu }^{(w)}_{i,k}$ and μ̲i,k(w)$\underline{\mu }^{(w)}_{i,k}$ are the Lagrange multipliers of the constraints 0≤wi,k≤$0\le w_{i,k}\le$ 1 which allow wi,k$w_{i,k}$ to take values between zero and one. μ¯i,k(v)$\bar{\mu }^{(v)}_{i,k}$ and μ̲i,k(v)$\underline{\mu }^{(v)}_{i,k}$ are the Lagrange multipliers of the constraints 0≤vi,k≤1$0\le v_{i,k}\le 1$, and μ¯i,k(u)$\bar{\mu }^{(u)}_{i,k}$ and μ̲i,k(u)$\underline{\mu }^{(u)}_{i,k}$ as the Lagrange multipliers for constraints 0≤ui,k≤1$0\le u_{i,k}\le 1$.Using the nodal prices ζn,k$\zeta _{n,k}$ derived in (7), the non‐negativity profit can be ensured by constraint (10).10∑kζk,nΛn,ipi,k−(CiPpi,k+Civvi,k+CiRri,k)≥0∀i,n.$$\begin{align} & \sum _{k} {\left(\zeta _{k,n}\Lambda _{n,i}p_{i,k} - (C^P_i p_{i,k} + C^v_i v_{i,k}+C^R_i r_{i,k})\right)}\nonumber \\ & \ge 0 \nobreakspace \nobreakspace \quad \forall i,n. \end{align}$$The bilinear term ζk,npi,k$\zeta _{k,n}p_{i,k}$ in (10) are linearized in (11) using discrete approximation technique.11a∑ρP̂ρ,ibi,ρ,k≤pi,k∀i,k$$\begin{align} &\sum _{\rho } \hat{P}_{\rho ,i}b_{i,\rho ,k} \le p_{i,k} \quad \forall i,k \end{align}$$11b∑ρP̂ρ,ibi,ρ,k≥pi,k−Pi¯Υ∀i,k$$\begin{align} &\sum _{\rho } \hat{P}_{\rho ,i}b_{i,\rho ,k} \ge p_{i,k}-\frac{\bar{P_{i}}}{\Upsilon } \quad \forall i,k \end{align}$$11c∑ρbi,ρ,k=1∀i,k$$\begin{align} &\sum _{\rho } b_{i,\rho ,k}=1 \quad \forall i,k \end{align}$$11dei,n,ρ,k≤X2Λn,i∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \le X_2\Lambda _{n,i}\quad \forall i,k,\rho , n \end{align}$$11eei,n,ρ,k≥−X2Λn,i∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \ge -X_2\Lambda _{n,i}\quad \forall i,k,\rho , n \end{align}$$11fei,n,ρ,k≤X2bi,ρ,k∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \le X_2b_{i,\rho ,k} \quad \forall i,k,\rho , n \end{align}$$11gei,n,ρ,k≥−X2bi,ρ,k∀i,k,ρ,n$$\begin{align} & e_{i,n,\rho ,k} \ge -X_2b_{i,\rho ,k} \quad \forall i,k,\rho , n \end{align}$$11hζk,n−ei,n,ρ,k≥−X2(1−bi,ρ,k)∀i,k,ρ$$\begin{align} &\zeta _{k,n}-e_{i,n,\rho ,k} \ge -X_2(1-b_{i,\rho ,k}) \quad \forall i,k,\rho \end{align}$$11iζk,n−ei,n,ρ,k≤X2(1−bi,ρ,k)∀i,k,ρ$$\begin{align} &\zeta _{k,n}-e_{i,n,\rho ,k} \le X_2(1-b_{i,\rho ,k}) \quad \forall i,k,\rho \end{align}$$11j∑k,ρ,nei,n,ρ,kP̂i,ρ−(CiPpi,k+Civvi,k)≥0∀i,$$\begin{align} &\sum _{k,\rho ,n} {\left(e_{i,n,\rho ,k}\hat{P}_{i,\rho } - (C^P_i p_{i,k} + C^v_i v_{i,k})\right)} \ge 0 \quad \forall i, \end{align}$$where P̂ρ,i$\hat{P}_{\rho ,i}$ is the discrete approximation of generation and bi,ρ,k$b_{i,\rho ,k}$ is associated binary variable. X2 is a suitably large constant. ei,n,ρ,k$e_{i,n,\rho ,k}$ shows the components of the generator‐node incidence matrix. Also, the factor 1Υ$\frac{1}{\Upsilon }$ is for dividing units' capacity to Υ discrete capacities. Obviously, smaller divisions result in more accurate approximation but it requires more binary variables and increases computational time.Now we form the MILP optimization problem (12) where the integrality constraints are enforced.12aMinimizeΩ2OFPP−OFDP$$\begin{align} &\underset{\Omega _2}{\text{Minimize}}\nobreakspace \nobreakspace OF^{PP} - OF^{DP} \end{align}$$12bsubjectto(1b)−(1k),(4a)−(4b),(SCs),(11)$$\begin{align} &\text{subject\nobreakspace to}\nobreakspace (1b)-(1k),(4a)-(4b), \nonumber \\ &(SCs), (11)\end{align}$$12cpi,k,ri,k,fk∈R+∀i,k$$\begin{align} & p_{i,k}, r_{i,k}, f_k \in \mathbb {R}_+ \quad \forall i,k \end{align}$$12dvi,k,wi,k,ui,k∈{0,1}∀i,k$$\begin{align} & v_{i,k}, w_{i,k}, u_{i,k} \in \lbrace 0,1\rbrace \quad \forall i,k \end{align}$$12eκ¯j,k,κ̲j,k,μ¯i,k,μ̲i,k,λ̲j,k∈R+,δ¯n,k,δn,k,σi,k,πj,k,λk,λ¯j,k∈R+,$$\begin{align} & \overline{\kappa }_{j,k}, \underline{\kappa }_{j,k},\overline{\mu }_{i,k}, \underline{\mu }_{i,k}, \underline{\lambda }_{j,k}\in \mathbb {R}_+ , \nonumber \\ & \overline{\delta }_{n,k}, \delta _{n,k} , \sigma _{i,k}, \pi _{j,k}, \lambda _{k}, \overline{\lambda }_{j,k} \in \mathbb {R}_+, \end{align}$$where SCs stands for stationary conditions. Ω2={pi,k$\Omega _2 = \lbrace