Model predictive control of electric power and reserve dynamic dispatch including demand response

Abstract

In this paper we formulate a dynamic economic emission and spinning reserve dispatch (DEESRD) problem which determines the optimal power and spinning reserve schedule by simultaneously minimizing the power and spinning reserve costs, and the amount of emission under some constraints. Demand response (DR) can improve the reliability and reduce the energy price. In this paper, we focus on Game Theory DR program which is one of the incentive-based DR programs. We incorporate DR into the DEESRD problem by formulating dynamic economic emission and spinning reserve dispatch with demand response (DEESRD-DR) problem. The objective of the DEESRD-DR is to minimize the energy and reserve costs, minimize the amount of emission and maximize the benefit of the generation company (GENCO). The optimal solutions of the DEESRD and DEESRD-DR problems can be obtained using artificial intelligent-based optimization techniques such as differential evolution, particle swarm optimization, artificial immune system, artificial bee colony, etc., however these methods give an open-loop optimal solution. The open-loop nature cannot deals with inaccuracies, modeling uncertainties and unexpected external disturbances where the power system components suffer from, therefore we have designed closed-loop solutions by model predictive control (MPC). The performance of the MPC has been investigated by applying the MPC strategy to the DEESRD and DEESRD-DR problems with test system consisting of five generating units and five customers.

Keywords

Computational intelligent technique dynamic dispatch emission optimization feedback control spinning reserve demand response

1 Introduction

Dynamic economic dispatch (DED) problem has received considerable attention during the recent years. The objective of the DED problem is to satisfy the predicted load customer’s demand over a certain period (e.g. 24 hours) at minimum generation cost taking into consideration the ramp rate limits of the thermal generating units [1 –14]. Several researchers have devoted their efforts to propose optimization methods and techniques for solving the DED problem with different objectives and constraints [15 –32]. To reduce the emission of gaseous pollutants produced from thermal units, dynamic economic emission dispatch (DEED) has been formulated with the purpose to minimize both fuel cost and the amount of emission and satisfy the system constraints (see e.g. [33 –47]).

In regulated and deregulated power systems, spinning reserve plays an important role in maintaining the power system reliability and security against sudden load changes and generation outages. In regulated systems, the spinning reserve is taken in the static or dynamic dispatch problems as a constraint. In contrast, in deregulated environment it is important to find the optimal spinning reserve requirement. In this case, both the energy and spinning reserve costs are minimized under a set of constraints. In [48 –53], the spinning reserve constraint is included into the static and dynamic dispatch problems. Joint generation and spinning reserve dispatch has been studied in [54 –59], where the spinning reserve is inserted in the objective function as another optimization variable in addition to the power output variable. To incorporate the spinning reserve into the DEED problem, dynamic economic emission and spinning reserve dispatch (DEESRD) can be formulated. DEESRD is a multi-objective optimization problem which simultaneously minimize the energy and reserve costs and the amount of emission while satisfying the load demand balance form both power and spinning reserve, ramp rate constraints and other constraints.

We note that, the DEESRD problem has been formulated in view point of the generation side. In this case, electricity supplier must satisfy the demand in real time. When the demand is increased during the peak, a power plant is required to increase their power generation to meet rising demand. GENCOs and utilities face some difficulties to satisfy the load demand during the peak hours. One of the easier and cheaper option is to apply demand response (DR) programs. In this case, instead of increase the generation to satisfy the demand, GENCOs and utilities pay energy users to reduce their consumption during the peak. There are two categories for DR programs [60].

Incentive-based programs as: Interruptible/Curtailable; Direct Load Control; Emergency DR; Demand Bidding/Buyback; Ancillary Services; Game Theory DR; Capacity Market;

Price based programs as: Time-of-Use; Critical Peak Pricing; Real-Time Pricing.

The main target from applying DR programs is to increase the power system efficiency and this give benefits for customers, reliability, whole market and market performance.

Recently, DR has been incorporated in the dynamic dispatch problem in [61 –63] by formulating dynamic economic dispatch with demand response (DED-DR) and dynamic economic emission dispatch with demand response (DEED-DR). In [61] and [62], the spinning reserve has not been considered, while in [63], the spinning reserve is added to the DED as a constraint. Therefore, dynamic economic emission and spinning reserve dispatch with demand response (DEESRD-DR) has not been formulated before. In this paper, we have considered Game Theory DR program and then formulated the dynamic economic emission and spinning reserve dispatch with demand response (DEESRD-DR). The objective of the DEESRD-DR is to minimize the energy and reserve costs, minimize the amount of gases emission and maximize the benefit of the customers and GENCOs. Both DEESRD and DEESRD-DR problems provide open-loop solutions. The open-loop nature cannot deals with inaccuracies, modeling uncertainties and unexpected external disturbances where the power system components suffer from. A good solution of such deficiency is to design a feedback control strategy using model predictive control (MPC) method. MPC provides closed-loop solutions which has the ability to deal with disturbances that arise from real systems. MPC has been applied in power system in [64 –70] and [72]. In [66, 67], the MPC has been applied for the optimal dynamic dispatch problems without taking into consideration the DR and spinning reserve. In [70], the MPC has been applied to the DEED-RD problem, however the spinning reserve has been neglected.

In this paper, we first formulate the DEESRD and DEESRD-DR problems, then we construct a feedback control by using the MPC strategy. The performance of MPC algorithm including convergence and robustness have been shown and the controller has been tested with test system consisting of five units and five customers.

2 Problem formulation

We devote this section to introduce the mathematical formulations of the DEESRD, DR and DEESRD-DR problems.

