Optimal dynamic emergency reserve activation using spinning, hydro and demand-side reserves

Abstract

This paper proposes an optimal dynamic reserve activation plan after the occurrence of an emergency situation (generator/transmission line outage, load increase or both). An optimal plan is developed to handle the emergency, using the coordinated action of fast and slow reserves, for secure operation with minimum overall cost. It considers the reserves supplied by the conventional thermal generators (spinning reserves), hydro power units and load demands (demand-side reserves). The optimal backing down of costly/fast reserves and bringing up of slow reserves in each sub-interval in an integrated manner is proposed. The proposed reserve activation approaches are solved using the genetic algorithm, and some of the simulation results are also compared using the Matlab optimization toolbox and the general algebraic modeling system (GAMS) software. The simulation studies are performed on the IEEE 30, 57 and 300 bus test systems. These results demonstrate the advantage of the proposed integrated/dynamic reserve activation plan over the conventional/sequential approach.

Keywords

demand-side reserves / dynamic reserve activation approach / hydro power units / post contingency / sequential reserve activation approach / spinning reserves

Introduction

Power system operator (SO) is responsible for the secure and reliable operation of the power system. To take care of load increase, and to remove constraint violations in base case or after contingency, SO needs adequate and spread of spinning reserves. The SO procures these spinning reserves in day-ahead market, and the optimum plans for utilization of these reserves are drawn up depending on the nature of the problem. This exercise is also repeated in real time operation. Implementation of the demand-side reserves will result in efficient system operation. Presently, reserves as ancillary service products are purchased centrally by the power system operator to ensure system reliability. Demand-side reserves are defined as one of the emergency reserves based on the rules of North America Electricity Reliability Council (NERC) [ 1].

Reserves must be present in the bulk electric power system, so that the operator can replace or redistribute generation, on short notice, in the event of failure of equipment, e.g. a generator or a transmission line or both, which cause near-instantaneous changes on the electrical power system. Such outages result in change of power flows that may cause overloading of lines, or voltages to cross limits [ 2]. Contingency/spinning reserves are called during such sudden/emergency events. Generally, these events are a large loss of supply either from large transmission lines or generating units, but more generally, loss of large blocks of load demand. Contingencies occur suddenly, and on most occasions, the reserves must act immediately. Post-contingency corrective rescheduling actions remove operational violations, with the lowest increase in operational costs. Corrective rescheduling can include fast re-dispatching of generation, curtailment of selected load demands, and transmission switching [ 3].

Spinning reserve is the idle capacity connected to the system to ensure reliable operations in case of equipment outages. The spinning reserve has an economic value, as it reduces the outage costs. In several electricity markets, demand-side reserve functions have been implemented to take into account the value of reserve in the electricity market clearing [ 4]. The reserves are of slow and fast types, depending on the nature of the resource. Conventional thermal generators respond slowly due to their lower generation rate constraints (GRC). The hydro reserves are very fast in response due to the relatively much higher GRC. Gas turbines have an intermediate speed of response. In the deregulated power markets, demand-side reserves are also considered important to tackle certain emergencies, due to their fastest response as well as considerable spread throughout the system. Obviously, these are the costliest, and are called upon only when other reserves are inadequate to tackle the emergency.

In the event of emergencies, independent system operators (ISOs) generally prefer to supply demand as much as possible by participating in generators. If this is not feasible, they use demand-side reserves to retain system security in case of emergencies. Generators participate in the reserve market by bidding for up and down reserves. Some customers (demand-side reserves) agree, in exchange for significant payments, to increase or curtail their load demand to help correct an emergency situation [ 5]. Demands can bid as demand-side reserves [ 6] for up and down changes in their loads. While choosing generators or demands for re-dispatching, options are picked up so as to minimize the total cost, or to maximize total social welfare/benefit.

A mathematical framework for the solution of economic dispatch problem with security constraints, which can take into account the system corrective capabilities after the outage is described [ 7]. An approach for the post-contingency corrective action in security constrained unit commitment (SCUC), considering corrective actions based on quick-start units, is presented [ 8]. A new particle swarm optimization (PSO) based corrective strategy to the alleviate overloads of transmission lines is presented [ 9], and a security constrained forward market clearing algorithm within which the inherent characteristics of load demand flexibility are acknowledged when the provision of reserve from the demand-side is considered [ 10].

A SCUC model for energy and ancillary services auction which can be used by an ISO to optimize reserve requirements in electricity markets is addressed [ 11]. A market model that includes demand response offers and where energy and reserves are jointly dispatched is proposed [ 12]. The basic energy and reserve dispatch optimization in setting of a centralized market is presented [ 13]. A new approach for aggregated joint energy and spinning reserves market clearing considering, system reserve requirements, AC power flow constraints and lost opportunity cost (LOC) considerations is presented [ 14]. The issues related to the security constrained optimal power flow (SCOPF) problem formulation, challenges to the techniques for solving the SCOPF, and the current practice of extending the SCOPF formulation to take into account increasing levels of uncertainty in the operation planning are discussed [ 15]. An algorithm is developed to optimize the dispatch combination of wind and hydro generation levels and demand-response to meet a firm commitment to an hourly dispatch schedule [ 16]. A model based on market prices to determine the optimal bidding strategies that enable the total profit maximization of a pumped-hydro unit is proposed [ 17] and an optimal bidding strategies for a pumped-storage generating unit in a competitive electricity market, in which the market clearing price is insensitive to the bid price of a single generator is investigated [ 18].