p_{i,k}$, ri,k$r_{i,k}$, vi,k$v_{i,k}$, wi,k$w_{i,k}$, ui,k$u_{i,k}$, zϱ,k,κ¯j,k,κ̲j,k,μ¯i,k,μ̲i,k,λk,λ¯j,k,λ̲j,k,δ¯n,k,δn,kσi,k,πj,k}$z_{\varrho ,k},\overline{\kappa }_{j,k}, \underline{\kappa }_{j,k},\overline{\mu }_{i,k}, \underline{\mu }_{i,k}, \lambda _{k}, \overline{\lambda }_{j,k}, \underline{\lambda }_{j,k},\overline{\delta }_{n,k}, \delta _{n,k}\, \sigma _{i,k}, \pi _{j,k} \rbrace$ is the set of decision variables.It is also noteworthy to mention that the mentioned methods for price formation are discussed in literature and investigation of the mathematical basis of the methods is out of the scope of this paper. However, further details can be found in previous works such as [37, 38].ILLUSTRATIVE CASE STUDYTo illustrate the results of proposed —UC model and the non‐convex pricing techniques in Section 4, the five‐node example system in Figure 1 is carefully analysed and discussed.1FIGUREThe single‐line diagram of the five‐node systemThe basic data is presented in Table 1 and the detailed data is provided in [45]. The load damping factor is assumed to be 0.85 [46]. Furthermore, Nadir requirement is set to be 49.4 Hz and maximum ROCOF to be 0.5 Hz/S. The following models are used and discussed:FA‐UC‐A1(‐A2 or ‐B) model which refers to standard UC model with FA adequacy constraints. The non‐convex pricing approach A1 (A2 or B) is used.UC‐A1(‐A2 or ‐B) model which is standard UC model without FA adequacy constraints. The non‐convex pricing approach A1 (A2 or B) is used here as well.All models are coded in GAMS platform and solved using the CPLEX solver [41]. It is also assumed that the lack of coherency in the frequency of different buses is negligible [40, 47]. We have also benchmarked our results with the case where FA adequacy constraints are not modelled (UC‐A1, UC‐A2 and UC‐B models).1TABLEThe generation data of the five‐node systemUnitSRMC ($/MWh)SUC ($)IC (MWSec/Hz)MC (MW)G125000140G210700170170G3301000350260G4401000450200G525800240300Abbreviations: SRMC, short run marginal cost; SUC, start‐up cost, IC, inertia constant; MC, maximum capacity.As can be noted in the Table 1, G1 is providing no inertia and also has a low marginal cost. This generator is modelling a typical renewable energy source (e.g. wind) here. It is to be noted that the price and inertia are the main factors which affect our study here. The optimal dispatch instructions under UC and FA‐UC models are reported in Table 2.2TABLEGeneration levels (MW) in both UC and FA‐UC models for the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6G1140140140140140140G267.5112.9140.1170143.8127G3000000G4000000G50023.035.218.70As can be seen in Table 2, both UC and FA‐UC models have led to same commitment and generations levels. Generators G3 and G4 produce 0 MWh in both cases. This is while G1 is dispatched at 140 MW in all hours because of having lowest marginal cost. Generators G2 and G5 also contribute in providing energy in some hours of the dispatch period. The nodal prices for the 6‐h period using FA‐UC‐A1 and FA‐UC‐A2 models are presented in Table 3. The nodal prices have also been depicted in Figure 3. The height and colour of the bars are showing the prices in different nodes and hours.3TABLENodal prices ($/MWh) from FA‐UC‐A1, FA‐UC‐A2, UC‐A1 and UC‐A2 models in the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6Node 1101010101010Node 2101017.118.717.010Node 3101016.318.416.110Node 4101016.318.416.110Node 51010252525102FIGUREImpact of FA adequacy constraints on payment to generators in five‐node system3FIGURENodal prices ($/MWH) from FA‐UC‐A1, FA‐UC‐A2, UC‐A1 and UC‐A2 models in the 5‐node systemAs shown in Table 3, the nodal prices for UC and FA‐UC models are the same. This is because the generator G3 (which is just committed for ensuring FA adequacy) does not provide energy (see Table 2). This means nodal prices under approaches A1 and A2 in FA‐UC model do not reflect the extra cost of FA adequacy constraints. The profit