2.1 DEESRD formulation

In this subsection we formulate the DEESRD problem as a multi-objective optimization which simultaneously minimize the energy and reserve costs and the amount of emission so as to meet the predicted power and spinning reserve load demands over a certain period under ramp rate limits and other constraints. Let $p_{i}^{t}$ and $s_{i}^{t}$ be the power output and spinning reserve contribution of unit i at time t, respectively. The cost and emission functions of unit are given, respectively, as [2, 43, 44, 66]: $C_{i} (p_{i}^{t}) = a_{i} + b_{i} p_{i}^{t} + c_{i} (p_{i}^{t})^{2},$

$E_{i} (P_{i}^{t}) = α_{i} + β_{i} p_{i}^{t} + γ_{i} (p_{i}^{t})^{2},$ where, a_i, b_i, c_i, α_i, β_i and γ_i are constants. Other forms for the fuel cost and emission functions are given in [2]. Let us define the following: N_G and N_T be the number of committed generating units and number of intervals in the dispatch period, respectively; $P^{t} = {p_{1}^{t}, . . ., p_{N_{G}}^{t}}$ , $S^{t} = {s_{1}^{t}, . . ., s_{N_{G}}^{t}}$ , PS^t = (P^t, S^t) , PS = (PS¹, PS²,. . . , PS^{N
_T}); r is forecasted probability that the reserve is actually called up; B_ij is ijth element of the transmission line loss coefficient matrix; UL_i and DL_i are the maximum ramp rate of unit i; D^t and ${SR}_{D}^{t}$ are the power and spinning reserve demands at time t; $p_{i}^{min}, p_{i}^{max}$ are the limits of power capacity of unit i. We assume that the power and spinning reserve demands are periodic with period N_T, then D^t = D^t+N_T, SR^t = SR^t+N_T.

There are two methods for solving the multi-objective problem. The first method is by finding the Pareto solutions and the second one is by combining the objectives into a single-objective function. The advantages of the second method include (i) it makes the multi-objective problem easy to solve; (ii) it gives the decision maker the ability to change the weighting factor according to the importance of each objective. The second method will be used in this paper. Let ω ∈ [0, 1] be a weighting factor, and $h_{i}^{t}$ be the price penalty factor at time t which blends the emission cost with the normal fuel cost. Let us define ${CE}_{i} (p_{i}^{t}) = ω C_{i} (p_{i}^{t}) + (1 - ω) h_{i}^{t} E_{i} (p_{i}^{t})$ .

The price penalty factor h^t can be determined for a particular demand D^t as follows [71]:

Evaluate the ratio between the maximum fuel cost and maximum emission for each unit as:

$h_{i}^{t} = \frac{C_{i} (p_{i}^{max})}{E_{i} (p_{i}^{max})}, i = 1, 2, . . ., N_{G},$

arrange $h_{i}^{t}, i = 1, 2, . . ., N_{G}$ in ascending order,

add the maximum capacity of each unit, $(P_{i}^{max})$ , one at a time, starting from unit having smallest $h_{i}^{t}$ until the demand is met as $\sum p_{i}^{max} > D^{t}$ ,

at this stage, $h_{i}^{t}$ associated with the last unit in the process is the price penalty factor, $h_{i}^{t}$ for the given demand D^t.

We formulate the DEESRD problem over dispatch interval [0, N_T] as:

$\begin{matrix} min_{PS} TC (PS) & = \sum_{t = 1}^{N_{T}} \sum_{i = 1}^{N_{G}} (1 - r) {CE}_{i} (p_{i}^{t}) \\ + {rCE}_{i} (p_{i}^{t} + s_{i}^{t}) \end{matrix}$ (1)

where r is forecasted probability that the reserve is actually called up.

Subject to:

Load-generation balance $\sum_{i = 1}^{N_{G}} p_{i}^{t} = D^{t} + {Loss}^{t}, t = 1, . . ., N_{T},$ (2) where ${Loss}^{t} = \sum_{i = 1}^{N_{G}} \sum_{j = 1}^{N_{G}} p_{i}^{t} B_{ij} p_{i}^{t}$ .

Spinning reserve and demand balance $\sum_{i = 1}^{N_{G}} s_{i}^{t} = {SR}_{D}^{t} t = 1, . . ., N_{T},$ (3)

Generating unit capacity limits $p_{i}^{min} \leq p_{i}^{t} \leq p_{i}^{max}, t = 1, . . ., N_{T}, i = 1, . . ., N_{G} .$ (4)

Ramp rate limits

$- {DL}_{i} \leq p_{i}^{t + 1} - p_{i}^{t} \leq - {UL}_{i}, t = 1, . . ., N_{T} - 1, i = 1, . . ., N_{G}, - {DL}_{i} \leq p_{i}^{1} - p_{i}^{N} \leq - {UL}_{i}, i = 1, . . ., N_{G},$ (5)

Generating spinning reserve capacity limits $0 \leq s_{i}^{t} \leq {UL}_{i}, t = 1, . . ., N_{T}, i = 1, . . ., N_{G},$ (6)

Unit power and spinning reserve coupling capacity limits $s_{i}^{t} + p_{i}^{t} \leq p_{i}^{max}, t = 1, . . ., N_{T}, i = 1, . . ., N_{G} .$ (7)

In this case we assume that the spinning reserve demand is 10% of the power demand. This means that ${SR}_{D}^{t} = 0.1 D^{t}, t = 1, . . ., N_{T}$ . The function TC is quadratic and then this optimization problem (1) can be solved by e.g., quadratic programming.

Now instead of solving DEESRD problem over the interval [0, N_T], we solve the problem over an arbitrary interval [k, k + N_T] for any k ≥ 0.

Let ${\bar{PS}}^{k} = ({PS}^{1 + k}, {PS}^{2 + k}, . . ., {PS}^{N_{T} + k})$ . Thus the mathematical model for DEESRD problem is given by $min_{{\bar{PS}}^{k}} F ({\bar{PS}}^{k})$

Subject to ${PS}^{t} \in Γ ({\bar{PS}}^{k}), t = k + 1, k + 2, . . ., k + N_{T}$ where the feasible domain $Γ ({\bar{PS}}^{k})$ is defined to be the set of $(p_{i}^{t}, s_{i}^{t} : i = 1, . . ., N_{G}, t = k + 1, . . ., k + N_{T})$ satisfying constraints (2)– (7). Since both the power and spinning reserve demands are periodic and all the parameters of the problem do not change over time, then PS^1+k = PS^1+k+N_T and $\begin{matrix} \begin{matrix} Γ ({\bar{PS}}^{k + 1}) = Γ ({PS}^{2 + k}, {PS}^{3 + k}, . . ., {PS}^{N_{T} + k}, {PS}^{N_{T} + 1 + k}) \\ = Γ ({PS}^{2 + k}, {PS}^{3 + k}, . . ., {PS}^{N_{T} + k}, {PS}^{1 + k}) \\ = Γ ({PS}^{1 + k}, {PS}^{2 + k}, . . ., {PS}^{N_{T} + k}) \\ = Γ ({\bar{PS}}^{k}) . \end{matrix} \end{matrix}$

Then $Γ ({\bar{PS}}^{k})$ is shift-invariant, which is required for applying the MPC method to the DEESRD problem [72].