From the above literature, it is clearly seen that the existing approaches for removing the overloads/constraints violations consider a single time-step optimization, using the reserves which can be deployed. The time-step corresponds to the period for which overloads can be tolerated. Starting from this operating point, the system then evolves to a better economic state through successive optimal power flows in real time. This is a sequential approach. In the present paper, the integrated problem of overload removal at minimum cost over an appropriate interval is considered. This interval is sub-divided into a number of sub-intervals. The sub-interval duration is the same as that of the single time step in the conventional approach. The coordinated decision making over these sub-intervals makes it a dynamic problem. The overloads are removed in the first sub-interval itself by utilizing fast reserves, which are generally costlier. In the remaining sub-intervals, the slower and cheaper reserves gradually replace the costlier, faster reserves. All decisions, in all the sub-intervals are coordinated, resulting in an overall better economical operation than that in the conventional sequential approach.

The optimum plans to tackle constraint violations due to load increase and/or a contingency are always drawn up in both day-ahead as well as in real time market clearing. The conventional procedure for tackling these emergencies is to first find out the nature and level of threat in terms of constraint limit violations. From the extent of limit violations, operators figure out how much time they have, to alleviate the threat. Based on this time and the GRC constraint, the limits on reserve can be found out which can be called into play. Optimal corrective control rescheduling, then provides the complete solution to make the system secure once again [ 19].

While making the system secure, the recourse might have been taken to costlier reserves due to time constraints. However, after reaching such a secure state, there is a need to move to a more economical schedule, as with more time, cheaper, slower reserves becoming available. The present practice is to do it in a ‘sequential’ manner. Intuitively, it can be seen that the entire process of first bringing in faster reserves, and then backing down these costlier, fast reserves and replacing them with cheaper and slower reserves is essentially a multi-stage decision making problem. To the best of the authors’ knowledge, there is no work on dynamic/integrated reserve activation. Hence, the motivation in this paper is to demonstrate that if optimal, dynamic emergency reserve activation is resorted to, over the scheduling interval, through an integrated multi-stage decision making process in each sub-interval, it is likely to achieve the objective at a lesser cost.

Problem formulation

In this paper, an optimal dynamic/integrated reserve activation model, including demand-side reserve offers is developed. Several reserve products are considered, including up and down spinning reserves supplied by the generators, and the load demands. It is assumed that the load demand forecast is available for every sub-interval. In the following mathematical model, it is assumed that there is an emergency situation due to some factors which are not specified. Two mathematical models, the dynamic/integrated optimization model, and the conventional/sequential optimization model, have been proposed for solving the problem.

In the mathematical models, the scheduling period (T), one hour (60 min) is divided into N_int of 10 min sub-intervals (t = 1, 2, ...,N_int).

Dynamic reserve activation: problem formulation

The proposed dynamic/integrated reserve activation problem considering the reserves from conventional thermal units and demand-side reserves is formulated as follows,

Minimize, total cost over the scheduling period ‘T’,

(1) {\displaystyle \sum }_{t=\mathrm{1}}^{N_{\mathrm{int}}}\left[{\displaystyle \sum }_{i=\mathrm{1}}^{N_G}{C}_{Gi}(P_{Gi}^t)+{\displaystyle \sum }_{k=\mathrm{1}}^{N_D}{C}_k(P_{\mathrm{shd},k}^t)\right],

where C_Gi (P_Gi^t) is the cost associated with conventional thermal generators, and is expressed as

(2) (P_{Gi}^t)=a_i+b_i(P_{Gi}^t)+c_i{(P_{Gi}^t)}^{\mathrm{2}}+|e_i\times \sin (f_i\times (P_{Gi}^{\min }-P_{Gi}^t))|,

where d_i, and e_i are the coefficients corresponding to valve point loading effects, C_k (P^t_shd,k) is the cost associated with demand-side reserve offers provided by load demands [ 20, 21], and is expressed as

(3) C_k(P_{\mathrm{shd},k}^t)=a_k^{\text{'}}+b_k^{\text{'}}(P_{\mathrm{shd},k}^t)+c_k^{\text{'}}{(P_{\mathrm{shd},k}^t)}^{\mathrm{2}}.

The proposed dynamic reserve activation problem formulation considering the reserves from conventional thermal generators, hydro power units and demand-side reserves is formulated as

Minimize, total cost over the scheduling period ‘T’,

(4) {\displaystyle \sum }_{t=\mathrm{1}}^{N_{\mathrm{int}}}\left[{\displaystyle \sum }_{i=\mathrm{1}}^{N_G}{C}_{Gi}(P_{Gi}^t)+{\displaystyle \sum }_{k=\mathrm{1}}^{N_D}{C}_k(P_{\mathrm{shd},k}^t)+{\displaystyle \sum }_{j=\mathrm{1}}^{N_H}{C}_{Hj}(P_{Hj}^t)\right],

where C_Hj (P_Hj^t) is the cost associated with hydro power units, and is expressed as

(5) C_{Hj}(P_{Hj}^t)=d_j(P_{Hj}^t),

Equations (1) and (4) are solved subjected to the following equality and inequality constraints.

Equality constraints

The nodal power balance (i.e., equality) constraints involve active and reactive power balances (typical load flow equations). In each sub-interval ‘t’, the sum of power output from generating units and the amount of power from demand-side reserves is equal to the sum of the total load demand and losses in the system [ 22, 23].

