Demand-Side Management and Demand Response for Smart Grid

Abstract

Demand-side management (DSM) and a market mechanism involving demand response (DR) receive significant attention. The DSM is an emerging initiative which is one of the key elements of restructured power systems. An objective of any DSM program could be peak load clipping instead of adding generation supply, by simply shifting timing from the peak load period to off-peak period. The DR seeks to adjust load demand instead of adjusting generation supply. Different types of load shaping objectives, such as peak clipping, valley filling, load shifting, produce the DR. A compensation for the DR is triggered by diverse policies, market mechanism and implementation models. The integration of DR resources in electric power system becomes worldwide due to advent of communication technologies and metering infrastructure. With the evolving restructured electricity market, aggregator as a mediator between market operator and end-user customers. This chapter discusses six major DSM aspects: (1) the DR resources, (2) possible DR program models, (3) enabler technology framework and policy, (4) role of DR exchange (DRX) market, (5) optimization algorithms used and (6) a few implementation issues like end-users engagement, privacy preservation, and DR rebounding. An optimization algorithm for specific DRX market structures and how the market participants interact is described in detail.

Appendix: Optimization Methods Revisit

Optimization is a method to obtain the optimal variables that suggest minimum cost or maximum welfare of an objective function. The variables in the optimization problem are subject to a set of constraints [40]. The variables may be scheduling consequences of the physical process. Constraints can be categorized as a hard or soft constraint. The first constraint is the condition that must be satisfied. The latter has some degree of flexibility to select the variable. It can penalize objective if the conditions set of variables are not satisfied [93]. Further, the optimization can be characterized based on polynomial nature of the objective function. If at least one of the objective function is nonlinear, the optimization is said to be a nonlinear optimization, otherwise linear one. If some of the variables are integers, the optimization is said to be a mixed integer optimization. The integer variables take care of yes/no decision on the concerned variable. Additionally, the variables in a problem may be deterministic or the stochastic. Accordingly, the optimization can be categorized into deterministic and stochastic optimization problems. A constraint optimization model involving the equality and inequality constraints is provided in the next section followed by a step by step solution process.

1.1 Formulation of an Optimization Problem

An optimization problem in general form is given by [88].

$${\text{Minimize}}\quad \, f\left( x \right)$$

(6.49)

$${\text{Subject to}}:\quad g_{i} \left( x \right) \le 0,\quad i = \, 1,\;2,\; \ldots ,\;m$$

(6.50)

$$h_{j} \left( x \right) = 0,\quad i = \, 1,\;2,\; \ldots ,\;p$$

(6.51)

where $x \in {\mathbb{R}}^{n}$ is a vector including n optimization variable. The objective function $f\left( x \right) : {\mathbb{R}}^{n} \to {\mathbb{R}}$ is differentiable convex functions. The $f\left( x \right)$ maps the variable x close to a real value depicting the desirability of a solution to the decision-maker. Usually, the $f\left( x \right)$ represents a cost function in minimization problem and a payoff in maximization problem. The $g_{i} \left( x \right) :{\mathbb{R}}^{n} \to {\mathbb{R}}$ and $h_{j} \left( x \right) :{\mathbb{R}}^{n} \to {\mathbb{R}}$, respectively, represent inequality and equality constraint of the problem. There are such m number of equality and p number of inequality constraints exist in the optimization. The simplest form of an optimization model is a linear programming problem. This is obtained when the objective functions (9.49) and the constraints (6.50) and (6.51) are linear. A linear programming problem can be reformulated as

$${\text{Minimize}}\quad c^{\text{T}} x$$

(6.52)

$${\text{Subject to}}:\quad A_{I} x \le b_{I}$$

(6.53)

$$A_{E} x = b_{E}$$

(6.54)

$$x_{l} \le x \le x_{u}$$

(6.55)

It is worthy to note that functions f(·), g(·) and h(·) are affine expressions involving b vectors and matrices A. In (6.52), the term $c \in {\mathbb{R}}^{n}$ is the cost coefficient of the optimization variable, x. The inequality matrix, $A_{I} \in {\mathbb{R}}^{p \times n}$, and $b_{I} \in {\mathbb{R}}^{m}$ define the m linear inequality constraints (6.53). The equality matrix, $A_{E} \in {\mathbb{R}}^{p \times n}$, and $b_{E} \in {\mathbb{R}}^{p}$ define the p equality constraints (6.54). The constraint (6.55) denotes the variable bonds within lower x_l and upper x_l limits. The linear programming deals with a wide variety of practical problems including economic dispatch, unit commitments, supply and demand-side bidding and so forth.

1.2 Duality in Linear Programming

Defining a new set of m variables $\mu \in {\mathbb{R}}^{m}$ for inequality (6.53) and set of p variables $\lambda \in {\mathbb{R}}^{p}$ for equality (6.54), one for each constraint, there is a corresponding dual problem associated with the primal problem (6.56)–(6.58) discussed earlier given by:

$${\text{Maximize}}\quad b_{I}^{\text{T}} \mu + b_{E}^{\text{T}} \lambda$$

(6.56)

$${\text{subject to}}:\quad A_{I}^{\text{T}} \mu + A_{I}^{\text{T}} \mu = c$$

(6.57)

$$\lambda \ge 0 .$$

(6.58)

The dual problem in (6.56)–(6.58) is a transposed form of the primal problem. Note that, the primal and dual are through deals objective function minimization. However, it holds for objective function maximization, by minimizing its negative.