2.2 DR formulation

We consider the following Game Theory DR program. Let $u_{j}^{t}$ and $v_{j}^{t}$ be the incentive given to customer j and the curtailed power at time t, respectively. The cost of customer j of type η_j of curtailing $v_{j}^{t}$ at time t is given by [73]: $Δ (η_{j}, v_{j}^{t}) = δ_{1, j} {(v_{j}^{t})}^{2} + δ_{2, j} v_{j}^{t} - η_{j} δ_{2, j} v_{j}^{t}$

The parameter η_j ∈ [0, 1] describe the willingness of each customer to shed load, where η_j = 0 for the “least willing” customer and η_j = 1 for the “most willing” customer. We can see that as the marginal cost $Δ (η_{j}, v_{j}^{t})$ is a decreasing function of η_j. The parameters δ_1,j and δ_2,j are the coefficient of the costumer cost function [73].

The benefit for customer j at time t by: $benefit = incentive - cos t$ ${CB}_{j}^{t} = u_{i}^{t} - Δ (η_{j}, v_{j}^{t}), t = 1, . . ., N_{T}, j = 1, . . ., N_{C},$ (8) where, N_C is the number of customers participate in the DR program. The total benefit of each customer over a dispatch period should be non-negative. This will encourage customers to participate in the DR program.

The benefit for GENCO at time t due to the DR program is defined as:

${GB}_{j}^{t} = θ_{j}^{t} v_{j}^{t} - u_{j}^{t}, t = 1, . . ., N_{T}, j = 1, . . ., N_{C},$ (9) where, $θ_{j}^{t}$ is the value of not delivering power to customer j at time t “the value of power interruptibility” [73]. Let us use the following notations: $\begin{matrix} U^{t} = {u_{1}^{t}, . . ., u_{N_{C}}^{t}}, V^{t} = {v_{1}^{t}, . . ., v_{N_{C}}^{t}}, \\ {UV}^{t} = (U^{t}, V^{t}), UV = ({UV}^{1}, {UV}^{2}, . . ., {UV}^{N_{T}}) . \end{matrix}$

To maximize the benefit of the GENCO over the dispatch period [0, N_T], we introduce the following optimization problem $max_{UV} \sum_{t = 1}^{N_{T}} \sum_{j = 1}^{N_{C}} {GB}_{j}^{t}$ (10)

Subject to $\sum_{t = 1}^{N_{T}} {CB}_{j}^{t} \geq 0, j = 1, . . ., N_{C},$ (11) $\sum_{t = 1}^{N_{T}} ({CB}_{j}^{t} - {CB}_{j - 1}^{t}) \geq 0, j = 1, . . ., N_{C},$ (12) $\sum_{t = 1}^{N_{T}} \sum_{j = 1}^{N_{C}} u_{j}^{t} \leq UB$ (13) $\sum_{t = 1}^{N_{T}} v_{j}^{t} \leq {CM}_{j}, j = 1, . . ., N_{C}$ (14) where UB is the GENCO’s total budget and CM_j is the customer j daily of curtailable power.

2.3 DEESRD-DR formulation

In this subsection, we integrate DR program into the DEESRD problem. We formulate DEESRD-DR problem with the objective to minimize the energy and spinning reserve costs, minimize the amount of emission and maximize the benefit of the GENCO. We convert the multi-objective optimization into single-objective one using the weight factors ω₁, ω₂, ω₃ where ω₁ + ω₂ + ω₃ = 1. We define the following $U^{t} = {u_{1}^{t}, . . ., u_{N_{C}}^{t}}, V^{t} = {v_{1}^{t}, . . ., v_{N_{C}}^{t}},$ Q^t = (P^t, S^t, U^t, V^t), $\bar{Q} = (Q^{1}, Q^{2}, . . ., Q^{N_{T}})$ . Let us define the objective function to be minimized as:

$F (\bar{Q}) = \sum_{t = 1}^{N_{T}} \sum_{i = 1}^{N_{G}} (\begin{matrix} (1 - r) (ω_{1} C_{i} (p_{i}^{t}) + ω_{2} h_{i}^{t} E_{i} (p_{i}^{t})) + \\ r (ω_{1} C_{i} (p_{i}^{t} + s_{i}^{t}) + ω_{2} h_{i}^{t} E_{i} (p_{i}^{t} + s_{i}^{t})) \\ - ω_{3} {GB}_{j}^{t} \end{matrix})$

Then the mathematical formulation of DEESRD-DR problem is given as: $min_{\bar{Q}} F (\bar{Q})$ (15) subject to constraints (2), (4)– (7) and (11)– (14) and $\sum_{i = 1}^{N_{G}} s_{i}^{t} = 0.1 (D^{t} - \sum_{j = 1}^{N_{C}} v_{j}^{t}), t = 1, . . ., N_{T}$ (16)

Problem (15) can be solved over an interval [k, k + N_T] for any k ≥ 0. Let ${\bar{Q}}^{k} = (Q^{1 + k}, Q^{2 + k}, . . ., Q^{N_{T} + k})$ . Thus the DEESRD-DR problem can be formulated as $min_{{\bar{Q}}^{k}} F ({\bar{Q}}^{k})$ (17)

Subject to $Q^{t} \in Θ ({\bar{Q}}^{k}), t = k + 1, k + 2, . . ., k + N_{T}$ , where, the feasible domain $Θ ({\bar{Q}}^{k})$ is defined to be the set of $(p_{i}^{t}, s_{i}^{t}, u_{i}^{t}, v_{i}^{t} : i = 1, . . ., N_{G}, t = k + 1, . . ., k + N_{T})$ satisfying constraints (2), (4)– (7), (11)– (14) and (16). Similar to the DEESRD problem one can easily show that $Θ ({\bar{Q}}^{k + 1}) = Θ ({\bar{Q}}^{k}) .$ Then $Θ ({\bar{Q}}^{k})$ is shift-invariant.