(6) P_{Gi}^t-[P_{Di}^t-P_{\mathrm{shd},i}^t]=V_i^t{\displaystyle \sum _{j=\mathrm{1}}^n{V}_j^t}(G_{ij}\cos {\ddot{a}}_{ij}^t+B_{ij}\sin {\ddot{a}}_{ij}^t)=P_i^t,

(7) Q_{Gi}^t-[Q_{Di}^t-Q_{\mathrm{shd},i}^t]=V_i^t{\displaystyle \sum _{j=\mathrm{1}}^n{V}_j^t}(G_{ij}\sin {\ddot{a}}_{ij}^t-B_{ij}\cos {\ddot{a}}_{ij}^t)=Q_i^t,

where i = 1,2,...,n, d_ij^t = d_i^t–d_j^t. Y_ij = G_ij + jB_ij is the (i, j) entry of the nodal admittance matrix. In demand-side reserve bidding, the load active power is adjusted and its reactive power usually varies at a constant power factor.

Inequality constraints

These inequality constraints involve the system operating limits.

Generator constraints

The generator active power outputs are restricted by their minimum and maximum power generation limits as

(8) P_{Gi}^{\min }\le P_{Gi}^t\le P_{Gi}^{\max },i\mathrm{=1,2,}\mathrm{...},N_G.

Due to ramp rate limits, the power output P_Gi^t is limited by its previous sub-interval power generation (P_Gi^t-1). The ramp rate limits are expressed as

(9) P_{Gi}^t-P_{Gi}^{t-\mathrm{1}}\le R_{Gi}^{\mathrm{up}},i\mathrm{=1,2,}\mathrm{...},N_G,

(10) P_{Gi}^{t-\mathrm{1}}-P_{Gi}^t\le R_{Gi}^{\mathrm{down}},i\mathrm{=1,2,}\mathrm{...},N_G.

After including the ramp rate limits in the generator operation constraint, Eq. (8) is modified as

(11) \max [P_{Gi}^{\min },P_{Gi}^{t-\mathrm{1}}-R_{Gi}^{\mathrm{down}}]\le P_{Gi}^t\le \min [P_{Gi}^{\max },P_{Gi}^{t-\mathrm{1}}+R_{Gi}^{\mathrm{up}}].

The generator reactive power is limited by the lower and upper reactive power generation as

(12) Q_{Gi}^{\min }\le Q_{Gi}^t\le Q_{Gi}^{\max },i\mathrm{=1,2,}\mathrm{...},N_G.

The generator voltage magnitudes in the sub-interval ‘t’ (V_Gi^t) are limited by

(13) V_{Gi}^{\min }\le V_{Gi}^t\le V_{Gi}^{\max },i\mathrm{=1,2,}\mathrm{...},N_G.

Constraints on conventional thermal generator spinning reserves

The possible spinning reserve capacity is dependent on the operating status of the generator. The sum of unit generation and spinning reserve must be less than or equal to the maximum capacity of the units. This is represented by [ 24]

(14) P_{Gi}^t+P_{\mathrm{SR}i}^{t,\max }\le P_{Gi}^{\max },

where P_SRi^t,max is the maximum available reserve capacity in the sub-interval ‘t’, and is given by [ 24]

(15) P_{\mathrm{SR}i}^{t,\max }=P_{Gi}^{\max }-P_{Gi}^t.

But, P_SRi^t depends on P_SRi^t,max and the ramp rate limits. Therefore, the spinning reserve limit is given by [ 25]

(16) \mathrm{0}\le P_{\mathrm{SR}i}^t\le \min (R_{Gi}^{\mathrm{up}},P_{\mathrm{SR}i}^{t,\max }).

Demand side reserve constraints

This constraint provides the relation between P^t_shd,k and P^t_D,k in each sub-interval ‘t’. The spinning reserve provided by the kth load/demand [ 3, 10, 12] is

(17) \mathrm{0}\le P_{\mathrm{shd},k}^t\le (P_{D,k}^t-P_{D,k}^{\min }),k=\mathrm{1},\mathrm{2},\mathrm{...},N_D,

that is

(18) \mathrm{0}\le P_{\mathrm{shd},k}^t\le P_{\mathrm{shd},k}^{t,\max },k=\mathrm{1},\mathrm{2},\mathrm{...},N_D,

where P_shd,k^t,max is the maximum demand response offer provided by the loads in the sub-interval ‘t’.

Transformer constraints

Transformer taps have minimum and maximum setting limits, which are expressed as

(19) T{T}_i^{\min }\le T{T}_i^t\le T{T}_i^{\max },i=\mathrm{1},\mathrm{2},\mathrm{...},N_T,

where TT_i stands for the transformer tap setting of the ith transformer in sub-interval ‘t’, N_T is the number of transformer branches, and TT_i^min and TT_i^max are minimum and maximum transformer tap limits.

Constraints on switchable VAR/shunt devices

The shunt admittance limits of switchable reactor/capacitor devices [ 26] are

(20) Y_{\mathrm{sh},j}^{\min }\le Y_{\mathrm{sh},j}^t\le Y_{\mathrm{sh},j}^{\max },j=\mathrm{1},\mathrm{2},\mathrm{...},N_{\mathrm{sh}},

where Y_sh,j^t is the shunt admittance of the jth switchable shunt device in sub-interval ‘t’, N_sh is the number of switchable shunt/reactive power source installation buses, and Y_sh,j^min and Y_sh,j^max are minimum and maximum shunt admittance limits.

Security constraints

Security constraints include the limits on the load bus voltage magnitudes and line flow limits [ 26].