of generators using these nodal prices for FA‐UC‐A1 model is set out in Table 4.4TABLEProfits ($) before uplift paymentsModelG1G2G3G4G5UC6720−70000−800FA‐UC6720−700−10000−800As in Table 4, generators G2 and G5 experience negative profits in UC, while in FA‐UC, generators G2, G3 and G5 are all experiencing negative profits. An interesting observation is that, in this case, the only generator with positive profit is G1, which does not contribute to FA adequacy.For calculating uplift payments under approaches A1 and A2, the α1 and α2 are, respectively, assumed to be 0.05 and 0.1. The uplift payments for these generators are reported in Table 5.5TABLEUplift payment ($) from FA‐UC‐A1 and FA‐UC‐A2 models in the five‐node systemModelG1G2G3G4G5UC‐A10773.500859.3UC‐A2077000880FA‐UC‐A10753.71032.50850.8FA‐UC‐A2077011000880The FA adequacy constraints have a clear impact on uplift payments (while it did not have any impact on nodal prices). Generator G2 receives $753.7 uplift payment in FA‐UC model and approach A1 (FA‐UC‐A1). This is while its uplift payment without )s (UC‐A1) is $773.5 (a decrease of 753.7−773.5773.5=−2.5%$\frac{753.7-773.5}{773.5}=-2.5 \%$). Also, when the FA adequacy constraints are considered, G3 needs to receive uplift payment of $1032.5, while it was not being paid any uplift in UC‐A1. This is because this generator is only committed to ensure the FA adequacy constraints and does not generate so much energy to cover its start‐up costs. Generator G5 receives slightly less uplift payment when the FA adequacy constraints are modelled ($859.3 as compared to $850.8 in FA‐UC‐A1 and UC‐A1 cases). For approach A2, the FA adequacy constraints do not change the uplift payment for generators G2 and G5. Generator G2 receives $770 and generator G5 receives $880 uplift payment with (FA‐UC‐A2) and without (UC‐A2) FA adequacy constraints. Generator G3 receives $1100 which is slightly higher than method A1. No uplift payment is received by G1 in these cases. Table 6 shows raw profit (profit without any extra uplift payment) and also uplift payment with and without FA adequacy constraints.6TABLERaw profit, uplift payment and profit (= raw profit + uplift payment) of all generators over different hours in five‐node systemPayment approachModelRaw profit ($)Uplift payment ($)Profit ($)A1UC52201632.86852.8FA‐UC422026376857Difference−1000 (−19.2%)1005 (62%)5 (0.1%)A2UC522016506870FA‐UC422027506970Difference−1000 (−19.2%)1100 (67%)100 (1.5%)As we can see from Table 6, raw profit is decreased while uplift payment and profit are increased when the FA adequacy constraints are explicitly modelled (−19.2%$-19.2 \%$ for raw profit, +62%$+62 \%$ for uplift payment and +0.1%$+0.1 \%$ for profit). The same pattern is observed under approach A2. The raw profit, uplift payment and profit are changed by −19.2%$-19.2 \%$, +67%$+67 \%$ and +1.5%$+1.5 \%$, respectively, when the FA adequacy constraints are considered. Approach B is also applied to the illustrative five‐node system. The dispatch levels of generators are provided in Table 7. The generation levels under FA‐UC‐B model are presented in Table 8.7TABLEGeneration levels (MW) in UC‐B model for the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6G1109140140140140140G298.5112.974.0138.5130.5116.9G3000000G4000000G50081.135.713.110.08TABLEGeneration levels (MW) in FAUC‐B model for the five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6G179.2105.5140140140140G260.12095.189.983.171G30117.420.322.923.916.1G4000000G568.210.039.753.444.639.9Furthermore, the nodal prices using the UC‐B and FA‐UC‐B models are reported in Tables 9 and 10. In order to illustrate the differences better, the results have been also plotted in Figures 4 and 5. As can be observed from the height and colour of the bars, the prices obtained from FA‐UC are generally higher than the prices from UC. This is because of the fact that in approach B, the differences in payments are reflected in prices. The