3 Model predictive control method for DEESRD-DR

In this section we show how to apply MPC strategy to the DEESRD and DEESRD-DR problems. It is enough to consider the DEESRD-DR problem. Consider the linear discrete time control system [67] $p_{i}^{t + 1} = p_{i}^{t} + x_{i}^{t}, s_{i}^{t + 1} = s_{i}^{t} + y_{i}^{t}, u_{j}^{t + 1} = u_{j}^{t} + z_{j}^{t},$

$\begin{matrix} v_{j}^{t + 1} = v_{j}^{t} + d_{j}^{t}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, \\ t = 1, . . ., N_{T} - 1, \end{matrix}$ (18) where, $(p_{i}^{t}, s_{i}^{t}, u_{j}^{t}, v_{j}^{t}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C},$ t = 1,. . . , N_T), are the state variables and $(x_{i}^{t}, y_{i}^{t}, z_{j}^{t}, d_{j}^{t}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, t = 1, . . ., N_{T} - 1)$ are the control inputs. Then, we have the following transformation $p_{i}^{t} = p_{i}^{1} + \sum_{m = 1}^{t - 1} x_{i}^{m}, s_{i}^{t} = s_{i}^{1} + \sum_{m = 1}^{t - 1} y_{i}^{m}, u_{j}^{t} = u_{j}^{1} + \sum_{m = 1}^{t - 1} z_{i}^{m},$

$\begin{matrix} v_{j}^{t} = v_{j}^{1} + \sum_{m = 1}^{t - 1} d_{j}^{m}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, \\ t = 2, . . ., N_{T} . \end{matrix}$ (19)

Substituting transformations (19) into the optimization problem (15) and constraints (2), (4)– (7), (11)– (14) and (16), we get the control version of the dynamic economic emission and spinning reserve dispatch with demand response (CV-DEESRD-DR) problem. Here, the decision variables are $(p_{i}^{1}, s_{i}^{1}, u_{j}^{1}, v_{j}^{1}, x_{i}^{t}, y_{i}^{t}, z_{j}^{t}, d_{j}^{t},$ . i = 1,. . . , N_G, j = 1,. . . , N_C, t = 1,. . . , N_T - 1). Assume that $(p_{i}^{1}, s_{i}^{1}, u_{j}^{1}, v_{j}^{1}, i = 1, . . ., N_{G} j = 1, . . ., N_{C})$ are given. Then, the decision variables of the CV-DEESRD-DR will be $(x_{i}^{t}, y_{i}^{t}, z_{j}^{t}, d_{j}^{t}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, t = 1, . . ., N_{T} - 1)$ The idea of the MPC is that, at time t = 1 we measure the current state of the system $(p_{i}^{1}, s_{i}^{1}, u_{j}^{1}, v_{j}^{1}, i = 1, . . ., N_{G} j = 1, . . ., N_{C})$ , then solve the optimization problem CV-DEESRD-DR over the interval [0, N_T - 1], we get the optimal controller as $({\bar{x}}_{i}^{t}, {\bar{y}}_{i}^{t}, {\bar{z}}_{j}^{t}, {\bar{d}}_{j}^{t}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, t = 1, . . ., N_{T} - 1)$ Then the first part of the optimal controller $({\bar{x}}_{i}^{1}, {\bar{y}}_{i}^{1}, {\bar{z}}_{j}^{1}, {\bar{d}}_{j}^{1}, i = 1, . . ., N_{G} j = 1, . . ., N_{C})$ is applied for the system on the interval [1, 2). $p_{i}^{2} = p_{i}^{1} + {\bar{x}}_{i}^{1} r s_{i}^{2} = s_{i}^{1} + {\bar{y}}_{i}^{1} r$

$\begin{matrix} u_{j}^{2} = u_{j}^{1} + {\bar{z}}_{j}^{1} r v_{j}^{2} = v_{j}^{1} + {\bar{d}}_{j}^{1} r i = 1 r . . . r N_{G} r \\ j = 1 r . . . r N_{C} r \end{matrix}$ (20)

At the time instant t = 2 the whole procedure is repeated.

The objective function of the DEESRD-DR are differentiable and quadratic, and $Θ ({\bar{Q}}^{k})$ is shift-invariant the constraints. Then, using Theorems 1–2 of 67], we obtain that, the solution of MPC algorithm converges to the solution of the DEESRD-DR problem. Moreover, MPC is robust against the disturbances and uncertainties happen in the execution of the controller. In this case, the system actually execute $p_{i}^{2} = p_{i}^{1} + {\bar{x}}_{i}^{1} + b_{p, i}^{1}, s_{i}^{2} = s_{i}^{1} + {\bar{y}}_{i}^{1} + b_{s, i}^{1},$

$\begin{matrix} u_{j}^{2} = u_{j}^{1} + {\bar{z}}_{j}^{1} + b_{u, j}^{1}, v_{j}^{2} = v_{j}^{1} + {\bar{d}}_{j}^{1} + b_{v, j}^{1}, \\ i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, \end{matrix}$ (21) where, the disturbances $(b_{p, i}^{t}, b_{s, i}^{t}, b_{v, j}^{t}, b_{u, j}^{t}, i = 1, . . ., N_{G}, j = 1, . . ., N_{C}, t = 1, . . ., N_{T} - 1)$ satisfy the following bound $∥ b_{p, i}^{t} ∥ \leq ɛ_{p, i}, ∥ b_{s, i}^{t} ∥ \leq ɛ_{s, i}, ∥ b_{v, i}^{t} ∥ \leq ɛ_{v, i} ∥ b_{u, i}^{t} ∥ \leq ɛ_{p, i}$ Since the demand is forecasted then, it may has some disturbances or uncertainties. Then the actual (disturbed) demand will be ${\tilde{D}}^{t}$ . It has been shown in [2] that if $∥ D^{t} - {\tilde{D}}^{t} ∥$ is small enough, then, the demand disturbances can take the form of (21). Since we assumed that the forecasted spinning reserve demand is 10% of the power demand, therefore Theorem 2 of [67] is still valid for this case.

4 Model predictive control method for DEESRD-DR

In this section, we first solve the DEESRD and DEESRD-DR problems. We present two test systems. The first for the DEESRD problem and consists of five generating units. The data of this system is taken from [36]. The second test consists of five generating units and five customers. The data of the five units in the second test system is taken from [36]. We have modified the data the five customers given in [70] (see Tables 1 and 2). We take UB = 25000$.