(21) V_{Dk}^{\min }\le V_{Dk}^t\le V_{Dk}^{\max },k=\mathrm{1},\mathrm{2},\mathrm{...},N_D,

(22) \left|L_{ij}^t\right|\le L_{ij}^{\max },,

where L_ij^t is the MVA flow in sub-interval ‘t’, and L_ij^max is the thermal limit of the line between bus i and bus j.

Equality and inequality constraints related to hydro power plant

Water to power conversion [ 27]

The head dependent water to power conversion function is given by

(23) {{\displaystyle P}}_k^t={{\displaystyle \eta }}_k{{\displaystyle u}}_k{{\displaystyle \eta }}_k,

where ƞ_k is the natural inflow (water to power conversion coefficient), w_k is the water discharge, and h_k is water head.

(24) {{\displaystyle P}}_k^t={{\displaystyle \eta }}_k{{\displaystyle u}}_k({{\displaystyle h}}_{\mathrm{0}k}+{{\displaystyle a}}_k{{\displaystyle v}}_k),

where h_0k and a_j are constant terms related to reservoir j, and v_k is reservoir volume.

Reservoir volume limits

(25) v_{\min ,k}^t\le v_k^t\le v_{\max ,k}^t.

Initial and terminal reservoir volume

(26) v_{k\mathrm{0}}^t=V_{\mathrm{0},k}^t,

(27) v_{k,\mathrm{NT}}^t=V_{\mathrm{NT},k}^t.

Water discharge limits

(28) {{\displaystyle u}}_{\min ,g,k}^t\le {{\displaystyle u}}_{g,k}^t\le {{\displaystyle u}}_{\max ,g,k}^t,

Generation reserve offered by a pumped storage unit

(29) {{\displaystyle P}}_{g,k}^t+{{\displaystyle SR}}_{g,k}^t+{{\displaystyle OR}}_k^t\le {{\displaystyle \eta }}_k{{\displaystyle u}}_{\max ,g,k}^t({{\displaystyle h}}_{\mathrm{0},k}+{{\displaystyle a}}_k{{\displaystyle v}}_k).

where n is the pumping state of the pumped-storage unit, NP_k is the number of pumping states of the pumped-storage hydro unit k, w_g,kt is the water discharge of the pumped-storage hydro unit k when generating at time t, and w_pn,kt is the water discharge of the pumped-storage hydro unit k when pumping at state n.

Water balance constraint

(30) {{\displaystyle v}}_{k,(t+\mathrm{1})}={{\displaystyle v}}_{kt}-{{\displaystyle u}}_{g,kt}+{\displaystyle \sum _{n=\mathrm{1}}^{{{\displaystyle NP}}_k}{{\displaystyle u}}_{pn,k}},

Hydro power generator constraint

The hydro generator power output is restricted by

(31) P_{hj}^{\min ,t}\le P_{hj}^t\le P_{hj}^{\max ,t}

Sequential reserve activation: Problem formulation

The conventional/sequential reserve activation is performed for each sub-interval separately (sequentially). The scheduled power output from each sub-interval is the input for the next sub-interval. In the next sub-interval, the scheduled power output will vary based on their GRC limits. The sequential reserve activation problem considering the reserves from conventional thermal generators and demand-side reserves is formulated as

Minimize, cost in each sub-interval ‘t’,

(32) {\displaystyle \sum }_{i=\mathrm{1}}^{N_G}{C}_{Gi}(P_{Gi}^t)+{\displaystyle \sum }_{k=\mathrm{1}}^{N_D}{C}_k(P_{\mathrm{shd},k}^t).

The total cost in this approach is the sum of all sub-intervals cost. The sequential reserve activation problem formulation considering the reserves from conventional thermal generating units, hydro power units and demand-side reserves is formulated as

Minimize, cost in each sub-interval ‘t’,

(33) {\displaystyle \sum }_{i=\mathrm{1}}^{N_G}{C}_{Gi}(P_{Gi}^t)+{\displaystyle \sum }_{j=\mathrm{1}}^{N_H}{C}_{Hj}(P_{Hj}^t)+{\displaystyle \sum }_{k=\mathrm{1}}^{N_D}{C}_k(P_{\mathrm{shd},k}^t)

The total cost in this approach is the sum of all sub-intervals cost. The equality and inequality constraints in every sub-interval, for the conventional/sequential reserve activation problem over the scheduling interval ‘T ’are similar to the proposed dynamic/integrated reserve activation problem.

Proposed approach/methodology

The assumption in the proposed approach is that the unit commitment decision will not change after the occurrence of an emergency situation. The synchronized generators and demand-side reserves with pickup in response to emergency situation. Here, an emergency is created, and it is assumed that it can be tolerated for first 10 min, i.e., overloads should be removed in this 10-min period. The optimization is performed in several time steps of 10 min until all costly/fast reserves are replaced by slow/least cost reserves.

•It is assumed that at t = 0^-, the system is in a normal operating condition. For any loss of generation, at t = 0⁺ automatic generation control (AGC) will bring the system to a steady-state. Post AGC picture is the starting point for these sequential and dynamic/integrated reserve activation problems.

•It is assumed that the outage of the generator/line creates an emergency at t = 0⁺, that can be assumed to be tolerated by the system for 10 min (i.e., overloads should be removed in this 10-min period).

•Generators based on their GRC, and demand-side reserves will participate, so that all the system variables remain within their limits, and cascading of events does not occur.