uplift payments have been implicitly included in these nodal prices such that the non‐negative profitability is ensured. This is in contrast to the explicit approaches (A1 and A2), where there are no higher prices to ensure inertia adequacy but the explicit out‐of‐market uplift payments are added.9TABLENodal prices ($/MWh) using UC‐B model in five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6Node 117.110.010.010.010.010.0Node 213.710.214.518.116.310.0Node 313.310.215.918.417.916.2Node 414.110.016.318.418.416.3Node 513.310.734.925.025.025.010TABLENodal prices ($/MWh) using FA‐UC‐B model in five‐node systemHour 1Hour 2Hour 3Hour 4Hour 5Hour 6Node 121.81010101010Node 221.116.317.018.619.422.6Node 319.938.630.030.030.030.0Node 419.415.422.323.623.721.6Node 536.925.025.025.025.025.04FIGURENodal prices ($/MWH) using UC‐B model in 5‐node system5FIGURENodal prices ($/MWH) using FA UC‐B model in 5‐node systemComparing Table 9 with Table 10, the impact of FA adequacy constraints on nodal prices can be seen. The nodes and hours which experience different prices because of FA adequacy constraints are highlighted in Tables 9 and 10. The total profit of generators using UC‐B and FA‐UC‐B models are presented in Table 11.11TABLEGenerators' profits ($) over different hours in five‐node systemModelG1G2G3G4G5UC‐B7091.10000FA‐UC‐B6710.90000Difference−380.2 (−5.2%)0000Figure 2 compares raw profit, uplift payment and profit of generators under different investigated models. The difference between bars for different approaches is caused by the FA adequacy constraints.NUMERICAL RESULTSFor further discussions, the Nordic 44‐node system is studied. All simulations are carried out on a computer with 8 GB of RAM and 2.6 GHz CPU. The GAMS platform and the CPLEX solver are used.Regarding the computational time, we should note that there is a trade‐off between computational time and employing detailed formulation to assess FA adequacy. It is clear that a more detailed formulation of FA such as the one in [32] can lead to more precise results but at the cost of higher computational complexity. However, improving the current formulation in our paper using more detailed models such as those in [32] is a good extension of the current paper. Besides the linearization techniques proposed in this paper, the computational complexity of our proposed modified UC model can be handled through decomposition algorithms such as Benders decomposition or Lagrangian Relaxation algorithms (which are common technique for dealing with computational complexity of UC models [48] and [49]).The GAMS platform and the CPLEX solver are used. In order to consider the worst‐case scenario for FA adequacy, we assume load damping factor to be zero. We also assume no transmission congestion for our case studies. The intertemporal system prices for a period of 12 h from FA‐UC‐A1 and FA‐UC‐A2 models are reported in Table 12. The benchmark prices from UC‐A1 and UC‐A2 are also reported. For hours 1–8, the system prices with and without FA adequacy constraints are the same. However, for hours 9–12, the FA adequacy constraints result in an increase in intertemporal system prices.12TABLESystem prices ($/MWh) for the Nordic 44‐node system from FA‐UC and UC modelsHourUCFA‐UCHourUCFA‐UC1–715415410154205815415411154205917620512154176The prices reported in Table 12 result in negative profit for several generators. The explicit and implicit approaches for non‐convex pricing are applied and the results for total profits are reported in Table 13.13TABLERaw profit, uplift payment and profit (= raw profit + uplift payment) under explicit and implicit non‐convex pricingPayment approachModelRaw profit ($)Uplift ($)Total profits ($)A1UC371,01067,950438,960FA‐UC369,430123,700493,130Difference−158055,75054,170 (13%)A2UC371,01090,189462,450FA‐UC369,430146,435515,865Difference−158056,24653,415 (11.6%)BUC‐‐567,420FA‐UC‐‐1,316,021Difference‐‐748,601(132%)Under approach A1, the