Table 1
Hourly values of power interruptibility

t $θ_{1}^{t}$ $θ_{2}^{t}$ $θ_{3}^{t}$ $θ_{4}^{t}$ $θ_{5}^{t}$

1 8.2830 8.4900 8.6370 8.0790 8.2800

2 8.8230 9.0210 9.1590 8.6370 8.8320

3 8.4720 8.6610 8.7840 8.2980 8.4960

4 8.0070 8.6280 8.7420 8.3220 8.4720

5 8.7030 9.6720 9.7920 9.3600 9.4980

6 10.188 11.0010 11.145 10.614 10.797

7 25.191 26.8470 27.195 25.713 26.310

8 24.330 24.8640 25.137 23.718 24.318

9 33.180 33.8790 34.233 32.316 33.132

10 22.236 22.6290 22.827 21.720 22.185

11 23.685 24.0570 24.195 23.187 23.679

12 35.055 35.2650 35.328 34.725 35.001

13 14.394 14.5740 14.589 14.130 14.379

14 20.046 20.3220 20.421 19.665 20.022

15 14.550 14.8050 14.988 14.223 14.541

16 14.763 15.0840 15.261 14.382 14.757

17 19.995 20.8080 21.087 19.815 20.313

18 18.447 19.9710 20.157 17.907 19.872

19 16.857 17.3010 17.475 16.344 16.959

20 21.876 22.9140 23.094 22.974 23.094

21 14.748 14.9580 15.108 14.493 14.688

22 16.200 16.3140 16.452 16.038 16.089

23 10.311 10.4010 10.488 10.194 10.263

24 9.0900 9.2130 9.3000 8.9670 9.0600

t	$θ_{1}^{t}$	$θ_{2}^{t}$	$θ_{3}^{t}$	$θ_{4}^{t}$	$θ_{5}^{t}$
1	8.2830	8.4900	8.6370	8.0790	8.2800
2	8.8230	9.0210	9.1590	8.6370	8.8320
3	8.4720	8.6610	8.7840	8.2980	8.4960
4	8.0070	8.6280	8.7420	8.3220	8.4720
5	8.7030	9.6720	9.7920	9.3600	9.4980
6	10.188	11.0010	11.145	10.614	10.797
7	25.191	26.8470	27.195	25.713	26.310
8	24.330	24.8640	25.137	23.718	24.318
9	33.180	33.8790	34.233	32.316	33.132
10	22.236	22.6290	22.827	21.720	22.185
11	23.685	24.0570	24.195	23.187	23.679
12	35.055	35.2650	35.328	34.725	35.001
13	14.394	14.5740	14.589	14.130	14.379
14	20.046	20.3220	20.421	19.665	20.022
15	14.550	14.8050	14.988	14.223	14.541
16	14.763	15.0840	15.261	14.382	14.757
17	19.995	20.8080	21.087	19.815	20.313
18	18.447	19.9710	20.157	17.907	19.872
19	16.857	17.3010	17.475	16.344	16.959
20	21.876	22.9140	23.094	22.974	23.094
21	14.748	14.9580	15.108	14.493	14.688
22	16.200	16.3140	16.452	16.038	16.089
23	10.311	10.4010	10.488	10.194	10.263
24	9.0900	9.2130	9.3000	8.9670	9.0600

Table 2

Customer cost function coefficients, customer type and daily customer power limit

Customer	δ _1,j	δ _2,j	η _j	CM _j
1	1.847	11.64	0	100
2	1.378	11.63	0.1734	140
3	1.079	11.32	0.4828	205
4	0.9124	11.5	0.7208	250
5	0.8794	11.21	1	350

4.1 Simulation results for DEESRD problem

First we present the optimal solutions of the DEESRD problem, with ω= 0.5. The optimal solution of this problem is given by Table 3. From the table we can show that that, all constraints are satisfied and our results are accurate.

Table 3
Hourly power and SR schedule obtained from DEESRD

t p1 p2 p3 p4 p5 Loss s1 s2 s3 s4 s5

1 46.3855 59.6700 116.8015 112.1502 78.4381 3.4452 4.5579 6.4774 9.4529 12.5435 7.9683

2 50.0050 63.3498 123.0169 119.6002 82.9110 3.8828 4.9083 6.8831 9.9698 13.2964 8.4423

3 54.6588 69.5984 132.4562 131.7907 91.1343 4.6385 5.3552 7.5357 10.8442 14.5714 9.1934

4 61.0826 78.2002 145.4675 148.5713 102.4684 5.7901 5.9603 8.4375 12.0407 16.3409 10.2206

5 64.3560 82.5847 152.1075 157.1282 108.2503 6.4267 6.2679 8.9002 12.6466 17.2414 10.7438

6 70.2291 90.4248 163.9694 172.4201 118.6050 7.6484 4.7709 10.8408 11.0306 21.0522 13.1055

7 72.3507 93.2442 168.2526 177.9300 122.3374 8.1149 2.6493 12.7898 6.7474 24.8402 15.5733

8 75.0000 97.7592 174.9999 186.7942 128.3164 8.8696 0.0000 15.7051 0.0001 30.4382 19.2566

9 75.0000 106.5168 175.0000 203.7534 139.6447 9.9150 0.0000 16.5994 0.0000 32.2068 20.1937

10 75.0000 109.9189 175.0000 210.3214 144.1001 10.3404 0.0000 15.0811 0.0000 34.0098 21.3091

11 75.0000 113.7909 175.0000 217.8048 149.2439 10.8396 0.0000 11.2091 0.0000 32.1952 28.5957

12 75.0000 118.6269 175.0000 227.1590 155.6973 11.4832 0.0000 6.3731 0.0000 22.8410 44.7859

13 75.0000 109.9195 175.0000 210.3211 144.0998 10.3404 0.0000 15.0805 0.0000 34.0098 21.3097

14 75.0000 106.5169 175.0000 203.7534 139.6446 9.9150 0.0000 16.5995 0.0000 32.2070 20.1935

15 75.0000 97.7591 175.0000 186.7942 128.3163 8.8696 0.0000 15.7052 0.0000 30.4383 19.2565