•The proposed dynamic optimization is performed with coordinated decision (sequence of decisions) for minimizing the cost of generation over the scheduling interval subject to GRCs of the generators. This optimization is performed till the demand-side/faster reserves are replaced by slower reserves.

In this paper, the genetic algorithm (GA) is used to solve the proposed reserve activation approaches. However, depending on the specific requirements of the problem, any other suitable algorithms may also be used, if deemed appropriate. In this paper, some of the simulation studies are also performed using the Matlab optimization toolbox (Fmincon function) and the general algebraic modeling system (GAMS) software (using nlp-MINOS solver). In GA, the encoded binary strings are used for independent variable representation. The string/chromosome length depends on the precision required. A fitness function will be given for evaluating the performance of the strings. In this paper, the objective function plus the penalty functions is defined as the fitness function. The GA is achieved by three operations, i.e., selection, crossover, and mutation. The string that is fitter will have a better chance to be selected to the next iteration with a pre-specified population size. The randomized couple of strings are selected to achieve crossover after the selection. The parent selection is achieved by using the Roulette Wheel selection technique. The mutation operation provides an opportunity to create new chromosomes and avoids normal evolution for only considering reproduction and crossover. The detailed information of GA can be found in Ref. [ 28], and the GA fitness function evaluation is described in Ref. [ 29].

Results and discussion

The reserve activation problem is solved by using the conventional/sequential approach considering each sub-interval at a time, and by using the proposed integrated/dynamic approach, considering all the sub-intervals simultaneously over the scheduling period ‘T ’. Here, the scheduling period (1 h) is divided into 6 sub-intervals of 10 min duration, based on the nature and the extent of limit violations. The line flow/thermal limits for the first, second and third sub-intervals is considered to be 20%, 10% and 5% higher than the steady-state flow limits. For the fourth, fifth and sixth sub-intervals, the line flow limits are equal to the steady-state flow limits. These figures may, however, change with the actual practical data availability.

GA is used to solve the sequential and dynamic reserve activation problem. Some of the meta-heuristic optimization techniques suffer in searching global optimum including GA. The reason behind this issue is that these techniques use more probability/random functions and many parameters. This can be avoided by selecting/tuning best suited parameters for the given problem. The most important issue is diversity concept, and it is considered in designing of the GA. Mutation probability (Pm) should be set relative small, usually (1/length of the chromosome). Starting with randomly created solution, and scattered over the search space reduces the possibility of getting trapped in local optima. If the search gets stuck in local optima, it can be disturbed by injecting new randomly created solutions to the population.

The GA tuning parameters selected in this paper are that the population size is 60, the maximum number of generations are 200, the uniform crossover probability is 0.95, the bit-wise mutation probability is 0.01, and the elitism strategy (with probability 0.15) is employed to reproduce the best string/chromosome in an iteration to the next iteration. As mentioned, the Roulette Wheel parent selection technique is used. The chromosome length is selected depending on the desired accuracy for the control variables and the type of approach (sequential or dynamic). The algorithm is stopped when all the chromosomes/strings assume similar fitness values or the maximum number of generations are reached. Thirty runs have been performed in all the cases examined. The simulation results reported in this paper are the best solution obtained over these 30 runs.

The control variables considered for this problem are generator active power outputs, load reduction powers (demand-side reserves), generator bus voltage magnitudes, transformer tap settings, and switchable shunt devices. The reactive power generation, slack bus active power generation, branch/line flow, and load bus voltage magnitude constraints are treated as quadratic penalty terms in the GA fitness function.

Here, the GA encoding is performed using different gene lengths for each set of control variables, depending on the desired accuracy. The gene length for generator power outputs and demand-side reserves is 12 bits, the generator voltage magnitude is 8 bits, and these are considered as continuous controls. The transformer-tap settings can take 17 discrete values, each one being encoded using 5 bits. The minimum and maximum limits are 0.9 p.u. and 1.1 p.u respectively, and the step size is 0.0125 p.u. The bus shunt susceptances can take 6 discrete values, each one being encoded using 3 bits, the minimum and maximum limits are 0.0 p.u. and 0.05 p.u., respectively, and the step size is 0.01 p.u. The minimum and maximum limits of load bus voltages are 0.95 p.u. and 1.05 p.u., respectively. The generator bus voltage magnitude lower and upper limits considered are 0.95 p.u and 1.1 p.u.

Both sequential and dynamic reserve activation programs are coded in Matlab and implemented on a PC-Core 2 Quad computer with 3.24 GB of RAM. The IEEE 30, 57 and 300 bus systems are used to establish the effectiveness of the proposed reserve activation approaches.

Results on IEEE 30 bus system

The IEEE 30 bus system [ 30] consists of 6 generator buses, 21 load demand buses and 41 branches/lines, among which 4 branches are tap setting transformer branches. Buses 10, 12, 15, 17, 20, 21, 23, 24 and 29 are [ 31] considered as shunt compensation buses. In the proposed approach, the emergency situation is created by taking line 36 out (connecting buses 27 and 28 in IEEE 30 bus system), and by increasing the loading to 120% of the base case. The thermal limits of the lines for the IEEE 30 bus system are presented in Ref. [ 32].

In this test system, 5 generator powers, 21 load reduction powers (demand-side reserves), 6 generator-bus voltage magnitudes, 4 transformer-tap settings, 9 bus shunt susceptances are selected as control variables. The string/bit length for sequential reserve activation approach is 407 (i.e., 5 × 12+ 21 × 12+ 6 × 8+ 4 × 5+ 9 × 3), and for dynamic reserve activation, the approach is 2442 (i.e., 407 × 6), because of 6 sub-intervals.