FA adequacy constraints decrease the total raw profit of all generators by $1580. The uplift payment is increased by $55, 750 (=123,700−67,950$=123,700-67,950$) and this results in 13% raise in the profit. Similar situation is observed in case A2 with the increase of 11.6% in profit of generators in case of FA‐UC‐A2 model as compared to UC‐A2 model. Under implicit non‐convex pricing approach B, the total profit is changed from $567, 420 in UC‐B model to $1316, 021 in FA‐UC‐B model (an increase of 132%). This significant raise in profit mainly comes from the fact that the generators with low marginal cost of energy, are being paid by system prices which are adjusted to cover the start‐up cost of FA providers.We have also reported the intertemporal system prices from FA‐UC‐B model. In hours 1, 3 and 8–12, the system prices with FA adequacy constraints are higher than those without FA adequacy constraints. In hour 2, the price with FA adequacy constraint is lower than the one without this adequacy constraint. In the rest of hours, the prices are the same with and without FA adequacy constraints.As in Table 14, we see a considerable difference in system prices in hours 1 and 9. This is due to the need to compensate the start‐up cost of extra units committed to ensure FA adequacy. Figure 6 compares the profits for the Nordic 44‐node system under explicit and implicit non‐convex pricing approaches. Figure 6 shows the difference in total payments. It can be seen that the two explicit uplift payment methods (A1 and A2) result in lower amount for total payments to generators, and as a result, lower cost for the operator to provide the energy, especially when considering FA adequacy concerns. However, the advantage of implicit method B over A1 and A2 is that it does not require out‐of‐market payments and this means the entire cost is reflected in prices.14TABLEIntertemporal system prices ($/MWh) from UC‐B and FA‐UC‐B modelsHourUCFA‐UCHourUCFA‐UC1169.3317.071541542163.6155.28154.3155.73154156.39154343.5415415410154.4206.6515415411154.5205.0615415412155.3205.06FIGUREImpact of FA adequacy constraints on payments to generators in Nordic 44‐node systemFURTHER DISCUSSIONOne of the points that can be a subject of discussion is that, what is the reason and motivation behind building up a complex formulation, as the one discussed here, to ensure FA adequacy, while one way to ensure FA adequacy currently is to consider the worst‐case scenario and commit plenty of synchronous units to provide adequate inertial response. The answer is, with a high rate of renewable integration, there might not be necessarily plenty inertia providers available among the committed units [50]. This important limitation has overlooked as inertia and ramping capability has been adequate up to now. There are also reports such as [51] and [52] which state that in a number of power systems such as Nordic system, the lack of frequency response contribution will be more critical in future. Accordingly, in a situation where there is a considerable share of renewable non‐synchronous generation, in order to calculate the FA cost properly and economically, inertia and ramping constraints need to be integrated in UC model and the cost of providing FA should be minimized along with the cost of providing energy. This modelling needs the dynamic equations of FA as we proposed in our paper. In other words, the proper modelling of FA which considers both inertia and ramping constraints adds the level of complexity proposed in our study to the standard UC formulation.In such formulation, due to the need for including non‐linear algebraic and differential equations as constraints, there will be added computational complexity which is unavoidable. Also, a number of other studies that consider FA constraints have a similar level of complexity, as previously reviewed in this paper [25, 26, 32, 40].However, we have tried to reduce the computational complexity as much