16 66.9267 86.0354 157.3384 163.8445 112.8058 6.9508 6.5034 9.2630 13.1040 17.9770 11.1526

17 64.3567 82.5845 152.1079 157.1279 108.2498 6.4267 6.2670 8.9007 12.6460 17.2415 10.7448

18 70.2283 90.4246 163.9694 172.4205 118.6056 7.6484 4.7717 10.8417 11.0306 21.0514 13.1045

19 75.0000 97.7587 175.0000 186.7942 128.3167 8.8696 0.0000 15.7058 0.0000 30.4385 19.2558

20 75.0000 109.9195 175.0000 210.3209 144.1000 10.3404 0.0000 15.0805 0.0000 34.0104 21.3092

21 75.0000 104.1014 175.0000 199.0675 136.4487 9.6177 0.0000 16.3587 0.0000 31.7072 19.9341

22 69.8758 89.9550 163.2556 171.5018 117.9838 7.5720 5.1242 10.5166 11.7444 20.4212 12.6936

23 60.7329 77.7306 144.7569 147.6549 101.8486 5.7239 5.9269 8.3869 11.9748 16.2445 10.1668

24 53.2615 67.7241 129.6225 128.1304 88.6662 4.4046 5.2222 7.3376 10.5812 14.1907 8.9684

Cost=42486 $ Emission=18393 lb Loss=188.0734 M W

t	p1	p2	p3	p4	p5	Loss	s1	s2	s3	s4	s5
1	46.3855	59.6700	116.8015	112.1502	78.4381	3.4452	4.5579	6.4774	9.4529	12.5435	7.9683
2	50.0050	63.3498	123.0169	119.6002	82.9110	3.8828	4.9083	6.8831	9.9698	13.2964	8.4423
3	54.6588	69.5984	132.4562	131.7907	91.1343	4.6385	5.3552	7.5357	10.8442	14.5714	9.1934
4	61.0826	78.2002	145.4675	148.5713	102.4684	5.7901	5.9603	8.4375	12.0407	16.3409	10.2206
5	64.3560	82.5847	152.1075	157.1282	108.2503	6.4267	6.2679	8.9002	12.6466	17.2414	10.7438
6	70.2291	90.4248	163.9694	172.4201	118.6050	7.6484	4.7709	10.8408	11.0306	21.0522	13.1055
7	72.3507	93.2442	168.2526	177.9300	122.3374	8.1149	2.6493	12.7898	6.7474	24.8402	15.5733
8	75.0000	97.7592	174.9999	186.7942	128.3164	8.8696	0.0000	15.7051	0.0001	30.4382	19.2566
9	75.0000	106.5168	175.0000	203.7534	139.6447	9.9150	0.0000	16.5994	0.0000	32.2068	20.1937
10	75.0000	109.9189	175.0000	210.3214	144.1001	10.3404	0.0000	15.0811	0.0000	34.0098	21.3091
11	75.0000	113.7909	175.0000	217.8048	149.2439	10.8396	0.0000	11.2091	0.0000	32.1952	28.5957
12	75.0000	118.6269	175.0000	227.1590	155.6973	11.4832	0.0000	6.3731	0.0000	22.8410	44.7859
13	75.0000	109.9195	175.0000	210.3211	144.0998	10.3404	0.0000	15.0805	0.0000	34.0098	21.3097
14	75.0000	106.5169	175.0000	203.7534	139.6446	9.9150	0.0000	16.5995	0.0000	32.2070	20.1935
15	75.0000	97.7591	175.0000	186.7942	128.3163	8.8696	0.0000	15.7052	0.0000	30.4383	19.2565
16	66.9267	86.0354	157.3384	163.8445	112.8058	6.9508	6.5034	9.2630	13.1040	17.9770	11.1526
17	64.3567	82.5845	152.1079	157.1279	108.2498	6.4267	6.2670	8.9007	12.6460	17.2415	10.7448
18	70.2283	90.4246	163.9694	172.4205	118.6056	7.6484	4.7717	10.8417	11.0306	21.0514	13.1045
19	75.0000	97.7587	175.0000	186.7942	128.3167	8.8696	0.0000	15.7058	0.0000	30.4385	19.2558
20	75.0000	109.9195	175.0000	210.3209	144.1000	10.3404	0.0000	15.0805	0.0000	34.0104	21.3092
21	75.0000	104.1014	175.0000	199.0675	136.4487	9.6177	0.0000	16.3587	0.0000	31.7072	19.9341
22	69.8758	89.9550	163.2556	171.5018	117.9838	7.5720	5.1242	10.5166	11.7444	20.4212	12.6936
23	60.7329	77.7306	144.7569	147.6549	101.8486	5.7239	5.9269	8.3869	11.9748	16.2445	10.1668
24	53.2615	67.7241	129.6225	128.1304	88.6662	4.4046	5.2222	7.3376	10.5812	14.1907	8.9684
			Cost=42486 $	Emission=18393 lb	Loss=188.0734 M W

Our second target in this section is to show that the solution of the MPC converge to the optimal solutions of the DEESRD problem. The initial power and spinning reserve are chosen such that $\sum_{i = 1}^{N_{G}} p_{i}^{1} = D^{1} + {Loss}^{1}, {SR}_{D}^{1} = 0.1 D^{1} .$

Figure 1, shows that the MPC solutions approach the optimal solution of the DEESRD problem, in a few hours. To show the inherent robustness properties of the MPC (IRP-MPC), we consider two types of disturbances:

(i) Execution of the controller with disturbances. For this case we take $\begin{matrix} b_{p, i}^{t} = - ɛ_{p, i} + 2 ɛ_{p, i} n (t), b_{s, i}^{t} = - ɛ_{s, i} + 2 ɛ_{s, i} n (t), \\ b_{u, j}^{t} = - ɛ_{u, j} + 2 ɛ_{u, j} n (t), b_{v, j}^{t} = - ɛ_{v, j} + 2 ɛ_{v, j} n (t), \end{matrix}$ $i = 1, . . ., 5, j = 1, . . ., 5, t = 2, 3, . . .,$ where n (t) ∈ [0, 1] is random variable taken from normal distribution. Let us choose two sets of bounds:

IRP-MPC-(I): $\begin{matrix} ɛ_{p, 1} = 3, ɛ_{p, 2} = 4, ɛ_{p, 3} = 2, ɛ_{p, 4} = 6, ɛ_{p, 5} = 3, \\ ɛ_{s, 1} = 2, ɛ_{s, 2} = 1, ɛ_{s, 3} = 2, ɛ_{s, 4} = 1, ɛ_{s, 5} = 2 . \end{matrix}$

IRP-MPC-(II): $\begin{matrix} ɛ_{p, 1} = 9, ɛ_{p, 2} = 12, ɛ_{p, 3} = 6, ɛ_{p, 4} = 18, ɛ_{p, 5} = 9, \\ ɛ_{s, 1} = 6, ɛ_{s, 2} = 3, ɛ_{s, 3} = 6, ɛ_{s, 4} = 3, ɛ_{s, 5} = 6 . \end{matrix}$ (ii) Demand with disturbances. Let us define the actual demand ${\tilde{D}}^{t}$ as: ${\tilde{D}}^{t} = {\begin{matrix} D^{t} if t = 1 \\ D^{t} + ɛ_{D} (1 - 2 n (t)) D^{t} if t = 2, 3, . . ., 24 \end{matrix}$ where ɛ_D (1 -2n (t)) D^t is the relative change between the actual demand ${\tilde{D}}^{t}$ and the forecasted demand D^t. Since the forecasted spinning reserve load depend on the power demand, therefore, the actual spinning reserve equal to $0.1 {\tilde{D}}^{t}$ . Now let us consider two values of ɛ_D: $IRP - MPC - (III) : ɛ_{D} = \frac{5}{100}$ $IRP - MPC - (IV) : ɛ_{D} = \frac{10}{100} .$

This means that the power demand is perturbed with 5% and 10% of the nominal one.

Fig. 1

Convergence of the MPC solutions to those of EESRD problem for the power and spinning reserve of unit 2.

We have tested the MPC strategy against IRP-MPC-(I), IRP-MPC-(II), IRP-MPC-(III) and IRP-MPC-(IV). In these cases the initial P¹ and S¹ for the MPC are chosen as the optimal solution of the DEESRD problem. It has been shown in Figs. 2 and 4 that the MPC can keep the solution near to the optimal solution of the DESRD, DEESRD and PDESRD problems. From Figs. 3 and 5, we can see that, in spite of increasing the disturbance, the MPC still has the robustness when applying to DEESRD problem.

Fig. 2

The power and reserve of unit 2 under DEESRD problem and IRP-MPC-(I).

Fig. 3

The power and reserve of unit 2 under DEESRD problem and IRP-MPC-(II).

Fig. 4

The power and reserve of unit 2 under DEESRD problem and IRP-MPC-(III).

Fig. 5

The power and reserve of unit 2 under DEESRD problem and IRP-MPC-(IV).

4.2 Simulation results for DEESRD-DR problem

In this part we first present the optimal solution of the proposed DEESRD-DR. Figure 6 shows that the effect of the demand response on the normal load demand. Tables 4 and 5 summarize hourly power, spinning reserve, customer power curtailed and customer incentive schedule obtained from DEESRD-DR respectively. In Table 6 we present a comparison between the DEESRD and DEESRD-RD problems. One can see that, demand response strategy reduce the cost, emission and transmission line losses. In Figs. 7 and 8, we show that the MPC algorithm converges to the optimal solution of the DEESRD-DR problem.

Fig. 6

Load profiles before and after demand response.

Fig. 7

Convergence of the MPC solutions to those of DEESRD-DR problem for the power and spinning reserve of unit 2.

Fig. 8

Convergence of the MPC solutions to those of DEESRD- DR problem for the optimal power curtailed from customer 2.

Table 4

Hourly power and SR schedule obtained from DEESRD-DR

t	p1	p2	p3	p4	p5	Loss	s1	s2	s3	s4	s5
1	41.1676	58.4048	111.4634	108.3294	78.2997	3.1873	4.6901	5.9402	9.0911	12.0650	7.6613
2	43.6858	62.0435	117.0622	115.5410	83.1937	3.5819	4.9687	6.2698	9.6143	12.8716	8.0700
3	48.1967	68.2438	126.6905	127.9110	91.4196	4.3127	4.8318	7.0550	10.5985	14.2373	9.0923
4	54.1782	76.6062	139.8199	144.7280	102.6048	5.4117	5.3116	7.9733	11.6647	16.1173	10.1857
5	56.9752	80.4569	145.9919	152.5578	107.9459	5.9693	5.5523	8.4835	12.1907	16.9074	10.6619
6	62.0754	87.6342	157.1105	166.8495	117.7345	7.0588	6.0290	9.2226	13.1723	18.4079	11.6028
7	60.3180	85.0569	153.2609	161.9654	114.2976	6.6697	5.8768	9.0406	12.8071	17.9332	11.1652
8	63.6532	90.0135	161.7726	170.9094	121.0038	7.4437	6.1962	9.8425	11.9280	19.6506	12.3735
9	65.5465	92.4848	164.9493	176.5401	124.3710	7.8568	6.8650	10.6012	10.0507	20.9521	13.1344
10	69.4324	98.4416	174.1376	188.2932	132.2870	8.8642	5.5676	13.9472	0.8624	27.5441	17.4515
11	71.1837	101.0586	174.7845	193.0773	135.4186	9.2184	3.8163	14.3043	0.2155	29.5858	18.7085
12	70.2895	100.2664	174.9681	190.8568	134.0200	9.0772	4.7105	13.8354	0.0319	29.2190	18.3356
13	71.5947	101.8101	175.0000	194.9960	136.7808	9.3476	3.4053	15.0291	0.0000	29.9297	18.7192
14	68.4459	97.0600	172.0329	185.4636	130.4171	8.6199	6.5541	12.8958	2.9671	25.7416	16.3215
15	66.0476	93.2924	166.3947	178.0666	125.5038	7.9937	7.0698	11.0051	8.6053	21.7826	13.6683
16	58.0965	82.0417	148.3390	155.6665	109.9872	6.1960	5.6236	8.6097	12.4483	17.2126	10.8993
17	54.3941	76.8196	140.1810	145.2065	102.9349	5.4452	5.2757	8.0370	11.7160	16.1372	10.2432
8.	60.0437	84.6729	152.7129	161.1367	113.8370	6.6116	5.8511	9.0597	12.8077	17.6565	11.2042
19	65.4991	92.4883	165.0737	176.5536	124.4166	7.8602	6.8708	10.6523	9.9263	21.0149	13.1529
20	69.2810	98.2466	173.8530	187.8689	132.0010	8.8283	5.7190	13.7961	1.1470	27.2969	17.2833
21	68.4311	97.3123	172.5594	185.9995	130.8353	8.6654	6.5689	13.0753	2.4406	26.1008	16.4617
22	60.4659	85.3260	153.5909	162.3406	114.5410	6.7016	5.8827	9.0297	12.8068	17.9960	11.2411
23	53.4075	75.5167	138.1078	142.5483	101.1918	5.2627	5.2147	7.8760	11.5791	15.7620	10.1191
24	46.7747	66.2110	123.5586	123.8769	88.6884	4.0669	4.6909	6.8757	10.2983	13.7872	8.8522
			Cost=40070 $		Emission=16063 lb		Loss=164.251 M W