The forecasted load assumed in each 10 min sub-intervals are 340.08 MW, 343.4808 MW, 346.8816 MW, 350.2824 MW, 353.6832 MW, and 357.084 MW, respectively. The two (sequential and dynamic/integrated) approaches are simulated next.

Case 1: Reserve activation by using conventional/sequential approach

Study 1: Sequential reserve activation approach using GA considering the reserves from conventional thermal generators and demand-side reserves

Here, it is assumed that at t = 0⁺, an emergency has occurred. As explained earlier, the system can withstand limit violations for the first 10 min. Table 1 shows the optimum generation schedules, the amount of load shed (sum of demand-side reserves), and the optimum cost in each sub-interval for conventional/sequential reserve activation approach using GA considering the reserves from the conventional thermal generators and demand-side reserve offers. In every sub-interval, the present generation and the GRCs of each generator having reserves will decide the minimum and maximum generation limits. The actual use of these reserves within the determined limits will, however, be decided by the optimization algorithm, based on cost and constraint considerations.

In each sub-interval, the system demand is determined by using the load forecast. In the first sub-interval, the assumed load forecast is 340.08 MW. Here, the generators alone cannot meet the load due to their GRC. Hence, demand-side reserves are utilized. In this sub-interval, the amount of load shed (sum of demand-side reserves) is 56.5867 MW, and the total power supplied by generators is 297.5578 MW. Therefore, the actual load served in the first sub-interval is 283.4933 MW. The optimum total cost incurred in the first sub-interval is 2268.1159 $/h which is the sum of the generation cost (1557.4305 $/h) and load reduction/demand-side reserve cost (710.6855 $/h). The generation schedules obtained from this first sub-interval are the starting point for the second sub-interval.

In the second sub-interval, the assumed load forecast is 343.4808 MW. Here, the generation supplied has increased to 332.2930 MW due to the availability of more spinning reserves, and the amount of load reduction (sum of all demand-side reserves) has decreased to 32.7925 MW. Hence, the actual load served is 310.6883 MW. In this sub-interval, the obtained optimum generation and demand-side reserve costs are 1729.1902 $/h, 434.0486 $/h, respectively. Hence, the optimum total cost is 2163.2388 $/h. The scheduled power output from this second sub-interval acts as input for third sub-interval.

This sequential optimization process will be repeated till all the faster/costly (demand-side) reserves are replaced by slower (generator spinning) reserves. In the sixth sub-interval, the total load is supplied by only generator/slower reserves. The optimum cost obtained in this sixth sub-interval is 1900.6101 $/h. In this sub-interval, the actual load served is equal to the forecasted load. The optimum total cost obtained in this sequential reserve activation approach for all 6 sub-intervals is 12274.8853 $/h. The computational time required for this sequential reserve activation approach using GA is 153.6645 s.

Study 2: Sequential reserve activation approach using Matlab optimization toolbox (Fmincon function) considering the reserves from conventional thermal generators and demand-side reserves

Table 2 shows the optimum generation schedules, the amount of load shed (sum of demand-side reserves), and optimum cost in each sub-interval for sequential reserve activation approach using Matlab optimization toolbox (Fmincon function). The optimum cost obtained in six sub-intervals are 2271.3156 $/h, 2168.2635 $/h, 2056.1722 $/h, 1989.5697 $/h, 1913.3379 $/h and 1902.9132 $/h, respectively. Therefore, the optimum total (sum of six sub-intervals) cost obtained in this case is 12301.5721 $/h. Hence, the total cost obtained using GA and Matlab optimization toolbox are 12274.8853 $/h and 12301.5721 $/h, respectively. From these results, it can be observed that the total cost obtained with both the methods (i.e., GA and Matlab optimization toolbox) is almost the same. But, the computational time required for this sequential reserve activation approach using Matlab optimization toolbox (Fmincon function) is 20.1592 s, which is less than the time required for GA.

The proposed sequential reserve activation approach is also implemented using the GAMS software (with nlp-MINOS solver). The optimum cost obtained in all six sub-intervals using GAMS are 2272.1132 $/h, 2168.5931 $/h, 2056.6215 $/h, 1990.0215 $/h, 1913.5113 $/h and 1903.0054 $/h, respectively. Therefore, the optimum total cost obtained in this sequential reserve activation approach for all six sub-intervals is 12303.866 $/h. The computational time required for this approach using the GAMS software is 18.1633 s, however, it is 153.6645 s using GA, and 20.1592 s using Matlab optimization toolbox. For the sake of simplicity, GA is used in this paper. However, any optimization algorithm can be used, if deemed appropriate.

Study 3: Sequential reserve activation approach using GA considering the reserves from conventional thermal generators, hydro power units and demand-side reserves

Table 3 shows the optimum generation schedules and optimum cost in each sub-interval for sequential reserve activation approach considering reserves from the conventional thermal generators, the hydro power units and demand-side reserves. The generator at bus 13 (P_H13) is considered as the hydro power generator and its maximum power generation limit is 40 MW. The optimum cost obtained in six sub-intervals are 2105.5233 $/h, 2015.5418 $/h, 1921.3881 $/h, 1907.5563 $/h, 1872.0831 $/h and 1847.8744 $/h, respectively. The optimum total (sum of 6 sub-intervals) cost obtained in this case is 11669.9670 $/h. From the simulation studies, it can be observed that the optimum total cost obtained in this case is less than the total cost obtained from Study 1 and Study 2 in Case 1 due to the activation of faster and cheaper hydro power reserves.