as possible by linearization of the formulation (improving the computational tractability of our proposed UC formulation by proposing an MILP model of its original MINLP model). Further reduction of computational complexity is out of the scope of this study but it can be a potential for future work and has been added in the conclusion and future work in Section 8.Also, on the importance and motivation behind building up a formulation that integrates dynamic equations inside UC optimization problem, we should state that since the total inertia available in the network depends on the online units, the optimal way to ensure FA adequacy is to integrate FA constraints in the energy optimization (UC) problem to make sure that we can satisfy FA requirements along with other technical constraints by commitment decisions. This approach avoids the suboptimality that may occur in sequential approach as discussed in references [25] and [26]. Our proposed joint optimization needs proper modelling of FA provision cost which in turn involves the dynamic equations (as we proposed in our paper) in the standard UC formulation.One can determine by a Digsilent/PSSE/PowerWorld simulation a set of generators to maintain adequacy, but that set of generators, in general, is a feasible but not necessarily optimal solution to the UC model. In other words, if we solve the FA adequacy problem based on worst‐case scenario approach and UC problem separately, we obtain suboptimum solution as compared to the case where we do joint optimization. More advantages of joint energy and ancillary service markets over the separate designs are discussed in [33–36].CONCLUSIONSThis paper proposes an MILP model for ensuring the FA adequacy constraints. First, Nadir and ROCOF measures are modelled through a set of bilinear constraints and differential equations and are included in standard UC formulation. The resulting model is a mixed‐integer bilinear program with differential equations. Through a series of linearization techniques, an MILP model is proposed. Due to the non‐convexity of proposed MILP model, some generators providing energy or inertia service might experience negative profit. To tackle the negative profit, the explicit and implicit non‐convex pricing techniques are proposed and implemented using our proposed MILP model. In the explicit pricing approaches, the generators with negative profit receive uplift payment. Under the implicit non‐convex pricing approach, all payments are embedded in the calculated nodal prices. The developed MILP model of FA‐UC and all approaches for explicit and implicit non‐convex pricing are applied to an illustrative five‐node system and Nordic 44‐node system and the results are carefully studied.Devising allocation mechanisms for distributing the FA adequacy costs among generators can be a good extension of this work. Furthermore, our proposed model can be improved by more detailed modelling of FA adequacy constraint and changing the assumption of frequency coherency used in our paper.Also, reduction of computational complexity of the proposed modified UC model can be addressed through decomposition algorithms such as Benders decomposition or Lagrangian decomposition (which are common techniques for dealing with computational complexity of UC models [49] and [48]).In addition, a sensitivity analysis on the effect of load damping factor on the overall system frequency response can be a subject for further study.AUTHOR CONTRIBUTIONSEhsan Davari Nejad: Conceptualization, data curation, investigation, methodology, visualization, writing ‐ original draft, writing ‐ review and editing. Mohammad Hesamzadeh: Conceptualization, methodology, resources, software, supervision, writing ‐ original draft, writing ‐ review and editing. 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"IET Generation, Transmission & Distribution"Wiley

Published: Dec 1, 2022

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