Table 5

Hourly customer power curtailed and customer incentive schedule obtained from DEESRD-DR

t	Hourly optimal customer power curtailed (xi MW)						Hourly optimal customer incentive (yi$)
	x1	x2	x3	x4	x5	Total xi	y1	y2	y3	y4	y5
1	0.0874	0.9353	3.0040	4.6893	6.8063	15.5223	54.4863	85.6238	124.6220	142.4540	156.2802
2	0.2740	1.1828	3.3158	5.0774	7.2058	17.0557	54.4864	85.6238	124.6220	142.4540	156.2802
3	0.2489	1.1459	3.2616	5.0331	7.1616	16.8512	54.4870	85.6238	124.6220	142.4540	156.2802
4	0.2185	1.2620	3.4057	5.2396	7.3487	17.4745	54.4863	85.6238	124.6219	142.4540	156.2802
5	0.4510	1.7005	3.9681	5.8972	8.0247	20.0415	54.4869	85.6238	124.6219	142.4540	156.2802
6	0.9354	2.2938	4.7367	6.7514	8.9376	23.6549	54.4864	85.6238	124.6219	142.4540	156.2802
7	4.9645	8.0101	12.1304	14.9665	17.6995	57.7710	54.4865	85.6239	124.6219	142.4540	156.2801
8	4.7872	7.3645	11.2715	13.9854	16.6828	54.0913	54.4867	85.6238	124.6219	142.4540	156.2801
9	7.2132	10.6817	15.5451	18.7615	21.7637	73.9652	54.4865	85.6238	124.6220	142.4540	156.2802
10	4.3354	6.7067	10.3968	13.1217	15.7118	50.2723	54.4866	85.6239	124.6220	142.4539	156.2802
11	4.7558	7.2631	11.0785	13.9802	16.6180	53.6956	54.4865	85.6238	124.6220	142.4540	156.2802
12	7.8198	11.3190	16.2229	20.2803	23.0343	78.6763	54.4864	85.6243	124.6220	142.4540	156.2802
13	2.2581	3.8402	6.6519	9.0516	11.3641	33.1660	54.4864	85.6244	124.6220	142.4541	156.2801
14	3.7224	5.8409	9.2456	11.9540	14.4375	45.2004	54.4863	85.6244	124.6219	142.4541	156.2802
15	2.1863	3.7709	6.6407	8.8731	11.2177	32.6887	54.4873	85.6242	124.6219	142.4540	156.2843
16	2.1075	3.6892	6.5334	8.6833	11.0516	32.0650	54.4864	85.6241	124.6219	142.4540	156.2845
17	3.4633	5.6879	9.1328	11.5394	14.0857	43.9091	54.4863	85.6241	124.6220	142.4540	156.2801
18	3.1361	5.5067	8.8584	10.6794	14.0278	42.2084	54.4863	85.6241	124.6221	142.4540	156.2801
19	2.8003	4.6642	7.7774	10.0153	12.5717	37.8289	54.4863	85.6243	124.6220	142.4540	156.2802
20	4.2348	6.8057	10.5148	13.8014	16.2210	51.5777	54.4869	85.6241	124.6220	142.4539	156.2802
21	2.2941	3.8992	6.7899	9.1299	11.4148	33.5278	54.4871	85.6242	124.6219	142.4539	156.2802
22	2.5353	4.1881	7.1524	9.6697	11.8916	35.4371	54.4866	85.6241	124.6218	142.4539	156.2845
23	0.8290	1.8888	4.1935	6.2392	8.3402	21.4906	54.4873	85.6243	124.6221	142.4540	156.2802
24	0.3932	1.3156	3.4615	5.3531	7.4340	17.9574	54.4869	85.6243	124.6221	142.4540	156.2802
Total	66.0513	110.9628	185.2892	242.7732	301.0526	906.1290	1307.7	2055.0	2990.9	3418.9	3750.7

Table 6

Comparison between DEESRD and DEESRD-DR problems

Problem	Cost	Emission	Loss
DEESRD	42486	18393l	188.0734
DEESRD-DR	40070	16063l	164.2507

5 Conclusions

In this paper, we have formulated dynamic economic emission and spinning reserve dispatch (DEESRD) problem which integrates the spinning reserve into the DEED problem. DEESRD determines the optimal power and spinning reserve allocation by simultaneously minimizing the power and spinning reserve costs, and the amount of emission under dynamic constraints and other constraints. This problem helps GENCOs to participate in the market by submitting bids for both energy and reserve. Implementation of DR programs can improve the reliability and reduce the energy price in the energy markets. We have integrated Game Theory DR program which is one of the incentive-based DR programs into the DEESRD problem.

We have formulated dynamic economic emission and spinning reserve dispatch with demand response (DEESRD-DR) problem which minimizes the energy and reserve costs, minimizes the amount of emission and maximizes the benefit of the GENCOs. In fact the DEESRD-DR model is useful not only for GENCOs but also for customers, since it increases the network reliability and customer’s benefit. Since the optimal solutions of the DEESRD and DEESRD-DR problem are open-loop, we have introduced a suitable version of MPC approach to construct closed-loop solutions. The performance of the MPC has been investigated by applying the MPC strategy to the DEESRD and DEESRD-DR problems with test system consisting of five generating units and five customers.

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