Case 2: Reserve activation by using integrated/dynamic approach

Study 1: Dynamic reserve activation approach using GA considering the reserves from conventional thermal generators and demand-side reserves

Table 4 shows the optimum generation schedules, the amount of load shed (sum of all demand-side reserves), and the cost in each sub-interval for the proposed integrated approach. In every sub-interval, the optimal backing down of fast/costly reserves and bringing up of slow reserves is done to get the minimum cost. In this dynamic/integrated approach, the optimization/control variables are 6 times more than the sequential approach, as there are six sub-intervals. In this approach, the objective function includes the sum of all sub-intervals cost.

The amount of load shed in the first sub-interval is 54.3075 MW and the total cost in this interval is 2281.4855 $/h. In the second sub-interval, the amount of load shed is 29.4779 MW, and hence the total cost in this interval is shed to 2064.4643 $/h. This process is repeated till all the costly reserves are replaced by slower/less costly reserves. Here, in sub-interval 6, all the costly/demand-side reserves are replaced by generator reserves, and the actual load served is equal to load forecast in sub-interval 6. The total cost obtained in sub-interval 6 is 1836.5079 $/h. The optimum total (sum of all sub-intervals) cost obtained in this approach is 11939.3018 $/h, which is 2.73% less than the cost obtained from the conventional sequential reserve activation approach (i.e., 12274.8853 $/h).

Study 2: Dynamic reserve activation approach using Matlab optimization toolbox (Fmincon function) considering the reserves from the conventional thermal generators and demand-side reserves

Table 5 tabulates the optimum generation schedules, the amount of load shed (sum of demand-side reserves) and the optimum cost in each sub-interval for the dynamic reserve activation approach using the Matlab optimization toolbox (Fmincon function) considering the reserves from the conventional thermal generators and demand-side reserves. The optimum cost obtained in six sub-intervals are 2289.9621 $/h, 2072.3611 $/h, 2026.2152 $/h, 1885.4746 $/h, 1868.6722 $/h and 1839.2706 $/h, respectively. The optimum total (sum of 6 sub-intervals) cost obtained in this case is 11981.9602 $/h, whereas the optimum total cost obtained using GA is 11939.3018 $/h. From these results, it can be observed that the total cost obtained using both the methods (i.e., GA and Matlab optimization toolbox) are in close agreement.

Study 3: Dynamic reserve activation approach using GA considering the reserves from the conventional thermal generators, hydro power units and demand-side reserves

Table 6 lists the optimum generation schedules and optimum cost in each sub-interval for the dynamic reserve activation approach considering the reserves from conventional thermal generators, hydro power units and demand-side reserves. The generator at bus 13 (P_H13) is considered as the hydro power generator and its maximum power limit is 40 MW. The optimum cost obtained in six sub-intervals are 2072.2658 $/h, 1960.1894 $/h, 1878.6348 $/h, 1854.2549 $/h, 1809.4620 $/h and 1772.3201 $/h, respectively. The optimum total (sum of 6 sub-intervals) cost obtained in this case is 11347.1270 $/h, which is less than the total cost obtained from Study 1 and Study 2 in Case 2 due to the activation of faster and cheaper reserves from the hydro power units. The total cost obtained from this Study 3 in Case 2 is 2.76% less than the total cost obtained from Study 3 in Case 1.

Results on IEEE 57 bus system

The IEEE 57 bus system [ 33] consists of 80 branches, 7 generator buses, and 42 load buses. Here, an emergency is created by taking line 66 out (connecting buses 13 and 49 in IEEE 57 bus system), and by increasing the load to 110%. 6 generator powers, 42 load reduction/demand-side reserve powers, 7 generator bus voltage magnitudes, 15 transformer tap settings, and 3 bus shunt susceptances are selected as control variables. The string/chromosome length in GA for the sequential reserve activation approach is 716 (i.e., 6 × 12+ 42 × 12+ 7 × 8+ 15 × 5+ 3 × 3), and for dynamic reserve activation approach is 4296 (i.e., 716 × 6), because of 6 sub-intervals. The forecasted load assumed in each 10 min sub-intervals are 1375.88 MW, 1389.6388 MW, 1403.3976 MW, 1417.1564 MW, 1430.9152 MW, and 1444.6740 MW, respectively.

Case 1: Reserve activation by using conventional/sequential approach

Table 7 shows the optimum generation schedules, the amount of load shed and optimum cost in each sub-interval using the sequential reserve activation approach for the IEEE 57 bus system. In the first sub-interval, the forecasted load is 1375.88 MW, but the generators cannot supply that load due to their ramp rate constraints. Hence, the demand-side reserves are also incorporated, to meet the load-generation balance. The amount of load shed (sum of demand-side reserves) in this sub-interval is 107.3479 MW, and the demand-side reserve cost is 4287.6276 $/h. Hence, the actual load served in this sub-interval is 1268.5321 MW. The optimum cost obtained in this sub-interval is 18085.0393 $/h, which is the sum of generation cost (13797.4117 $/h) and demand-side reserve cost (4287.6276 $/h). The generation schedules obtained from the first sub-interval are given as the input for the next sub-interval (sub-interval 2).