1.3 Lagrangian Function

Assuming m = p = 0, the problem is said to be unconstrained and the optimal solution of $f\left( x \right)$ simply occurs at a point $x^{*}$ if $\nabla f\left( {x^{*} } \right) = 0$, i.e. at those $x^{*}$, where the first derivative of the objective vanishes. This is called first-order necessary conditions [93]. In a constrained optimization, the decision variable $x \in {\mathbb{R}}^{n}$ is said to be feasible, if it satisfies the bound constraints (6.53), (6.54) and (6.55). Additionally, amid the set of possible variables, the one produces the minimum value of the function (6.52) is said to be optimal. In this case, first-order necessary conditions for optimality written by adding weighted sum of the constraints to the objective give the Lagrangian in the following form [93, 88].

$$\text{L} (x ,\alpha ,\beta )= \text{ }f (x )+ \sum\limits_{i = 1}^{m} {\mu_{i} g_{i} } (x )+ \sum\limits_{j = 1}^{p} {\lambda_{j} h_{j} } (x )$$

(6.59)

The weighting elements of $\mu \in {\mathbb{R}}^{m}$ and $\lambda \in {\mathbb{R}}^{p}$ are collectively named as dual variables of Lagrangian function.

1.4 Karush–Kuhn–Tucker (KKT) Conditions

Assuming some regularity conditions for problem (6.52)–(6.55), if the optimal $x^{*} = \, (x_{1}^{*} , x_{2}^{*} , \ldots ,x_{n}^{*} )$ minimize objective f (x) in (6.52), subject to the constraints (6.53) and (6.54) then there exist some dual optimal $\mu^{*} = \left( {\mu_{1} ,\;\mu_{2} ,\; \ldots ,\;\mu_{m} } \right) \ge 0$ and $\lambda^{*} = \left( {\lambda_{1}^{*} ,\;\lambda_{2}^{*} ,\; \ldots ,\;\lambda_{p}^{*} } \right) \ge 0$ such that

$$\nabla f (x^{*} )+ \sum\limits_{i = 1}^{m} {\mu_{i} \nabla g_{i} } (x )+ \sum\limits_{j = 1}^{p} {\lambda_{j} \nabla h_{j} } (x )= 0$$

(6.60)

$${\text{g}}_{i} (x^{*} )\le 0\quad \forall i = 1 ,\; 2 ,\; \ldots ,\;m$$

(6.61)

$$h_{j} (x^{*} )= 0\quad \forall j = 1 ,\; 2 ,\; \ldots ,\;p$$

(6.62)

$$\mu_{i} \ge 0\quad \forall i = 1 ,\; 2 ,\; \ldots ,\;m$$

(6.63)

$$\mu_{i} g_{i} (x^{*} )= 0\quad \forall i = 1 ,\; 2 ,\; \ldots ,\;m$$

(6.64)

The first set of KKT in (6.60) is known as stationarity condition found by differentiating the Lagrangian (6.59) concerning the relevant variables and then equating to zero. Constraints (6.61) and (6.62) enforce feasibility of the primal variables, while the constraint in (6.63) is feasibility of the Lagrangian multipliers. The constraint in (6.64) enforces complementary slackness which is also known as KKT complementarity. Complementary slackness can be rewritten in many equivalent ways. One way is the pair of conditions given by

$$\mu_{i}^{*} > 0 { } \Rightarrow \, g_{i} (x^{*} )\; = \; 0 ,\quad \forall i = 1 ,\; 2 ,\; \ldots ,\;m$$

(6.65)

$$g_{i} (x^{*} )\; < \; 0 { } \Rightarrow \, \mu_{i}^{*} = 0 ,\quad \forall i = 1 ,\; 2 ,\; \ldots ,\;m$$

(6.66)

Another way, the notion in (6.65), (6.66) can be compacted in the following form given by (6.67)

$$0 \le \mu_{i}^{*} \bot g_{i} (x^{*} )\; \ge \; 0 { ,}\quad \forall i = 1 ,\; 2 ,\; \ldots ,\;m$$

(6.67)

The orthogonality sign ⊥ in (6.67) of the form $0 \le \mu_{i}^{*} \bot g_{i} (x^{*} )\; \ge \; 0$ indicates, at most one between the dual, $\mu \in {\mathbb{R}}^{m}$ or the constraint, g associated with the dual $\mu \in {\mathbb{R}}^{m}$ can take a strictly nonzero value [93].

1.5 Economic Interpretation of the Dual Variables

It is worthy to mention that the dual variables $\mu \in {\mathbb{R}}^{m}$ and $\lambda \in {\mathbb{R}}^{p}$ have key to an economic explanation. In economics, it refers to a marginal worth of any resources [88]. These are also known as shadow price. Indeed, shadow price penalizes objective function marginally for unit variation in the variable value. In minimization problem, dual variable non-negative μ ≥ 0; while for a maximization problem, it is negative, μ ≥ 0. In fact, a marginal change of any component of the inequality vector $b_{I} \in {\mathbb{R}}^{m}$ would yield a narrower solution space, thereby achieve an inferior value of the objective function.