In sub-interval 2, the generation schedules vary based on schedules obtained from sub-interval 1, and their GRC. The amount of load shed (sum of demand-side reserves) in this sub-interval is shed to 60.2117 MW, and hence the actual load served is increased to 1329.4271 MW. The optimum total cost obtained in this sub-interval is 18010.9719 $/h. This process is repeated till the actual load served is equal to the forecasted load demand in that sub-interval. Here, in sub-interval 6, all the loads are served by only generators. The optimum cost obtained in this sub-interval is 15790.6474 $/h. The total cost over the scheduling period (1 h) is the sum of all 6 sub-intervals cost. In this sequential approach, the total cost obtained is 100897.0868 $/h.

Case 2: Reserve activation by using integrated/dynamic approach

Table 8 shows the optimum generation schedules, the amount of load shed, and the optimum cost in each sub-interval using the integrated reserve activation approach for the IEEE 57 bus system. Here, the objective function contains the cost of all 6 sub-intervals. This dynamic optimization is performed with coordinated decision for minimizing the total cost over the scheduling period.

In the first sub-interval, the amount of load shed is 112.3994 MW. The total cost obtained in this first sub-interval is 18376.1417 $/h, which is the sum of generation cost (13375.275 $/h) and demand-side reserve cost (5000.8665 $/h). The total cost obtained from all 6 sub-intervals is 96092.8272 $/h, which is 4.76% lower compared to the cost obtained from the sequential reserve activation approach.

Results on IEEE 300 bus system

The IEEE 300 bus system [ 30] consists of 411 branches, 69 generating units, and 198 load buses. Here, an emergency is created by increasing the load to 115%. 68 generator powers, 198 demand-side reserve powers, 69 generator bus voltage magnitudes, 62 transformer tap settings, and 12 bus shunt susceptances are selected as control variables. The chromosome length in GA for the sequential reserve activation approach is 4366, and for the dynamic reserve activation approach is 26196. The forecasted load assumed in each 10 min sub-intervals are 26733.8890 MW, 26966.3576 MW, 27198.8262 MW, 27431.2948 MW, 27663.7634 MW, and 27896.2320 MW, respectively.

Case 1: Reserve activation by using conventional/sequential approach

Table 9 gives the optimum generation and load reduction costs in each sub-interval using the sequential reserve activation approach for the IEEE 300 bus system. The total cost over the scheduling period (1h) is the sum of all 6 sub-intervals cost. In this sequential approach, the total cost obtained is 5964946.2674 $/h.

Case 2: Reserve activation by using integrated/dynamic approach

Table 10 shows the optimum generation and load reduction cost in each sub-interval using the dynamic reserve activation approach for the IEEE 300 bus system. Here, the objective function contains cost of all 6 sub-intervals. This dynamic optimization is performed with coordinated decision for minimizing the total cost over the scheduling period. The total cost obtained from all 6 sub-intervals is 5698459.7579 $/h, which is 4.47% lower compared to the cost obtained from the sequential reserve activation approach.

Table 11 presents the computational times required for the IEEE 30, 57, and 300 bus systems considering the sequential and dynamic reserve activation approaches using GA and Matlab optimization toolbox. The computational times required for IEEE 30, 57 and 300 bus systems for the conventional/sequential reserve activation approach using GA are 153.6645 s, 347.5862 s, and 57.4162 min, respectively. By using the proposed dynamic reserve activation approach, the computational times required are 281.6094 s, 640.2503 s, and 106.0572 min, respectively, which are 82.87%, 84.2% and 84.7% higher than the conventional/sequential reserve activation approach.

As mentioned earlier, the simulation results are also compared with the Matlab optimization toolbox (Fmincon function). The computational times required for the IEEE 30, 57 and 300 bus test systems for the sequential reserve activation approach using the Matlab optimization toolbox are 20.1592 s, 43.7143 s. and 7.2296 min, respectively, whereas the computational times for the dynamic reserve activation approach are 38.0036 s, 81.6824 s and 13.432 min, respectively. As mentioned earlier, these computational times presented in this paper are based on the PC-Core 2 Quad computer with 3.24 GB of RAM. However, these computational times are much less for the practical cases as the capacity of computer/hardware used by the system operator is much higher compared to the one used in this paper. Therefore, all the approaches proposed in this paper can be easily implemented in real time for all practical applications. The proposed multi-stage dynamic optimization can be easily implemented with increased computer hardware or software capabilities, and efficient optimal power flow (OPF) packages.

From the above case studies, the advantage of the proposed integrated/dynamic reserve activation approach over the conventional sequential reserve approach can be observed.

Conclusions

In this paper, an optimal integrated/dynamic post-contingency reserve activation plan has been developed by utilizing fast and slow reserves from generating units and demands. In this approach, an attempt is made to remove overloads, following a contingency, using coordinated action of fast and slow reserves, in order to restore secure operation at minimum overall cost. This paper considers the reserves supplied by the conventional thermal generators (spinning reserves), hydro power units and load demands (demand-side reserves). The optimization is performed for a one-hour scheduling period, with sub-intervals of 10-min duration. The optimal backing down of fast/costly reserves and bringing up of slow reserves is performed for each sub-interval simultaneously, and sequentially. The proposed reserve activation approaches are solved using GA, and some of the simulation results are also compared with the Matlab optimization toolbox and the GAMS software. The simulation studies on the IEEE 30, 57 and 300 bus test systems demonstrate the advantage of the proposed integrated/dynamic reserve activation approach over the conventional sequential approach. The overall cost benefit obtained with the proposed dynamic reserve activation approach compared to the sequential approach for the IEEE 30, 57 and 300 bus test systems are 2.73%, 4.76%, and 4.47%, respectively. Developing an optimal dynamic reserve activation plan considering the battery storage is the scope for future research.

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